Local Theorem Research Articles

The impact of fear and harvesting on plankton-fish system dynamics incorporating harmful phytoplankton in the contaminated environment

In this study, an aquatic system with fish, zooplankton, and harmful phytoplankton interacting with one another through the food chain in a polluted environment was proposed. It was thought about how the dynamics of the food chain were affected by fear and harvesting. An alternate food source was included for fish to eat to make the model more realistic. The characteristics of every solution were investigated. Both a local and global analysis of the system's stability was conducted. The system's persistence was investigated. The effects of changing the system's parameters were investigated using the local bifurcation theorem. To validate the discovered theoretical result and comprehend the effect of the system's parameters, a numerical example was then provided. It has been found that the system responds very quickly to changing the parameter values and exhibits a variety of attractors, including stable limit cycles and bi-stable behavior.

Asynchronous domain decomposition methods for nonlinear PDEs

One- and two-level parallel asynchronous methods for the numerical solution of nonlinear systems of equations, especially those arising from (nonlinear) partial differential equations, are studied. The proposed methods are based on domain decomposition techniques. Local convergence theorems are presented in several cases, with appropriate hypotheses. Computational results on a shared memory multiprocessor machine for various problems exhibiting nonlinearities are reported, illustrating the potential of these asynchronous methods, especially for heterogeneous clusters.

The Einstein-Vlasov system in maximal areal coordinates---Local existence and continuation

<p style='text-indent:20px;'>We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system in maximal areal coordinates. The latter coordinates have been used both in analytical and numerical investigations of the Einstein-Vlasov system [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b8">8</xref>,<xref ref-type="bibr" rid="b18">18</xref>,<xref ref-type="bibr" rid="b19">19</xref>], but neither a local existence theorem nor a suitable continuation criterion has so far been established for the corresponding nonlinear system of PDEs. We close this gap. Although the analysis follows lines similar to the corresponding result in Schwarzschild coordinates, essential new difficulties arise from to the much more complicated form which the field equations take, while at the same time it becomes easier to control the necessary, highest order derivatives of the solution. The latter observation may be useful in subsequent investigations.</p>

Inflation and the Local Flatness Theorem

Inflation and the Local Flatness Theorem

Solution manifold and its statistical applications

A solution manifold is the collection of points in a d-dimensional space satisfying a system of s equations with s<d. Solution manifolds occur in several statistical problems including missing data, algorithmic fairness, hypothesis testing, partial identifications, and nonparametric set estimation. We theoretically and algorithmically analyze solution manifolds. In terms of theory, we derive five useful results: smoothness theorem, stability theorem (which implies the consistency of a plug-in estimator), convergence of a gradient flow, local center manifold theorem and convergence of the gradient descent algorithm. We propose a Monte Carlo gradient descent algorithm to numerically approximate a solution manifold. In the case of the likelihood inference, we design a manifold constraint maximization procedure to find the maximum likelihood estimator on the manifold.

Local regularity of weak solutions of the hypodissipative Navier-Stokes equations

We consider the 3D incompressible hypodissipative Navier-Stokes equations, when the dissipation is given as a fractional Laplacian (−Δ)s for s∈(34,1), and we provide a new bootstrapping scheme that makes it possible to analyse weak solutions locally in space-time. This includes several homogeneous Kato-Ponce type commutator estimates which we localize in space, and which seems applicable to other parabolic systems with fractional dissipation. We also provide a new estimate on the pressure, ‖(−Δ)sp‖H1≲‖(−Δ)s2u‖L22. We apply our main result to prove that any suitable weak solution u satisfies ∇nu∈Llocp,∞(R3×(0,∞)) for p=2(3s−1)n+2s−1, n=1,2. As a corollary of our local regularity theorem, we improve the partial regularity result of Tang-Yu (2015) [26], and obtain an estimate on the box-counting dimension of the singular set S, dB(S∩{t≥t0})≤13(15−2s−8s2) for every t0>0.

Inverse Result of Approximation for the Max-Product Neural Network Operators of the Kantorovich Type and Their Saturation Order

In this paper, we consider the max-product neural network operators of the Kantorovich type based on certain linear combinations of sigmoidal and ReLU activation functions. In general, it is well-known that max-product type operators have applications in problems related to probability and fuzzy theory, involving both real and interval/set valued functions. In particular, here we face inverse approximation problems for the above family of sub-linear operators. We first establish their saturation order for a certain class of functions; i.e., we show that if a continuous and non-decreasing function f can be approximated by a rate of convergence higher than 1/n, as n goes to +∞, then f must be a constant. Furthermore, we prove a local inverse theorem of approximation; i.e., assuming that f can be approximated with a rate of convergence of 1/n, then f turns out to be a Lipschitz continuous function.

Extinction and stationary distribution of a stochastic COVID-19 epidemic model with time-delay

We reformulate a stochastic epidemic model consisting of four human classes. We show that there exists a unique positive solution to the proposed model. The stochastic basic reproduction number R0s is established. A stationary distribution (SD) under several conditions is obtained by incorporating stochastic Lyapunov function. The extinction for the proposed disease model is obtained by using the local martingale theorem. The first order stochastic Runge-Kutta method is taken into account to depict the numerical simulations.

The conformal Einstein field equations with massless Vlasov matter

We prove the stability of de Sitter space-time as a solution to the Einstein-Vlasov system with massless particles. The semi-global stability of Minkowski space-time is also addressed. The proof relies on conformal techniques, namely Friedrich's conformal Einstein field equations. We exploit the conformal invariance of the massless Vlasov equation on the cotangent bundle and adapt Kato's local existence theorem for symmetric hyperbolic systems to prove a long enough time of existence for solutions of the evolution system implied by the Vlasov equation and the conformal Einstein field equations.

Consumer Optimization and a First-Order PDE with a Non-Smooth System

We study a first-order nonlinear partial differential equation and present a necessary and sufficient condition for the global existence of its solution in a non-smooth environment. Using this result, we prove a local existence theorem for a solution to this differential equation. Moreover, we present two applications of this result. The first concerns an inverse problem called the integrability problem in microeconomic theory and the second concerns an extension of Frobenius’ theorem.

Normal form of equivariant maps in infinite dimensions

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.

Rigidity theorems for holomorphic curves in a complex Grassmann manifold G(3,6)

In this paper, we prove some local rigidity theorems of holomorphic curves in a complex Grassmann manifold [Formula: see text] by moving frames. By applying our rigidity theorems, we also give a characterization of all homogeneous holomorphic two-spheres in [Formula: see text] classified by the second author.

Extension of Killing vector fields beyond compact Cauchy horizons

We prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth vacuum spacetime is a smooth Killing horizon. The novelty here is that the Killing vector field is shown to exist on both sides of the horizon. This generalises classical results by Moncrief and Isenberg, by dropping the assumption that the metric is analytic. In previous work by Rácz and the author, the Killing vector field was constructed on the globally hyperbolic side of the horizon. In this paper, we prove a new unique continuation theorem for wave equations through smooth compact lightlike (characteristic) hypersurfaces which allows us to extend the Killing vector field beyond the horizon. The main ingredient in the proof of this theorem is a novel Carleman type estimate. Using a well-known construction, our result applies in particular to smooth stationary asymptotically flat vacuum black hole spacetimes with event horizons with constant non-zero surface gravity. As a special case, we therefore recover Hawking's local rigidity theorem for such black holes, which was recently proven by Alexakis-Ionescu-Klainerman using a different Carleman type estimate.

Local and global existence of a nonlocal equation with a singular integral drift term

We study an initial value problem with fractional Laplacian and a singular integral drift term. This equation quantifies fractal interfaces in statistical mechanics. The singularity of the drift term is a generalization of existing results. Making use of some important boundedness properties of Calder\'on-Zygmund operator in \(L_p\) and Lipschitz spaces, we obtain local and global existence theorems.

RESULT OF LOCAL EXISTENCE OF SOLUTIONS OF NONLOCAL VISCOELASTIC SYSTEM WITH RESPECT TO THE NONLINEARITY OF SOURCE TERMS

This work deals with the proof of local existence theorem of solutions for coupled nonlocal singular viscoelastic system with respect to the nonlinearity of source terms by using the Faedo–Galerkin method together with energy methods. This work makes a new contribution, since most of the previous works did not address the proof of the theorem of the local existence of solutions. It is also a completed study of Boulaaras et al. [Adv. Differ. Equ. 2020 (2020) 310].

5D model of pancreatic cancer: Key features of ultimate dynamics

In this paper, we study various types of ultimate dynamics for the 5-dimensional pancreatic cancer model without treatment. This model has been created by Hu et al. and describes interactions between pancreatic cancer cells (PCCs); pancreatic stellate cells (PSCs); immune cells and two types of cytokines, tumour–promoting and tumour–suppressing. Using iteratively the localization theorem of compact invariant sets we derive ultimate upper bounds for all cell populations and establish the property of the existence of the attracting set. Next, we find several conditions under which our system demonstrates various types of convergence dynamics: to equilibrium points located in the intersection of PCC-free plane with the PSC-free plane; equilibrium points with positive values of PSC density located in PCC-free plane; ω-limit sets with positive values of PCC density. Further, we explore the dynamics within the PSC-free plane and find attraction conditions to ω-limit sets with positive values of PCC density. Our assertions are expressed by means of iterative sequence of algebraic inequalities respecting parameters of the model. Our theoretical studies are supplied by results of numerical simulation. Locations of attracting equilibrium points have been found with help of the iteration procedure containing from 2 steps up to 23 steps in the dependence of parameter conditions and the location with respect to the PSC-free plane.

Exterior square gamma factors for cuspidal representations of $$\mathrm {GL}_n$$: simple supercuspidal representations

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.

Stable surfaces and free boundary marginally outer trapped surfaces

We explore various notions of stability for surfaces embedded and immersed in spacetimes and initial data sets. The interest in such surfaces lies in their potential to go beyond the variational techniques which often underlie the study of minimal and CMC surfaces. We prove two versions of Christodoulou–Yau estimate for \({\mathbf {H}}\)-stable surfaces, a Cohn-Vossen type inequality for non-compact stable marginally outer trapped surface (MOTS), and a global theorem on the topology of \({\mathbf {H}}\)-stable surfaces. Moreover, we give a definition of capillary stability for MOTS with boundary. This notion of stability leads to an area upper bound inequality and a local splitting theorem for free boundary stable MOTS. Finally, we establish an index estimate and a diameter estimate for free boundary MOTS. These are straightforward generalizations of Chen–Fraser–Pang and Carlotto–Franz results for free boundary minimal surfaces, respectively.

Implications of Local Friendliness Violation for Quantum Causality.

We provide a new formulation of the Local Friendliness no-go theorem of Bong et al. [Nat. Phys. 16, 1199 (2020)] from fundamental causal principles, providing another perspective on how it puts strictly stronger bounds on quantum reality than Bell’s theorem. In particular, quantum causal models have been proposed as a way to maintain a peaceful coexistence between quantum mechanics and relativistic causality while respecting Leibniz’s methodological principle. This works for Bell’s theorem but does not work for the Local Friendliness no-go theorem, which considers an extended Wigner’s Friend scenario. More radical conceptual renewal is required; we suggest that cleaving to Leibniz’s principle requires extending relativity to events themselves.

On the Convergence of a New Family of Multi-Point Ehrlich-Type Iterative Methods for Polynomial Zeros

In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.