One of the crucial and major issues in engineering, particularly in signal processing, is signal reconstruction. Frames are flexible tools that helps to reconstruct signals/vectors in a stable way. Frame theory was introduced for Hilbert spaces by Duffin et al. in 1952. If it happens that we only have the intensity measurements or the phaseless measurements of the lost signal then reconstructing the original signal becomes difficult. In such cases, phase retrieval sequences play essential roles to reconstruct or regain the signal from its intensity measurements or phaseless measurements. In 2006, Balan et al. introduced phase retrievable frames for Hilbert spaces. Phase retrieval has received significant attention in various fields, including image processing and signal reconstruction. Similar to phase retrieval, norm retrieval sequences help to regain the norm of the original signal. In 2015 Bahmanpour et al. introduced norm retrieval sequences for Hilbert spaces. We provide a thorough overview of phase retrieval and norm retrieval by vectors and subspaces in separable Hilbert space. We also highlight and discuss the recent developments in phase retrieval and norm retrieval of partially lost signals.

This paper proves the existence of nontrivial solution for two classes of quasilinear systems of the type { − ∆ Φ 1 u = F u ( x , u , v )+ λ R u ( x , u , v ) in Ω − ∆ Φ 2 v = − F v ( x , u , v ) − λ R v ( x , u , v ) in Ω u = v = 0 on ∂ Ω where λ>0 is a parameter, Ω is a bounded domain in R N ( N≥2) with smooth boundary ∂Ω. The first class we drop the ∆ 2 -condition of the functions Φ ̵̃ i ( i=1 ,2) and assume that F has a double criticality. For this class, we use a linking theorem without the Palais-Smale condition for locally Lipschitz functionals combined with a concentration–compactness lemma for nonreflexive Orlicz-Sobolev space. The second class, we relax the ∆ 2 -condition of the functions Φ i ( i=1 ,2). For this class, we consider F=0 and λ=1 and obtain the proof based on a saddle-point theorem of Rabinowitz without the Palais-Smale condition for functionals Fréchet differentiable combined with some properties of the weak ∗ topology.

In this paper, we use a stochastic partial differential equation (SPDE) as a model for the density of a population under the influence of random external forces/stimuli given by the environment. We study statistical properties for two crucial parameters of the SPDE that describe the dynamic of the system. To do that we use the Galerkin projection to transform the problem, passing from the SPDE to a system of independent SDEs; in this manner, we are able to find the Maximum likelihood estimator of the parameters. We validate the method by using simulations of the SDEs. We prove consistency and asymptotic normality of the estimators; the latter is showed using the Malliavin-Stein method. We illustrate our results with numerical experiments.

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti-) η-Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti-) η-Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented.

In this note, we consider the initial boundary value problem for a parabolic equation with logarithmic nonlinearity, which has been studied by Chen et al. (J. Math. Anal. Appl. 2015, 422, 84-98) and Han (J. Math. Anal. Appl. 2019, 474, 513-517). On the one hand, we not only prove the existence of doubly exponential decay solutions, but also find its threshold, and obtain the solutions with ∥ u 0 ∥ 2 2 → 0 + is always zero. On the other hand, we also prove the existence of doubly exponential growth solutions. The reseach results in this note extend previous results from both decay and growth.

In this article, we generalized Mittag-Leffler-type functions F ̵̄ A ( 3 ) , F ̵̄ B ( 3 ) , F ̵̄ C ( 3 ) and F ̵̄ D ( 3 ) , which correspond, respectively, to the familiar Lauricella hypergeometric functions F A ( 3 ) , F B ( 3 ) , F C ( 3 ) and F D ( 3 ) of three variables. Among the various properties and characteristics of these three-variable Mittag-Leffler-type function F ̵̄ D ( 3 ) , which we investigate in this article, include their relationships with other extensions and generalizations of the classical Mittag-Leffler functions, their three-dimensional convergence regions, their Euler-type integral representations, their Laplace transforms, their connections with the Riemann-Liouville operators of fractional calculus, and the systems of partial differential equations which are associated with them.

We investigate the motion of closed, smooth non-self-intersecting curves that evolve in space R 3 . The geometric evolutionary equation for the evolution of the curve is accompanied by a parabolic equation for the scalar quantity evaluated over the evolving curve. We apply the direct Lagrangian approach to describe the geometric flow of 3D curves resulting in a system of degenerate parabolic equations. We prove the local existence and uniqueness of classical Hölder smooth solutions to the governing system of nonlinear parabolic equations. A numerical discretization scheme has been constructed using the method of flowing finite volumes. We present several numerical examples of the evolution of curves in 3D with a scalar quantity. In this paper, we analyze the flow of curves with no torsion evolving in rotating and parallel planes. Next, we present examples of the evolution of curves with initially knotted and unknotted curves.

In this paper, we study the initial-boundary value problem of three-dimensional viscous, compressible, and heat conductive magnetohydrodynamic equations. Local existence and uniqueness of strong solutions is established with any such initial data that the initial compatibility conditions do not be required. The analysis is based on some suitable prior estimates for the strong coupling term u·∇ H and strong nonlinear term curl H × H . Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

The goal of this note is to study the spectrum of a self-adjoint convolution operator in L 2 ( R d ) with an integrable kernel that is perturbed by an essentially bounded real-valued potential tending to zero at infinity. We show that the essential spectrum of such operator is the union of the spectrum of the convolution operator and of the essential range of the potential. Then we provide several sufficient conditions for the existence of a countable sequence of discrete eigenvalues. For operators having non-connected essential spectrum we give sufficient conditions for the existence of discrete eigenvalues in the corresponding spectral gaps.

Pulse injection of insulin analogues is an important strategy to control blood glucose concentrations and can be combined with α– glucosidase inhibitors to reduce adverse effects to improve glucose control. We propose a novel mathematical models with pulse injection insulin and α– glucosidase inhibitors , eating in the form of pulse blood glucose injection. The existence and uniqueness of the positive periodic solution is confirmed In type 1 diabetes, which is globally asymptotically stable. Further, the permanence of the system is given in type 2 diabetes. The numerical analysis verified the correctness of the theoretical calculation results, and show that the period and the dose of insulin injections and α– glucosidase inhibitors are crucial for insulin therapies. In addition, we systematically evaluated a reasonable strategy to treat diabetes combined with α– glucosidase inhibitors, which can provide more reasonable clinical strategies.

In this paper, we introduce some explicit formulas for the positive integer order q-derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between the commutative algebras and the properties of q-operators. Especially, the third formula is a beautiful one involving the determinant of the functions and their q-derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, the last formula is used for proving of some interesting identities. MSC CLASSIFICATION: 05A30, 26A24

In this article, a two prey-one predator model in which prey and predator disperse simultaneously in a heterogeneous environment with n patches is proposed and analyzed. We prove that the solution of the system is positive and uniformly ultimately bounded. Meanwhile, we use the monotonic theory of spectral bounds to investigate the effect of the dispersal rate on population dynamics. To be precise, we discuss the stability behaviour for the trivial equilibrium and semitrivial equilibrium as well as the uniform persistence of the system. Furthermore, we prove the global asymptotic stability of the positive equilibrium by constructing a global Lyapunov function which applies the results from graph theory. Some numerical simulations are provided to show the effectiveness of the theoretical results.

The paper aims to apply the trigintaduonion spaces to explore the physical properties of four interactions simultaneously, including the electromagnetic fields, gravitational fields, weak nuclear fields, and strong nuclear fields. J. C. Maxwell first applied the algebra of quaternions to study the physical properties of electromagnetic fields. It inspired some subsequent scholars to introduce the quaternions, octonions, sedenions, and trigintaduonions to research the electromagnetic fields, gravitational fields, weak nuclear fields, strong nuclear fields, quantum mechanics, gauge fields, and curved spaces and so forth. The algebra of trigintaduonions is able to discuss the physical quantities of four interactions, including the field potential, field strong, field source, linear momentum, angular momentum, torque, and force. In the field theories described with the algebra of trigintaduonions, the weak nuclear field is composed of three types of fundamental fields. These three fundamental fields, related to weak nuclear fields, can describe the physical properties of weak nuclear fields collectively. This is consistent with the conclusion of the electroweak theory. Meanwhile the strong nuclear field consists of three types of fundamental fields. These three fundamental fields relevant to strong nuclear fields may investigate the physical properties of strong nuclear fields mutually. It is coincident with the deduction of quark theory. According to the properties of trigintaduonions, one can deduce the Yang-Mills equation related to the gauge fields. It means that the electromagnetic field occupies a quaternion space. The gravitational field owns one different quaternion space. The weak nuclear fields occupy three mutually independent quaternion spaces. The properties of weak nuclear fields are different from those of electromagnetic fields or gravitational fields. According to the multiplicative closure, the strong nuclear fields also own three quaternion spaces independent of each other. These explorations further deepen the understanding of the physical properties of weak and strong nuclear fields.

In this paper, we study the stability and convergence of a conservative Crank-Nicolson finite difference scheme applied to the Korteweg-De Vries (KdV) equation endowed with initial data. We design a three-point average scheme associated to the convective term and the dispersion term is discretized by certain discrete operators along with the Crank-Nicolson scheme for the temporal discretization to establish that the proposed scheme is L 2 -conservative. The convergence analysis reveals that utilizing inherent Kato’s local smoothing effect, the proposed scheme converges to a classical solution for sufficiently regular initial data u 0 ∈ H 3 ( R ) and to a weak solution in L 2 ( 0 , T ; L loc 2 ( R ) ) for non-smooth initial data u 0 ∈ L 2 ( R ) . Optimal convergence rates in both space and time for the devised scheme are derived. The theoretical results are justified through several numerical illustrations.

This paper presents the dynamic response of cantilever beam on fractional-order nonlinear viscoelastic foundation subjected to Gaussian white noise. The control equations of the cantilever beam on viscoelastic foundation are established using Hamilton's principle. The problem is solved using the shifted Chebyshev polynomial algorithm, and the control equations are transformed into a system of nonlinear algebraic equations. Numerical examples analyse the correction error and the second norm error, confirming the effectiveness and accuracy of the algorithm in solving such problems. Furthermore, the algorithm's robustness was verified by comparing the responses of Gaussian white noise and non-Gaussian white noise cantilever beam on the viscoelastic foundation. The study examined the impact of various loads and parameters on the cantilever beam, as well as the effect of different harmonic loads on its stress. The research results are in line with the existing literature. These studies offer valuable guidance for practical engineering and enhance comprehension of the dynamic response of cantilevers on foundation in complex environments.

In this paper, we consider a predator-prey model with Cosner type functional response and combined harvesting. First, we explore the existence and stability of the equilibria. Then using the center manifold theorem and normal form theory, we investigate codimension one and codimension two bifurcations of the model. The analysis shows that the system has a variety of bifurcation phenomena including transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation and homoclinic bifurcation. Our findings indicate that the dynamics with harvesting are significantly richer than the system without harvesting. Finally, numerical simulations are provided to support the analytical results.

We address the problem of existence, uniqueness and approximation of the solution of a large class of fractional integral differential equations with boundary conditions. Based on fixed point techniques with Boyd-Wong type conditions, the existence and uniqueness of the solution is investigated. This, together with the use of certain types of biorthogonal systems, allows us to propose a method to approximate the solution which is tested with several examples.

In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with Transmission Blocking Drugs (TBDs). The paper’s main aim is to demonstrate the impact the chosen model for demographic growth has on the disease’s transmission and the effect of its treatment with TBDs. We calculate the model’s control reproduction number and equilibria, and perform a global stability analysis of the disease-free equilibrium point. The mathematical analysis reveals that, depending on the model’s demography, the model can exhibit forward, backward and even some unconventional types of bifurcation, where disease elimination can occur for both small and large values of the reproduction number. We also conduct a numerical analysis to explore the short-time behavior of the model. A key finding is that for one type of demographic growth, the population experienced a significantly higher disease burden than the others, and when exposed to high levels of treatment with TBDs, only this population succeeded in effectively eliminating the disease within a reasonable timeframe.