In this paper, we prove the well-posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained by Frota-Goldstein [18] and Limaco-Clark-Frota-Medeiros [26] to allow weak internal dampings and removing the dimensional restriction 1≤ n≤4. The method we use is based on a finite-dimensional approach by combining the Faedo-Galerkin method with suitable energy estimates and multiplier techniques.

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.

In this paper, we consider the asymptotic behavior of the Boussinesq equation with nonlocal weak damping when the nonlinear function is arbitrary polynomial growth. We firstly prove the well-posedness of solution by means of the monotone operator theory. At the same time, we obtain the dissipative property of the dynamical system (E ,S( t)) associated with the problem in the space H 0 2 ( Ω ) × L 2 ( Ω ) and D ( A 3 4 ) × H 0 1 ( Ω ) , respectively. After that, the asymptotic smoothness of the dynamical system (E ,S( t)) and the further quasi-stability are demonstrated by the energy reconstruction method. Finally, different from [21] we show not only existence of the finite global attractor but also existence of the generalized exponential attractor.

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

By protein structure prediction (PSP) we refer to the prediction of the 3-dimensional (3D) folding of a protein, known as tertiary structure, starting from its amino acid sequence, known as primary structure. The state-of-the-art in PSP is currently achieved by complex deep learning pipelines that require several input features extracted from amino acid sequences. It has been demonstrated that features that grasp the relative orientation of amino acids in space positively impacts the prediction accuracy of the 3D coordinates of atoms in the protein backbone. In this paper, we demonstrate the relevance of Geometric Algebra (GA) in instantiating orientational features for PSP problems. We do so by proposing two novel GA-based metrics which contain information on relative orientations of amino acid residues. We then employ these metrics as an additional input features to a Graph Transformer (GT) architecture to aid the prediction of the 3D coordinates of a protein, and compare them to classical angle-based metrics. We show how our GA features yield comparable results to angle maps in terms of accuracy of the predicted coordinates. This is despite being constructed from less initial information about the protein backbone. The features are also fewer and more informative, and can be (i) closely associated to protein secondary structures and (ii) more readily predicted compared to angle maps. We hence deduce that GA can be employed as a tool to simplify the modeling of protein structures and pack orientational information in a more natural and meaningful way.

In this paper, we show that any displacement of a plate is the sum of a Kirchhoff-Love displacement and two terms, one for shearing and one for warping. Then, the plate is loaded in order to obtain that the bending and shearing contribute the same order of magnitude to the fiber rotations.

The present paper is devoted to the group classification of magnetogasdynamics equations in which dependent variables in Euler coordinates depend on time and two spatial coordinates. It is assumed that the continuum is inviscid and nonthermal polytropic gas with infinite electrical conductivity. The equations are considered in mass Lagrangian coordinates. Use of Lagrangian coordinates allows reducing number of dependent variables. The analysis presented in this article gives complete group classification of the studied equations. This analysis is necessary for constructing invariant solutions and conservation laws on the base of Noether’s theorem.

In this paper, we investigate the nonlinear Schödinger equations with cubic interactions, arising in nonlinear optics. To begin, we prove the existence results for normalized ground state solutions in the L 2 -subcritical case and L 2 -supercritical case respectively. Our proofs relies on the Concentration-compactness principle, Pohozaev manifold and rearrangement technique. Then, we establish the nonexistence of normalized ground state solutions in the L 2 -critical case by finding that there exists a threshold. In addition, based on the existence of the normalized solutions, we also establish the blow-up results are shown by using localized virial estimates, and a new blow-up criterion which is related to normalized solutions.

Many challenges are still faced in bridging the gap between Mathematical modeling and biological sciences. Measuring population immunity to assess the epidemiology of health and disease is a challenging task and is currently an active area of research. However, to meet these challenges, mathematical modeling is an effective technique in shaping the population dynamics that can help disease control. In this paper, we introduce a Susceptible-Infected-Recovered (SIR) model and a Susceptible-Infected-Recovered-Exposed-Deceased (SEIRD) model based on conformable space-time PDEs for the Coronavirus Disease 2019 (COVID-19) pandemic. As efficient analytical tools, we present new modifications based on the fractional exponential rational function method (ERFM) and an analytical technique based on the Adomian decomposition method for obtaining the solutions for the proposed models. These analytical approaches are more efficious for obtaining analytical solutions for nonlinear systems of partial differential equations (PDEs) with conformable derivatives. The interesting result of this paper is that it yields new exact and approximate solutions to the proposed COVID-19 pandemic models with conformable space-time partial derivatives

We prove the existence of periodic travelling wave solutions for general discrete nonlinear Klein-Gordon systems, considering both cases of hard and soft on-site potentials. In the case of hard on-site potentials we implement a fixed point theory approach, combining Schauder’s fixed point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for non-trivial solutions to exist, energy (norm) thresholds for their existence and upper bounds on their velocity. In the case of soft on-site potentials, the proof of existence of periodic travelling wave solutions is facilitated by a variational approach based on the Mountain Pass Theorem. The proof of the existence of travelling wave solutions satisfying Dirichlet boundary conditions establishes rigorously the presence of compactons in discrete nonlinear Klein-Gordon chains. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.

Some transformations acting on radially symmetric solutions to the following class of non-homogeneous reaction-diffusion equations | x | σ 1 ∂ t u = ∆ u m + | x | σ 2 u p , ( x , t ) ∈ R N × ( 0 , ∞ ) , which has been proposed in a number of previous mathematical works as well as in several physical models, are introduced. We consider here m≥1, p≥1, N≥1 and σ 1 , σ 2 real exponents. We apply these transformations in connection to previous results on the one hand to deduce general qualitative properties of radially symmetric solutions and on the other hand to construct self-similar solutions which are expected to be patterns for the dynamics of the equations, strongly improving the existing theory. We also introduce mappings between solutions which work in the semilinear case m=1.

Formulas to calculate multivector exponentials in a basis-free representation and orthonormal basis are presented for an arbitrary Clifford geometric algebra , . The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results are generalised to arbitrary functions of multivector and may be useful in the quantum circuits or in the problems of analysis of evolution of the entangled quantum states.

We show that the gradient of a strongly differentiable function at a point is the limit of a single coordinate-free Clifford quotient between a multi-difference pseudo-vector and a pseudo-scalar, or of a sum of Clifford quotients between scalars (as numerators) and vectors (as denominators), both evaluated at the vertices of a same non-degenerate simplex contracting to that point. Such result allows to fix a issue with a defective definition of pseudo-scalar field in Sobczyck’s Simplicial Calculus. Then, we provide some consequences and conjectures implied by the foregoing results.

This work is concerned with the global existence of large solutions to the three-dimensional dissipative fluid-dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the incompressible Navier–Stokes–Poisson–Nernst–Planck equations. Making full use of the algebraic structure of the system, we obtain the global existence of solutions without smallness assumptions imposed on the third component of the initial velocity field and the summation of initial densities of charged species. More precisely, we prove that there exist two positive constants c 0 , C 0 such that if the initial data satisfies ( ∥ u 0 h ∥ B _ p , 1 − 1 + 3 p + ∥ N 0 − P 0 ∥ B _ q , 1 − 2 + 3 q ) exp { C 0 ( ∥ u 0 3 ∥ B _ p , 1 − 1 + 3 p 2 + ( ∥ N 0 + P 0 ∥ B _ r , 1 − 2 + 3 r + 1 ) exp { C 0 ∥ u 0 3 ∥ B _ p , 1 − 1 + 3 p } + 1 ) } ≤ c 0 , then the incompressible Navier–Stokes–Poisson–Nernst–Planck equations admits a unique global solution.

A Coronavirus Disease 2019 (COVID-19) epidemiological model incorporating a boosted infection-acquired immunity and heterogeneity in infection-acquired immunity among recovered individuals is designed. The model is used to investigate whether incorporating these two processes can induce new epidemiological insights. Analytical findings reveal co-existence of multiple endemic equilibria on either regions divided by the fundamental threshold (control reproduction number). Numerical findings conducted to validate analytical results show that heterogeneity in infection-acquired immunity among recovered individuals can induce various bifurcation structures such as reversed backward bifurcation, forward bifurcation, backward bifurcation and reversed hysteresis effect. Moreover, numerical results show that reversed backward bifurcation is annihilated or switches to the usual forward bifurcation if infection-acquired immunity among recovered individuals with strong immunity is assumed to be everlasting. However, this is only possible if primary infection is more likely than reinfection. In case reinfection is more likely to occur than primary infection, reversed backward bifurcation structure switches to a backward bifurcation phenomenon. Further, longer duration of infection-acquired immunity does lead to COVID-19 decline over time but does not lead to flattening of the COVID-19 peak.

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.

This paper is concerned with a diffusive predator-prey model with prey-taxis and prey-structure under the homogeneous Neumann boundary condition. The stability of the unique positive constant equilibrium of the predator-prey model is derived. Hopf bifurcation and steady state bifurcation are also concluded.

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.