The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients related to each other. A unique relation between the Euler-Frobenius polynomial and B-spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and finite sums are derived. Moreover, A few recurrence relations containing these particular finite sums are given. Using these recurence relations, we give a solution of the problem which was given by Charalambides7, Exercise 30, p. 273. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.

We study the ground state solutions for the following p\&q-Laplacain equation \[ \left\{ \begin{array}{ll} -\Delta_pu-\Delta_qu+V(x) (|u|^{p-2}u+|u|^{q-2}u)=\lambda K(x)f(u)+|u|^{q^*-2}u,~x\in\R^N, \\ u\in W^{1,p}(\R^N)\cap W^{1,q}(\R^N), \end{array} \right. \] where $\lambda>0$ is a parameter large enough, $\Delta_ru = \text{div}(|\nabla u|^{r-2}\nabla u)$ with $r\in\{p,q\}$ denotes the $r$ Laplacian operator, $1

We present a framework to model and provide numerical evidence for compartmentalization in the yeast endoplasmic reticulum. Measurement data is collected and an optimal control problem is formulated as a regularized inverse problem. To our knowledge, this is the first attempt in the literature to introduce a PDE-constrained optimization formulation to study the kinetics of fluorescently labeled molecules in budding yeast. Optimality conditions are derived and a gradient descent algorithm allows accurate estimation of unknown key parameters in different cellular compartments. For the first time, the numerical results support the barrier index theory suggesting the presence of a physical diffusion barrier that compartmentalizes the endoplasmic reticulum by limiting protein exchange between the mother and its growing bud. We report several numerical experiments on real data and geometry, with the aim of illustrating the accuracy and efficiency of the method. Furthermore, a relationship between the size ratio of mother and bud compartments and the barrier index ratio is provided.

The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations, but also are physically relevant and mathematically important. Unfortunately, it is not often well understood for many ranges of fractional powers. This paper focuses on a system of the 3D Boussinesq equations with fractional horizontal ( − ∆ h ) α u and ( − ∆ h ) β θ dissipation and proves that if an initial data ( u 0 , θ 0 ) in the Sobolev space H 3 ( R 3 ) close enough to the hydrostatic balance state, respectively, the equations with α , β ∈ ( 1 2 , 1 ] then always lead to a steady solution.

This paper is devoted to solve the backward problem for radially symmetric time-fractional diffusion-wave equation under Robin boundary condition. This problem is ill-posed and we apply an iterative regularization method to solve it. The error estimates are obtained under the a priori and a posteriori parameter choice rules. Numerical results show that the proposed method is efficient and stable.

Precision in measurement of glucose level in artificial pancreas is a challenging task and mandatory requirement for the proper functioning of artificial pancreas. A suitable machine learning technique for the measurement of glucose level in artificial pancreas may play crucial role in the management of diabetes. Therefore in the present work, a comparison has been made among few machine learning (ML) techniques for measurement of glucose levels in artificial pancreas because the machine learning is an astounding technology of artificial intelligence, and widely applicable in various fields such as medical science, robotics, environmental science, etc. The models namely decision tree (DT), random forest (RF), support vector machine (SVM), and K-nearest neighbours (KNN), based on supervised learning, are proposed for the dataset of Pima Indian to predict and classify the diabetes mellitus. Ensuring the predictions and accuracy up to the level of DMT2, the comparative behavior of all four models has been discussed. The machine learning models developed here stratifies and predicts whether an individual is diabetic or not based on the features available in the data set. Dataset passes through pre-processing and machine learning algorithms are fitted to train the dataset, and then the performance of the test results has been discussed. Error matrix (EM) has been generated to measure the accuracy score of the models. The accuracies in prediction and classification of DMT2 models are 71%, 77%, 78%, and 80% for DT, SVM, RF and KNN algorithms respectively. The KNN model has shown a more precise result in comparison to other models. The proposed methods have shown astounding behaviour in terms of accuracy in the prediction of diabetes mellitus as compared to previously developed methods.

In this work we propose the construction of discrete-time systems with two time scales in which infectious diseases dynamics are involved. We deal with two general situations. In the first, we consider that individuals affected by the disease move between generalized sites on a faster time scale than the dynamics of the disease itself. The second situation includes the dynamics of the disease acting faster together with another slower general process. Once the models have been built, conditions are established so that the analysis of the asymptotic behavior of their solutions can be carried out through reduced models. This is done using known reduction results for discrete-time systems with two time scales. These results are applied in the analysis of two new models. The first of them illustrates the first proposed situation, being the local dynamics of the SIS-type disease. Conditions are found for the eradication or global endemicity of the disease. In the second model, a case of co-infection with a primary disease and an opportunistic disease is treated, the latter acting faster than the former. Conditions for eradication and endemicity of co-infection are proposed.

We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method including one dimensional periodic potentials of interest in quantum physics. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.

This paper firstly finishes off the exact solutions of Riemann-Liouville fractional differential time-delay oscillatory system of order ρ∈(1 ,2) by using two newly defined delayed perturbations of Mittag-Leffler matrix functions and constant variation method. In the light of the exact solutions, we explore the finite time stability of the nonhomogeneous fractional oscillatory differential equations with pure delay. Ultimately, an example is cited to verify the rationality of the results. Through our method, the public problems left by Mahmudov in 2022 were partially solved.

In this paper we consider the Catalan triangle numbers ( B n , k ) n ≥ 1 , 1 ≤ k ≤ n and ( A n , k ) n ≥ 1 , 1 ≤ k ≤ n + 1 to define powers of Catalan generating function C( T) where T is a linear and bounded operator on a Banach space X. When the operator 4 T is of power-bounded operator, the Catalan generating function is given by the Taylor series C ( T ) : = ∑ n = 0 ∞ C n T n , where c = ( C n ) n ≥ 0 is the Catalan sequence. Note that the operator C( T) is a solution of the quadratic equation T Y 2 − Y + I = 0 . We obtain new formulae which involves Catalan triangle numbers ( B n , k ) n ≥ 1 , 1 ≤ k ≤ n and ( A n , k ) n ≥ 1 , 1 ≤ k ≤ n + 1 . As element in the Banach algebra ℓ 1 ( N 0 , 1 4 n ) , we describe the spectrum of c ∗ j for j≥1, and the expression of c −∗ j in terms of Catalan polynomials. In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future.

In this paper, we study two inverse problems for the nonlinear Boussinesq system for incompressible viscoelastic non-isothermal Kelvin-Voigt fluids. The studying inverse problems consist of determining an intensities of density of external forces and heat source under given integral overdetermination conditions. Two types of boundary conditions for the velocity v are considered: sticking and sliding conditions on boundary. In both cases of these boundary conditions, the local and global in time existence and uniqueness of weak and strong solutions are established under suitable assumptions on the data. The large time behavior of weak solutions is also studied.

a) | k ( w ) | ≥ 0 is a piecewise continuous and bounded function in R = ( - ∞ , ∞ ) . The coefficients b ( w ) and q ( w ) are continuous functions in R and can be unbounded at infinity. The operator L admits closure in the space L 2 ( Ω ) and the closure is also denoted by L. Taking into consideration certain constraints on the coefficients b ( w ) q ( w ) , apart from the above-mentioned conditions, the existence of a bounded inverse operator is proved in this paper; a condition guaranteeing compactness of the resolvent kernel is found; and we also obtained two-sided estimates for singular numbers ( s-numbers). Here we note that the estimate of singular numbers ( s-numbers) shows the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. It is given an example of how the obtained estimates for the s-numbers enable to identify the estimates for the eigenvalues of the operator L. We note that the above results are apparently obtained for the first time for a mixed-type operator in the case of an unbounded domain with rapidly oscillating and greatly growing coefficients at infinity.

In this paper, we propose a unified non-conforming least-squares spectral element ap- proach for solving Stokes equations with various non-standard boundary conditions. Ex- isting least-squares formulations mostly deal with Dirichlet boundary conditions and are formulated using ADN theory based regularity estimates. However, changing boundary conditions lead to a search for parameters satisfying supplementing and complimenting con- ditions [4] which is not easy always. Here we have avoided ADN theory based regularity estimates and proposed a unified approach for dealing with various boundary conditions. Stability estimates and error estimates have been discussed. Numerical results displaying exponential accuracy have been presented for both two and three dimensional cases with various boundary conditions.

In this paper, we design two parametric classes of iterative methods without memory to solve nonlinear systems, whose convergence order is four and seven, respectively. From their error equations and to increase the convergence order without performing new functional evaluations, memory is introduced in these families of different forms. That allows us to increase from four to seven the convergence order in the first family and from seven to eleven in the second one. We perform some numerical experiments with big size systems for confirming the theoretical results and comparing the proposed methods along other known schemes.

We consider the scattering of a plane wave by a locally perturbed periodic (with respect to x_1) medium. If there is no perturbation it is usually assumed that the scattered wave is quasi-periodic with the same parameter as the incident plane wave. As it is well known, one can show existence under this condition but not necessarily uniqueness. Uniqueness fails for certain incident directions (if the wavenumber is kept fixed), and it is not clear which additional condition has to be assumed in this case. In this paper we will analyze three concepts. For the Limiting Absorption Principle (LAP) we replace the refractive index n=n(x) by n(x)+iε in a layer of finite width and consider the limiting case when ε tends to zero. This will give an unsatisfactory condition. In a second approach we require continuity of the field with respect to the incident direction. This will give the same satisfactory condition as the third approach where we approximate the incident plane wave by an incident point source and let the location of the source tend to infinity.

A new fractional difference equation 2D-TFCDM based on Caputo derivative is proposed. Using the bifurcation diagram, the maximum Lyapunov exponent and the phase diagram, the numerical solutions of the fractional difference equations are obtained, and the chaotic behavior is observed numerically. After encrypting the key with elliptic curve cryptosystem, the fractional map is developed as an encryption algorithm and applied to color image encryption. Finally, the proposed encryption system is systematically analyzed from five main aspects, and the results show that the proposed encryption system has a good encryption effect. In respective of application, we apply the proposed discrete fractional map into color image encryption with the secret keys ciphered by Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC). Finally, the image encryption algorithm is analysed in 4 aspects that indicates the proposed algorithm is superior to others.

In this paper, we investigate pattern dynamics in a reaction-diffusion-chemotaxis food chain model with predator-taxis, which enriches previous studies about diffusive food chain models. By virtue of diffusion semigroup theory, we first show the global classical solvability and uniform boundedness of the considered model over a bounded domain Ω ⊂ R N ( N ≥ 1 ) with smooth boundary. Then the linear stability analysis for the considered model shows that chemotaxis can induce the unique positive spatially homogeneous steady state loses its stability via Turing bifurcation and Turing-spatiotemporal Hopf bifurcation, which results in the formation of two kinds of important spatiotemporal patterns: stationary Turing pattern and oscillatory pattern. Simultaneously, the threshold values for Turing bifurcation and Turing-spatiotemporal Hopf bifurcation are given explicitly. In addition, the existence and stability of non-constant positive steady state that bifurcates from the positive constant steady state is investigated by the abstract bifurcation theory of Crandall-Rabinowitz and eigenvalue perturbation theory. Finally, numerical simulations are performed to verify our theoretical results, and some interesting non-Turing pattern are found in temporal Hopf parameter space by numerical simulation.

In this paper we consider the equilibrium problem of interaction of three elastic bodies of different elastic properties. The main body is the unit cube. On top of it is a thin layer/quboid of thickness $\eps$ of material whose stiffness is of order $\frac{1}{\eps}$ that in the middle contains another cuboid which is of width and thickness $\eps$ that is made of material with elasticity coefficients of order $\frac{1}{\eps^q}$ for $q>0$. We show that the family of solutions of linearized elasticity problems, when $\eps$ tends to zero, converges to a solution of a problem that is posed only on the unit cube with possibly additional elastic terms on the boundary related to the plate/rod energy of the thin elastic parts. It turns out that there are five different regimes related to different values of $q$ ($q\in \langle 0, 2\rangle, \{2\}, \langle 2, 4\rangle,\{4\},\langle 4, \infty\rangle$) with different limit problems. We further formulate a model posed on the unit cube that has the same asymptotics when $\eps$ tends to zero as the full 3d problem posed on the union of the unit cube and thin cuboids. This model then can be used as the approximating model in all regimes.