In this study, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings in [17]. Firstly we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of a class of hyper-modules, then we give some relevant properties of the hypermodule in this topological hyperspace.

In this research first, a sequence of properties called delta is assigned to each prime number and then examined. Deltas are only dependent on the distribution of prime numbers, so the results obtained for the delta distribution can be considered as a proxy for the distribution of prime numbers. The first observation was that these properties are not unique and different prime numbers may have the same value of delta of a given order. It was found that a small number of deltas cover a large portion of prime numbers, so by recognizing repetitive deltas, the next prime numbers can be predicted with a certain probability, but the most important observation of this study is the normal distribution of deltas. This research has not tried to justify the obtained observations and instead of answering the questions, it seeks to ask the right question.

Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that P\bigcap S=\emptyset In this article, we introduce several results concerning graded S-prime ideals. Then we introduce the concept of graded weakly S-prime ideals which is a generalization of graded weakly prime ideals. We say that P is a graded weakly S-prime ideal of R if there exists s\in S such that for all x, y\in h(R), if 0\neq xy\in P, then sx\in P or sy\in P. We show that graded weakly S-prime ideals have many acquaintance properties to these of graded weakly prime ideals.

In this paper, we study the symmetry between the supersingular prime number according to smallest sphenic number. With this symmetry, we show that the elements of sporadic group generates all prime numbers with the order by a simple application.

We talk about random when it is not possible to determine a pattern on the observed outcomes. A computer follows a sequence of fixed instructions to give any of its output, hence the difficulty of choosing numbers randomly from algorithmic approaches. However, some algorithms based on mathematical formulas like the Linear Congruential algorithm and the Lagged Fibonacci generator appear to produce "true" random sequences to anyone who does not know the secret initial input [1]. Up to now, we cannot rigorously answer the question on the randomness of prime numbers [2, page 1] and this highlights a connection between random number generator and the distribution of primes. From [3] and [4] one sees that it is quite naive to expect good random reproduction with prime numbers. We are, however, interested in the properties underlying the distribution of prime numbers, which emerge as sufficient or insufficient arguments to conclude a proof by contradiction which tends to show that prime numbers are not randomly distributed. To achieve this end, we use prime gap sequence variation. The algorithm that we produce makes possible to deduce, in the case of a binary choice, a uniform behavior in the individual consecutive occurrence of primes, and no uniformity trait when the occurrences are taken collectively.

The Ramanujan primes are the least positive integers Rn having the property that if m ≥ Rn, then π_m − π_(m/2) ≥ n. This document develops several bounds related to the Ramanujan primes, sharpening the currently known results. The theory presented is by no means exhaustive, however it provides insights for future research work. Alternatively, we may say that it is a road map which may be followed to make further discoveries.

Twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the part “twin prime” is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture.

We consider mixture experiments in which the proportions of the components must be non-negative and their sum must equal one. Thus, the experimental region for a mixture of components is a simplex. Li and Zhang (2017) made the conjecture that the pseudo component transformation of the lattice points in the simplex has a special property. In this paper, we show that this conjecture is not true in general. Furthermore, we refine this conjecture and prove the refined conjecture.

The Goldbach Conjecture, which is frequently termed as “1 + 1”, has been a fascinating goal for many mathematicians over centuries. A Chinese mathematician, Dr. Jingrun Chen, proved 1 + 2, which is a great success and the best result so far achieved. Although there were several attempts proving the conjecture, these attempts are either tediously long, complicated, or logically imperfect, thus not widely accepted. Taking advantage of the periodicity of primes, here the author provides a straight forward rigorous proof for the Conjecture.

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid disproof for twin prime conjecture.

The maximal prime gaps upper bound problem is one of the major mathematical problems to date. The objective of the current research is to develop a standard which will aid in the understanding of the distribution of prime numbers. This paper presents theoretical results which originated with a researchin the subject of the maximal prime gaps. the document presents the sharpest upper bound for the maximal prime gaps ever developed. The result becomes the Supremum bound on the maximal prime gaps and subsequently culminates with the conclusive proof of the Firoozbakht's Hypothesis No 30. Firoozbakht's Hypothesis implies quite a bold conjecture concerning the maximal prime gaps. In fact it imposes one of the strongest maximal prime gaps bounds ever conjectured. Its truth implies the truth of a greater number of known prime gaps conjectures, simultaneously, the Firoozbakht's Hypothesis disproves a known heuristic argument of Granville and Maier. This paper is dedicated to a fellow mathematician, the late Farideh Firoozbakht.

It is shown that the four fundamental physical forces, i.e. weak interaction, strong interaction, electromagnetism and gravity, all have their origin in the quark as the single true elementary particle.This requires conceiving the quark as a Dirac particle in a pseudo-tachyon mode, which possesses two real dipole moments: the common one associated with its angular momentum and a second one that is polarisable in a scalar field. This Dirac particle carries a regular charge magnetic monopole without Dirac’s string, theorized by Comay. The boson carrier of its field of energy is the gluon showing an exponential decay of its spatial range because of the influence of an omni-present energetic background field, known as the Higgs field, in this article interpreted as the Lambda in Einstein’s Field Equation.

Non-standard diesel blends can be harmful to the environment and human health. In this context, a simple analytical method to estimate the biodiesel mixture ratio in diesel was developed based on the impedance spectroscopy (IS) associated with the interdigitated sensors. In this article, four different interdigitated sensors, variable comb spacing (G), were simulated using the COMSOL Multiphysics software. Based on finite element simulations, four interdigitated electrode architectures by manufactured and evaluated. According to the X-ray powder diffraction technique, the deposition of the conductive layer (Au°) over the surface of the dielectric substrate (SiO2) did not alter its phase composition. In the analysis of AFM and SEM, it was possible to observe irregular edges on the electrodes, possibly related to thin layers' manufacturing process and mechanical stability. Another characteristic observed in the AFM images was the height of the step of the gold layer of the sensor. Several cross-sections were obtained, and the mean step value is 225.71 ± 0.0032 nm. Although there are differences in the roughness, the whole sensor has nanometric roughness. Based on the finite element method simulation performed, it can be assumed that the geometric parameters more suitable for the manufacturing of the electrode are: W = 20 µm, L = 1000 µm, G = 50 µm e N = 40 digits. The electrical characterization performed by impedance spectroscopy showed that we could differentiate between biodiesel and diesel fuels and their pseudo-binary mixtures in the low-frequency region.

Let G be a finite group. The prime graph Γ(G) of G is defined as follows: The set of vertices of Γ(G) is the set of prime divisors of |G| and two distinct vertices p and p' are connected in Γ(G), whenever G has an element of order pp'. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ(G)=Γ(P), G has a composition factor isomorphic to P. In [4] proved finite simple groups ²D_n(q), where n ≠ 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups ²D_2k(q), where k ≥ 9 and q is a prime power less than 10⁵.

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.

Prime number-related issues can be viewed from drastically different perspectives by examining the close connections between prime numbers and composite numbers. We think that multiple perspectives are the pillars on the path to solutions so we have created this study. As a result of the study, we proposed two new formulas by presenting three theorems and one proof for each theorem, a total of three proofs. We proved that the formula p · n + p returns a composite number in the first of the theorems, which is the preliminary theorem. Our first theorem except the preliminary theorem is that the formula p · n + p returns all composite numbers, and we proved that too. Finally, we created Theorem II using Theorem I to use in our other work and proved that the formula 2 · n · p + p returns all odd composite numbers, which is Theorem II. Afterward, we presented the similarities of the 2 · n · p + p formula we put forth with another known formula.

The efficiency of a new composite material of the layered double hydroxide (LDH) of ZnMgAl and rice husk biochar (RHB) for the removal of Cu(II) and Pb(II) ions from synthetic wastewater was investigated in this study. The images of scanning electron microscope showed extremely fine crystalline LDH particles decorated on the rough surface of the RHB while the succesful formation of the composite adsorbent (LDH/RHB) was confirmed by the corresponding energy dispersive X-ray and the Fourier-transform infrared spectroscopy. An equilibrium contact time of 30 and 15 min for Cu2+ and Pb2+, respectively, was proposed for the optimum performance of the batch adsorption process. The dose of the LDH/RHB adsorbent was optimized at 0.4 g yielding maximum adsorption capacities of 117 and 124 mg g-1 for Cu2+ and Pb2+, respectively, with corresponding maximum removal efficiencies of nearly 94% and 99%. The initial meta concentrations was optimized at 50 or 60 mg L-1 corresponding to maximum removal efficiencies or adsorption capacities. For the changing initial metal’s concentration, the removal efficiencies remained unchanged for up to 50 mg L-1 and decreased by about 50% by increasing the initial concentrations from 50 to 100 mg L-1. The adsorption capacities observed a linear increasing trend by increasing the initial concentrations from 5 to 60 mg L-1 and remained unchanged afterward up to 100 mg L-1. A solution pH of 6.0 yielded optimum results with an increasing trend in adsorption capacities and percentage removal by changing the solution pH from 2.0 to 7.0. Based on the best-fit of the pseudo second-order kinetic model to the experimental data, the chemisorption was suggested to be the controlling mechanism of adsorption. The fitting of the Langmuir model suggested a monolayer sorption of Cu2+ and Pb2+ in addition to a heterogeneous adsorption as supported by the fitting of the Temkin isotherm. The application of the Dubinin–Radushkevich isotherm proposed a physical adsorption (mean free energy of adsorption less than 8 kJ mol−1) of both heavy metal ions on the surface of the LDH/RHB adsorbent.

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.

Let's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}-\log \log x-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. Let's also define $S(x) = \theta(x) - x$, where $\theta(x)$ is the Chebyshev function. It is known that $S(x)$ changes sign infinitely often. We define the another function $\varpi(x) = \left(\sum_{{q\leq x}}{\frac{1}{q}}-\log \log \theta(x)-B \right)$. We prove that when the inequality $\varpi(x) \leq 0$ is satisfied for some number $x \geq 3$, then the Riemann hypothesis should be false. The Riemann hypothesis is also false when the inequalities $\delta(x) \leq 0$ and $S(x)\geq 0$ are satisfied for some number $x \geq 3$ or when $\frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log \theta(x)} \leq 1$ is satisfied for some number $x \geq 13.1$.

Robin criterion states that the Riemann Hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. Let $q_{1} = 2, q_{2} = 3, \ldots, q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann Hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n > 5040$ such that Robin inequality does not hold and $n < (4.48311)^{m} \times N_{m}$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$.

Let {$\Gamma(k)$} denote {$(k-1)!$}, and let {$\Gamma_{n}(k)$} denote {$(k-1)!$}, where {$n \in \{3,\ldots,16\}$} and {$k \in \{2\} \cup [2^{\textstyle 2^{n-3}}+1,\infty) \cap \mathbb N$}. For an integer {$n \in \{3,\ldots,16\}$}, let $\Sigma_n$ denote the following statement: if a system of equations {${\mathcal S} \subseteq \{\Gamma_{n}(x_i)=x_k:~i,k \in \{1,\ldots,n\}\} \cup \{x_i \cdot x_j=x_k:~i,j,k \in \{1,\ldots,n\}\}$} with $\Gamma$ instead of $\Gamma_{n}$ has only finitely many solutions in positive integers {$x_1,\ldots,x_n$}, then every tuple {$(x_1,\ldots,x_n) \in (\mathbb N \setminus \{0\})^n$} that solves the original system ${\mathcal S}$ satisfies {$x_1,\ldots,x_n \leqslant 2^{\textstyle 2^{n-2}}$}. Our hypothesis claims that the statements {$\Sigma_{3},\ldots,\Sigma_{16}$} are true. The statement {$\Sigma_6$} proves the following implication: if the equation {$x(x+1)=y!$} has only finitely many solutions in positive integers $x$ and $y$, then each such solution {$(x,y)$} belongs to the set {$\{(1,2),(2,3)\}$}. The statement {$\Sigma_6$} proves the following implication: if the equation {$x!+1=y^2$} has only finitely many solutions in positive integers $x$ and $y$, then each such solution {$(x,y)$} belongs to the set {$\{(4,5),(5,11),(7,71)\}$}. The statement {$\Sigma_9$} implies the infinitude of primes of the form {$n^2+1$}. The statement {$\Sigma_9$} implies that any prime of the form {$n!+1$} with {$n \geqslant 2^{\textstyle 2^{9-3}}$} proves the infinitude of primes of the form {$n!+1$}. The statement {$\Sigma_{14}$} implies the infinitude of twin primes. The statement {$\Sigma_{16}$} implies the infinitude of Sophie Germain primes.

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S-primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S-primary ideal if there exists s∈S such that whenever 0≠ab∈I for a,b∈R, then sa∈√I or sb∈I. The relationships among S-prime, S-primary, weakly S-primary and S-n-ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S_{r}-primary ideals are charctarized where S={1,r,r²,⋯}. Several properties, characterizations and examples concerning weakly S-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S-decomposable ideals and S-weakly Laskerian rings which are generalizations of S-decomposable ideals and S-Laskerian rings are introduced.

Initial periods of adsorption kinetics play an important role in estimating the initial adsorption rate and rate constant of adsorption process. Several adsorption processes rapidly occur, and the experimental data of adsorption kinetics under the initial periods can contain potential errors. The pseudo-second-order (PSO) kinetic model has been popularly applied in the field of adsorption. The use of the nonlinear optimization method to obtain the parameters of the PSO model can minimize error functions during modelling compared to the linear method. However, the nonlinear method has limitations in that it cannot directly recognize potential errors in the experimental points of time-dependent adsorption, especially under the initial periods. In this study, for the first time, the different linear types (Types 1–6) of the PSO model are applied to discover the error points under the initial periods. The result indicated that the fitting method using its linear equations (Types 2–5) is really helpful for identifying the error (doubtful) experimental points from the initial periods of adsorption kinetics. The imprecise points lead to low adjusted R2 (adj-R2), high reduced χ2 (red-χ2), and high Bayesian information criterion (BIC) values. After removing those points, the experimental data were adequately fitted with the PSO model. statistical analyses demonstrated that the nonlinear method must be used for modelling the PSO model because its red-χ2 and BIC were lower than the linear method. Type 1 has been extensively applied in the literature because of its very high adj-R2 (0.9999) and its excellent fitting to experimental points. However, its application should be limited because the potential errors from experimental points were not identified by this type. For comparison, the other kinetic models (i.e., pseudo-first-order, pseudo-nth-order, Avrami, and Elovich) are applied. The modelling result using the nonlinear forms of those models indicated that the fault experimental points from the initial periods were not detected.

We study the Gaudin models with gl(1|1) symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation gl(1|1)[t]-modules, extending all the results of to the twisted case. Namely, we give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation gl(1|1)[t]-modules and show that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces.

In the framework of a multiplicative metric space. We have discuss some Properties of convex structures in pseudo multiplicative metric space and have applied these properties toiobtain alfixed point results in complete pseudo. It would be interesting to prove some further results in such metric spaces with completeness property.

In this paper, we study the fractional pseudo-parabolic equations u_t + (−△)^s u + (−△)^s u_t = u log |u|. Firstly, we recall the relationship between the fractional Laplace operator (−△)^s and the fractional Sobolev space H^s and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of global weak solution: for the low initial energy J(u₀) < d, the solution is global in time with I(u₀) > 0 or ∥u₀∥X_0(Ω) = 0 and blows up +∞ with I(u₀) < 0; for the critical initial energy J(u₀) = d, the solution is global in time with I(u₀) ≥ 0 and blows up at +∞ with I(u₀) < 0. The decay estimate of the energy functional for the global solution is also given.

For each non-prime odd number as F=pq , if we consider m/n as an approximation for q/p and choose k=mn , then by proving some lemmas and theorems, we can compute the values of m and n. Finally, by using Fermat’s factorization method for F and 4kF as difference of two non-consecutive natural numbers, we should be able to find the values of p and q. Then we introduce two new and powerful sieves for separating composite numbers from prime numbers.

Let $S=\left(s_1, \ldots, s_n\right)$ be a finite sequence of integers. Then $S$ is a Gilbreath sequence of length $n$, $S\in\mathbb{G}_n$, iff $s_1$ is even or odd and $s_2, \ldots, s_n$ are respectively odd or even and $\min\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\leq s_{m+1}\leq\max\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\forall m\in\left[\left.1, n\right)\right.$. This, applied to the order sequence of prime number $P$, defines Gilbreath polynomials and two integer sequences A347924 \cite{oeisA347924} and A347925 \cite{oeisA347925} which are used to prove that Gilbreath conjecture $GC$ is implied by $p_n-2^{n-1}\leqslant\mathcal{P}_{n-1}\left(1\right)$ where $\mathcal{P}_{n-1}\left(1\right)$ is the $n-1$-th Gilbreath polynomial at 1.

In this paper, we investigated the trajectory of the massive particle in the vicinity of a general spherical symmetric black hole. Also, in the framework of general sphericalily symmetric black hole, Pseudo-Newtonian potential (PNP) and effective potentials has been investigated. As an example, static spherically symmetric black hole in f(R) gravity is considered and presented the brief discussion on the structure of spacetime and horizons. We calculated energy and angular momentum in the framework of general relativity as well as in Pseudo-Newtonian theory. A graphical comparison of angular momentum in this both framework has been studied.

Background: Biological processes are based on complex networks of cells and molecules. Single cell multi-OMICs is a new tool aiming to provide new incites in the complex network of events controlling the functionality of the cell.; Methods: Since single cell technologies provide many sample measurements, they are the ideal environment for the application of deep learning and machine learning approaches. An autoencoder (AE) is composed of an encoder and a decoder sub-model. AE are very powerful in data compression and noise removal. However, the decoder model remains a black box from which is impossible to depict the contribution of the single input elements. We have recently developed a new class of autoencoders, called Sparsely Connected Autoencoders (SCA), which have the advantage of providing a controlled association among the input layer and the decoder module. This new architecture has the benefit that the decoder model is no anymore a black box and it can be used to depict new biologically interesting features from single cell data; Results: In this paper, we show that SCA hidden layer can grab new information usually hidden in single cell data, like as providing clustering on meta-features difficult, i.e. transcription factors expression, or impossible, miRNA expression, to depict in single cell RNAseq data. Furthermore, a SCA representation of cell clusters has the advantage of simulating a conventional bulk RNAseq, which is a data transformation allowing the identification of similarity among independent experiments; Conclusions: In our opinion, SCA represent the bioinformatics version of a “Swiss Army knife” for the extraction of hidden knowledgeable features from single cell OMICs data.

In this paper, we investigate the effects of certain variants of the commutative laws on properties of several families of algebras which are in general not commutative, such as groups, linear algebras, or actually quite anti-commutative, such as $BCK$-algebras and $d$-algebras among others. From results obtained it becomes clear that by considering these variants in the presence of yet other axioms it is to be expected that a quite rich and varied set of results may be obtained both in the general and the particular setting of which what has been accomplished in this paper is a substantial sample.

In this paper, we study the semilinear pseudo-parabolic equations $\displaystyle u_{t} - \triangle_{\mathbb{B}}u - \triangle_{\mathbb{B}}u_{t} = \left|u\right|^{p-1}u$ on a manifold with conical singularity, where $\triangle _{\mathbb{B}}$ is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary $x_{1} = 0$. Firstly, we discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence and nonexistence of global weak solution: for the low initial energy $J(u_{0})<d$, the solution is global in time with $I(u_{0}) >0$ or $\displaystyle\Vert \nabla_{\mathbb{B}}u_{0}\Vert_{L_{2}^{\frac{n}{2}}(\mathbb{B})} = 0$ and blows up in finite time with $I(u_{0}) < 0$; for the critical initial energy $J(u_{0}) = d$, the solution is global in time with $I(u_{0}) \geq0$ and blows up in finite time with $I(u_{0}) < 0$. The decay estimate of the energy functional for the global solution and the estimates of the lifespan of local solution are also given.

We demonstrate a new quantitative method to the sieve of Eratosthenes, which is an alternative to the sieve of Legendre. In this method, every element of a given set is sifted out once only, and therefore, this method is free of the Mobius function and of the parity barrier. Using this method, we prove that every sufficiently large even number is the sum of two primes, and that every even number is the difference of two primes in infinitely many ways.

Integer is either a composite number or a prime number. Therefore, detecting composite numbers is important for solving prime numbers. The study of prime numbers, apart from satisfying human curiosity, can be very important. In this article, the order of composite numbers has been detected. And explained with a simple method and a simple function. And, a method has been developed in which all composite numbers and therefore prime numbers can be determined by using the specified methods, functions and formulas.

Natural products (NPs) are a rich source of structurally novel molecules, and the chemical space they encompass is far from being fully explored. Over history, NPs have represented a significant source of bioactive molecules and have served as a source of inspiration for developing many drugs on the market. On the other hand, computer-aided drug design (CADD) has contributed to drug discovery research, mitigating costs and time. In this sense, compound databases represent a fundamental element for the CADD. This work reviews the progress toward developing compound databases of natural origin, particularly databases developed in Latin America, and their practical applications in the drug discovery area. We also survey the computational methods, emphasizing chemoinformatic approaches to profile natural product databases.

Upon inbreeding, the architecture of the inbreeding load shifts as selection purges strongly deleterious recessive mutations and drift fixes many milder ones. Most small inbred populations show limited genetic variation while crosses between such populations commonly express pronounced heterosis, confirming fixation. In contrast, purging appears to be limited in that inbred populations often retain substantial inbreeding depression. In addition we have the enigma Darwin noted: purely selfing taxa are unknown. Because both purging and fixation reduce inbreeding depression and load, another mechanism must exist to sustain these. Background selection and the associations that develop among alleles in small inbred populations will shift the architecture of the load potentially creating blocks of recessive mutations linked in repulsion. This would generate pseudo-overdominance that could sustain these “PODs” and inbreeding load. Recombination and crosses between lineages could erode PODs. Crosses between populations fixed for different mutations would generate high pseudo-overdominance, enhancing heterosis and potentially POD formation. New recessive mutations arising within PODs would reinforce overdominance. PODs should generate clear genetic signatures including genomic hotspots of heterozygosity and linkage disequilibrium containing alleles at intermediate frequency generating segregating load. Results from several simulation and empirical studies match these predictions. Further simulations and comparative genomic analyses are needed to rigorously test whether PODs exist in sufficient strength and number to generate persistent inbreeding depression and load in inbred lineages.

Based on recent popular money priming research results, which conclude that money 4 makes self-sufficient (e.g. less interest in other people), we assumed that people are less interested 5 in finding out whether others are lying or telling the truth. In a laboratory experiment, 163 students 6 (85 women, 78 men, MAge = 23.08, ranging from 18 to 36 years) were primed by actively handling 7 money (versus paper sheets). Afterwards, they classified 24 video statements as true or deceptive 8 (senders describing their most/least favorite movie), rated their classification confidence for each 9 decision and then answered control questions. Results revealed no influence of priming condition 10 on judgmental bias, classification accuracy, and classification confidence. Also the level of self-re-11 ported motivation to find out who lied or told the truth did not differ between conditions. Higher 12 motivation was correlated to higher classification confidence. Additionally, and in line with Rein-13 hard (2010) and Reinhard et al. (2011), higher classification accuracy correlated to a higher use of 14 verbal content cues for classification decisions. So, while we were able to replicate these findings, 15 our results contradict the assumption of a money prime influence on lie detection ability. Conclud-16 ing, our results make self-sufficiency in this context questionable and offer next steps for research.

In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. For the spirals only produced by the prime numbers in the 6-spiral pattern plotting, the spiral 2 and spiral 3 each has only one spiral point produced by prime number 2 and 3, respectively, all other spiral points produced by other prime numbers are located on the spiral 1 and spiral 5. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P₁ = 1 + 6 * n (n > 0) P₅ = 5 + 6 * n (n ≥ 0) In other words, every prime number greater than 3 is either a P₁ prime number or a P₅ prime number, no exception. Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Hope such sufficient condition can be found in the future. The number of P₁ prime numbers roughly equal the number of P₅ prime number in the first 2 billion prime numbers. The galactic spiral equations with golden angle can duplicate Vogel’s result for the simulation of sunflower seed head pattern, and a pinwheel pattern can be produced also with galactic spiral equations and 1 degree more than golden angle.

The behaviour of the pseudo-protic ionic liquids TOAH+Cl- and TODAH+Cl- in the facilitated transport of indium(III) from HCl solutions is described. Metal transport is investigated as a function of different variables: hydrodynamic conditions, HCl concentration in the feed phase and indium and carrier concentrations. The experimental data are used to model both transport systems and to estimate diffusional parameters associated to them.

Ofloxacin is a highly efficient and widely used antibiotic drug. It is classified as a refractory pollutant due to its poor biodegradability. Consequently, it is commonly found in water sources, requiring efficient methods for its removal. Advanced Oxidation Processes (AOPs) offer efficient alternatives since those yield complete degradation not achieved in adsorption or membrane processes. Previous studies suggest ofloxacin degradation follows a pseudo-first or -second order processes, whereas for full removal of refractory pollutants – lower pseudo-orders are required. Monitoring the actual “pseudo-order” degradation kinetics of ofloxacin is needed to evaluate any proposed AOP process. This study presents a simple procedure to evaluate pseudo-orders of AOPs. Photolysis of 20 mM ofloxacin solutions follow pseudo-zero order kinetics, with half-life times (t_1/2) of approx. 60 min. TiO₂ heterogenous catalyst show to have no influence at low concentration (0.2 mg L^-1) but a significant reduction of half-life time (t_1/2 = 20 min) and increase in pseudo-order (0.8) is measured at 2.0 mg L^-1. Similar results are obtained with homogenous catalysis by 2.0 mg L^-1 H₂O₂. The combination of H₂O₂ and TiO₂ catalysts shows additional reduction in half-time life with increase in the pseudo-order to 1.2. The conclusions are (1) heterogenous and homogenous photocatalysis can effectively degrade ofloxacin, (2) combined photocatalysis yields higher pseudo-order, being less prone to achieve full removal, (3) analysis of specific pseudo-orders in AOPs of refractory pollutants helps to further elucidate the efficiency of the processes.

New series of compounds containing both heterocycle moieties and pseudo-symmetric hydroxyethylamine core were obtained using a simple synthetic path that can provide a library of compounds in few steps and high yields. Furthermore, diversity-oriented synthesis was studied to change different functionalities according to needs. The in vitro inhibition activity against recombinant HIV-1 protease was evaluated. A beneficial effect of this class of compounds can be obtained either for the presence of a bis-benzyl group into the core and for the heterocyclic moiety in P1, specifically the indole ring. Docking analysis was also reported.

Our method for density elimination is generalized to the non-commutative substructural logic GpsUL∗. Then the standard completeness of HpsUL∗follows as a lemma by virtue of previous workbyMetcalfeandMontagna. This result shows that HpsUL∗ is the logic of pseudo-uninorms and their residua and answered the question posed by Prof. Metcalfe, Olivetti, Gabbay and Tsinakis.

This paper presents a two-stage ADC based on pseudo-differential operational transconductance amplifier (OTA), which is designed for the readout circuit of X-ray linear array sensor. This hybrid ADC employs an incremental sigma-delta ADC and a cyclic ADC, achieving a good trade-off between accuracy and conversion speed. The two stages share the same hardware to reduce power consumption and die area. A common-mood feedback module is used to suppress the influence of charge injection, and the effectiveness is demonstrated by detailed theoretical analysis. A test chip of 14-bit ADC is fabricated in 0.35μm CMOS technology. The measured root mean square (RMS) value of DNL is 0.254 LSB, and the maximum value of INL is -0.776/+1.56 LSB. The measured effective number of bits (ENOB) is 13.43 bits.

In this paper, based on the pseudo-totem pole (PTP) circuit, a family of five-level PTP dual boost converters (PDBC) is proposed. Dual boost converter has some unique advantages such as no bridge arm shoot-through risk and no switch body diode reverse recovery problem, so that it has a good application prospect. Compared with conventional three-level PTP converters, PDBCs has higher power density, higher efficiency, and low voltage and current stress. The topological derivation process, working principle, modulation and strategy of the topology are analyzed firstly. Further, the number of power devices, the switches voltage and current stress of the proposed topology is analyzed. Finally, a representative five-level PDBC experimental prototype is designed with AC input 220V/50Hz, DC output 400V/1kW, and peak efficiency of 98.27%. The experimental results show that the five-level PDBC proposed in this paper has higher efficiency and verify the correctness of the topology.

This article presents a review of the distribution system state estimation (DSSE) algorithms from the model-based approach to the recent data-driven methodologies. The insufficient measurements which result in low observability have motivated the need to shift from the conventional algorithm to data-driven approaches that can successfully estimate states despite the DSSE challenges. This article discusses the nonlinearity in the DSSE problem formulation and how different model-based methods have been proposed to mitigate the problems of robustness, ill-conditioning, and complexity. Moreover, approximate DSSE to obviate nonlinearity were discussed: complex linearization, small angle approximation, convexification and compressed sensing. Furthermore, probabilistic DSSE methods were also discussed in the need to quantify the uncertainty associated with the state estimation results. Also, data-driven methods applicable to DSSE, pseudo measurement generations, and topology identification were also discussed using machine and deep learning methods. Lastly, a recent approach that employs a hybrid of model-based and data-driven methods using matrix and tensor completion is surfacing because they work in a low observable condition of the network and can estimate the states satisfactorily. With this review, researchers can look further into developing and improving on the model-based data-driven methods less susceptible to the barriers in conventional DSSE algorithms.

Abiotic stresses, including drought, salinity, cold, heat, and heavy metals, extensively reduce global agricultural production. Approaches such as conventional breeding and transgenic breeding have been widely used to cope with these environmental stresses. The clustered regularly interspaced short palindromic repeat- Cas (CRISPR/Cas) based gene-editing tool has revolutionized due to its simplicity, accessibility, adaptability, flexibility, and wide applicability. This system has a great potential to build up crop varieties with enhanced tolerance against abiotic stresses. In this review, we summarize the most recent findings on understanding the mechanism of abiotic stress response in plants and the application of CRISPR/Cas mediated gene-editing system towards enhanced tolerance to drought, salinity, cold, heat, and heavy metals stresses. Furthermore, in this review, we highlighted the recent advancements in prime editing and base editing tools for crop improvement.

Therapeutic vaccine studies should be designed to elicit durable, high magnitude, and efficacious T cell responses, all of which can be impacted by the choice of vaccination schedule. Here, we compare different prime-boost intervals (PBI) in a human papillomavirus (HPV) model using HPV16E6E7 Venezuelan equine encephalitis virus replicon particle (VRP) vaccination to address the optimal boosting schedule, quality of immune response, and overall in vivo efficacy. Six different vaccine regimens were tested with each group receiving booster vaccinations at different time intervals. Analysis of T-cell responses demonstrated a significant HPV16 E7 specific CD8+T cell response with minimally a one-week PBI between antigen re-exposure. Significant E7-specific in vivo cytotoxicity was also observed with longer PBIs. Additionally, longer PBIs led to an enhanced memory recall response to tumor challenge, which correlated with differential expansion of T cell memory subsets. Our findings imply that when using alphavirus vector platforms as a vaccination strategy, a one-week PBI is sufficient to induce high magnitude effector T cells with potent anti-tumor activity. However, longer PBIs lead to enhanced long-term protective anti-tumor immunity. These findings have implications for therapeutic vaccine clinical trials in which shorter intervals of prime-boost regimens may lead to suboptimal durable immune responses.

The acquired immunodeficiency syndrome (AIDS) caused by the human immunodeficiency virus (HIV) continues to be a public health problem. In 2020, 680,000 people died from HIV-related causes, and 1.5 million people were infected. Antiretrovirals are only a way to control HIV infection but not to cure AIDS. As such, effective treatment must be developed to control AIDS. Developing a drug is not an easy task, and there is an enormous amount of work and economic resources invested. For this reason, it is highly convenient to employ computer-aided drug design methods, which can help generate and identify novel molecules. Using the de novo design, new novel molecules can be developed using fragments as building blocks. In this work, we develop a virtual-focused compound library of HIV-1 viral protease inhibitors from natural product fragments. Natural products are characterized by a large diversity of functional groups, many sp3 atoms, and chiral centers. Pseudo-natural products are a combination of natural products fragments that keep the desired structural characteristics from different natural products. An interactive version of chemical space visualization of virtual compounds focused on HIV-1 viral protease inhibitors from natural product fragments is freely available at https://figshare.com/s/ceb58d58e8f5585ce67e.

Generative models for images, audio, text and other low-dimension data have achieved great success in recent years. Generating artificial human movements can also be useful for many applications, including improvement of data augmentation methods for human gesture recognition. The object of this research is to develop a generative model for skeletal human movement, allowing to control the action type of generated motion while keeping the authenticity of the result and the natural style variability of gesture execution. We propose to use a conditional Deep Convolutional Generative Adversarial Network (DC-GAN) applied to pseudo-images representing skeletal pose sequences using Tree Structure Skeleton Image format. We evaluate our approach on the 3D-skeleton data provided in the large NTU RGB+D public dataset. Our generative model can output qualitatively correct skeletal human movements for any of its 60 action classes. We also quantitatively evaluate the performance of our model by computing Frechet Inception Distances, which shows strong correlation to human judgement. Up to our knowledge, our work is the first successful class-conditioned generative model for human skeletal motions based on pseudo-image representation of skeletal pose sequences.

In this paper, we give explicit asymptotic formulas for some sums over primes involving generalized alternating hyperharmonic numbers of types I, II and III. Analogous results for numbers with $k$-prime factors will also be considered.

Sickle cell disease is characterized by stiff, “sickled” red blood cells that have difficulty moving through the bloodstream and do not efficiently carry oxygen. It is an inherited disease with severely limited treatment options, and is caused by a point mutation. Its prevalence in black and brown communities makes the already limited treatment options even less accessible. Base editing and prime editing are two relatively recent discoveries in the field of genome editing and were developed after the groundbreaking discovery of the CRISPR Cas9 system. While not fully tested, they hold a lot of promise in providing alternative treatment options for sickle cell disease. Both editing systems are able to install individual point mutations in the beta globin gene, which is where the sickle cell mutation occurs, and can thus cure sickle cell disease (in theory). In this paper we outline the mechanisms of CRISPR-Cas9 systems and base and prime editing, and provide insight into how to apply them to treat SCD. Further investigation should be done on specific editing systems and designs to use to ensure optimal treatment of SCD.

Elucidating physicochemical processes in the degradation of pollutants may optimize their removal from water sources. This work presents a set of elementary steps in the photocatalytic degradation of carbamazepine (CBZ), assuming a steady state approximation in an Advanced Oxidation Process (AOP) combining short-wave ultraviolet radiation (UVC), homogeneous reagent (H₂O₂) and heterogeneous (TiO₂) catalyst. Elementary steps include excitation of both reagent/catalysts by UVC photons, adsorption of CBZ on the excited TiO₂, or its oxidation by hydroxyl radicals. Assuming the steady state approximation on the intermediate products (excited TiO₂, CBZ- excited TiO₂ complex, and hydroxyl radicals), leads to rate laws for degradation of CBZ, in which UVC radiation, TiO₂, and H₂O₂ are pseudo first order at all concentrations or intensities having no direct influence on CBZ pseudo-order, whereas CBZ shifts from pseudo first order at low concentrations to pseudo-zero order at high concentrations. Several experiments to test the mechanism were conducted, by varying CBZ, H₂O₂, and TiO₂ concentrations, and UVC radiation intensities. Measured results indeed fit the suggested mechanism for the first three, but irradiation intensity appears to shift CBZ influence from pseudo-second to pseudo-first order with increased intensities. Corrected elementary steps are proposed to fit the results.

To study the strengthening effect on recycled concrete columns by carbon fiber composite materials (CFRP) under different levels of seismic damage, four column specimens were designed for pseudo-static tests. The four specimens were respectively non-destructive without strengthening (prototype), non-destructive strengthening, medium seismic damage strengthening and severe seismic damage strengthening, according to the replacement rate of recycled aggregates and the level of seismic damage. The characteristics of the deformation damage and seismic performance indicators of each specimen were compared and analyzed. A decrease was observed in the initial stiffness of the seismically damaged recycled concrete column specimens strengthened with CFRP, while the ductility, peak bearing capacity and energy dissipation capacity of the specimens were improved. On the other hand, with the reduction of seismic damage and the increase of recycled aggregate replacement rate, the ductility and energy dissipation capacity of the reinforced seismic damaged recycled concrete column specimens were enhanced to different levels. In particular, the cumulative energy consumption of the strengthened specimens under medium seismic damage increased most significantly by 32.5%. In general, the hysteretic curves of the strengthened specimens were full, and the average ductility coefficients were 4.1–6.8. CFRP strengthening was more effective in restoring and enhancing the performance of the recycled concrete column specimens with medium and lower seismic damage levels (displacement ratio ≤ 3%).

Abstract: Coal bed methane (CBM) reservoirs are complex systems whose properties differ from those of conventional reservoirs. Coal seams are dual-porosity systems that comprise the porosities of the matrix and cleat system. Gas in the coal seams can be stored as free gas in the cleat system and as adsorbed gas in the porous medium. The flow mechanisms of the natural gas through the formation include desorption, diffusion, and Darcy’s flow regimes. The permeability of CBM reservoirs is more sensitive to pressure variations than conventional gas reservoirs. To study the flow behavior of CBM reservoirs it is mandatory to use a model that considers their unique characteristics. The objective of this study was to propose a physical and mathematical model of production performance for horizontal wells in CBM reservoirs whose permeability is dependent on pressure. A solution for the model was obtained by applying Pedrosa´s transformation, perturbation theory, Laplace transformation, the point source method, and Sthefest´s algorithm. The solution to this problem was validated with previous work thoroughly. The type curves of the model were built and the pressure transient behavior of the model was analyzed and discussed. The effects of several parameters on pressure behavior were also discussed.

In this paper, firstly, some questionable formulas and conceptual oversights of previous reduced sigma set unscented transformation (UT) methods are revised through theoretical analysis. Then the revised UT methods based Kalman filters are used in a GPS/INS tightly-coupled system. The Kalman filter flows are the kind of square-root, since the square-root unscented Kalman filters (SRUKFs) can guarantee the stability of the system. By using the reduced sigma set SRUKFs (which contain simplex sigma set square-root unscented Kalman filter (S-SRUKF), spherical simplex sigma set square-root unscented Kalman filter (SS-SRUKF) and minimum sigma set square-root unscented Kalman filter (M-SRUKF)), the computation cost is greatly saved compared with the standard SRUKF, while the accuracy of the GPS/INS tightly-coupled system still maintained. The structure of the GPS/INS tightly-coupled system is in the form of error state, and the time updates of the state and the state covariance of SRUKFs are directly estimated without using UT, thus the computational time is also greatly saved. The pseudo-satellite is introduced to aid the system when the observation information is deficient, for example, when the GPS signal is deficient in the maneuver environment. By using the pseudo-satellite, the optimal performance of the system is guaranteed. Experiment of unmanned aerial vehicle (UAV) showed that the pseudo-satellite aided mechanism worked well.

Conventional plant breeding methods exploit already existing genomic variation in plants to develop a variety in 8 to 10 years, which can decrease the genetic variability of the plant’s genome. The ever-increasing food demand for cereals crops cannot be met by traditional breeding methods. In order to increase food production in less time, there is a dire need to improve breeding methods. Several conventional and molecular breeding methods are being used to improve the crops traits. Molecular researchers have developed new genome editing tools like CRISPR/Cas9, CRISPR/Cpf1, prime editing, base editing, dcas9 epigenetic modification, and several other transgene-free genomes editing approaches. These genome editing tools can improve the desired traits precisely and efficiently. Moreover, a newly developed breeding method “Speed Breeding” has revolutionized the agriculture by shortening the crop cycle. It can produce 5-6 generations of cereals in a year. In this review, we have summarized all these conventional and molecular breeding approaches to improve cereal crops.

This study investigated live animal performance and carcass characteristics of Australian prime lambs fed oil based polyunsaturated fatty acid (PUFA) enriched pellets in a feedlot system. The tested hypothesis was that supplementation of lambs with a variety of dietary oil based PUFA enriched pellets would enhance growth and carcass characteristics compared with the control lambs on lucerne only. Seventy-two, 6 months old White Suffolk x Corriedale first-cross prime lambs with an average liveweight (LWT) of 35.7 ± 0.9 kg were allocated to six treatment groups in a completely randomised experimental design. The treatments were: (1) control: lucerne hay only; or lucerne hay plus wheat-based pellets infused with 50 ml/kg DM of oils from (2) rice bran (RBO); (3) canola (CO); (4) rumen protected (RPO); (5) flaxseed (FO) and (6) safflower (SO) dietary sources. All lambs had ad libitum access to lucerne hay and clean fresh water. Supplemented lambs were fed 1kg of pellet/head/day for 10 weeks. Feed intake, final LWT, average daily gain (ADG), body conformation and carcass characteristics of lambs in the supplemented groups were all greater than for the control group. SO lambs had the lowest ADG of 190.3 g/day. RBO and CO treatments had the lowest feed cost per unit gain of AU$ 3.0/kg. Supplemented lambs had similar over the hooks (OTH) incomes that were all higher than that of the control group. This empirical evidence-based data demonstrated that supplementation of lambs with RBO and CO had comparatively lower feed costs without compromising ADG, carcass characteristics and OTH income.

In this paper, we will extend the expected value of the function w.r.t the uniform probability measure on sets measurable in the Caratheodory sense to be finite for a larger class of functions, since the set of all measurable functions with infinite or undefined expected values forms a prevalent subset of the set of all measurable functions, which means "almost all" measurable functions have infinite or undefined expected values. Before we define the specific problem in section 2, we will outline some preliminary definitions. We'll then define the specific problem (along with a partial solution in section 3) to visualize the complete solution. Along the way, we will ask a series of questions to clarify our understanding of the paper.

In this paper, by extending some results of informational genomics, we present a new randomness test based on the empirical entropy of strings and some properties of repeatability and unrepeatability of substrings of certain lengths. We give the theoretical motivations of our method and some experimental results of its application to a wide class of strings: decimal representations of real numbers, roulette outcomes, logistic maps, linear congruential generators, quantum measurements, natural language texts, and genomes. It will be evident that the evaluation of randomness resulting from our tests does not distinguish among the different sources of randomness (natural, or pseudo-casual).

This Paper intends to introduce the notion of prime ideals of Intuitionistic fuzzy normed Rings and to establish basic properties related to it. It investigates these notions and shown anew Result using intuinistic fuzzy points and non membership function incorparating with t-norm and s-norm to show aome results of fuzzy prime ideal.

The RNA-guided endonuclease system CRISPR-Cas9 has been extensively modified since its discovery, allowing its capabilities to be extending far beyond double-stranded cleavage to high fidelity insertions, deletions, and single base edits. Such innovations have been possible due to the modular architecture of CRISPR-Cas9 and the robustness of its component parts to modifications and the fusion of new functional elements. Here, we review the broad toolkit of CRISPR-Cas9-based systems now available for diverse genome editing tasks. We provide an overview of their core molecular structure and mechanism and distil the design principles used to engineer their diverse functionalities. We end by looking beyond the biochemistry and towards the societal and ethical challenges that these CRISPR-Cas9 systems face if their transformative capabilities are to be deployed in a safe and acceptable manner.

This work is divided into two parts. In the first one the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived, together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs, as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed and new integral lower and upper bounds of π(x) are found.

Euler’s product formula over the primes and Euler’s zeta function equate to enshrine the Fundamental Theorem of Arithmetic that every integer > 1 is the product of a unique set of primes. The product formula has no zero, and with a domain ≤1 Euler’s zeta diverges. Dirichlet’s eta function η(s), negates alternate terms of zeta, permitting convergence when s∈C and Re(s) < 1, and its non-trivial zeros {ρ}, have a deep relationship with the distribution of the primes. The Riemann Hypothesis is that all the non-trivial zeros have Re(ρ) = 1/2. This work examines the symmetries in a partial Euler’s zeta series with a complex domain equating it to the difference between two finite vector series whose matched terms have mirror-image arguments, but whose magnitudes differ when Re(s) ≠ 1/2. Analytical continuation generates a modified eta series η_l(s), in which every l^th term is multiplied by (1-l). If the integer l is appropriately determined by the Im(s), similar paired finite vector series have a difference that closely follows η_l (s) and their terminal vectors intersect in a unique way permitting zeros only when Re(s) = 1/2. Furthermore, those vectors tracking the derivatives of the series, have a special relationship permitting zeros of the differential only when Re(s) > 1/2.

Bacteria develop various defense mechanisms against stresses, including the bacteriophage infection. The giant phiKZ phage infection induced the appearance of a pseudo-nucleus inside the bacterial cytoplasm. Here, we used FISH, electron tomography and analytical electron microscopy to study the morphology of this unique nucleus-like shell and to demonstrate the distribution of phiKZ and bacterial DNA in infected P. aeruginosa cells. The maturation of the pseudo-nucleus was traced in short intervals for 40 min after infection. This study was accompanied by the identification of phiKZ and bacterial DNA by real-time RCR. We demonstrated that phage DNA that isolated from the cytoplasm during all infection stages were compacted within the pseudo-nucleus in a specific structure. Bacterial DNA was diminished in the course of infection, but did not completely degrade until at least 40 min after phage application. The content of the total phage DNA, on the other hand, increased. EDX analysis confirmed these results and revealed that, during the infection, Sulfur content in the bacterial cytoplasm increased, which suggests the increase of DNA-binding Met-reach proteins synthesis, which could protect bacterial DNA from stress.

Statistical modeling of lifetime data plays an essential role in a wide range of practical fields, such as health and engineering. There have been a lot of studies done to develop statistical models that can better describe health data than traditional models. For the first time, we pioneer a novel family of continuous probability distributions called the generalized odd beta prime generalized (GOBP-G) family of distributions. The cumulative distribution and probability density functions of the new family are presented. A new generalization of the Weibull distribution called "generalized odd beta prime-Weibull" (GOBPW) is proposed using the pioneered GOBP-G family. The mixture representations of the new distribution are defined and derived. Some formal statistical properties of the GOBPW distribution, such as the moments, moment generating function, incomplete moments, information generating function, entropies, stress-strength function, quantile function, and order statistics, are derived. The estimation of the parameters of the proposed distribution is evaluated using the maximum likelihood estimation approach. Different cancer disease data sets, such as the bladder, head and neck, acute bone, and blood cancers, are used to illustrate the applicability and usefulness of the new model and were compared using several statistical accuracy measures with that of well-established extended Weibull distributions, which are the beta modified Weibull distribution, Kumaraswamy modified Weibull distribution, gamma generalized modified Weibull distribution, gamma log-logistic Weibull distribution, and beta log-logistic Weibull distribution. The results show that the proposed model gives better results than the competitive models. This study could guide the relevant stakeholders in choosing a suitable statistical model for the health data instead of relying on traditional models to enhance decision-making.

One of the rare diseases throughout the world is Glycogen Storage Disease, which appears due to problems in glycogen metabolism. Among various subtypes of GSD, GSD Type 1a is the most abundant one of GSD Type 1, seen in approximately 80% and caused by different kinds of mutations in the Glucose-6-Phosphatase Catalytic Subunit (G6PC) gene in human chromosome 17q21. G6PC gene encodes for glucose-6-phosphatase (G6Pase) protein, which cleaves glucose-6-phosphate into glucose and inorganic phosphate (Pi), and GSD Type 1a patients fail to breakdown glucose-6-phosphate due to several mutations in the G6PC gene. In our study, we aim to create new therapeutic approaches for GSD 1a. We collected mutation data of 57 GSD Type 1a patients from Turkey. According to the data, 16 types of mutations were observed in the G6PC gene. Allele frequencies of these mutations are calculated as 59% for R83C/H, 11% for W160*, 7% for G270V, and 28% for others which have less frequency. Up to now, the tertiary protein structure of G6Pase has not been structured yet. To understand the possible impacts of these mutations, we statistically obtained possible tertiary structure predictions of G6Pase by running 5 different tools. At the end of the study, we suggest two effective and promising gene therapy methods for GSD Type 1a, Prime Editing for R83C/H mutations, and mRNA delivery for other mutations, in addition to a promising, commercially available drug suggestion for patients with W160*, W86*, and S15* mutations, although the drug belongs to another disease.

This study describes all the properties of high T_c cuprates by introducing rotating holes that are created by angular momentum conservations on a 2D CuO₂ surface, and which have a different mass from that of a normal hole because of the magnetic field energy induced by the rotation. This new particle called a macroscopic Boson describes the doping dependences of pseudo-gap temperature and the transition temperature at which an anomaly metal phase appears and describes the origin of the pseudo-gap. Furthermore, this study introduces a new model to handle many-body interactions, which results in a new statistic equation. This statistic equation describing many-body interactions accurately explains why high T_c cuprates have significantly high critical temperatures. Moreover a partition function of macroscopic Bosons describes all the properties of anomaly metal phase, which sufficiently agree with experiments, using the result from our previous study [1] that analytically describes the doping dependence of T_c. By introducing a macroscopic Boson and the new statistical model for many-body interactions, this study uncovered the mystery of high T_c cuprates, which have been a challenge for many researchers. An important point is that, in this study, pure analytical calculations are consistently conducted, which agree with experimental data well (i.e., they do not use numerical calculations or fitting methods but use many actual physical constants).

This article presents six probability distributions from the Gamma family with three parameters, for the flood frequency analysis in hydrology. The choice of the Gamma family of statistical dis-tributions was driven by its frequent use in hydrology. In the Faculty of Hydrotechnics, the im-provement of the estimation of maximum flows and including the methodological bases for the realization of a regionalization study with the linear moments method with the corrected pa-rameters was researched, being an element of novelty. The linear moments method is better than MOM because it avoids the choice of skewness depending on the origin of the flows, practiced in Romania. The L-moments method conforms to the current trend for estimating the parameters of statistical distributions. Observed data from hydrometric stations are of relatively short length, so the statistical parameters that characterize them are of a sample that requires correction. The correction of the statistical parameters is proposed, using the method of least squares based on the inverse functions of the statistical distributions expressed with the frequency factor for L-moments. All the necessary elements for their use are presented like, quantile functions, the exact and ap-proximate relations for estimating parameters and frequency factors. A flood frequency analysis case study was carried out for the Ialomita river, to verify the proposed methodology. The per-formance of this distributions is evaluated using Kiling-Gupta and Nash-Sutcliff coefficients.

Current prenatal genetic evaluation showed a significantly increase in non-invasive screening and the reduction of invasive diagnostic procedures. To evaluate the diagnostic efficacy on detecting common aneuploidies, structural chromosomal rearrangements and pathogenic copy number variants (pCNV), we performed a retrospective analysis on a case series initially analyzed by aneuvysion fluorescence in situ hybridization (FISH) and karyotyping then followed by array comparative genomic hybridization (aCGH). Of the 386 cases retrieved from the past decade, common aneuploidies were detected in 137 cases (35.5%), other chromosomal structural rearrangements were detected in four cases (1%), and pCNV were detected in five cases (1.3%). The relative frequencies for common aneuploidies suggested a under detection of sex chromosome aneuploidies. Approximately 9.5% of cases with common aneuploidies showed a mosaic pattern. Inconsistent results between FISH and karyotyping were noted in cases with pseudo-mosaicism introduced by culture artifact or variable cellular proliferation from cells with mosaic karyotypic complements under in vitro cell culture. Based on findings from this case series, cell-based FISH and karyotyping should be performed to detect common aneuploidies, structural chromosomal abnormalities, and mosaic pattern. DNA-based aCGH and reflex FISH should be performed to detect and confirm genomic imbalances and pCNV. Practice points to ensure the diagnostic accuracy and efficacy were summarized.