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Abstract: Lyapunov exponents characterize the asymptotic behavior of long matrix products. In this work we introduce a new technique that yields quantitative lower bounds on the top Lyapunov exponent in terms of an efficiently computable matrix sum in ergodic situations. Our approach rests on two results from matrix analysis—the n-matrix extension of the Golden–Thompson inequality and an effective version of the Avalanche Principle. While applications of this method are currently restricted to uniformly hyperbolic cocycles, we include specific examples of ergodic Schrödinger cocycles of polymer type for which outside of the spectrum our bounds are substantially stronger than the standard Combes–Thomas estimates. We also show that these techniques yield short proofs of quantitative stability results for the top Lyapunov exponent which are known from more dynamical approaches. We also discuss the problem of finding stable bounds on the Lyapunov exponent for almost-commuting matrices. PubDate: 2022-01-12

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Abstract: The main purpose of this paper is to established the modulus of derivative of rational functions r(z) having all its zeros in \( z \le k \le 1,\) except two zeros \(z_{0}\) and \(z_{1}\) of multiplicity \(\mu \) and \(\nu \) respectively and some other related inequalities. The obtained results generalize and sharpen some well-known inequalities for the derivative of rational functions with prescribed poles and in turn produces some results besides the refinements of some polynomial inequalities as well. PubDate: 2022-01-10

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Abstract: A new mathematical model for complex neural networks of the partly diffusive Hindmasrh-Rose equations with boundary coupling is proposed. Through analysis of absorbing dynamics for the solution semiflow, the asymptotic synchronization of the complex neuronal networks at a uniform exponential rate is proved under the condition that stimulation signal strength of the ensemble boundary coupling exceeds a quantitative threshold expressed by the biological parameters. PubDate: 2022-01-09

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Abstract: In this paper we study the impulsive Dirac operator with discontinuity and prove uniqueness theorems from introduced new supplementary data. It is shown that the potential on the whole interval can be uniquely determined by these given data. PubDate: 2022-01-08

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Abstract: We consider a system of Schrödinger equations on conical spaces. We first rewrite the iterative reconstruction algorithms for two kinds of average Schrödinger functionals and prove their convergence. Then the asymptotic pointwise error estimates are presented for both algorithms under the case that the average samples are corrupted by noise. PubDate: 2022-01-06

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Abstract: We develop a critical embedding theorem from fractional variable-order Sobolev space to variable exponent Lebesgue spaces over the whole space \({\mathbb {R}}^{N}\) , which is used to prove the existence and asymptotic behavior of the solution for a critical Kirchhoff systems driven by a fractional variable-order \( p(\cdot ) \& q(\cdot )\) Laplace operator. The main difficulty and features of this paper are the treatment of variable exponent critical nonlinear terms, \( p(\cdot ) \& q(\cdot )\) growth of fractional variable-order Laplace operator, and the system can be degenerate. As far as we know, these results are new even in bounded domains and when the Kirchhoff model is non-degenerate. PubDate: 2022-01-06

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Abstract: Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation. PubDate: 2022-01-05

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Abstract: For the modified complex short pulse equation on the line with zero boundary conditions, a Riemann–Hilbert approach is presented. A parametric representation of the solution to the related Cauchy problem is obtained. The explicit formulae are worked for the one-soliton solutions and the two-soliton solutions, which, depending on the parameters, may be either smooth solitons, cuspons or breathers. PubDate: 2022-01-04

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Abstract: In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr’s inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions. PubDate: 2022-01-04

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Abstract: In this paper, we present four modified inertial projection and contraction methods to solve the variational inequality problem with a pseudo-monotone and non-Lipschitz continuous operator in real Hilbert spaces. Strong convergence theorems of the proposed algorithms are established without the prior knowledge of the Lipschitz constant of the operator. Several numerical experiments and the applications to optimal control problems are provided to verify the advantages and efficiency of the proposed algorithms. PubDate: 2021-12-31

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Abstract: We give a new proof of Dennis Hejhal’s theorem on the nondegeneracy of the matrix that appears in the identity relating the Bergman and Szegő kernels of a smoothly bounded finitely connected domain in the plane. Mergelyan’s theorem is at the heart of the argument. We explore connections of Hejhal’s theorem to properties of the zeroes of the Szegő kernel and propose some ideas to better understand Hejhal’s original theorem. PubDate: 2021-12-27

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Abstract: In this paper, using the recently discovered notion of the S-spectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be \(-1\) ). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor. PubDate: 2021-12-27

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Abstract: We consider three bodies moving under their mutual gravitational attraction in spaces with constant Gaussian curvature \(\kappa \) . In this system, two bodies with equal masses form a relative equilibrium solution, these bodies are known as primaries, the remaining body of negligible mass does not affect the motion of the others. We show that the singularity due to binary collision between the negligible mass and the primaries can be regularized local and globally using hyperbolic functions. We show some numerical examples of orbits for the massless particle. PubDate: 2021-12-24

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Abstract: We study the Bochner–Schrödinger operator \(H_{p}=\frac{1}{p}\varDelta ^{L^p\otimes E}+V\) on high tensor powers of a positive line bundle L on a symplectic manifold of bounded geometry. First, we give a rough asymptotic description of its spectrum in terms of the spectra of the model operators. This allows us to prove the existence of gaps in the spectrum under some conditions on the curvature of the line bundle. Then we consider the spectral projection of such an operator corresponding to an interval whose extreme points are not in the spectrum and study asymptotic behavior of its kernel. First, we establish the off-diagonal exponential estimate. Then we state a complete asymptotic expansion in a fixed neighborhood of the diagonal. PubDate: 2021-12-22

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Abstract: Logarithmic capacity is shown to be minimal for a planar set having N-fold rotational symmetry ( \(N \ge 3\) ), among all conductors obtained from the set by area-preserving linear transformations. Newtonian and Riesz capacities obey a similar property in all dimensions, when suitably normalized linear transformations are applied to a set having irreducible symmetry group. A corollary is Pólya and Schiffer’s lower bound on capacity in terms of moment of inertia. PubDate: 2021-12-21

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Abstract: In this paper, we discuss the following Kirchhoff-type problem with convolution nonlinearity $$\begin{aligned} -\left( 1+ b\int _{ \mathbb {R}^{3}} \nabla u ^{2} dx \right) \triangle u+ V(x)u=(I_{\alpha }*F(u))f( u),~x\in \mathbb {R}^{3},~u\in H^{1}(\mathbb {R}^{3}), \end{aligned}$$ where \(b>0\) , \(I_{\alpha }:\mathbb {R}^{3}\rightarrow \mathbb {R}\) , with \(\alpha \in (0,3)\) , is the Riesz potential, V is differentiable, \(f\in \mathbb {C}(\mathbb {R},\mathbb {R})\) and \(F(t)=\int ^{t}_{0}f(s)ds\) . Let f satisfies some relatively weak conditions in the absence of the usual Ambrosetti-Rabinowitz or monotonicity conditions. We get two classes of ground state solutions under the general “Berestycki–Lions conditions” on the nonlinearity f and we also give a minimax characterization of the ground state energy. PubDate: 2021-12-17

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Abstract: The well-posedness problem of a nonlinear parabolic equation with double variable exponents is studied in this paper. This kind of nonlinear parabolic equation includes the non-Newtonian fluids equation, the polytropic filtration equation and the so-called electro-rheological fluid equation. One of the important characteristics is that there is a diffusion coefficient a(x, t) in the equation. Unlike the usual assumption \(a(x,t)>a>0\) , the paper only assumes \(a(x,t)\ge 0\) . If \(a(x,t) _{x\in \partial \Omega }=0\) , by choosing a(x, t) as a test function, the stability result of positive solutions can be established without the boundary value condition. PubDate: 2021-12-17

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Abstract: In this paper, we study the global existence and blow up for the Cauchy problem for some hyperbolic system $$\begin{aligned} u_{ktt}+\delta u_{kt}-\phi \Delta u_{k}+f_k(u_1,u_2)=\lambda u_k ^{\beta -1}u_k. \quad k=1,2. \end{aligned}$$ Under certain conditions we prove the global existence of solutions by adapting the method of modified potential well in a functional setting of generalized Sobolev spaces, and we prove that the solution decays exponentially by introducing an appropriate Lyapunov function. By the concave method, we discuss the blow-up behavior of weak solution with certain conditions and give some estimates for the lifespan of solutions. PubDate: 2021-12-12

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Abstract: In this paper, we consider the following non-autonomous Schrödinger–Bopp–Podolsky system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + V(x) u + q^2\phi u = f(u)\\ -\Delta \phi + a^2 \Delta ^2 \phi = 4\pi u^2 \end{array}\right. } \hbox { in }{\mathbb {R}}^3. \end{aligned}$$ By using some original analytic techniques and new estimates of the ground state energy, we prove that this system admits a ground state solution under mild assumptions on V and f. In the final part of this paper, we give a min-max characterization of the ground state energy. PubDate: 2021-12-10

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Abstract: In this work, we investigate two-dimensional nonlinear Prandtl equations on the half plane and prove the local existence of solutions by energy methods in an exponential weighted Sobolev space. We use the skill of cancellation mechanism and construct a new unknown function to overcome some difficulties respectively. PubDate: 2021-12-09