Uniqueness Criteria Research Articles

Modeling and analysis of a fractional-order prey-predator system incorporating harvesting

In the present paper, we propose and study a fractional-order prey-predator type ecological model with the functional response of Holling type II and the effect of harvesting. In addition, we have also considered the influence of a super-predator on the conventional predator. The complex dynamical behaviour of the proposed model system including existence and uniqueness criteria, finiteness and nonnegativity of the solutions have studied rigorously. In addition, we have determined existence criteria of several equilibria and analyzed the asymptotic nature of those equilibria. We have enriched our analysis with the inclusion of fractional Hopf bifurcation and the criteria of global stability of the equilibrium points. Finally, some numerical simulation works have been incorporated to validate the analytical analysis.

Weak-strong uniqueness in weighted L2 spaces and weak suitable solutions in local Morrey spaces for the MHD equations

We consider here the magneto-hydrodynamics (MHD) equations on the whole space. For the 3D case, in the setting of the weighted L2 spaces we obtain a weak-strong uniqueness criterion provided that the velocity field and the magnetic field belong to a fairly general multipliers space. On the other hand, we study the local and global existence of weak suitable solutions for intermittent initial data, which is characterized through a local Morrey space. This large initial data space was also exhibit in a contemporary work [4] in the context of 3D Navier-Stokes equations. Finally, we make a discussion on the local and global existence problem in the 2D case.

Can Individual Movement Characteristics Across Different Throwing Disciplines Be Identified in High-Performance Decathletes?

Although the individuality of whole-body movements has been suspected for years, the scientific proof and systematic investigation that individuals possess unique movement patterns did not manifest until the introduction of the criteria of uniqueness and persistence from the field of forensic science. Applying the criteria of uniqueness and persistence to the individuality of motor learning processes requires complex strategies due to the problem of persistence in the learning processes. One approach is to examine the learning process of different movements. For this purpose, it is necessary to differentiate between two components of movement patterns: the individual-specific component and the discipline-specific component. To this end, a kinematic analysis of the shot put, discus, and javelin throwing movements of seven high-performance decathletes during a qualification competition was conducted. In total, joint angle waveforms of 57 throws formed the basis for the recognition task of individual- and discipline-specific throwing patterns using a support vector machine. The results reveal that the kinematic throwing patterns of the three disciplines could be distinguished across athletes with a prediction accuracy of up to 100% (57 of 57 throws). In addition, athlete-specific throwing characteristics could also be identified across the three disciplines. Prediction accuracies of up to 52.6% indicated that up to 10 out of 19 throws of a discipline could be assigned to the correct athletes, based on only knowing these athletes from the kinematic throwing patterns in the other two disciplines. The results further suggest that individual throwing characteristics across disciplines are more pronounced in shot put and discus throwing than in javelin throwing. Applications for training and learning practice in sports and therapy are discussed. In summary, the chosen approach offers a broad field of application related to the search of individualized optimal movement solutions in sports.

Existence and uniqueness criteria for the higher-order Hilfer fractional boundary value problems at resonance

This investigation is devoted to the study of a certain class of coupled systems of higher-order Hilfer fractional boundary value problems at resonance. Combining the coincidence degree theory with the Lipschitz-type continuity conditions on nonlinearities, we present some existence and uniqueness criteria. Finally, to practically implement the obtained theoretical criteria, we give an illustrative application.

On uniqueness ofp-adic period morphisms, II

We prove equality of the various rational$p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the$K$-theoretical uniqueness criterion we had found earlier for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the$h$-cohomology period morphism agrees with the syntomic and almost étale period morphisms whenever the latter morphisms are defined (and up to a change of Hyodo–Kato cohomology). We do it by lifting the syntomic and almost étale period morphisms to the$h$-site of varieties over a field, where their equality with the$h$-cohomology period morphism can be checked directly using the Beilinson Poincaré lemma and the case of dimension$0$. This also shows that the syntomic and almost étale period morphisms have a natural extension to the Voevodsky triangulated category of motives and enjoy many useful properties (since so does the$h$-cohomology period morphism).

Fixed Point Theorem Based Solvability of 2-Dimensional Dissipative Cubic Nonlinear Klein-Gordon Equation

The purpose of this article is to establish the solvability of the 2-Dimensional dissipative cubic nonlinear Klein-Gordon equation (2DDCNLKGE) through periodic boundary value conditions (PBVCs). The analysis of this study is founded on the Galerkin’s method (GLK) and the Leray-Schauder’s fixed point theorem (LS). First, the GLK method is used to construct some uniform priori estimates of approximate solution to the corresponding equation of 2DDCNLKGE. Finally, the LS fixed point theorem is applied to obtain the efficient and straightforward existence and uniqueness criteria of time periodic solution to the 2DDCNLKGE.

Generalized norm for existence, uniqueness and stability of Hopfield neural networks with discrete and distributed delays

In this paper, the existence, uniqueness and stability criteria of solutions for Hopfield neural networks with discrete and distributed delays (DDD HNNs) are investigated by the definitions of three kinds of generalized norm (ξ-norm). A general DDD HNN model is firstly introduced, where the discrete delays τpq(t) are asynchronous time-varying delays. Then, {ξ,1}-norm, {ξ,2}-norm and {ξ,∞}-norm are successively used to derive the existence, uniqueness and stability criteria of solutions for the DDD HNNs. In the proof of theorems, special functions and assumptions are given to deal with discrete and distributed delays. Furthermore, a corollary is concluded for the existence and stability criteria of solutions. The methods given in this paper can also be used to study the synchronization and μ-stability of different DDD NNs. Finally, two numerical examples and their simulation figures are given to illustrate the effectiveness of these results.

Nonlinear characterizations of stochastic completeness

We prove that conservation of probability for the free heat semigroup on a Riemannian manifold M (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on M of the form ut=Δϕ(u), ϕ being an arbitrary concave, increasing, positive function, regular outside the origin and with ϕ(0)=0. Either property is also equivalent to nonexistence of nonnegative, nontrivial, bounded solutions to the elliptic equation ΔW=ϕ−1(W), with ϕ as above. As a consequence, explicit criteria for uniqueness or nonuniqueness of bounded solutions to fast diffusion-type equations on manifolds are given, these being the first results on such issues.

Uniqueness criteria in multi-energy CT

Multi-energy computed tomography (ME-CT) is a medical imaging modality aiming to reconstruct the spatial density of materials from the attenuation properties of probing x-rays. For each line in two- or three-dimensional space, ME-CT measurements may be written as a nonlinear mapping from the integrals of the unknown densities of a finite number of materials along said line to an equal or larger number of energy-weighted integrals corresponding to different x-ray source energy spectra. ME-CT reconstructions may thus be decomposed as a two-step process: (i) reconstruct line integrals of the material densities from the available energy measurements; and (ii) reconstruct densities from their line integrals. Step (ii) is the standard linear x-ray CT problem whose invertibility is well-known, so this paper focuses on step (i). We show that ME-CT admits stable, global inversion provided that (a well-chosen linear transform of) the differential of the transform in step (i) satisfies appropriate orientation constraints that makes it a P-matrix. We introduce a notion of quantitative P-function that allows us to derive global stability results for ME-CT in the determined as well as over-determined (with more source energy spectra than the number of materials) cases. Numerical simulations based on standard material properties in imaging applications (of bone, water, contrast agents) and well accepted models of source energy spectra show that ME-CT is often (always in our simulations) either (i) non-globally injective because it is non-injective locally (differential not of full rank), or (ii) globally injective as soon as it is locally injective (differentials satisfy our proposed constraints).

Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations

This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical L2-based Morrey space; (2) initial and eventual regularity of local energy solutions to the Navier-Stokes equations with initial data sufficiently small at small or large scales; (3) small-large uniqueness of local energy solutions for data in the critical L2-based Morrey space. A number of interesting corollaries are included, including eventual regularity in familiar Lebesgue, Lorentz, and Morrey spaces, a new local generalized Von Wahl uniqueness criteria, as well as regularity and uniqueness for local energy solutions with small discretely self-similar data.

A nonlocal boundary-value problem for a fourth-order mixed-type equation

The criterion of uniqueness of a solution of the problem with periodicity and nonlocal and boundary conditions is established by the spectral analysis for a fourth-order mixed-type equation in a rectangular region. When constructing a solution in the form of the sum of a series, we use the completeness in the space L_2, the system of eigenfunctions of the corresponding problem orthogonally conjugate. When proving the convergence of a series, the problem of small denominators arises. Under some conditions imposed on the parameters of the data of the problem and given functions, the stability of the solution is proved.

A well-balanced central-upwind scheme for the thermal rotating shallow water equations

We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal temperature and/or density gradients—the thermal rotating shallow water (TRSW). The scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free central-upwind scheme to the rewritten system. We ensure that the resulting method is well-balanced by switching off the numerical diffusion when the computed solution is near (at) thermo-geostrophic equilibria.The designed scheme is successfully tested on a series of numerical examples. Motivated by future applications to large-scale motions in the ocean and atmosphere, the model is considered on the tangent plane to a rotating planet both in mid-latitudes and at the Equator. The numerical scheme is shown to be capable of quite accurately maintaining the equilibrium states in the presence of nontrivial topography and rotation. Prior to numerical simulations, an analysis of the TRSW model based on the use of Lagrangian variables is presented, allowing one to obtain criteria of existence and uniqueness of the equilibrium state, of the wave-breaking and shock formation, and of instability development out of given initial conditions. The established criteria are confirmed in the conducted numerical experiments.

Uniqueness and non-uniqueness of solutions for critical Hardy-Hénon parabolic equations

We consider the Hardy-Hénon parabolic equation ∂tu=Δu+|x|−γ|u|αu, t>0, x∈RN∖{0}, N≥1, 0<γ<min⁡(2,N) and α>0. The existence of solutions in C([0,T],Lq(RN)), 1<q<∞, q≥qc:=Nα/(2−γ) with initial data u0∈Lq(RN) has been established recently. It is also known that uniqueness holds in C([0,T],Lq(RN)) for q>qc, q>Qc:=N(α+1)/(N−γ), while for the other values of q it holds under additional restrictions. In this paper, we consider the problem of uniqueness for the limiting case q=max⁡(qc,Qc). Our objective is to see whether or not we can remove the additional restrictions. Here two different situations arise:–A single critical case: when q=qc>Qc and u0∈Lq(RN) or q=Qc>qc and u0 in the Lorentz space Lq,α+1(RN), we prove that uniqueness holds in C([0,T],Lq(RN)).–The double-critical case: when q=qc=Qc, that is N≥3, 0<γ<2, α=(2−γ)/(N−2) and q=N/(N−2). We prove non-uniqueness in C([0,T],LNN−2(RN)) for any initial value u0∈LNN−2(RN). We also give a uniqueness criterion for the double-critical case and we show that it is optimal in the sense that it characterizes the largest space in which uniqueness holds. The uniqueness criterion as well as the uniqueness results involve Lorentz spaces and show that the second index in Lorentz spaces plays a crucial role. The proofs rely on some estimates for the heat kernel in Lorentz spaces and use the singular solutions known for the stationary Dirichlet problem on the unit ball.

Uniqueness of the solutions of the Hermite – Pade problems

New concepts are introduced in the present work. They are a quite normal index and a quite perfect system of functions. Using these concepts, the uniqueness criterion for solution of two Hermite – Pade problems is proved, the explicit determinant representations of type I and II Hermite – Padé polynomials for an arbitrary system of power series are obtained. The results obtained complement and generalize the well-known result in the theory of Hermite – Padé approximations.

Exponential Stability Analysis of Mixed Delayed Quaternion-Valued Neural Networks Via Decomposed Approach

With the application of quaternion in technology, quaternion-valued neural networks (QVNNs) have attracted many scholars' attention in recent years. For the existing results, dynamical behavior is an important studying side. In this paper, we mainly research the existence, uniqueness and exponential stability criteria of solutions for the QVNNs with discrete time-varying delays and distributed delays by means of generalized 2-norm. In order to avoid the noncommutativity of quaternion multiplication, the QVDNN system is firstly decomposed into four real-number systems by Hamilton rules. Then, we obtain the sufficient criteria for the existence, uniqueness and exponential stability of solutions by special Lyapunov-type functional, Cauchy convergence principle and monotone function. Furthermore, several corollaries are derived from the main results. Finally, we give one numerical example and its simulated figures to illustrate the effectiveness of the obtained conclusion.

Nonlocal Inverse Problem of Finding Unknown Multipliers in the Right-Hand Part of Lavrentiev-Bitsadze Equation

We consider an equation of mixed elliptic-hyperbolic type. The right-hand part of the equation is represented as a product of two functions, each which only depends on one of variables. We study an inverse problem for this equation; it consists in finding the unknown multipliers. We establish a uniqueness criterion for this problem. The solution is constructed as a series in eigenfunctions for the corresponding one-dimensional spectral problem. Under some conditions on the boundary of domain, boundary values, and the unknown multipliers, we establish separation from zero of small denominators of ratios containing in the coefficients of the series; the existence of solution and its stability are proved as well.

Dirichlet Problem for Mixed Type Equation with Characteristic Degeneration and Singular Coefficient

For a mixed-type equation of the second kind with a singular coefficient by the spectral expansions uniqueness criterion is installed for solving the first boundary-value problem. The solution was constructed as the sum of a Fourier–Bessel series. In justifying the uniform convergence appeared the problem of small denominators. In connection with this evaluation is set on a small separation from zero denominator with the corresponding asymptotic behavior that allowed to justify the convergence of the series in the class of regular solutions.

Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States

The basic aim of this paper is to provide a fundamental tool, the resolvent decomposition theorem, in the construction theory of denumerable Markov processes. We present a detailed analytic proof of this extremely useful tool and explain its clear probabilistic interpretation. We then apply this tool to investigate the basic problems of existence and uniqueness criteria for denumerable Markov processes with instantaneous states to which few results have been obtained even until now. Although the complete answers regarding these existence and uniqueness criteria will be given in a subsequent paper, we shall, in this paper, present part solutions of these very important problems that are closely linked with the subtle Williams S and N conditions.

POTENSI MAKANAN TRADISIONAL BALI YANG BERBASIS MASYARAKAT SEBAGAI DAYA TARIK WISATA DI PASAR UMUM GIANYAR

The purpose of this research was to determine the potential of gastronomic,examines the efforts undertaken by the market managers, and to determine the extentof involvement of local community towards the development of Balinese traditionalfood sold at common market of Gianyar. In this research, the data were analyzed withdescriptive qualitative method approach using multiple informants were selectedpurposively and techniques of data collection is done by the method of observation,in-depth interviews and documentation with gastronomic theory and the theory ofproduct development. The results showed that the potential of Balinese traditionalfood gastronomic sold at common market of Gianyar meet the criteria of uniqueness,originalities, authenticity, diversity, as well as an attractive appearance for touristsso decent offer as a tourist attraction. Efforts undertaken by the market managersin the development of Balinese traditional food that sold in the common market ofGianyar is: providing and managing the area for traders, involving the communitiestraditional village of Gianyar to conserve Balinese traditional food heritage, andalways maintain the cleanliness of the market area which impact on the quality ofthe food. Forms of community involvement of the indigenous village of Gianyar inthe development Balinese traditional foods sold at common market of Gianyar is: asthe market manager, as a particular merchant that sells Balinese traditional food, asfood suppliers such as suckling pig (babi guling) and Balinese pastries (jaje Bali)especially visitors who buy Balinese traditional food. Synergy among the traders andmarket management, governments, and local community involvement, in collaborationwith the tourism entrepreneurs was needed to raise the value of Balinese traditionalfood and to achieve the goal of Balinese traditional food development.

THE CORRECTNESS OF A DIRICHLET TYPE PROBLEM FOR THE DEGENERATE MULTIDIMENSIONAL HYPERBOLIC-ELLIPTIC EQUATIONS

Multidimensional hyperbolic-elliptic equations describe important physical, astronomical, and geometric processes. It is known that the oscillations of elastic membranes in space according to Hamiltons prism can be modeled by multidimensional degenerate hyperbolic equations. Assuming that the membrane is in equilibrium in half the bend, from Hamiltons principle we also obtain degenerate elliptic equations. Consequently, vibrations of elastic membranes in space can be modeled as multidimensional degenerate hyperbolic-elliptic equations. When studying these applications, it is necessary to obtain an explicit representation of the investigated boundary value problems. The author has previously studied the Dirichlet problem for multidimensional hyperbolic-elliptic equations, where the unique solvability of this problem is shown, which essentially depends on the height of the cylindrical domain under consideration. However, the Dirichlet problem in a cylindrical domain for multidimensional degenerate hyperbolic-elliptic equations has not been studied previously. In this paper, the Dirichlet problem is studied for a class of degenerate multidimensional hyperbolic-elliptic equations. Moreover, the existence and uniqueness of the solution depends on the height of the considered cylindrical domain and on the degeneration of the equation. A uniqueness criterion for a regular solution is also obtained.