We introduce a new generalized inverse which is a natural generalization of (pseudo) right core inverse in a Banach *-algebra. We characterize this generalized inverse by using right core and quasi-nilpotent decomposition. We then present its polar-like characterization and investigate related algebraic properties. Finally, the right core-EP inverse is characterized by certain new ways.

In this paper, we introduce the notion of weighted generalized group inverse in a Banach algebra with proper involution. This is a natural generalization of weighted weak group inverse for a complex matrix and Hilbert space operator. We present several characterizations and representations of this generalized inverse. In addition, a new partial order on elements in a Banach *-algebra is investigated by using the weighted generalized group inverse and some known results are thus generalized.

In this paper, we introduce the notion of the generalized $w$-group inverse in a Banach algebra with proper involution. This is a natural generalization of generalized (weak) group inverse in Banach *-algebras. Many elementary properties of such new weighted generalized inverse are presented. We further establish the equivalent relations between generalized weighted group inverse and weighted generalized Drazin inverse. Related properties of generalized (weak) group inverse are extended to the wider cases.

This article shows how to establish expansion formulas for calculating the mixed operations $(A^{\dag})^{\#}$, $(A^{\#})^{\dag}$, $((A^{\dag})^{\#})^{\dag}$, $((A^{\#})^{\dag})^{\#}$, $\ldots$ of generalized inverses, where $(\cdot)^{\dag}$ denotes the Moore--Penrose inverse of a matrix and $(\cdot)^{\#}$ denotes the group inverse of a square matrix. As applications of the formulas obtained, the author constructs and classifies some groups of matrix equalities involving the above mixed operations, and derives necessary and sufficient conditions for them to hold.

We investigate the existence and representation of the generalized core-EP inverse of some triangular matrices over a Banach algebra. Further, the general representations of the generalized core-EP inverse of a triangular matrix over a $C^*$-algebra are presented. As applications, the generalized core-EP inverses of some block operator matrices over Hilbert spaces are given.

Input reconstruction problem is usually encountered in the areas such as virtual sensing, image restoration, sensor linearization, and communications. The input reconstruction problem can be seen as an inverse problem. Given a nominal system, two types of approaches are commonly used for solving the inverse problem. The first one is to directly invert the nominal system, and the second one is to derive an inversion of the nominal system in an indirect way. However, for the first type of approaches, system inversion cannot be directly conducted under some conditions such as there exist nonminimum-phase zeros in the nominal system, while for the second type of approaches, simultaneously guaranteeing stability and causality of the obtained inversion is still not solved well. In order to avoid the drawbacks existing in the two types of approaches, an alternative approach is proposed for input reconstruction. In this approach, the input signal is first modeled as the output of a state-space model, afterwards two Kalman filters for the model resulted by combining the input signal model and the nominal system model are implemented alternatively such that the input signal can be sequentially reconstructed in an infinite horizon. The proposed input reconstruction approach is applied in two examples, and simulation results can illustrate the effectiveness of the proposed approach.

In this paper, the entire downhole fluid sucker rod pump system is replaced with a viscoelastic vibration model, namely a third-order differential equation with an inhomogeneous forcing term. Both Kelvin and Maxwell viscoelastic models can be implemented along with the dynamic behaviors of a mass point attached to the viscoelastic model. By employing the time-dependent polished rod force measured with a dynamometer as the input to the viscoelastic dynamic model, we have obtained the displacement responses, which match closely with the experimental measurements in actual operations, through an iterative process. The key discovery of this work is the feasibility of the so-called inverse optimization procedure which can be utilized to identify the equivalent scaling factor and viscoelastic system parameters. The proposed Newton-Raphson iterative method with some terms in the Jacobian matrix expressed with averaged rates of changes based on perturbations of up to two independent parameters provides a feasible tool for optimization issues related to complex engineering problems with mere information of input and output data from either experiments or comprehensive simulations. The same inverse optimization procedure is also implemented to model the entire fluid delivery system of a very viscous non-Newtonian polymer modeled as a first-order ordinary differential system similar to the transient entrance developing flow. The convergent parameter reproduces transient solutions which match very well with those from full-fledged computational fluid dynamics models with the required inlet volume flow rate and the outlet pressure conditions.

We provide representations for the generalized Drazin inverse of an anti-triangular matrix of the form $\left( \begin{array}{cc} a&b\\ 1&0 \end{array} \right)$ in a Banach algebra $\mathcal{A}$, under the condition that $ab=ba$. Specifically, we present the representation of Drazin inverse for these types of anti-triangular matrices in Banach algebras.

Given a square matrix $A$, we are able to construct numerous equalities that involve reasonable mixed operations of $A$ and its conjugate transpose $A^{\ast}$, Moore--Penrose inverse $A^{\dag}$, and group inverse $A^{\#}$. Such kind of equalities can be generally represented in the equation form $f(A, \, A^{\ast}, A^{\dag}, A^{\#}) =0$. In this article, the author constructs a series of simple or complicated matrix equalities, as well as matrix rank equalities involving the mixed operations of the four matrices. As applications, we give a sequence of necessary and sufficient conditions for a square matrix to be range-Hermitian.

It is a classic topic in algebras to construct and verify equalities that are composed of various algebraic operations of elements and their inverses or generalized inverses. In this note, the author constructs a matrix equality $(AB)^{\dag} = B^{\ast}(A^{\ast}ABB^{\ast})^{\#}A^{\ast}$ (called a mixed reverse-order law), where $A$ and $B$ are two matrices of appropriate sizes, $(\cdot)^{\ast}$, $(\cdot)^{\dag}$, and $(\cdot)^{\#}$ denote the conjugate transpose, the Moore--Penrose inverse, and the group inverse of a matrix, respectively, and shows that the mixed reverse-order law always holds. A list of variation forms of this matrix equality are also given, and necessary and sufficient conditions are obtained for them to hold. Especially, we show a remarkable fact that the two reverse-order laws $(AB)^{\dag} = B^{\dag}A^{\dag}$ and $(A^{\ast}ABB^{\ast})^{\#} = (BB^{\ast})^{\#}(A^{\ast}A)^{\#}$ are equivalent.

This study delves into the Schrödinger equation's new invariants, shedding light on their crucial implications for theoretical physics and beyond. Specifically, it explores the behavior of scattering amplitudes and bound states concerning the choice of coordinate systems. The research unveils that while the energy of bound states remains invariant under coordinate transformations, the phase of scattering amplitudes undergoes variations, underscoring its pivotal role in theories reliant on phase normalization. Moreover, these findings have played a pivotal role in constructing estimates for three-dimensional Navier-Stokes equations, enhancing our ability to model complex fluid dynamics with greater precision and reliability. Additionally, in seismic exploration, tomography, and ultrasound imaging, these properties allow researchers to optimize phase selection, facilitating the most effective interpretation of measurement results. This strategic phase manipulation ensures clearer insights into subsurface structures, tissue composition, and fluid dynamics, with the flexibility to seamlessly transition back to the original coordinate system post-interpretation. This interdisciplinary synergy underscores the profound impact of fundamental theoretical research on advancing practical applications across diverse scientific domains.

There are many problems based on solving non-autonomous differential equations of the form x¨(t)+ω2(t)x(t)=0, where x(t) represents the coordinate of a material point and ω is the angular frequency. The inverse problem involves finding the bounded coefficient ω. Continuity of the function ω(t) is not required. The trajectory x(t) is unknown, but the initial and final values of the phase variables are given. The variation principle of minimum time for the entire dynamic process allows for the determination of the optimal solution {x(t),ω(t)}. Thus, the inverse problem is an optimal control problem. No simplifying assumptions were made.

Subglottal Impedance-Based Inverse Filtering (IBIF) allows for the continuous, non-invasive estimation of glottal airflow from a surface accelerometer placed over the anterior neck skin below the larynx, which has been shown to be advantageous for the ambulatory monitoring of vocal function. However, during long-term ambulatory recordings over several days, conditions may drift from the laboratory environment where the IBIF parameters were initially estimated due to sensor positioning, skin attachment, and temperature, among other factors. Observation uncertainties and model mismatch may result in significant deviations in the glottal airflow estimates, but are very difficult to quantify in ambulatory conditions due to a lack of a reference signal. To address this issue, we propose a Kalman filter implementation of the IBIF filter, which allows for both estimating the model uncertainty and adapting the airflow estimates to correct for signal deviations. One-way ANOVA results from laboratory experiments using the Rainbow Passage indicate a an improvement on amplitude-based measures for PVH subjects compared to IBIF which shows a statistically difference with respect to the reference oral airflow (p=0.02,F=4.1). MFDR from PVH subjects is slightly different to the oral airflow when compared to IBIF (p=0.04, F=3.3). Other measures did not have significant differences with either Kalman or IBIF, with the exception of H1H2, whose performance deteriorates for both methods. Overall, both methods show similar flottal airflow measures, with the advantage of Kalman by improving amplitude estimation. Moreover, Kalman filter deviations from the IBIF output airflow might suggest a better representation of some fine details in the ground-truth glottal airflow signal. Other applications may take more advantage from the adaptation offered by the Kalman filter implementation.

Let G[F,Vk, Huv] be the graph with k pockets, where F is a simple graph of order n ≥ 1,Vk= {v1,v2,··· ,vk} is a subset of the vertex set of F and Hvis a simple graph of order m ≥ 2,v is a specified vertex of Hv. Also let G[F,Ek, Huv] be the graph with k edge pockets, where F is a simple graph of order n ≥ 2, Ek= {e1,e2,···ek} is a subset of the edge set of F and Huvis a simple graph of order m ≥ 3, uv is a specified edge of Huvsuch that Huv− u is isomorphic to Huv− v. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of G[F,Vk, Hv] and G[F,Ek, Huv] in terms of the resistance distance and Kirchhoff index F, Hv and F, Huv, respectively.

Predicting ground reaction forces (GRFs) and ground reaction moments (GRMs) through a biomechanical model-based approach offers advantages for biomechanical analysis, particularly in situations where the application of a statistical model is limited by insufficient training data. However, the current prediction method is unsuitable for clinical applications due to its long computational time. The present study developed a new optical motion capture (OMC)-based method to predict GRFs, GRMs, and joint torques. The proposed approach performed the estimation process by distributing external forces, as determined by the equation of motion, between the left and right feet based on foot deformations, thereby predicting the GRFs and GRMs without relying on machine learning or optimization techniques. We investigated the prediction accuracy during walking by comparing a conventional analysis using a force plate with the OMC results. The comparison revealed excellent or strong correlations between the prediction and measurement for all GRFs, GRMs, and lower limb joint torques. The proposed method, which provides practical estimation with low computational cost, facilitates efficient biomechanical analysis and rapid feedback of analysis results, thereby increasing its applicability for kinetics estimation in clinical settings.

The inverse system identification toolbox named INVSID 1.0 for MATLAB, which is used to identify the inversion of single-input single-output systems, is developed. The complete process of developing the toolbox is demonstrated. Afterwards, a numerical example is illustrated to describe how the toolbox can be used to solve inverse identification problems. Simulation results demonstrate the effectiveness of the toolbox.

This paper addresses the Collatz Conjecture, an open question in mathematics that postulates all positive integers will eventually reach one when a pair of specific operations are repeatedly applied. Despite its apparent simplicity, the conjecture lacks a formal proof. To tackle this enigma, we introduce Algebraic Inverse Trees (AITs), data structures that trace inverse operations of the Collatz sequence. This new approach not only elaborates our unique methodology but also sheds light on the underlying complexities of the Collatz Conjecture.

The Collatz conjecture (or 3n+1 problem) has been explored for about 86 years. In this article, we prove the Collatz conjecture. We will show that this conjecture holds for all positive integers by applying the Collatz inverse operation to the numbers that satisfy the rules of the Collatz conjecture. Finally, we will prove that there are no positive integers that do not satisfy this conjecture.

In this paper, the author derives an explicit form of Heaviside Step Function, which evidently constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics.In particular, this special function is exhibited in a very simple manner as a summation of four inverse tangent functions. The novelty of this work is that the proposed exact formulae are not performed in terms of miscellaneous special functions, e.g. Bessel functions, Error function, Beta function etc and also are neither the limit of a function, nor the limit of a sequence of functions with point – wise or uniform convergence.Hence, this formula may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

Karst aquifers show dissolution/precipitation processes of the minerals present in the carbonate rocks. The Ring of Cenotes (RC) extends along the edge of the Chicxulub crater, in the limestone platform of the Yucatan Peninsula (YP), where groundwater shows preferential flow paths toward the coast near Celestun and Dzilam Bravo towns. This study aimed to describe the regional hydrogeochemical evolution of groundwater of the RC, and its association with the dissolution/precipitation of the minerals present along its path to the ocean. To achieve this aim, we: a) characterized groundwater's hydrogeochemistry; b) determined the calcite, dolomite, and gypsum saturation indexes (reaction phases with the groundwater) in the study area; c) proposed a hydrogeochemical model developed through PHREEQC using an inverse modelling approach. The model predictions confirmed that there are two evolution pathways of the groundwater consistent with the preferential flow paths suggested in a previous regionalization of the RC. On the western path, where groundwater flows towards Celestun, an important marine intrusion influences the hydrogeochemical processes and represents a risk for the prevalence of freshwater. On the eastern path, where groundwater flows toward Dzilam Bravo, the hydrogeochemistry in the sinkholes correlates well with rainfall, suggesting a higher vulnerability during droughts than during rainy periods.

This paper describes the development of a controller that enables trajectory control and vibration control. The controller performance was verified the using a 3D 2-link, flexible manipulator. On trajectory control using inverse kinematics, it was confirmed that the deflection due to its own weight deteriorated the track following performance. The vibration component of the resonance frequency of the flexible manipulator was generated, and the tip position accuracy is deteriorated. Using the results of control experiments based on the inverse kinematics, the system is identified and then created an inverse system for simultaneous control of trajectory control and vibration control. The target trajectories were the three joint angles. Finally, it was demonstrated through experiments on actual manipulator, that the system could sufficiently follow the ideal trajectory and suppress link vibrations.

As many if not most ligands at G protein-coupled receptor antagonists are inverse agonists, we have systematically reviewed inverse agonism at the nine adrenoceptor subtypes. Except for β3-adrenoceptors, inverse agonism has been reported for each of the adrenoceptor subtypes, most often for β2-adrenoceptors, including endogenously expressed receptors in human tissues. As with other receptors, detection and degree of inverse agonism depends on the cells and tissues under investigation, i.e. is greatest when the model has a high intrinsic tone/constitutive activity for the response being studied. Accordingly, it may differ between parts of a tissue, for instance atria vs. ventricles of the heart, and within a cell type between cellular responses. The basal tone of endogenously expressed receptors often is low, leading to less consistent detection and smaller extent of observed inverse agonism. The extent inverse agonism depends on specific molecular properties of a compound but clusters by chemical class. While inverse agonism is a fascinating facet in attempts to mechanistically understand observed drug effects, we are skeptical whether an a priori definition of extent of inverse agonism in the target product profile of a developmental candidate is a meaningful option in drug discovery and development.

This paper investigates an inverse problem of option pricing in the extended Black--Scholes model. We identify the model coefficients from the measured data and attempt to find arbitrage opportunities in financial markets using a Bayesian inference approach. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by Markov Chain Monte Carlo (MCMC), which explores the posterior state space. The efficient sampling strategy of MCMC enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown drift and volatility coefficients from the measured data.

This paper investigates the inverse option problems (IOP) in the extended Black--Scholes model arising in financial markets. We identify the volatility and the drift coefficient from the measured data in financial markets using a Bayesian inference approach, which is presented as an IOP solution. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by a Markov Chain Monte Carlo (MCMC) algorithm, which exploits the posterior state space. The efficient sampling strategy of the MCMC algorithm enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown trend and volatility coefficients from the measured data.

Ultrafast pump-probe spectroscopic studies allow for deep insights into the mechanisms and timescales of photophysical and photochemical processes. Extracting valuable information from these studies, such as reactive intermediates' lifetimes and coherent oscillation frequencies, is an example of the inverse problems of chemical kinetics. This article describes a consistent approach for solving this inverse problem that avoids the common obstacles of simple least-squares fitting that can lead to unreliable results. The presented approach is based on the regularized Markov Chain Monte Carlo sampling for the strongly nonlinear parameters, allowing for a straightforward solution of the ill-posed nonlinear inverse problem. The software to implement the described fitting routine is introduced and the numerical examples of its application are given. We will also touch on critical experimental parameters, such as the temporal overlap of pulses and cross-correlation time and their connection to the minimal reachable time resolution.

Introduction: Depending on the kind of cancer and neurological disorders, there might be a direct or inverse link between neurodegeneration and cancer. The inverse interaction between cancer and Alzheimer's pathology has garnered substantial interest in medical research. While this inverse interaction between Alzheimer's pathology and cancer presents a fascinating scientific puzzle, there is still limited information and studies on the potential mechanisms involved in this inverse rela-tionship. Aim: This study aims to identify potential genes from Alzheimer's disease that may protect against cancer. Method: The Gene Expression Omnibus (GEO) database provided the Alzheimer's pathology dataset, and GEO 2R was utilized to examine genes that were differentially expressed. The expression of upregulated Alzheimer's pathology genes in breast, colorectal, and prostate cancer was compared using VENN 2.1, colorectal, and prostate cancer was compared using VENN 2.1, and colorectal and prostate cancers were compared using VENN 2.1. Results: Among the upregulated genes associated with Alzheimer's disease, 25 genes are shared among breast cancer, colorectal cancer, and prostate cancer, while 28 of these genes were exclusive to Alzheimer's dis-ease. Gene expression profiling interactive analysis among the shared genes revealed that AQP1, C4A, CSRP2, CRYAB, MACF1, HSPB2, and NEAT1 were downregulated in breast, colorectal, and prostate cancers. These genes have varying roles in cancer progression and suppression. Functional analysis of Alzheimer's disease-related upregulated genes revealed that C1orf56 and HIST1H2AC control cell proliferation, with HIST1H2AC playing a role in necroptosis. Additionally, C1orf56 worked as an oncogenic modifier to support DNMT3B's tumor suppressor properties. Conclusion: Given the role of HIST1H2AC and C1orf56 in the negative control of cell proliferation and their upregulation in Alzheimer's disease, there is a need to further investigate HIST1H2AC and C1orf56 as potential tumor suppressors responsible for Alzheimer's disease and cancer inverse association.

Using geometric considerations, we provide a clear derivation of the integral representation for the error function, known as the Craig formula. We calculate the corresponding power series expansion and prove the convergence. The same geometric means finally help to systematically derive handy formulas that approximate the inverse error function. Our approach can be used for applications in e.g. high-speed Monte Carlo simulations where this function is used extensively.

Diffuse optical tomography (DOT) is a medical imaging procedure using light to measure the geometric and working properties of cells, for instance, oxygen consumption, water content, and fat percentage in the tissue by performing three-dimensional visualization of the tissue. This paper aims to explain the theory behind diffuse optical tomography imaging and how the technology works. The paper explains how photon migration techniques based on diffusion theory can be used to image the optical properties of tissue. There are several reasons why near-infrared (NIR) imaging is the most effective method in terms of recovering optical parameters quantitatively in the near-infrared region. The author discusses the methods in detail. This research also presents various advantages, practical uses, and potential problems that have been related to DOT in this work. There is also a brief discussion of current research developments in medical imaging using near-infrared wavelengths, and what the future holds for that area.

Today, mechanical tracking systems have been downsized to allow them to be used in the field of airborne laser communications and in the military domain. Risley systems are used for this purpose, which work by directing a beam of light to a given target point, this procedure is commonly known as the inverse problem. In this paper, an analytical method, the geometric method, has been designed and developed to determine the beam steering in a Risley system and solve the inverse problem. The method focuses on different geometric shapes, like circumference or ellipse, that are described when the beam passes through the second prism. The accuracy and efficiency of the geometric method has been analysed and found to be faster than the two-step method. Furthermore, the geometric method has been implemented in an iterative process and an accuracy of up to 1 pm has been achieved. This high accuracy would allow the geometric method to be applied in fields such as lithography, stereolithography or 3D printers.

Sometimes a pavement deflects only because of seasonal volume change of expansive soils in the pavement subgrade. Engineering practitioners expect an implementable and straightforward analysis method for a geosynthetic-reinforced pavement subjected to the swelling/shrinkage issue of expansive clayey subgrade, in an effort to find the bending moment, shear force and tension force distributions through the reinforced pavement, which are induced from the volumetric changes of subgrade soils. The virtual load method (VLM) was proposed in the past following the Timoshenko beam theory to analyze geosynthetic-reinforced pavement on expansive soils. In the VLM, the unknown virtual distributed load was obtained in the way by applying the inverse theory for the identification of material parameters of the pavement-foundation system. It was seen that the selection of the number of material parameters to obtain virtual load significantly affects the accuracy of the structural properties of the pavement and tensile properties of the geosynthetics if the linear least square method is used. In this paper, a unique numerical scheme was proposed in the hopes of solving the issue. After a forward problem was solved numerically, the Timoshenko beam deflection was taken as a start-up for the inverse problem to back analyze the load applied to the pavement. Solutions from forward/backward examples have explicitly shown the accuracy achieved related to the bending moment in the pavement and tension in the geosynthetic reinforcements. The proposed methodology can be applied for an in-depth understanding of the geosynthetic function for the mitigation of longitudinal cracks on pavements caused by heave/shrinkage of expansive soils.

One important goal of research on heterogeneous catalysis for Fischer Tropsch synthesis (FTS) is to understand the connection linking catalyst morphology and its activity. Development of these effective catalysts highlights the potential for understanding the effectiveness of promoted supported catalysts and revealing their complex and dynamic effects in FTS. The metal-support interaction (MSI) is influenced by the physicochemical and textural features provided by the modified support and can directly influence the sizes of active metal crystallites, reactants and/or products mass transfer, dispersion, mechanical strength, stability and consequently the adsorption-desorption characteristics of the catalyst. Herein, this article explores the general strategies for modifying silica supports (achieved through doping, coating, or promoting the support or support precursor) and how these strategies can be manipulated to further alter catalyst performance and product selectivity by enhancing the amount of active metal species and improving the intrinsic turnover frequency (TOF). Furthermore, the review emphasises a new strategy (the inverse model of the silica support modification) aiming at overcoming the challenges posed by active metal coating or coverage, as well as pore blockages by promoters that occur during conventional preparation of promoted silica supported FT catalysts.

The spread of high-performance personal computers, frequently equipped with powerful Graphic Processing Units (GPU), raised the interest on a set of techniques able to extract models of electromagnetic phenomena (and devices) directly from available examples of the desired behavior. Such approaches are collectively referred to as Machine Learning (ML). A typical representative ML approach is the so called “Neural Network” (NN). Using such data-driven models allows evaluating the output in a much shorter time when a theoretical model is available, or allows predicting the behavior of the systems and devices when no theoretical model is available. With reference to a simple yet representative benchmark electromagnetic problem, some of the possibilities and the pitfalls of the use of NN for the interpretation of measurements (inverse problem) or to obtain the required measurements (optimal design problem) are discussed. The investigated aspects include the choice of the NN model; the generation of the dataset(s); the selection of hyperparameters (hidden layers, training paradigm). Finally, the capabilities in the handling of ill-posed problems are critically revised.

We propose a deep reinforcement learning based manipulator path tracking method to solve the computationally difficult and non-unique problem of manipulator path tracking methods based on inverse kinematics. By transforming the path tracking task into a sequence decision problem, our method adopts an end-to-end learning method for closed-loop control and avoids the process of finding the inverse solution. We first explored the feasibility of the deep reinforcement learning method in the path tracking of the manipulator. After verifying the feasibility, the path tracking of the multi-degree-of-freedom(multi-DOF) manipulator was realized by combining the maximum entropy deep reinforcement learning algorithm. The experimental results show that our method has a good effect on the path tracking of the manipulator, which not only avoids the process of finding the inverse kinematics solution, but also requires no dynamic model. Therefore, we believe that our method has great significance in the study of manipulator path tracking.

We propose an algorithm for generating a panoramic image of a pipe’s inner surface based on inverse perspective mapping (IPM). For efficient crack detection, we used only perspective images of the pipe interior that can be obtained with a single lens camera instead of those taken with a 360-degree camera or a 3D depth camera. Frontal images taken while passing through the pipe were converted to images of the inner surface of the pipe using IPM. We derived a generalized IPM formula that considers the slope of the image plane to correct image distortion caused by the tilt of the plane; this IPM formula was derived based on the vanishing point of the perspective image, which was detected using optical flow techniques. Finally, the multiple transformed images with overlapping areas were combined via image stitching to create a panoramic image of the inner pipe surface. To validate our proposed algorithm, we restored images of pipe inner surface using a 3D pipe model and used these images for crack detection.

In this paper, an analytical form of the Unit Step Function (or Heaviside Step function) is presented. This important function constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics. In particular, Heaviside Step Function is performed in a very simple manner by the use of a finite number of standard operations, as the summation of six inverse tangent functions. The novelty of this work when compared with other analytical expressions, is that the proposed exact formula contains two arbitrary single - valued continuous functions which satisfy only one restriction. In addition, the proposed representation is not exhibited in terms of miscellaneous special functions, e.g. Bessel functions, Beta function, Logistic function. Besides, it is neither the limit of a function, nor the limit of a sequence of functions with point – wise or uniform convergence. Hence, this formula may be much more practical, flexible and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

Ultrasound based haptic feedback is a potential technology for human-computer interaction (HCI) with the advantages of low cost, low power consumption and controlled force. In this paper, the phase optimization for multipoint haptic feedback based on ultrasound array is investigated and the corresponding experimental verification is provided. A mathematical model of acoustic pressure is established for the ultrasound array and then a phase optimization model for an ultrasound transducer is constructed. We propose a pseudo-inverse (PINV) algorithm to accurately determine the phase contribution of each transducer in the ultrasound array. By controlling the phase difference of the ultrasound array, the multipoint focusing forces are formed leading to various shapes such as geometries and letters that can be visualized. Because the unconstrained PINV solution results in unequal amplitudes for each transducer, a weighted amplitude iterative optimization is deployed to further optimize the phase solution, by which the uniform amplitude distributions of each transducer are obtained. For the purpose of experimental verifications, a platform of ultrasound haptic feedback consisting of a Field Programmable Gate Array (FPGA), an electrical circuit and an ultrasound transducer array is prototyped. The haptic performances of single point, multiple points and dynamic trajectory were verified by controlling the ultrasound force exerted on the liquid surface. The experimental results demonstrate that the proposed phase optimization model and theoretical results are effective and feasible, and the acoustic pressure distribution is consistent with the simulation results.

In this note we generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.

Accurate monthly runoff estimation is fundamental in water resources management, planning, and development, resulting in preventing and reducing water-related problems, such as flooding and drought. This article evaluates the performance of the monthly hydrological rainfall-runoff model, GR2M model, in Thailand's southern basins. The GR2M model requires only two parameters, and no prior research has been reported on its application in this region. The 37 runoff stations, which are distributively located in three sub-watersheds of Thailand's southern region, namely; Thale Sap Songkhla, Peninsular-East Coast, and Peninsular-West Coast, were selected as study cases. The available monthly hydrological data of runoff, rainfall, air temperature from the Royal Irrigation Department (RID) and the Thai Meteorological Department (TMD) were collected and analyzed. Thornthwaite method was utilized for the determination of evapotranspiration. The model's performance was conducted using three statistical indices: Nash-Sutcliffe Efficiency (NSE), Correlation Coefficient (r), and Overall Index (OPI). The model's calibration results for 37 runoff stations gave the average of NSE, r, and OPI of 0.637, 0.825, and 0.757, and those values for verification of 0.465, 0.750, and 0.639, respectively. It indicated a model's acceptable performance and could apply the GR2M model for determining monthly runoff variation in this region. The spatial distribution of X1 and X2 values was conducted by using IDW method. It was susceptible to the X1 value and X2 value of approximately more than 0.90 gave the higher model's performance.

We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0<\alpha<1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.

Background: Underreporting of Visceral Leishmaniasis (VL) in India remains a problem to public health controls. Effective and reliable surveillance systems are critical for monitoring disease outbreaks and public health control programs. However, in India, government surveillance systems are affected by levels of scarcity in resources and therefore, uncertainty surrounds the true incidence of asymptomatic and clinical cases, affecting morbidity and mortality rates. The State of Bihar alone contributes up to the 40\% of the worldwide VL cases. The inefficiency of surveillance systems occurs because of multiple reasons including delay in seeking health care, accessing non-authentic health care clinics, and existence of significant asymptomatic self healing infectious cases. This results in a failure of the system to adequately report true transmission rates and number of symptomatic cases that have sought medical advice (thus, high underreporting of cases). Objectives and Methods: There are several methods to estimate the extent of underreporting in the surveillance system. In this research, we use a mathematical dynamic model and two different types of data sets, namely, monthly incidence for 2003-2005 and yearly incidence from 2006-2012 from the Bihar's 21 most VL affected districts out of its 38 districts. The goals of the study are to estimate critical metrics to measure level of transmission and to evaluate the estimation process between the two data sets and 21 districts. In particularly, our focus is on (i) estimating infection transmission potential, underreporting level in incidence and proportion of self-healing cases, (ii) quantifying reproduction number of the$R_0$, and (iii) comparing underreporting incidence levels and proportion of self-healing cases between the two periods 2003-2005 and 2006-2012 and between 21 districts. Results: Our research suggests that the number of asymptomatic individuals in the population who eventually self-heal may have a significant effect on the dynamics of VL spread. The estimated mean self-healing proportion (out of all infected) is found to be $\sim 0.6$ with only 7 out of 21 affected districts having self-healing proportion less than $0.5$ for both data sets. The estimated mean underreporting level is at least $64$\% for the state of Bihar. The estimates of the basic reproduction numbers obtained are similar in magnitude for most of the districts, being in the range of (0.88, 2.79) and (0.98, 1.01) for 2003-2005 and 2006-2012, respectively. Conclusions: The estimates for the two types (monthly and yearly) of temporal data suggest that monthly data are better for estimation if less number of data points are available, however, in general, using such data set results in larger variances in parameters as compared to estimates obtained through aggregated yearly data. Estimated values of transmission related metrics are lower than those obtained from earlier analyses in the literature, and the implications of this for VL control are discussed. The spatial heterogeneity in these control metrics increases the risk of epidemics and makes the control strategies more complex.

Model-based image reconstruction has brought improvements in terms of contrast and spatial resolution to imaging applications such as magnetic resonance imaging and emission computed tomography. However, their use for pulse-echo techniques like ultrasound imaging is limited by the fact that model-based algorithms assume a finite grid of possible locations of scatterers in a medium -- which does not reflect the continuous nature of real world objects and creates a problem known as off-grid deviation. To cope with this problem, we present a method of dictionary expansion and constrained reconstruction that approximates the continuous manifold of all possible scatterer locations within a region of interest. The expanded dictionary is created using a highly coherent sampling of the region of interest, followed by a rank reduction procedure based on a truncated singular value decomposition. We develop a greedy algorithm, based on the Orthogonal Matching Pursuit (OMP), that uses a correlation-based non-convex constraint set that allows for the division of the region of interest into cells of any size. To evaluate the performance of the method, we present results of 2-dimensional ultrasound image reconstructions with simulated data in a nondestructive testing application. Our method succeeds in the reconstructions of sparse images from noisy measurements, providing higher accuracy than previous approaches based on regular discrete models.

Optimal control is a critical tool for mechanical robotic systems, facilitating precise manipulation of dynamic processes. These processes are described through differential equations governed by a control function, addressing a time-optimal problem with bilinear characteristics. Our study utilizes the classical approach complemented by Pontryagin's Maximum Principle (PMP) to explore this inverse optimal problem. The objective is to develop an exact piecewise control function that effectively manages trajectory control while considering the effects of viscous friction. Our simulations demonstrate that the proposed control law markedly diminishes oscillations induced by boundary conditions. This research not only aims to delineate the reachability set but also strives to determine the minimal time required for the process. The findings include an exact analytical solution for the stated control problem.

This article describes an approach for optimising sensor-controlled systems through minimal intervention, utilising Fuzzy Linear Systems of Equations (FLSE). Staring with a generalised model of the system behaviour, incorporating an array of control units, environmental sensors, and an expert knowledge base. The described problems of detecting the level of intervention needed to change the system state to another is handled with the help of developed methods for solving the inverse problem for FLSE. By achieving minimal intervention, we ensure that system adjustments effective, economically optimal and non-intrusive. A MATLAB-based implementation is presented.

A human eye has about 120 million rod cells and 6 million cone cells. This huge number of light sensing cells inside a human eye will continuously produce a huge quantity of visual signals which flow into a human brain for daily processing. However, the real-time processing of these visual signals does not cause any fatigue to a human brain. This fact tells us the truth which is to say that human-like vision processes do not rely on complicated formulas to compute depth, displacement, and colors, etc. On the other hand, a human eye is like a PTZ camera. Here, PTZ stands for pan, tilt and zoom. We all know that in computer vision, each set of PTZ parameters (i.e., coefficients of pan, tilt and zoom) requires a dedicated calibration to determine a camera’s projection matrix. Since there is an infinite number of PTZ parameters which could be produced by a human eye, it is unlikely that a human brain stores an infinite number of calibration matrices for each human eye. Therefore, it is an interesting question for us to answer, which is to say whether simpler formulas of computing depth and displacement exist or not. Moreover, these formulas must be calibration friendly (i.e., easy process on the fly or on the go). In this paper, we disclose an important discovery of a new solution to 3D projection in a human-like binocular vision system. The purpose of doing 3D projection in binocular vision is to undertake forward and inverse transformations (or mappings) between coordinates in 2D digital images and coordinates in a 3D analogue scene. The formulas underlying the new solution are accurate, easily computable, easily tunable (i.e., to be calibrated on the fly or on the go) and could be easily implemented by a neural system (i.e., a network of neurons). Experimental results have validated the discovered formulas.

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics and others. Due to its application in control theory, the matrix equation AXB=C has been extensively studied. However, there is currently limited information on matrix equation AXB=C over the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of the dual quaternion matrix equation AXB=C, and present the expression for the general solution when it is solvable. As an application, we derive the ϕ-Hermitian solutions for the dual quaternion matrix equation AXAϕ=C, where the ϕ-Hermitian extends the concepts of Hermiticity and η-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper.

In the broader landscape of cryptocurrency risk management, this study delves into the nuanced estimation of Value at Risk (VaR) for a uniformly weighted portfolio of cryptocurrencies, employing the bivariate Normal Inverse Gaussian distribution renowned for its semi-heavy tails. Utilizing high-frequency data spanning from January 1st, 2017, to October 25th, 2022, with a primary focus on Bitcoin and Ethereum, our research seeks to accentuate the resilience of VaR methodology as a paramount risk assessment tool. The essence of our investigation lies in advancing the comprehension of VaR accuracy by quantitatively comparing observed returns of both cryptocurrencies with their corresponding estimated values, with a central theme being the endorsement of the Normal Inverse Gaussian distribution as a potent model for risk measurement, particularly in the domain of high-frequency data. To bolster the statistical reliability of our results, we adopt a forward test methodology, showcasing not only a contribution to the evolution of risk assessment techniques in Finance but also underscoring the practicality of sophisticated distributional models in econometrics. Our findings not only contribute to the refinement of risk assessment methods but also highlight the applicability of such models in precisely modeling and forecasting financial risk within the dynamic realm of cryptocurrencies, epitomized by the case study of Bitcoin and Ethereum.

Hydroxyapatite and b-tricalcium phosphate have been clinically applied as artificial bone materials due to their high biocompatibility. The development of artificial bones requires the verification of safety and efficacy through animal experiments; however, from the viewpoint of animal welfare, there is necessary to reduce the number of animal experiments. In this study, we utilized machine learning to construct a model that estimates the bone-forming ability of bioceramics from material fabrication conditions, material properties, and in vivo experimental conditions. We succeeded in constructing the two models as follows: ‘Model 1’, which predicts material properties from their fabrication conditions, and ‘Model 2’, which predicts the bone-formation rate from material properties and in vivo experimental conditions. By an inverse analysis of the two models, we proposed candidates for material fabrication conditions to achieve target values of bone-formation rate. Under the proposed conditions, the material properties of the fabricated material were consistent with the estimated material properties. Furthermore, a comparison between bone-formation rates after 12 weeks of implantation in the porcine tibia and the estimated bone-formation rate. This result showed that the actual bone-formation rates existed within the error range of the estimated bone-formation rates, indicating that machine learning consistently predicts the results of animal experiments using material fabrication conditions. We believe that these findings will lead to the establishment of alternative animal experiments to replace animal experiments in the development of artificial bones.

Nanoparticles are of great importance for various scientific and technological applications, drug discovery, ecology, molecular biology, etc. Size is a basic characteristic of any particle, and dynamic light scattering is beyond competition when measuring particle size in dispersions ranging from nanometers to micrometers. Unfortunately, this method has a serious problem in processing and interpreting experimental data. Instrument noise and polydispersity of the sample lead to the situation when one initial autocorrelation function corresponds to several different particle size distributions. Sometimes these different solutions of the inverse task can be found by changing parameters in the software. Sometimes the scientist cannot be suspicious of possible other solutions, having one result from the program. In this article I demonstrate the problem on model and experimental data using three known programs: CONTIN, the Malvern Zetasizer and DynaLS. I also present my own free program Autocor, which allows to test our own hypotheses about the modality and shape of the particle size distribution by optimizing the parameters of our own models with a small number of degrees of freedom.

Mercury (Hg (II)) contamination is indefatigable global hazard which causes severe permanent damages to human health. Extensive research has been carried out to produce mercury adsorbents however they are still facing certain challenges limiting their upscaling. Herein we reported the synthesis of novel amine impregnated inverse vulcanized copolymer for effective mercury removal. Poly(S-MA) was prepared using sulfur and methacrylic acid employing inverse vulcanization method following by functionalization. The polyethylenime (PEI) was impregnated on poly(S-MA) to increase the adsorption active sites. The developed adsorbent was then characterized using Fourier transform infrared spectroscopy (FTIR) and scanning electron microscopy (SEM). FTIR confirmed formation of copolymer and successful impregnation of PEI and SEM revealed composite porous morphology of the copolymer. Amine impregnated copolymer (amine@poly(S-MA)) outperformed poly(S-MA) in mercury as it showed 20 % superior performance with 44.7 mg/g maximum adsorption capacity. The adsorption data best fits the pseudo second order which indicates that the chemisorption is most influential mechanism in this case indicating the involvement of NH2 in mercury removal. The adsorption is mainly monolayer on homogenous surface as indicated by 0.76 value of Redlich-Peterson exponent (g) which describes the adsorption nature advent from R2 value of 0.99.

This article proposes application of a new mathematical model of spots for solving inverse problems using a learning method, which is similar to using the deep learning. In general, the spots represent vague figures in abstract “information spaces” or crisp figures with lack of in-formation about their shapes and are adequate for representation human mental images and reasoning in Artificial Intelligence (AI). However, crisp figures are regarded as a special and limiting case of spots. A basic mathematical apparatus, basing on L4 numbers, has been developed for the representation and processing of qualitative information of elementary spatial relations between spots. Also, we defined L4 vectors, L4 matrices, and mathematical operations on them. Developed apparatus can be used in AI, in particular, for knowledge representation and for modeling qualitative reasoning and learning. Another application area is the solution of inverse problems by learning. For example, this can be applied to image reconstruction using ultrasound, X-ray, magnetic resonance, or radar scan data. The introduced apparatus was verified by solving problems of reconstruction of images, utilizing only qualitative data of its elementary relations with some scanning figures. This article also demonstrates application of spot-based inverse Radon algorithm for binary image reconstruction.

Model predictive control (MPC) is a multi-objective control technique that can handle 2 system constraints. However, the performance of an MPC controller highly relies on a proper 3 prioritization weight for each objective, which highlights the need for a precise weight tuning 4 technique. In this paper, we propose an analytical tuning technique by matching the MPC 5 controller performance with the performance of a linear quadratic regulator (LQR) controller. The 6 proposedmethodology derives the transformation of LQRweightingmatrixwith a fixedweighting 7 factor using discrete algebraic Riccati equation (DARE) and designs MPC controller using the 8 idea of discrete time linear quadratic tracking problem (LQT) in the presence of constraints. 9 The proposed methodology ensures the optimal performance between unconstrained MPC and 10 LQR controller and provides a sub-optimal solution while the constraints are active during 11 transient operations. The resultingMPC behaves as the discrete time LQR by selecting appropriate 12 weighting matrix in the MPC control problem and ensures the asymptotic stability of the system. 13 In this paper, the effectiveness of the proposed technique is investigated in the application of 14 a novel vehicle collision avoidance system which is designed in the form of linear inequality 15 constraints within MPC. The simulation results confirm the potency of the proposed MPC control 16 technique in performing a safe, feasible and collision free path while respecting the inputs, states 17 and collision avoidance constraints.

This note addresses a fundamental problem in matrix theory on establishing and characterizing range equalities for matrix expressions that involve generalized inverses. We first establish a group of necessary and sufficient conditions for the matrix range equality ${\rm range}(D_1 - C_1A_1^{\dag}B_1) = {\rm range}(D_2 - C_2A_2^{\dag}B_2)$ to hold, where $(\cdot)^{\dag}$ denotes the Moore--Penrose inverse of matrix. We then give several groups of range equalities with extrusion properties for multiple matrix products associated with two matrices and their conjugate transposes and Moore--Penrose inverses.

Crystallization is a significant procedure in the manufacturing of many pharmaceutical and solid food products. In-situ Electrical Resistance Tomography (ERT) is a novel Process Analytical Tool (PAT) to provide a cheap and quick way to test, visualize, and evaluate the progress of crystallization processes. In this work, the spatial accuracy of the non-conductive phantoms in low conductivity solutions was evaluated. Gauss-Newton, Linear Back Projection, and iterative Total Variation reconstruction algorithms were used to compare the phantom reconstructions for tap water, industrial-grade saturated sucrose solution, and demineralized water. Cylindrical phantom measuring 10 mm in diameter and a cross-section area of 1.5 % of the total beaker area was detected at the center of the beaker. Two phantoms with a 10 mm diameter were visualized separately in non-central locations. The quantitative evaluations were done for the phantoms with radii ranging from 10 mm to 50 mm in demineralized water. Multiple factors such as ERT device and sensor development, FEM mesh density and simulations, image reconstruction algorithms, number of iterations, segmentation methods, and morphological image processing methods were discussed and analyzed to achieve spatial accuracy. The development of ERT imaging modality for the purpose of monitoring crystallization in low conductivity solutions was performed satisfactorily.

This note shows that the well-known reverse order law $(AB)^{\dag} = B^{\dag}A^{\dag}$ for the Moore--Penrose inverse of matrix product is equivalent to many other equalities for that are composed of multiple products $(AB)^{\dag}$ and $B^{\dag}A^{\dag}$ by means of the definition of the Moore--Penrose inverse and orthogonal projector theory.

In the present work, we evaluate the prediction capability of six Pedotransfer functions (PTFs), reported in the literature, for the saturated hydraulic conductivity estimations (Ks). We used a database with 900 measured samples obtained from the Irrigation District 023, in San Juan del Rio, Queretaro, Mexico. Additionally, six new PTFs were construct for Ks from clay percentage, bulk density and saturation water content data. The results show, for the evaluated models, that one model present an overestimation for Ks>0.5 cm h-1 values, three models have a underestimation for Ks>1.0 cm h-1 and two models have a good correlation (R2>0.98) but are necessary more than three parameters. Nevertheless, the last two models requires from three to four parameters in order to get the optimization. By other hand, the models proposed in this work have a similar correlation with a less number of parameters: the fit is seen to be much better than using the existing ones, achieving a correlation of R2 = 0.9822 with only one variable and a R2 = 0.9901 when we use two.

One of the typical forms of linear matrix expressions (linear matrix-valued functions) is given by $A + B_1X_1C_1 + \cdots + B_kX_kC_k$, where $X_1, \ldots, X_k$ are independent variable matrices of appropriate sizes, which include almost all matrices with unknown entries as its special cases. The domain of the matrix expression is defined to be all possible values of the matrix expressions with respect to $X_1, \ldots, X_k$. I this article, we approach some problems on the relationships between the domains of two linear matrix expressions by means of the block matrix method (BMM), the matrix rank method (MRM), and the matrix equation method (MEM). As application, we discuss some topics on the relationships among general solutions of some linear matrix equations and their reduced equations.

The development and generalization of Digital Volume Correlation (DVC) on X-ray computed tomography data highlight the issue of long term storage. The present paper proposes a new model-free method for pruning the DVC data. The size of the remaining sampled data can be user-defined, depending on the needs concerning storage space. The data pruning procedure is deeply linked to hyper-reduction techniques. The DVC data of a resin-bonded sand tested in uniaxial compression is used as an illustrating example. The relevance of the pruned data is tested afterwards for model calibration. A new Finite Element Model Updating (FEMU) technique coupled with an hybrid hyper-reduction method is used to successfully calibrate a constitutive model of the resin bonded sand with the pruned data only.

We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schrödinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions when using different numbers of width-optimized IMQ functions. The effects of the ratio of the number of collocation points to the number of basis functions and of the choice of the IMQ exponent are studied. We show that the IMQ basis can be used with parameters where the IMQ function is not integrable. We find that the quality of the spectrum with IMQ basis functions is somewhat lower that that with a Gaussian basis when the basis size is large and for a range of IMQ exponents. The IMQ functions are, however, advantageous when a small number of functions is used or with a small number of collocation points e.g. when using square collocation.

The active power-assist function greatly expands the potential applications of exoskeleton robots, yet the motion intention estimation (MIE) for active power-assist strategy is quite problematic. Through the analysis of the conduction path and the different stage manifestations of motion intention in human body, we confirmed that the joint torque of human body meets the basic requirements of MIE for the active power-assist that we suggest, namely: (i) it reflects the direction and intensity of the wearer’s efforts; (ii) it precedes the human limb motion; (iii) it generates real-time and continuous output. Thus, an online calculation method of human joint torque was proposed. The sensing system integrated in exoskeleton robots was designed to perceive motion data and foot contact force of a human body. A special inverse dynamics with a parameterized model of the human body was proposed. Contrast experiments were carried out with the motion capture system, which results’ accuracy and similarity were evaluated via the root mean square error and correlation coefficient. The comparative analysis of two synchronous results shows good accuracy of the proposed MIE method, which lays the foundation for the realization of active power-assist.

The correct selection of the value of p is a complex and iterative procedure that requires experience in the interpretation of the obtained interpolated maps. Inverse Distance Weighting is a method applied to the porosities of the K and L hydrocarbon reservoirs, discovered in the Neogene (Lower Pontian) subsurface sandstones in the northern Croatia (Pannonian Basin System). They represent small and large data samples. Also a standard statistical analysis of the data was made, followed by the qualitative-quantitative analysis of the maps, based on the selection of different values of the power distance exponent (p-value) for the K and L reservoirs maps. According to the qualitative analysis, for a small data set, p-value could be set on 1 or 2 giving the most optimal result, while for a large data set, a p value of 3 and 4 could be applied. By quantitative analysis, for the case of small data p=2 is recommended, resulting in a root mean square error value of 0.03458, a mean absolute error of 0.02013 and a median absolute deviation of 0.00546. Oppositely, a p-value of 3 and 4 is selected as appropriate for a large data set, with root mean square errors of 0.02435 and 0.02437, mean square errors of 0.01582 and 0.01509 and median absolute deviations 0.00896 and 0.00444. Eventually for a small data set, it is recommended to use a p-value of 2, and for a large data set, a p-value of 3 or 4.

Applications of gradient method for nonlinear optimization in development of Gradient Neural Network (GNN) and Zhang Neural Network (ZNN) are investigated. Particularly, the solution of the matrix equation AXB=D which changes over time is studied using the novel GNN model, termed as GGNN(A,B,D). The GGNN model is developed applying GNN dynamics on the gradient of the error matrix used in the development of the GNN model. The convergence analysis shows that the neural state matrix of the GGNN(A,B,D) design converges asymptotically to the solution of the matrix equation AXB=D, for any initial state matrix. It is also shown that the convergence result is the least square solution which is defined depending on the selected initial matrix. A hybridization of GGNN with analogous modification GZNN of the ZNN dynamics is considered. The Simulink implementation of presented GGNN models is carried out on the set of real matrices.

Application of the Ramsey Theory to the set of the points placed in the plane is discussed. Consider a set of N points located in the same plane. The straight lines connecting these points are yikx=αikx+βik i,k=1...N. Listed below values of slopes αik are possible: αik&gt;0; αik=0; αik is not defined; αik&lt;0. Following coloring procedure is introduced: we connect the pairs of points for which αik&gt;0, αik=0 or αik is not defined, with the red links, and the pairs of points for which αik&lt;0 place with green links. The suggested coloring procedure enables building of the complete bi-colored graph for any set containing N points located in the same plane. We apply the Ramsey theorem to the complete graph emerging from the suggested coloring. For the set containing N=6 points at least one monochromatic triangle will necessarily appear in the graph. The values of the slopes αik depend on the chosen coordinate system. The rotation of coordinate axes changes the coloring of the graph; however, at least one monochromatic triangle will be present for N=6. The introduced coloring procedure diminishes the order of the symmetry group of the regular hexagon, irrespectively to the orientation of the coordinate axes. We hypothesized that this will be true for arbitrary regular n-polygon, independently on the orientation of coordinate axes. The inverse bi-color Ramsey graphs arise, when we replace red links appearing in the source graph with red ones, and vice versa. The total number of triangles in the “direct” and “inverse” Ramsey graphs is the same. We considered the particular case of the set of points for which αik&gt;0 or αik&lt;0 is true in the given coordinates. In this particular case, the rotation of the Cartesian coordinate axes to the angle θ=π2 yields the inverse complete graph. Generalization of this coloring for an arbitrary number and arbitrary location of the source points is introduced.

In this note we study the computation of the minimum polynomial of a matrix $A$ and how we can use it for the computation of the matrix $A^n$. We also describe the form of the elements of the matrix $A^{-n}$ and we will see that it is closely related with the computation of the Drazin generalized inverse of $A$. Next we study the computation of the exponential matrix and finally we give a simple proof of the Leverrier - Faddeev algorithm for the computation of the characteristic polynomial.

We present an improved algorithm for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithm is much more accurate than the famous fast inverse square root algorithm and has a similar computational cost. The presented modification concern Newton-Raphson corrections and can be applied when the distribution of these corrections is not symmetric (for instance, in our case they are always negative).

In this research, a novel method to investigation the transient heat transfer coefficient in a channel is suggested experimentally, in which the water flow, itself, is considered both just liquid phase and liquid-vapor phase. The experiments were designed to predict the temporal and spatial resolution of Nusselt number. The inverse technique method is non-intrusive, in which time history of temperature is measured, using some thermocouples within the wall to provide input data for the inverse algorithm. The conjugate gradient method is used mostly as an inverse method. The temporal and spatial changes of heat flux, Nusselt number, vapor quality, convection number, and boiling number have all been estimated, showing that the estimated local Nusselt numbers of flow for without and with phase change are close to those predicted from the correlations of Churchill and Ozoe (1973) and Kandlikar (1990), respectively. This study suggests that the extended inverse technique can be successfully utilized to calculate the local time-dependent heat transfer coefficient of boiling flow.

Here are analysed data taken in two hydrocarbon fields ("A" and "B"), located in the western part of Sava Depression (North Croatia). They are in the secondary phase of production. The selected reservoirs "L" (in the “A” Field) and "K" (“B”) are of the Lower Pontian (Upper Miocene) age and belong to Kloštar-Ivanić Formation. Due to strong tectonics, there are numerous tectonic block, relatively rarely sampled with well and laboratory tests. Here are selected two variables for interpolation - reservoirs permeabilities and the injected volumes of field water. The following interpolation methods are described, compared and applied: Nearest Neighbourhood, Natural Neighbour (the first time in the Sava Depression) and Inverse Distance Weighting. The last one has been proven as the most appropriate for datasets with size lower than 20 points.

Different process parameters can alter the temperature during machining. Consequently, selecting process parameters that lead to a desirable cutting temperature would help to increase the tool life, decrease the tensile residual stress, and controls the microstructure evolution of the workpiece. An inverse computational methodology is proposed to design the process parameters for specific cutting temperature. A physics-based analytical model is used to predict the temperature induced by cutting forces. To calculate the temperature induced by the deformation in the shear zone, a moving point heat source approach is used. The shear deformation and chip formation model is implemented to calculate machining forces as functions of process parameters, material properties, and etc. The proposed model uses the analytical model to predict the cutting temperatures and applies a variance-based recursive method to guide the inverse analysis. In order to achieve the cutting process parameters, an iterative gradient search is used to adaptively approach the specific temperature by the optimization of process parameters such that an inverse reasoning can be achieved. Experimental data are used to illustrate the implementation and validate the viability of the computational methodology.

Schrödinger dynamics is a nonlocal process. Not only does local perturbation affect instantaneously the entire space, but the effect decays slowly. When the wavefunction is spectrally bounded, the Schrödinger equation can be written as a universal set of ordinary differential equations, with universal coupling between them, which is related to Euler’s formula. Since every variable represents a different local value of the wave equation, the coupling represents the dynamics’ nonlocality. It is shown that the nonlocal coefficient is inversely proportional to the distance between the centers of these local areas. As far as we know, this is the first time that this inverse square law was formulated.

The natural compounds PSK and PSP have antitumor and immunostimulant properties. These pharmacological benefits have been documented in vitro and in vivo, although there is not information in silico which describes the action mechanisms at molecular level. In this study, the inverse docking method was used to identify the interactions of PSK and PSP with two local databases: BPAT with 66 antitumor proteins, and BPSIC with 138 surfaces and intracellular proteins. This led to the identification interactions and coincidences of PSK and AB680 inhibitor in the active site of CD73. It was also found that PSK bonds to CD59 interacting with amino acids APS22 and PHE23, which coincide with the rlLYd4 internalization inhibitor. With the isoform of K-RAS protein, PSK bonded to the TYR32 amino acid at switch 1, while with BAK it bonded to the region of the α1 helix, while PSP bonded to the activation site and the C-terminal and N-terminal ends of that helix. In Bcl-2, PSK interacted at the binding site of the Venetoclax inhibitor, showing coincidences with amino acids ASP111, VAL133, LEU137, MET115, PHE112, and TYR108; while PSP had coincidences with THR132, VAL133, LEU137, GLN118, MET115, APS111, PHE112, and PHE104.

In this paper, we propose a system of nonlinear mixed variational inequality problems, which consists of two elliptic mixed variational inequality problems on Banach spaces. Under suitable assumptions, using the Kakutani-Ky Fan fixed point theorem and Minty techniques, we prove the solution set to the system of nonlinear mixed variational inequality problem is nonempty, weakly compact and unique. Additionally, we suggest a stability result for the system of nonlinear mixed variational inequality problem by perturbing the duality mappings. Furthermore, we present an optimal control problem governed by the system of nonlinear mixed variational inequality problems and establish a solvability result.

Living systems must solve the problem of realizing their environment to act appropriately to maintain external relationships and internal order for survival and reproduction. This problem is a biological version of the philosophical enigma of how the self can escape solipsism. This study develops a comprehensive model to solve this adaptation problem. LSs must be composed of material entities that have the property of detecting their external states. The detection of the outside is conceptualized as “cognition,” a related state change that occurs at three levels of nested hierarchy: physical, chemical, and semiotic cognitions. Any entity that cognizes the surroundings is a “cognizer.” The hierarchical extension of cognition to all levels of the world provides a monistic view that only cognizers exist. Semiotic cognition is essential for the emergence and evolution of life. Living systems invented semiotic cognitions based on physical and chemical cognitions to manage the probability distribution of events that occur to them. This study proposes a theoretical model in which semiotic cognition is an adaptive process in which inverse causality operations produce particular internal states as symbols that signify hidden external states. This operation makes the living systems aware of the external world.

Digital twin, an innovative technology propelled by data and models, plays a seminal role in the digital transformation and intelligent upgrade of ships. This study introduces a digital twin methodology for real-time monitoring of ship structure deformation fields, based on finite discrete strain data, and a visualization tool framework is developed using virtual reality technology. First, the inverse Finite Element Method (iFEM) is employed to derive the deformation field of the ship structure in real time via sensor strain data. Secondly, the deformation field data obtained based on the iFEM algorithm is converted into general visualization data conducive to interpretation within VR applications. Lastly, a digital twin software tool is built to enable synchronous response and interaction between the virtual scene and the physical scene, directly superposing particular virtual objects (data acquired by sensors, CAD virtual models, and deformation field cloud images) onto the physical scene in real-time. The digital twin tool embodies a virtual reality visualization system framework integrating physical data measurement, reconstruction, analysis, expression, storage, rendering and interaction of deformation field data. Through practical application, the flexibility, effectiveness and compatibility of the developed prototype tool are verified. According to the results, the system can enhance the efficiency of scientific communication, model validation, and interdisciplinary sharing during the analysis and evaluation of mechanical properties of ship structures.

Inverse optimal control is a method for recovering the cost function used in an optimal control problem in expert demonstrations. Most studies on inverse optimal control have focused on building the unknown cost function through the linear combination of given features with unknown cost weights, which are generally considered to be constant. However, in many real-world applications, the cost weights may vary over time. In this study, we propose an adaptive online inverse optimal control approach based on a neural-network approximation to address the challenge of recovering time-varying cost weights. We conduct a well-posedness analysis of the problem and suggest a condition for the adaptive goal, under which the weights of the neural network generated to achieve this adaptive goal are unique to the corresponding inverse optimal control problem. Furthermore, we propose an updating law for the weights of the neural network to ensure the stability of the convergence of the solutions. Finally, simulation results for an example linear system are presented to demonstrate the effectiveness of the proposed strategy. The proposed method is applicable to a wide range of problems requiring real-time inverse optimal control calculations

Diffuse optical tomography (DOT) is an optical medical imaging method that can assess the structural properties of tissues and their functional characteristics, such as hemoglobin concentration, water content, as well as lipid concentration, via three-dimensional image reconstruction. This paper presents the theoretical basis and working principle of diffuse optical tomography. The paper explains how the optical properties of tissue can be imaged by photon migration techniques based on diffusion theory. The author presents the reasoning for near-infrared (NIR) imaging as the most effective technique in terms of quantitative recovery of spectroscopic optical parameters. This work also listed various advantages, applications, and challenges of DOT. The paper also briefly discusses current progress in near-infrared medical imaging and its future direction.

One of matrix equalities composed of multiple products of matrices and their generalized inverses is given by $A_1B_1^{-}A_2B_2^{-} \cdots A_kB_k^{-}A_{k+1}= A$ where $A_1$, $B_1$, $A_2$, $B_2$, $\ldots$, $A_k$, $B_k$, $A_{k+1}$, and $A$ are given matrices of appropriate sizes, and $B_1^{-}$, $B_2^{-}$, $\cdots$, $B_k^{-}$ are generalized inverses of matrices. The cases for $k = 1, 2$ and their special forms were properly approached in the theory of generalized inverses of matrices. In this note, the author presents an algebraic procedure to derive explicit necessary and sufficient conditions for the equality with $k = 3$ to always hold using certain rank equalities for the block matrices constructed by the given matrices, and then mention a key step of extending the previous work to a general situation.

Automatic control to any of robot manipulators, some kind of issues are being observed. A numerical method for solution generation to the inverse kinematics problem of redundant robotic manipulators is presented to obtain the smoothest algorithm as possible, leading to a robust iterative method. After the primary objective of the reachability of end-effectors to the target point is achieved, the aim is set to resolve the redundant degrees of freedom of redundant manipulator. This method is numerically stable since it converges to the correct answer with virtually any initial approximation, and it is not sensitive to the singular configurations of the manipulator. In addition, this technique is computationally effective and able to apply for serial manipulators with any DOF applications. A planar 3R-DOF serial link redundant manipulator is considered as exemplar problem for solving. Also, the continuum approach for resolving more complex structure with variable DoF is illustrated here and their brief applicability to support surgeries and adaptive use of artificial linkage moments is also calculated.

Reverse order laws for generalized inverses of matrix products is a classic object of study in the theory of generalized inverses. One of the well-known reverse order laws for a matrix product $AB$ is $(AB)^{(i,\ldots,j)} = B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$, where $(\cdot)^{(i,\ldots,j)}$ denotes an $\{i,\ldots, j\}$-generalized inverse of matrix. Because $\{i,\ldots, j\}$-generalized inverse of a singular matrix is unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. This paper provides a thorough coverage of the reverse order laws for $\{i,\ldots, j\}$-generalized inverses of $AB$, from the development of background and preliminary tools to the collection of miscellaneous formulas and facts on the reverse order laws in one place with cogent introduction and references for further study. We begin with the introduction of a linear mixed model $y = AB\beta + A\gamma + \epsilon$ and the presentation of two least-squares methodologies to estimate the fixed parameter vector $\beta$ in the model, and the description of connections between the two types of least-squares estimators and the reverse order laws for generalized inverses of $AB$. We then prepare some valued matrix analysis tools, including a general theory on linear or nonlinear matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$, as well as necessary and sufficient conditions for $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$ to be invariant with respect to the choice of $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$. We then present a unified approach to the 512 matrix set inclusion problems associated with the above reverse order laws for the eight commonly-used types of generalized inverses of $A$, $B$, and $AB$ through use of the definitions of generalized inverses, the block matrix method (BMM), the matrix rank method (MRM), the matrix equation method (MEM), and various algebraic calculations of matrices.

Interpolation is procedure that depends on spatial and/or statistical properties of analysed variable(s). It is special challenging task for data that included low number of samples, like dataset with less than 20 data. This problem is especially emphasized in the subsurface geological mapping, i.e. in the cases where data are taken solely from wells. Successful solutions of such mapping problems ask for knowledge about interpolation methods designed primarily for small datasets and dataset itself. Here are compared two methods, namely Inverse Distance Weighting and Modified Shepard’s Method, applied for three variables (porosity, permeability, thickness) measured in the Neogene sandstone hydrocarbon reservoirs (Northern Croatia). The results showed that pure cross-validation is not enough condition for appropriate map selection, but also geometrical features need to be considered, for datasets with less than 20 points.

Based on the Graph Model of Conflict Resolution (GMCR), a two-stage decision framework is developed to reveal the essence of brownfield incidents and facilitate the resolution of brownfield conflicts caused by the incidents. More particularly, the forward GMCR is utilized in Stage I, the negotiation stage, to simulate the evolution of a brownfield conflict (BC) and predict its potential resolution via stability analysis. If no acceptable equilibrium can be obtained, the BC progresses into Stage II, the third-party-intervention stage, where the inverse GMCR is used to assist a third party in intervening the conflict to achieve a desirable outcome. To illustrate the practicality of this framework, a recent BC occurred in Changzhou, China, is taken as a case study. Invaluable insights are provided through the computation and investigation of the corresponding preference relationships.

The article presents a cyber-physical system for acquiring, processing and reconstructing images from measurement data. The technology was based on process tomography, intelligent measurement sensors, machine learning, Big Data, Cloud Computing, Internet of Things as a solution for Industry 4.0. Industrial tomography enables observation of physical and chemical phenomena without the need of internal penetration and allows real-time monitoring of production processes. The application includes specialized intelligent devices for tomographic measurements and dedicated algorithms for solving the inverse problem. The work focuses mainly on electrical tomography and image reconstruction using deterministic methods and machine learning, the reconstruction results were compared, different measurement models were used. The researches were carried out for synthetic data and laboratory measurements. The main advantage of the proposed system is the possibility of spatial data analysis and their high processing speed. The presented research results show that the process tomography gives the possibility to analyse the processes taking place inside the facility without disturbing the production, analysis and detection of obstacles, defects and various anomalies. Knowing the characteristics of a given solution, the application allows you to choose the appropriate method to reconstruct the image.

The article presents a non-destructive test system based on electrical impedance tomography for monitoring flood embankments. The technology of cyber-physical systems and the Internet of Things with the use of electrical impedance tomography enables real-time monitoring of flood embankments. This solution provides a visual analysis of damage and leaks, which allows for quick and effective intervention and possible prevention of danger. A dedicated solution based on the IT structure, dedicated laboratory models and a dedicated measurement system with various types of sensors and machine learning algorithms for image reconstruction has been developed. The system includes specialized intelligent devices for tomographic measurements. The application contains the analysis of anomalies occurring in the structure of the object as a result of damage or danger and breaking the shaft during the flood. The presented solution enables ongoing monitoring of objects by collecting measurement results, forecasts and simulations. The main advantage of the proposed system is the spatial ability to analyse shafts, high accuracy of imaging and high speed of data processing. The use of tomographic techniques in conjunction with image reconstruction algorithms allow for non-invasive and very accurate spatial assessment of humidity and damages of flood embankments. The presented results show the effectiveness of the presented research.

The article presents the results of research on a new method of spatial analysis of walls and buildings moisture. Due to the fact that destructive methods are not suitable for historical buildings of great architectural significance, a non-destructive method based on electrical tomography has been adopted. A hybrid tomograph with special sensors was developed for the measurements. This device enables the acquisition of data, which are then reconstructed by appropriately developed methods enabling spatial analysis of wet buildings. Special electrodes that ensure good contact with the surface of porous building materials such as bricks and cement were introduced. During the research, a group of algorithms enabling supervised machine learning was analyzed. They have been used in the process of converting input electrical values into conductance depicted by the output image pixels. The conductance values of individual pixels of the output vector made it possible to obtain images of the interior of building walls, both flat intersections (2D) and spatial (3D) images. The presented group of algorithms has a high application value. The main advantages of the new methods are: high accuracy of imaging, low costs, high processing speed, ease of application to walls of various thickness and irregular surface. By comparing the results of tomographic reconstructions, the most efficient algorithms were identified.

The paper proposes an approach to solving the inverse epidemiological problem, written in terms of the “mean field” theory. Finding the coefficients of the epidemiological SIR mean field model is reduced to solving an optimization problem, for the solution of which only zero-order methods can be used. The Nelder-Mead method was chosen as such a method. The algorithm for solution of the inverse coefficient problem is proposed. Computational experiments were carried out to compare the obtained solutions with respect to synthetic and real data. The results of computational experiments have shown the efficiency of this approach. Ways to further improve the approach have also been determined.

Laboratory experiments and investigations into natural phenomena in this research series have unveiled the presence of a gravitational repulsion force that permeates our environment on both microscopic and macroscopic scales, contingent upon the thermal energy present. This paper presents a study exploring novel properties of gravitational forces among gas molecules, employing principles of thermodynamics. A model has been devised based on the interactions between pairs of gas molecules. While traditional models treat gravitational interaction as a singular force, our experimental approach validates it as the composite of two distinct forces: gravitational repulsion and attraction. By utilizing established experimental data on gas thermodynamics, our model demonstrates robust performance both analytically and experimentally. It verifies the coexistence of gravitational repulsion and attraction forces among gas molecules, showcases their Inverse-Cube relationship with distance, and elucidates the direct proportionality of the repulsion force to absolute temperature. This bridges a crucial gap between energy and fundamental forces. The order of magnitudes of gravitational repulsion and attraction forces are found to be considerably large, contrasting with the low values predicted by classical theory, which results from their amalgamation. Recognizing these forces as substantial in magnitude promises unprecedented outcomes and advancements.

The available direct gas dynamic tables provide the area ratio and the Prandtl-Meyer(P-M) angle of isentropic expansion of gas as functions of the Mach number. But finding the value of the Mach number in terms of area ratio or P-M angle requires an inverse function. Currently, the Mach number is determined by either interpolation from the direct gas dynamics tables, or the application of non-linear equation solvers using numerical methods. Both methods are approximate. In this paper inverse functions for Mach number in the form of Taylor’s series expansions are reported to develop the so-called inverse tables. The advantage of this approach is that the Mach number with a greater accuracy, referred to as nearly exact can be obtained by reading from the inverse table. In the presentation, the necessary equations and new tables that can be used by the scientific community will be disseminated. The data and computer programs are made available on GitHub through https://github.com/anillals/Inverse-gas-dynamic-tables- and-codes. A paper describing the complete mathematical steps in the derivation of the equations can be found at https://rdcu.be/dgrlf

In this paper, we introduce as a pedagogical strategy an internal division-free, straightforward, and symmetrically progressing algorithm in manually computing matrix inverse and solving systems of linear equations by revisiting the application of elementary row operations in the Gauss-Jordan reduction method and connecting it to the determinant method. The proposed cross-multiplication-flip (CMF) algorithm employs cross-multiplication similar to the butterfly movement in computing determinants as a strategic application of elementary row operations to efficiently reduce the rows and then applies flipping of rows and entries to put an upper triangular matrix into lower triangular form to continue the reduction process.

In this paper, under the symmetric entropy and the scale squared error loss functions, we consider the maximum likelihood (ML) estimation and Bayesian estimation of the Shannon entropy and Rényi entropy of the two-parameter inverse Weibull distribution. In the ML estimation, the dichotomy is used to solve likelihood equation. In addition, the approximation confidence interval is given by the Delta method. Because the form of estimation results is more complex in the Bayesian estimation, the Lindley approximation method is used to achieve the numerical calculation. Finally, Monte Carlo simulations and a real data set are used to illustrate the results derived. By comparing the mean square error between the estimated value and the real value, it can be found that the performance of ML estimation of Shannon entropy is better than that of Bayesian estimation, and there is no significant difference between the performance of ML estimation of Rényi entropy and that of Bayesian estimation.

Inverse problems of recovering surface fluxes on the boundary of a domain from pointwise observations are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness conditions on the data, geometric conditions on the location of measurement points, and the boundary of a domain. The proof relies on asymptotics of fundamental solutions to the corresponding elliptic problems and the Laplace transform. The problem is reduced to a linear algebraic system with a nondegerate matrix.

As an implementation of inverse Radon transform, the Second-order Divided-difference Back Projection (SDBP) technique has been proposed in this paper, by which the accurate CT reconstruction can be theoretically realized. A computational model for CT reconstruction filter expressions has been derived on the basis of SDBP technique. The method reveals the correlation between the convolution kernel for data restoration and the filter expression. By substituting kernel functions into the computational model, the amplitude sequences of a variety of filters can be calculated. On the proposed filter construction method, the decomposition form of filters has been discovered, where a series of basic filters have been acquired. The properties of any filter depend on the decomposed basic filters. Basic filters can be used to compose filters in actual needs.

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0&lt;ρ&lt;1, 0&lt;t≤T), u(ξ)=αu(0)+φ (α is a constant and 0&lt;ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann-Liouville derivative; naturally, in the case of the Riemann - Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

Often, when solving forward, inverse or data assimilation problems, only a part of the solution is needed. As a model, we consider the stationary diffusion problem. We demonstrate an algorithm that can compute only a part or a functional of the solution, without calculating the full inversion operator and the complete solution. It is a well-known fact about partial differential equations that the solution at each discretisation point depends on the solutions at all other discretisation points. Therefore, it is impossible to compute the solution only at one point, without calculating the solution at all other points. The standard numerical methods like a conjugate gradient or Gauss elimination compute the whole solution and/or the complete inverse operator. We suggest a method which can compute the solution of the given partial differential equation 1) at a point; 2) at few points; 3) on an interface; or a functional of the solution, without computing the solution at all points. The required storage cost and computational resources will be lower as in the standard approach. With this new method, we can speed up, for instance, computation of the innovation in filtering or the likelihood distribution, which measures the data misfit (mismatch). Further, we can speed up the solution of the regression, Bayesian inversion, data assimilation, and Kalman filter update problems. Applying additionally the hierarchical matrix approximation, we reduce the cubic computational cost to almost linear $\mathcal{O}(k^2n \log^2 n)$, where $k\ll n$ and $n$ is the number of degrees of freedom. Up to the hierarchical matrix approximation error, the computed solution is exact. One of the disadvantages of this method is the need to modify the existing deterministic solver.

This note reconsiders a matrix equality $A_1A_2^{-}A_3A_4^{-}A_5 = A$ composed by six matrices of appropriate sizes, where $A_2^{-}$ and $A_4^{-}$ are generalized inverses of $A_2$ and $A_4$, respectively, and solves a selection of matrix set inclusion problems associated with various mixed reverse order laws for generalized inverses of products of two, three, and four matrices by means of this equality and its variations.

The application of instrumented indentation to assess material properties like Young’s modulus and micro-hardness has become a standard method. In recent developments, indentation experiments and simulations have been combined to inverse methods, from which further material parameters as yield strength, work hardening rate, and tensile strength can be determined. In this work, an inverse method is introduced by which material parameters for cyclic plasticity, i.e. kinematic hardening parameters, can be determined. To accomplish this, cyclic Vickers indentation experiments are combined with finite element simulations of the indentation with unknown material properties, which are then determined by inverse analysis. To validate the proposed method, these parameters are subsequently applied to predict the uniaxial stress-strain response of a material with success. The method has been validated successfully for a quenched and tempered martensitic steel and for technically pure copper, where an excellent agreement between measured and predicted cyclic stress-strain-curves has been achieved. Hence, the proposed inverse method based on cyclic nanoindentation, as a quasi-non-destructive method, could complement or even substitute the resource-intensive conventional fatigue testing in the future for some applications.

Despite their obvious differences, biological systems at different scales tend to exhibit common organizational patterns. Unfortunately, these commonalities are usually obscured by the parcelled terminology employed by various scientific sub-disciplines. To explore these commonalities, this papers a comparative study of diverse applications of the maximum entropy principle, ranging from amino acids up to societies. By presenting these studies under a common language, this paper establishes a unified view over seemingly highly heterogeneous biological scenarios.

In laser-assisted milling, higher temperature in shear zone softens the material potentially resulting in a shift of mean residual stress, which significantly affects the damage tolerance and fatigue performance of product. In order to guide the selection of laser and cutting parameters based on the preferred mean residual stress, inverse analysis is conducted by predicting residual stress based on guessed process parameters, which is defined as the forward problem, and applying iterative gradient search to find process parameters for next iteration, which is defined as the inverse problem. An analytical inverse analysis is therefore proposed for the mean residual stress in laser-assisted milling. The forward problem is solved by analytical prediction of mean residual stress after laser-assisted milling. The residual stress profile is predicted through the calculation of thermal stress, by treating laser beam as heat source, and plastic stress by first assuming pure elastic stress in loading process, then obtaining true stress with kinematic hardening followed by the stress relaxation. The variance-based recursive method is applied to solve inverse problem by updating process parameters to match the measured mean residual stress. Three cutting parameters including depth of cut, feed per tooth, and cutting speed, and two laser parameters including laser-tool distance and laser power, are updated with respected to the minimization of resulting residual stress and measurement in each iteration. Experimental measurements are referred on the laser-assisted milling of Ti-6Al-4V grade 5 and ELI. The percentage difference between experiments and predictions is less than 5% for both materials, and the selection is completed within 50 loops.

In this paper, the neural network is developed for solving the inverse photoacoustic problem aiming to estimate thermo-elastic properties and thickness of a semiconductor sample. The idea was that these sample properties be estimated from the phase characteristic because the phase measurements are more sensitive. The neural network has been trained on a large basis of photoacoustic phases calculated from a theoretical Si n-type model in the range of 20Hz to 20kHz, on which random Gaussian noise has been applied. It is shown that high accuracy and precision could be reached in solving of inverse photoacoustic problem using only phase measurement.

The growing expansion of the economy and population has resulted in an increased inclination towards the utilization of groundwater resources. Conducting a geophysical survey is a widely employed method for subsurface mapping and the detection of groundwater. A geophysical study was conducted in Nanshan township, Huarong County, located in the Hunan province of the South-Central region of China. The investigation involved the utilization of a 2D Electrical Resistivity Imaging technique employing forward and inverse pole-dipole electrode array. A total of six survey lines were established, each with an electrode distance from the nearest measuring point to exceeding 800 m. The maximum current electrode separation was utilized in this setup. The spacing between the electrical resistivity sounding points was established at regular intervals. The findings from the exploration indicate the existence of multiple faults within the surveyed region. The study additionally identified two regions of structural failure, which occurred due to the convergence of faults oriented in different directions. This convergence led to the fracturing of rocks, an increase in water content, and a decrease in resistivity. The findings from the exploration were utilized in the formulation of five verification target boreholes. The results of this study offer significant insights that can inform future investigations into groundwater exploration endeavors within the region.

To identify the exchange coefficients of a nonlinear polynomial tricompartmental general system of α+β , we follow the procedure; firstly, the recommended solution is to introduce an adequate time t*&gt;0 in a determinable manner. That is, after injecting quantity a into the main compartment, wait a moment for the exchange in the polynomial α+β order nonlinear general system to settle, and then compare this compartment to compartment 2 at t*. Secondly, applying the Taylor formula will linearize the system and identify the exchange coefficients. Finally, we will prove that the linearization method is stable.

Australia is a unique continent surrounded by the ocean and the majority of catchments flow to the coast. Some of these catchments are gauged and others are ungauged. There are 405 gauged catchments covering 2,549,000 km2 across the coastal regions of 12 drainage divisions in Australia. Whereas there are 771 catchments conceptualised as ungauged covering additional 835,000 km2. The mean annual rainfall and PET and its spatial and temporal distribution vary significantly from one drainage division to another. We developed a continuous daily streamflow time series of all gauged and ungauged catchments from 1993 onwards. We applied the GR4J lumped conceptual model to these catchments. The performance of gauged catchments was analysed through (i) visual inspection of daily hydrographs, flow duration curves and daily scatter plots, and (ii) performance metrics including NSE and PBias. Based on the NSE and PBias, performance ratings of 80% and 96% of the models respectively were found 'good'. There was no relationship found between the catchment area and model performance. The ungauged catchments were divided into four categories based on distance from potential donor catchments where observed data are available for GR4J model calibration, and Köppen climate zone. The total ungauged catchments represent 24.7% of the total drainage division areas. The streamflow from ungauged catchments was estimated using GR4J model based on the parameters of their donor catchments. Overall, runoff ratios from ungauged catchments were found to be higher compared to their donor gauged catchments, likely driven by their higher rainfall and less PET. This tendency was particularly evident in two drainage divisions – the Carpentaria Coast (CC) and Tanami-Timor Sea Coast (TTS) where ungauged areas comprised 51% and 43% respectively. The mean gauged annual streamflow varied significantly across drainage divisions – 230 GL from South Australian Gulf (SAG) to 146,150 GL in TTS. The streamflow from all ungauged catchments was estimated at 232,200 GL per year. Overall, the average streamflow from all drainage divisions, including gauged and ungauged areas, across the coastal regions of Australia was estimated at 419,950 GL per year. This nationwide estimate of streamflow dataset could potentially enhance our understanding of coastal processes and improvements of marine modelling systems and tools.

In the broad field of Inverse Analysis and Structural Identification, it is nowadays of a large interest the study of Gaussian Processes, as a reliable and efficient optimisation method, particularly helpful toward the identification of a global optimum point, under the conditions of complicated functions to be optimised. In the present contribution, a specific case study is considered, focusing on a historical road reinforced concrete arched bridge, located in Northern Italy, employing dynamic modal properties, deciphered from in-situ measurements, previously acquired under operational traffic conditions, by a standard wired accelerometer system, placed at the deck level. Aiming at the identification and diagnosis of the bridge structure, three main methodological steps are herein considered: the adoption of a FEM model of the structure (in the linear dynamic framework), the definition of an appropriate discrepancy function, based on measured and numerically computed quantities (natural frequencies and mode shapes of the bridge), and the investigation of such a discrepancy function, toward a consistent selection of an optimisation strategy for the identification of sought parameters (Young’s moduli and mass densities of diverse elements of the structure). The presented developments, within the framework of methodological optimisation approach by Gaussian Processes, and achieved results display a rather efficient perspective, with reference to the considered case study, toward inverse analysis for structural diagnosis, in the context of strategic (historical) infrastructures.