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The virus SARS-CoV-2 has resulted in numerous modelling approaches arising rapidly to understand the spread of the disease COVID-19 and to plan for future interventions. Herein, we present an SEIR model with a spatial spread component as well as four infectious compartments to account for the variety of symptom levels and transmission rate. The model takes into account the pattern of spatial vulnerability in South Africa through a vulnerability index that is based on socioeconomic and health susceptibility characteristics. Another spatially relevant factor in this context is level of mobility throughout. The thesis of this study is that without the contextual spatial spread modelling, the heterogeneity in COVID-19 prevalence in the South African setting would not be captured. The model is illustrated on South African COVID-19 case counts and hospitalisations.

In this paper any superalgebra is generalized to a new form, namely ultra algebra. We study some algebraic structure results for ultra algebras and we have defined some maps on ultra algebras to know ultra derivation and Jordan ultra derivation. Moreover on certain assumptions we have proved that Jordan ultra derivation is an ultra derivation.

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.

In this article, we introduce the concept of soft intervals, soft ordering and sequences of soft real numbers, and some of their structural properties are studied. The notion of soft Lebesgue measure on the soft real numbers has been introduced. Also, a correspondence relationship has been established between the soft Lebesgue measure and the classical Lebesgue measure. Furthermore, we have studied some exciting results and relations between the soft Lebesgue measure and the Lebesgue measure of soft real sets.

The following system of equations {x_1 \cdot x_1=x_2, x_2 \cdot x_2=x_3, 2^{2^{x_1}}=x_3, x_4 \cdot x_5=x_2, x_6 \cdot x_7=x_2} has exactly one solution in ({\mathbb N}\{0,1})^7, namely (2,4,16,2,2,2,2). Hypothesis 1 states that if a system of equations S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,7}} \cup {2^{2^{x_j}}=x_k: j,k \in {1,...,7}} has at most five equations and at most finitely many solutions in ({\mathbb N}\{0,1})^7, then each such solution (x_1,...,x_7) satisfies x_1,...,x_7 \leq 16. Hypothesis 1 implies that there are infinitely many composite numbers of the form 2^{2^{n}}+1. Hypotheses 2 and 3 are of similar kind. Hypothesis 2 implies that if the equation x!+1=y^2 has at most 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)}. Hypothesis 3 implies that if the equation x(x+1)=y! has at most finitely many solutions in positive integers x and y, then each such solution (x,y) belongs to the set {(1,2),(2,3)}. We describe semi-algorithms sem_j (j=1,2,3) that never terminate. For every j \in {1,2,3}, if Hypothesis j is true, then sem_j endlessly prints consecutive positive integers starting from 1. For every j \in {1,2,3}, if Hypothesis j is false, then sem_j prints a finite number (including zero) of consecutive positive integers starting from 1., 10 pages

Build a special identical equation, use its calculation characters to prove and search for solution of any odd converging to 1 equation through (*3+1)/2^k operation, change the operation to (*3+2^m-1)/2^k, and get a solution for this equation, which is exactly same with that got from calculating directly. Then give a specific example to verify. Thus prove the Collatz Conjecture is true. Furthermore, analysis the sequences produced by iteration calculation during the procedure of searching for solution, build a weight function model, prove it decrease progressively to 0, build a complement weight function model, prove it increase to its convergence state. Build a (*3+2^m-1)/2^k odd tree, prove if odd in (*3+2^m-1)/2^k long huge odd sequence can not converge, the sequence must outstep the boundary of the tree after infinite steps of (*3+2^m-1)/2^k operation.

We approximate the trigonometric function sine and cosine on the interval . This analysis provides two formulas to approximate sine and cosine. At first, we try to derive the formula which involves a square root, and then we derive another formula that does not require any use of a square root. Nevertheless, after deriving the procedure which requires no square root, we further try to increase its accuracy and then derive another formula that approximates trigonometric functions more accurately on the interval [0, pi/2]. So, this analysis provides mainly two types of procedures. One uses square roots, whereas the other does not. We also focus on ensuring the accuracy of these trigonometric functions in the interval [0,pi/2]. This accuracy analysis is portrayed using the graph. This graph shows the difference between the values generated by the functions defined here and the actual value of these functions. So, these graphs also indicate the error of these functions on that interval. Finally, we compare our approximation with the approximation formula of the 7th-century Indian Mathematician Bhaskara I.

Recent improvements in deepfake creation made deepfake videos more realistic. Open-source software has also made deepfake creation more accessible, which reduces the barrier to entry for deepfake creation. This could pose a threat to the public privacy. It is a potential danger if the deepfake creation techniques are used by people with an ulterior motive to produce deepfake videos of world leaders to disrupt the order of the countries and the world. Research into automated detection for deepfaked media is therefore essential for public safety. We propose in this work the use of upper body language analysis for deepfake detection. Specifically, a many-to-one LSTM network was designed and trained as a classification model is trained for deepfake detection. Different models trained using various hyper-parameters to build a final model with benchmark accuracy. We achieve 94.39% accuracy on a test deepfake set. The experimental results show that upper body language can effectively provide identification and deepfake detection.

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.

This paper ensures an extensive survey of the generalization of the various hybrid numbers and hybrid polynomials especially as part of its enhancing importance in the disciplines of mathematics and physics. In this paper, by using the Horadam polynomials, we define the Horadam hybrid polynomials called Horadam hybrinomials. We obtain some special cases and algebraic properties of the Horadam hybrinomials such as recurrence relation, generating function, exponential generating function, Binet formula, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity, respectively. Moreover, we give some applications related to the Horadam hybrinomials in matrices.

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.

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

The 3x+1 problem asks the following: Suppose we start with a positive integer, and if it is odd then multiply it by 3 and add 1, and if it is even, divide it by 2. Then repeat this process as long as you can. Do you eventually reach the integer 1, no matter what you started with? Collatz conjecture (or 3n+1 problem) has been explored for about 85 years. In this paper, we convert an integer number from decimal to binary number, and convert the Collatz function to binary function, which is multiplication and division of two binary numbers. Finally the iternation of the Collatz function, eventually reach the integer 1, thus we solve the 3x+1 problem completely.

The 3x+1 problem asks the following: Suppose we start with a positive integer, and if it is odd then multiply it by 3 and add 1, and if it is even, divide it by 2. Then repeat this process as long as you can. Do you eventually reach the integer 1, no matter what you started with? Collatz conjecture (or 3n+1 problem) has been explored for about 85 years. In this article, we prove the Collatz conjecture by modifying Sharkovsky ordering of positive integers and denote the composition of the collatz function as a algebraic formula about $\frac{3^{m}}{(2^{r}}$, convert the problem to a algebraic problem, we can solve it completely.

Proof of the Riemann hypothesis is obtained using the properties of the Gamma function, the Bernoulli function, and limitations imposed by the rational relations of the complex roots of the Riemann zeta function, and the rational relations of the complex Gamma functions.

The calculation of the exact value of the square root of 2 is requested. In order to obtain its infinite value, A new concept is proposed where the accumulations of the infinitely many of finity is indicated by the change in direction which means that there is a jump from finity to infinity .The meaningless for an infinite number with a decimal point is indicated by this jump because any decimal number only have meanings within a finite range values and there is only an infinite integer quantity that can not be operated by algorithms like operations of multiplication，division,addition, and subtraction . The final result of the change in direction is two quantity where the second quantity and the first quantity extend in parallel line and never intersect and the second quantity represent the size of the first quantity. The first quantity is the infinitely great that can't be talked about anything outside of it and can compress any quantities outside of it to nothing and it is the exact value of the square root of 2 .

Collatz conjecture (or 3n+1 problem) has been explored for about 85 years. In this article, we prove the Collatz conjecture. We will show that this conjecture is valid for all positive integers by performing the Collatz inverse operation on the numbers that comply with the rules of the Collatz conjecture. Finally, it will be proved that there are no positive integers that do not comply with this conjecture.

In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become $\left( m,n\right) $-rings. Second, we introduce a possible $p$-adic analog of the residue class modulo a $p$-adic integer. Then, we find the relations which determine, when the representatives form a $\left( m,n\right) $-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models., Comment: 9 pages, amslatex, the journal version: small corrections and changes, Example 2.1 and Conclusions added, Bibliography updated; https://www.mdpi.com/2073-8994/14/12/2591

In this paper, we define and investigate modified p-Leonardo, p-Leonardo-Lucas and p-Leonardo sequences as special cases of the generalized Leonardo sequence. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there are close relations between modified p-Leonardo, p-Leonardo-Lucas, p-Leonardo numbers and Fibonacci, Lucas numbers.

This paper asserts that the falsity of Goldbach Conjecture results in the breakdown of Trichotomy Law. Assuming that Goldbach Conjecture breaks down at an even integer 2n, where n ≥ 3, we note that the differences of 2n with the odd primes in the interval (1, n) are composite integers. We form a product δ of the collection of the least prime factors of these composite integers. We show that for a base case this product breaks Trichotomy Law. With the inductive hypothesis that such is the case with 2n, we prove that for all even integers the breakdown of Goldbach Conjecture entails the breakdown of Trichotomy Law.

Dilation of contractions on Hilbert space and Banach space is classical. Recently, dilations theory has been put in the setting of sets, vector spaces, p-adic Hilbert spaces and modules. In this paper, we derive important dilation results for self-adjoint morphisms on indefinite inner product modules over *-rings of characteristic 2. More precisely, we prove indefinite inner product versions of Halmos dilation, Egervary N-dilation and Sz.-Nagy dilation.

We evaluate the nested sum $\sum_{a_{n - 1} = c}^{a_n } {\sum_{a_{n - 2} = c}^{a_{n - 1} } { \cdots \sum_{a_0 = c}^{a_1 } {x^{a_0 } } } }$ where $a_n$ and $c$ are any integers and $x$ is a real or complex variable. Consequently, we evaluate multiple sums involving the terms of a general second order sequence, the Horadam sequence $(W_j(a,b;p,q))$, defined for all non-negative integers $j$ by the recurrence relation $W_0 = a,\,W_1 = b;\,W_j = pW_{j - 1} - qW_{j - 2}\, (j \ge 2)$; where $a$, $b$, $p$ and $q$ are arbitrary complex numbers, with $p\ne 0$, $q\ne 0$., 16 pages, no figures or tables

Modern agent-based models (ABM) and other simulation models require evaluation and testing of many different parameters. Managing that testing for large scale parameter sweeps (grid searches), as well as storing simulation data, requires multiple, potentially customizable steps that may vary across simulations. Furthermore, parameter testing, processing, and analysis are slowed if simulation and processing jobs cannot be shared across teammates or computational resources. While high-performance computing (HPC) has become increasingly available, models can often be tested faster with the use of multiple computers and HPC resources. To address these issues, we created the Distributed Automated Parameter Testing (DAPT) Python package. By hosting parameters in an online (and often free) “database”, multiple individuals can run parameter sets simultaneously in a distributed fashion, enabling ad hoc crowdsourcing of computational power. Combining this with a flexible, scriptable tool set, teams can evaluate models and assess their underlying hypotheses quickly. Here, we describe DAPT and provide an example demonstrating its use.

Based on the Hadamard product $\xi(s)= \xi(0)\prod_{\rho}(1-\frac{s}{\rho})$, a new expression of $\xi(s)$ is obtained by paring $\rho$ and $\bar{\rho}$ $$\xi(s)=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{d_{i}}$$ where $\xi(0)=\frac{1}{2}$, $\rho_i=\alpha_i+j\beta_i$ and $\bar{\rho}_i=\alpha_i-j\beta_i$ are complex conjugate zeros of $\xi(s)$, $0

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 consequential 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.

The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2 and https://philpapers.org/rec/PENFLT-3) demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart.

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

In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.

A generalization of ?-Ricci solitons is considered involving an additional metric and functions as soliton coefficients. The soliton potential is torse-forming and orthogonal to the contact distribution of the almost contact B-metric manifold. Then such a manifold can also be considered as an almost Einstein like manifold, a generalization of an ?-Einstein manifold with respect to both B-metrics and functions as coefficients. Necessary and sufficient conditions are found for a number of properties of the curvature tensor and its Ricci tensor of the studied manifolds. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.

In this part of the series of two papers, we extend the theorems discussed in part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta function, Riemann zeta function, Polylogarithm and Fibonacci numbers. In continuation, we obtain some zeros of the newly developed zeta functions and explain their behaviour using plots in complex plane. Furthermore, we provide particular cases for the theorems and corollaries which show that our results generalise the currently available functions and series such as the Riemann zeta function and the geometric series. Finally, we provide four miscellaneous examples to showcase the vast scope of the developed theorems.

In gravitational theory and astrophysical dynamics, singular initial value problems (IVPs) are frequently encountered. Finding the solutions to this class of IVPs can be challenging due to their complex nature. This study strives to circumvent the complexity by proposing a numerical method for solving such problems. The approach proposed in the current research seeks solutions to the IVP by partitioning the domain [0,L] of the problem into two intervals and solving the problem on each domain. The study seeks a closed-form solution to the IVP in the interval containing the singular point. A linearization technique and piecewise partitioning of the domain not containing the singularity are applied to the nonlinear IVP. The resulting linearized differential equation is solved using the Chebyshev spectral collocation method. Some examples are presented to illustrate the efficiency of the proposed method. Numerical analysis of the solution and residual errors are shown to ascertain convergence and accuracy. The results suggest that the technique gives accurate convergent solutions using a few collocation points.

Maze navigation using one or more robots has become a recurring challenge in scientific literature and real life practice, with fleets having to find faster and better ways to navigate environments such as a travel hub, airports, or for evacuation of disaster zones. Many methodologies have been explored to solve this issue, including the implementation of a variety of sensors and other signal receiving systems. Most interestingly, camera-based techniques have become more popular in this kind of scenarios, given their robustness and scalability. In this paper, we implement an end-to-end strategy to address this scenario, allowing a robot to solve a maze in an autonomous way, by using computer vision and path planning. In addition, this robot shares the generated knowledge to another by means of communication protocols, having to adapt its mechanical characteristics to be capable of solving the same challenge. The paper presents experimental validation of the four components of this solution, namely camera calibration, maze mapping, path planning and robot communication. Finally, we showcase some initial experimentation in a pair of robots with different mechanical characteristics. Further implementations of this work include communicating the robots for other tasks, such as teaching assistance, remote classes, and other innovations in higher education.

In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identitiy of these numbers.

A generalization of the sum of divisors function involves a recursive definition. This leads to variants of superabundant numbers, colossally abundant numbers, and Gronwall's theorem (relevant to the Riemann hypothesis).

Malware analysis is fundamental for defending against prevalent cyber security threats and requires a means to deploy and study behavioural software traits as more sophisticated malware is developed. Traditionally, virtual machines are used to provide an environment that is isolated from production systems so as to not cause any adverse impact on existing infrastructure. Malware developers are fully aware of this and so will often develop evasion techniques to avoid detection within sandbox environments. In this paper, we conduct an investigation of anti-evasion malware triggers for uncovering malware that may attempt to conceal itself when deployed in a traditional sandbox environment. To facilitate our investigation, we developed a tool called MORRIGU that couples together both automated and human-driven analysis for systematic testing of anti-evasion methods using dynamic sandbox reconfiguration techniques. This is further supported by visualisation methods for performing comparative analysis of system activity when malware is deployed under different sandbox configurations. Our study reveals a variety of anti-evasion traits that are shared amongst different malware families, such as sandbox &ldquo, wear-and-tear&rdquo, and Reverse Turing Tests (RTT), as well as more sophisticated malware samples that require multiple anti-evasion checks to be deployed. We also perform a comparative study using Cuckoo sandbox to demonstrate the limitations of adopting only automated analysis tools, to justify the exploratory analysis provided by MORRIGU. By adopting a clearer systematic process for uncovering anti-evasion malware triggers, as supported by tools like MORRIGU, this study helps to further the research of evasive malware analysis so that we can better defend against such future attacks.

Motivated by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim, in the present paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Frobenius-Genocchi polynomias equal a linear combination of the classical Frobenius-Genocchi polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim, we introduce the unipoly-Frobenius-Genocchi polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.

The 3x+1 problem is a problem of continuous iteration for integers. According to the basic theorem of arithmetic and the way of iteration, we derive a general formula for continuous iteration for odd integers. Through this formula, we can construct a loop iteration equation and obtain the result of the equation: the equation has only one positive integer solution. In addition, this general formula can be converted into a linear indeterminate equation. The process of solving this equation shows that the relationship between the iteration result and the odd number being iterated is linear. Extending this result to all positive even numbers, we get the answer to the 3x + 1 question.

Coronavirus disease (COVID-19) is a pandemic infectious disease that has a severe risk of spreading rapidly. The quick identification and isolation of the affected persons is the very first step to fight against this virus. In this regard, chest radiology images have been proven to be an effective screening approach of COVID-19 affected patients. A number of AI based solutions have been developed to make the screening of radiological images faster and more accurate in detecting COVID-19. In this study, we are proposing a deep learning based approach using Densenet-121 to effectively detect COVID-19 patients. We incorporated transfer learning technique to leverage the information regarding radiology image learned by another model (CheXNet) which was trained on a huge Radiology dataset of 112,120 images. We trained and tested our model on COVIDx dataset containing 13,800 chest radiography images across 13,725 patients. To check the robustness of our model, we performed both two-class and three-class classifications and achieved 96.49% and 93.71% accuracy respectively. To further validate the consistency of our performance, we performed patient-wise k-fold cross-validation and achieved an average accuracy of 92.91% for three class task. Moreover, we performed an interpretability analysis using Grad-CAM to highlight the most important image regions in making a prediction. Besides ensuring trustworthiness, this explainability can also provide new insights about the critical factors regarding COVID-19. Finally, we developed a website that takes chest radiology images as input and generates probabilities of the presence of COVID-19 or pneumonia and a heatmap highlighting the probable infected regions. Code and models' weights are availabe.

Multi-objective optimization (MOO) is an optimization involving minimization or maximization of several objective functions more than the conventional one objective optimization, which is useful in many fields. Many of the current methodologies addresses challenges and solutions that attempt to solve simultaneously several Objectives with multiple constraints subjoined to each. Often MOO are generally subjected to linear inequality, equality and or bounded constraint that prevent all objectives from being optimized at once. This paper reviews some recent articles in area of MOO and presents deep analysis of Random and Uniform Entry-Exit time of objectives. It further break down process into sub-process and then provide some new concepts for solving problems in MOO, which comes due to periodical objectives that do not stay for the entire duration of process lifetime, unlike permanent objectives which are optimized once for the entire process duration. A methodology based on partial optimization that optimizes each objective iteratively and weight convergence method that optimizes sub-group of objectives are given. Furthermore, another method is introduced which involve objective classification, ranking, estimation and prediction where objectives are classified based on their properties, and ranked using a given criteria and in addition estimated for an optimal weight point (pareto optimal point) if it certifies a coveted optimal weight point. Then finally predicted to find how far it deviates from the estimated optimal weight point. A Sample Mathematical Tri-Objectives and Real world Optimization was analyzed using partial method, ranking and classification method, the result showed that an objective can be added or removed without affecting previous or existing optimal solutions. Therefore suitable for handling time governed MOO. Although this paper presents concepts work only, it’s practical application are beyond the scope of this paper, however base on analysis and examples presented, the concept is worthy of igniting further research and application.

In this paper, we examine the weak abc conjecture and the strong abc conjecture using some functions. we attempt to discuss the possibilty that the negative consequence of the weak abc conjecture. Namely, we derive that the weak abc conjecture is not true. Additionally, we show that the strong abc conjecture is not true.

The paper is a continuation of another paper (https://philpapers.org/rec/PENFLT-2) published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense (https://dx.doi.org/10.2139/ssrn.3656179), in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.

In this paper, we present sums and partial sums of Horadam sequences via generating functions which extends a recent result of Prodinger.

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.

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.

Let a,b and n>1 be three positive integers such that a and ∑j=0n−1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of N generated by {∑j=0n−1bj}∪{∑j=0n−1bj+a∑j=0i−2bj∣i=2,…,n} is determinantal. Moreover, we prove that for n>3, the ideal I has a unique minimal system of generators if and only if a

For discrete big data which have a limited range of values, Conventional machine learning methods cannot be applied because we see clutter and overlapping of classes in such data: many data points from different classes overlap. In this paper we introduce a solution for this problem through a novel heuristics method. By applying a running average (with a window-size= d) we could transform Discrete data to broad-range, Continuous values. When we have more than 2 columns and one of them is containing data about the tags of classification (Class Column), we could compare and sort the features (Non-class Columns) based on the R2 coefficient of the regression for running averages. The parameters tuning could help us to select the best features (the non-class columns which have the best correlation with the Class Column). “Window size” and “Ordering” could be tuned to achieve the goal. This optimization problem is hard and we need an Algorithm (or Heuristics) for simplifying this tuning. We demonstrate a novel heuristics, Called Simulated Distillation (SimulaD), which could help us to gain a somehow good results with this optimization problem.

In this paper, we derive Von Koch's inequality without using the Riemann Hypothesis. As well known, Von Koch's inequality is equivalent to the Riemann Hypothesis. Namely, we show that the Riemann Hypothesis is true. Furthermore, we show that the upper limit of its can be lowered.

Multiagent technologies provide a new way for studying and controlling complex systems. Local interactions between agents often lead to group synchronization, also known as clusterization (or clustering), which is usually a more rapid process in comparison with relatively slow changes in external environment. Usually, the goal of system control is defined by the behavior of a system on long time intervals. As is well known, a clustering procedure is generally much faster than the process of changing in the surrounding (system) environment. In this case, as a rule, the control objectives are determined by the behavior of the system at large time intervals. If the considered time interval is much larger than the time during which the clusters are formed, then the formed clusters can be considered to be &ldquo, new variables&rdquo, in the &ldquo, slow&rdquo, time model. Such variables are called &ldquo, mesoscopic&rdquo, because their scale is between the level of the entire system (macro-level) and the level of individual agents (micro-level). Detailed models of complex systems that consist of a large number of elementary components (miniature agents) are very difficult to control due to technological barriers and the colossal complexity of tasks due to their enormous dimension. At the level of elementary components of systems, in many applications it is impossible to verify the models of the agent dynamics with the traditionally high degree of accuracy, due to their miniaturization and high frequency of control actions. The use of new mesoscopic variables can make it possible to synthesize fewer different control inputs than when considering the system as a collection of a large number of agents, since such inputs will be common for entire clusters. In order to implement this idea, the &ldquo, clusters flow&rdquo, framework was formalized and used to analyze the Kuramoto model as an attracting example of a complex nonlinear networked system with the effects of opportunities for the emergence of clusters. It is shown that clustering leads to a sparse representation of the dynamic trajectories of the system, which makes it possible to apply the method of compressive sensing in order to obtain the dynamic characteristics of the formed clusters. The essence of the method is as follows. With the dimension N of the total state space of the entire system and the a priori assignment of the upper bound for the number of clusters s, only m integral randomized observations of the general state vector of the entire large system are formed, where m is proportional to the number s that is multiplied by logarithm N/s. A two-stage observation algorithm is proposed: first, the state space is limited and discretized, and compression then occurs directly, according to which reconstruction is then performed, which makes it possible to obtain the integral characteristics of the clusters. Based on these obtained characteristics, further, it is possible to synthesize mesocontrols for each cluster while using general model predictive control methods in a space of dimension no more than s for a given control goal, while taking the constraints obtained on the controls into account. In the current work, we focus on a centralized strategy of observations, leaving possible decentralized approaches for the future research. The performance of the new framework is illustrated with examples of simulation modeling.

This paper studies the goodness of fit test for the bivariate Hermite distribution. Specifically, we propose and study a Cram\'er-von Mises-type test based on the empirical probability generation function. The bootstrap can be used to consistently estimate the null distribution of the test statistics. A simulation study investigates the goodness of the bootstrap approach for finite sample sizes., Comment: 21 pages

Locally finiteness of some varieties of nonassociative coalgebras is studied and the Gelfand-Dorfman construction for Novikov coalgebras and the Kantor construction for Jordan super-coalgebras are given. We give examples of a non-locally finite differential coalgebra, Novikov coalgebra, Lie coalgebra, Jordan super-coalgebra, and right-alternative coalgebra. The dual algebra of each of these examples satisfies very strong additional identities. We also constructed examples of an infinite dimensional simple differential coalgebra, Novikov coalgebra, Lie coalgebra, and Jordan super-coalgebra over a field of characteristic zero., 19 pages