Descriptive geometry has indispensable applications in many engineering activities, some of which are presented in the first chapter of this paper in order to place the development presented here among them. As a result of the continuous variability of the technological environment according to various optimization aspects, the engineering activities must also be continuously adapted to the changes, which an appropriate approach and formulation are required from the practitioners of descriptive geometry, and can even lead to improvement in the field of descriptive geometry. The imaging procedures are always based on the methods and theorems of descriptive geometry. Resolving contradictions in spatial geometry reconstruction research is a constant challenge, to which a possible answer in many cases is the search for the right projection direction. A special method of enumerating the possible infinite viewpoints for the representation of the surface edge curves is presented in another part of the paper. The procedure for determining the correct directions in a mathematically exact way is also presented through examples. The analysis and some of the results of the Monge mapping, which is suitable for the solution of a mechanical engineering task to be solved in a specific technical environment, are also presented.

This work presents an approach for detecting mathematical formulas in scanned document images. The proposed approach is end-to-end trainable. Since many OCR engines cannot reliably work with the formulas, it is essential to isolate them to obtain the clean text for information extraction from the document. Our proposed pipeline comprises a hybrid task cascade network with deformable convolutions and a Resnext101 backbone. Both of these modifications help in better detection. We evaluate the proposed approaches on the ICDAR-2017 POD and Marmot datasets and achieve an overall accuracy of 96% for the ICDAR-2017 POD dataset. We achieve an overall reduction of error of 13%. Furthermore, the results on Marmot datasets are improved for the isolated and embedded formulas. We achieved an accuracy of 98.78% for the isolated formula and 90.21% overall accuracy for embedded formulas. Consequently, it results in an error reduction rate of 43% for isolated and 17.9% for embedded formulas.

This paper presents a novel architecture for detecting mathematical formulas in document images, which is an important step for reliable information extraction in several domains. Recently, Cascade Mask R-CNN networks have been introduced to solve object detection in computer vision. In this paper, we suggest a couple of modifications to the existing Cascade Mask R-CNN architecture: First, the proposed network uses deformable convolutions instead of conventional convolutions in the backbone network to spot areas of interest better. Second, it uses a dual backbone of ResNeXt-101, having composite connections at the parallel stages. Finally, our proposed network is end-to-end trainable. We evaluate the proposed approach on the ICDAR-2017 POD and Marmot datasets. The proposed approach demonstrates state-of-the-art performance on ICDAR-2017 POD at a higher IoU threshold with an f1-score of 0.917, reducing the relative error by 7.8%. Moreover, we accomplished correct detection accuracy of 81.3% on embedded formulas on the Marmot dataset, which results in a relative error reduction of 30%.

The thymus hosts the development of a specific type of adaptive immune cells called T cells. T cells orchestrate the adaptive immune response through recognition of antigen by the highly variable T-cell receptor (TCR). T-cell development is a tightly coordinated process comprising lineage commitment, somatic recombination of Tcr gene loci and selection for functional, but non-self-reactive TCRs, all interspersed with massive proliferation and cell death. Thus, the thymus produces a pool of T cells throughout life capable of responding to virtually any exogenous attack while preserving the body through self-tolerance. The thymus has been of considerable interest to both immunologists and theoretical biologists due to its multiscale quantitative properties, bridging molecular binding, population dynamics and polyclonal repertoire specificity. Here, we review mathematical modelling strategies that were reported to help understand the flexible dynamics of the highly dividing and dying thymic cell populations. Furthermore, we summarize the current challenges to estimating in vivo cellular dynamics and to reaching a next-generation multiscale picture of T-cell development.

The description of the microscopic world in quantum mechanics is very different from that in classical physics, and there are some points of view that are contrary to intuition and logic. The first is the loss of reality, the behavior of micro particles shows randomness and hopping. The second is the loss of certainty, the conjugate physical variables of a system cannot be determined synchronously, they satisfy the Heisenberg uncertainty principle. The third is the non-local correlation. The measurement of one particle in the quantum entanglement pair will change the state of the other entangled particle simultaneously. In this paper, some concepts related to quantum entanglement, such as EPR correlation, quantum entanglement correlation function, Bell's inequality and so on, are analyzed in detail. Analysis shows that the mystery and confusion in quantum theory may be caused by the logical problems in its basic framework. Bell's inequality is only a mathematical theorem, but its physical meaning is actually unclear. The Bell state of quantum entangled pair may not satisfy the dynamic equation of quantum theory, so it cannot describe the true state of microscopic particles. In this paper, the correct correlation functions of spin entanglement pair and photonic entanglement pair are strictly derived according to normal logic. Quantum theory is a more fundamental theory than classical mechanics, and they are not parallel relation in logic. However, there are still some unreasonable contents in the framework of quantum theory, which need to be improved. In order to disclose the real relationship between quantum theory and classical mechanics, we propose some experiments which show the wave-particle duality simultaneously and provide intuitionistic teaching materials for the new interpretation of quantum theory.

In the evolving landscape of vibration analysis, the integration of machine learning technologies for semi-automated analytical processes presents challenges in establishing clear cause-and-effect relationships in decision-making formulas. This research addresses this gap by exploring analytical diagnosis through residual signals, enabling the identification of anomalies and underperformance in machinery. Vibration signature analysis, focusing on distinctive vibration patterns, proves instrumental in detecting bearing defects and optimizing maintenance for enhanced operational efficiency and safety. Mathematical modeling, particularly worst-case analysis, provides a structured framework for decision-making under uncertainty. The research employs descriptive methods to analyze bearing wear data, transforming it into quantitative data for multiple regression equations. The derived equations reveal the dominance of harmonics in determining bearing wear stages, with PeakVue and overall vibration further indicating damage progression. The findings emphasize the significance of harmonic detection and high-frequency vibration analysis in predicting bearing health, offering a comprehensive approach to enhance machinery reliability. The study concludes with practical recommendations for bearing damage diagnosis based on the developed mathematical models, providing valuable insights for predictive maintenance strategies.

The study shows that the volumes and surfaces of $n$-balls, $n$-simplices, and $n$-orthoplices are holomorphic functions of $n$, which makes those objects omnidimensional, that is well defined in any complex dimension. Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about $n$-balls reveal previously unknown properties of these geometric objects. In particular, for $0 < n < 1$, the volumes of the omnidimensional polytopes are larger than those of circumscribing $n$-balls, and both their volumes and surfaces are smaller than those of inscribed $n$-balls. The surface of an $n$-simplex circumscribing unit diameter $n$-ball is spirally convergent to zero with real $n$ approaching negative infinity but first has a local maximum at $n=-3.5$. The surface of an $n$-orthoplex circumscribing unit diameter $n$-ball is spirally divergent with real $n$ approaching negative infinity but first has a local minimum at $n=-1.5$, where its real and imaginary parts are equal to each other; similarly, as its volume, where the similar local minimum occurs at $n=-3.5$. Reflection functions for volumes and surfaces of these polytopes inscribed in and circumscribed about $n$-balls are proposed. symmetries of products and quotients of volumes in complex dimensions $n$ and $-n$ and of surfaces in complex dimensions $n$ and $2-n$ are shown to be independent of the metric factor and the gamma function. Specific symmetries also hold between volumes and surfaces in dimensions $n = -1/2$ and $n = 1/2$.

This paper offers an integrated approach for correlated storage assignment strategy (CSAS) to optimize travel distance by considering item correlation, picking frequency, zoning, and slot constraints to improve operational efficiency in distribution centers. We simplify the mixed integer non-linear programming (MINLP) model and incorporate integrated heuristic procedures for faster convergence. We introduce positive and negative centroid deviations as techniques to guide the model convergence and explore different scenarios. In the second stage of item assignment, we prioritize items within zones using a ranking formula that is optimal for both single-item and multiple-item orders. By analyzing the distribution of correlated item across zones and the impact of their picking frequency on optimizing travel distance, we propose an improved CSAS that minimize travel distance by strategically placing items based on their proximity to the depot. Our model reduces trips in distant zones by prioritizing larger average order sizes, while maximizing trips for frequently ordered smaller lists in closer zones. This method significantly reduces travel distance by minimizing the need for pickers in distant zones to frequently travel to the depot. Simulation results show reductions of up to 36.75% for fitted data and 33.59% for recent data.

A calibrated mathematical model of antiviral immune response to SARS-CoV-2 infection is developed. The model considers the innate and antigen-specific responses to SARS-CoV-2 infection. Recently published data sets from human challenge studies with SARS-CoV-2 were used for parameter estimation. Understanding the regulation of multiple intertwined reaction components of the immune system is necessary for linking the clinical phenotypes of COVID-19 with the kinetics of immune responses. Consideration of multiple immune reaction components in a single calibrated mathematical model allowed us to address some fundamental issues related to pathogenesis of COVID-19, i.e. sensitivity of the peak viral load to parameters characterizing the specific response components, the kinetic coordination of the individual responses, and the factors favoring a prolonged viral persistence. The model provides a tool for predicting the infectivity of patients, i.e. the amount of virus which is transmitted via droplets from the person infected with SARS-CoV-2, depending on the time of infection. The thresholds in the relative unbalance between innate and adaptive response parameters which lead to a prolonged persistence of SARS-CoV-2 due to the loss of a kinetic response synchrony/coordination were identified.

Higher traffic density in Russian cities and less uniform traffic due to congestions are reasons for increased emissions of pollutants contained in exhaust gases of vehicles. The paper presents the modeling results of pollutant dispersion in Magnitogorsk. The analysis of these results shows that now many blocks of flats and social infrastructure of the city are in a zone with an excess of maximum allowable concentration of pollutants. Having summarized the methodological framework formulated by Russian and foreign researchers, the authors systemized the existing methods used to decrease pollutants produced by vehicles. It is proposed to choose an optimal combination of such methods applying a developed mathematical model. By adopting an optimal combination of the methods, it is possible to ensure minimum air pollutant emissions from vehicles, allocating limited financial resources. The mathematical modeling of the solutions on changing traffic density and structure and implementing methods used to decrease pollutants on one of the Magnitogorsk crossroads with the heaviest traffic is used to draw a conclusion about efficiency of the designed model used as a tool for strategic planning of development and transformation of street networks.

A method for calibrating models of agricultural production and resource use presented by Howitt [1] for policy analysis is proposed to leverage multidisciplinary agricultural research at the National Center for Alluvial Aquifer Research (NCAAR). An illustrative example for Sunflower County, MS is presented to show how plot-level research can be extended to draw systemic region or basin wide implications. A hypothetical improvement in yields for dryland soybean varieties is incorporated to the model and shown to have a positive impact on aquifer outcomes and producer profits. The example illustrates that a change in one practice-crop combination can have system-wide impacts as evidenced by the change in acreages for all crops and practices.

Advanced methods of treatment are needed to fight the threats of virus-transmitted deseases and pandemics. Often they are based on an improved biophysical understanding of virus replication strategies and processes in their host cells. For instance, an essential component of the replication of the Hepatitis C virus (HCV) proceeds under the influence of non-structural HCV proteins (NSPs) that are anchored to the endoplasmatic reticulum (ER), such as the NS5a protein. The diffusion of NSPs has been studied by in-vitro fluorescence recovery after photobleaching (FRAP) experiments. The diffusive evolution of the concentration field of NSPs on the ER can be described by means of surface partial differential equations (sufPDE). Knodel et al, Viruses 2018, 10, 28 estimated the diffusion coefficient of the NS5a protein by minimizing the discrepancy of an extended set of sufPDE simulations and experimental FRAP time series data. Here, we provide a scaling analysis of the sufPDEs that describe the diffusive evolution of the concentration field of NSPs on the ER. This analysis provides an estimate of the diffusion coefficient that is based only on the ratio of the membrane surface area in the FRAP region and its contour length. The quality of this estimate is explored by comparison to numerical solutions of the sufPDE for a flat geometry and for ten different 3D embedded 2D ER grids that are derived from fluorescence z-stack data of the ER. Finally, we apply the new data analysis to the experimental FRAP time-series data analyzed in our previous paper, and we discuss the opportunities of the new approach.

On February 24, 2022 Russia started the full-scale invasion of Ukraine, which created an unprecedented number of refugees. To estimate the influence of this humanitarian disaster on the COVID-19 pandemic dynamics, the averaged daily numbers of cases and reproduction numbers in Ukraine, the UK, Poland, Germany, the Republic of Moldova, and in the whole world were calculated for the period February-April 2022. The registered numbers of cases were compared with ones calculated with the use of the generalized SIR-model and corresponding parameter identification procedure for the previous epidemic waves in Ukraine, Poland, Germany, and the world. Since before February 24, 2022 the estimation of the number of infectious persons per capita in Ukraine 3.6 times exceeded the global figure, the increase of the number the new cases and the pandemic duration was expected. In March 2022 the increase of the averaged number of new cases in the UK, Germany, and worldwide was visible. A simple formula to estimate the effective reproduction number based on the smoothed accumulated numbers of cases is proposed. The results of calculations agree with the figures presented by John Hopkins University and demonstrate a short-term growth of the reproduction number in the UK, Poland, Germany, Moldova, and worldwide in March 2022.

This paper offers a novel geometric and visual approach to the renowned Four Color Map Theorem and K5 non-planarity problem while unveiling their profound connection to the "kissing number" problem in 2D. We represent planar graphs through circles and tangents, simplifying complex structures and shedding light on these classic problems. Our proofs by contradiction, rooted in the kissing number concept, reveal that both the Four Color Map Theorem and K5 non-planarity are fundamentally linked, as they pivot around the concept of coloring. This study bridges the realms of geometry and graph theory, providing fresh insights and emphasizing the significance of the kissing number problem in various fields.

In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a Lévy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.

Imaginary dimensions in physics require an imaginary set of base Planck units and some negative parameter $c_n$ corresponding to the speed of light in vacuum $c$. The second, negative fine-structure constant $\alpha_2^{-1} \approx -140.178$ present in Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene leads to the imaginary Planck units. Furthermore, it sets $c_n \approx -3.06 \times 10^8~\text{[m/s]}$. It follows that electric charges are the same in real and imaginary dimensions. Neutron stars and white dwarfs, objects emitting perfect black-body radiation, need energy exceeding their mass-energy equivalence ratios. Complex energies are defined in terms of real and imaginary Planck units. Their imaginary parts, inaccessible for direct observation, store the excess of these energies. It follows that black holes are fundamentally uncharged, charged micro neutron stars and white dwarfs with masses lower than $5.7275 \times 10^{-10}~[\text{kg}]$ are unphysical, and the radii of white dwarfs' cores are limited to $R_{\text{WD}} < 6.7933~G M_{\text{WD}}/c^2$. It is conjectured that the maximum atomic number $Z=238$. A black-body object is in the equilibrium of complex energies of masses, charges, and wavelengths if its radius $R_\text{eq} \approx 2.7665~G M_{\text{BBO}}/c^2$, which corrects the value of the photon sphere radius $R_{\text{ps}}=3G M/c^2$, taking into account the value(s) of the fine-structure constant(s), which is otherwise neglected in general relativity.