We establish an asymptotic property of Toeplitz operators on the Dirichlet space and on a weighted Bergman space. Additionally, we determine a sufficient condition for a self-commutator inequality to hold in a specific case.
We study Li-Yorke and distributional chaos for weighted shifts on directed trees. We give a complete characterization of Li-Yorke chaotic weighted shifts on rooted directed trees. Also we show some sufficient conditions for distributional chaos of these operators. Our results generalize the previous results on Li-Yorke and distributionally chaotic unilateral and bilateral weighted shifts. It also has relations with some papers in which other important dynamical properties were investigated for weighted shifts on trees.
We provide a complete proof of a sharp Gagliardo-Nirenberg-type inequality on the unit interval, in its additive-invariant form inf (c is an element of R )|| f - c || L infinity <= (3/2)(1/3)( inf (c is an element of R )|| f - c || (L2))(2/3 ) ( )|| f' || (1/3)(L infinity) which is motivated by the quantitative stability analysis in optimal transport theory. The proof employs the one-dimensional area formula together with rather elementary calculations.
We apply proof mining techniques to obtain quantitative and qualitative results on asymptotic and T-asymptotic regularity for the inexact generalized Halpern iteration, a viscosity-type extension of an iteration recently studied by Kanzow and Shehu. Specializing our results to the Kanzow-Shehu iteration and the sequential averaging method (SAM) yields analogous results for these iterations. Furthermore, we compute rates of (T-)asymptotic regularity for particular choices of the parameter sequences, and for one of them, we obtain linear rates as an application of a lemma due to Sabach and Shtern.
In this paper, we consider the Phi-Laplacian problem with Dirichlet boundary condition, -div(Phi(|del u|del u/del u) = lambda g(.)Phi(u) in Omega,lambda is an element of R and u|partial derivative Omega and u|(partial derivative Omega) = 0. The term Phi is a real odd and increasing homeomorphism, g is a nontrivial, nonnegative function in L infinity(Omega) and Omega subset of R-N is abounded connected domain. We examine the asymptotic behavior of sequences of eigenvalues of the differential equation. The treatment is based solely on the asymptotic homogeneity of Phi plus an additional condition on the domain (the segment property). We choose a sequence of eigenfunctions tending either to zero or infinity (in the sense of the norm). The core result of these notes shows that the liminf of the associated sequence of eigenvalues coincides with the first eigenvalue of the usual p-Laplace operator, and that the weak-* limit of the corresponding eigenfunctions is an associated ground state.
In this paper, we obtain explicit bounds for the real part of the logarithmic derivative of the Riemann zeta-function on the line Re s = 1, assuming the Riemann hypothesis. The proof combines the Guinand-Weil explicit formula with extremal bandlimited majorants and minorants for the Poisson kernel. As an application, we revisit the classical estimates of Littlewood for the modulus of the Riemann zeta-function and of its reciprocal on the line Re s = 1, and derive a slight refinement of the bounds of Lamzouri, Li, and Soundararajan. In addition, we establish an explicit bound for the modulus of the logarithmic derivative of the Riemann zeta-function on the line Re s = 1 under the Riemann hypothesis, improving the lower-order term in a result of Chirre, Hagen, and Simonic.
Let G be a simple graph and k is an element of N. Two important graph operations are: the k-th graph power of G, denoted by G(k), where G(k) is the graph obtained from G by adding an edge between every pair of vertices that have a distance at most k, and the complement graph of G, denoted by G(sic). In this paper, we studied the relation between the independence numbers of G(k)(sic) and G(k)(sic).
We prove the existence of local solutions of any second order quasilinear elliptic system with prescribed 1-jet and present some applications in Riemannian geometry.
A coupled cell network is a type of ordinary differential equation \dot x(t)=f(x(t)) , with structural constraints on the vector field f , encoded in a directed graph, whose cells and arrows are labeled by type. The generated dynamics can model, for example, those of neural networks or ecological systems. These systems and the synchrony patterns observed in their solutions have been intensely studied, particularly by Golubitsky, Stewart, and their coauthors. In the present article, we show that, for a generic vector field f , the synchrony patterns of the solutions of \dot x(t)=f(x(t)) are always balanced. This roughly means that for almost all f , the observed synchrony patterns, such as synchronization in two different cells, are inherited from the structural symmetries imposed by the graph and the cell types. Any other synchronization, not directly imposed by the geometry of the graph and the cell types, cannot occur. By doing so, we are completing the proof of several conjectures, including the rigid synchrony conjecture, the full oscillation conjecture and the observation of constant states. This article is the published version of the results stated by the second author in his PhD thesis.
We consider a nodal curve C in the complex projective plane whose irreducible components C_i are smooth. A minimal set of generators G for the first and second syzygy modules of the Jacobian ideal of C are described, using recent results by Th. Kahle, H. Schenck, B. Sturmfels and M. Wiesmann on the likelihood correspondence. The elements of G have explicit formulas in terms of the equations f_i=0 of the irreducible components C_i of C.
We obtain estimates on the decay of correlations, Central Limit Theorem and Large Deviations for dynamical systems admitting an induced weak Gibbs–Markov map, for larger classes of observables with weaker regularity than Hölder, characterized by suitable moduli of continuity.
The sigma-convergence concept has been up to now used to derive macroscopic models in full space dimensions. In this work, we generalize it to thin heterogeneous domains given rise to phenomena in lower space dimensions. More precisely, we provide a new approach of the sigma-convergence method that is suitable for the study of phenomena occurring in thin heterogeneous media. This is made through a systematic study of the sigma-convergence method for thin heterogeneous domains. Assuming that the thin heterogeneous layer is made of microstructures that are distributed inside in a deterministic way including as special cases the periodic and the almost periodic distributions, we make use of the concept of algebras with mean value to state and prove the main compactness results. As an illustration, we upscale a Darcy-Lapwood-Brinkmann micro-model for thin flow. We prove that, according to the magnitude of the permeability of the porous domain, we obtain as effective models, the Darcy law in lower dimensions. The effective models are derived through the solvability of either the local Darcy-Brinkmann problems or the local Hele-Shaw problems.
In this paper, we consider normalized solutions for fractional NLS systems with Sobolev critical nonlinearities and linear couplings. Under mass supercritical condition, there are two cases discussed. First, we use the minimax method to get the existence of positive normalized solution when the exponent is Sobolev subcritical. Second, if it is Sobolev critical, we prove the nonexistence of positive normalized solution. Scaling transformation and concentration-compactness principle are used in this paper.
We give a generalization of the mean value theorem to several variables that retains the structure of the original result. In one variable the mean value theorem says that if g is continuous in a closed interval [ a,b] and the derivative g^{\prime}(x) exists for all points x\in( a,b) then there exists a point \xi \in( a,b) such that g^{\prime}( \xi) =\frac{g( b) -g( a) }{b-a} . We first discuss a version of the idea of “being the derivative of a continuous function at a point” in several variables, the notion of proper values . Then we give a version of the mean value theorem in several variables.We establish that if g is continuous in a region of \mathbb{R}^{d} that contains the d -rectangle R given as a_{j}\leq x_{j}\leq b_{j} , 1\leq j\leq d , and f=\nabla_{1}\cdots\nabla_{d}g is a locally integrable function with proper values at all interior points of R then there exists a point ( \xi_{1},\ldots,\xi_{d}) \in\operatorname*{Int}R such that f( \xi_{1},\ldots,\xi_{d}) =\frac{1}{\prod_{j=1}^{d}( b_{j}-a_{j}) }\sum_{\mathbf{v}\in\mathfrak{V}}( -1)^{\sigma( \mathbf{v}) }g( \mathbf{v}) , where \mathfrak{V} is the set of vertices of the d -rectangle and \sigma( \mathbf{v}) is the number of indices j for which v_{j}=a_{j} . In two variables it means that if the rectangle R:a_{1}\leq x_{1}\leq b_{1} , a_{2}\leq x_{2}\leq b_{2} , is contained in a region U\subset\mathbb{R}^{2} and f=\nabla_{1}\nabla_{2}g is a locally integrable function with proper values at all points, then there exists a point ( \xi_{1},\xi_{2}) \in\operatorname*{Int}R such that f( \xi_{1},\xi_{2}) =\frac{g( b_{1},b_{2}) -g( b_{1},a_{2}) -g( a_{1},b_{2}) +g( a_{1}%,a_{2}) }{( b_{1}-a_{1}) ( b_{2}-a_{2}) }.
This paper studies the metrical theory of \vartheta -expansions, a generalization of regular continued fractions. We focus on the Hausdorff dimension of two classical types of exceptional sets. First, we extend Jarník’s results on the dimension of sets of numbers with bounded partial quotients to the \vartheta -expansion setting, obtaining new bounds that improve upon the classical ones in the special case of regular continued fractions. Second, if \mathcal{F}_{n}(x) is the largest partial quotient of the \vartheta -expansion, we prove that for all \beta \geq 0 , the set of numbers x for which (\mathcal{F}_{n}(x)\log\log n) / n converges to \beta has full Hausdorff dimension. This result complements a previous almost everywhere law and generalizes the work of Philipp (1975/76), Okano (2002), Wu and Xu (2009) to \vartheta -expansions.
In this article, we prove that the weak and strong Lefschetz properties hold, and, moreover, are equivalent, for any quotient ring at least 2-dimensional. Furthermore, we prove that the weak Lefschetz property holds for any dimension 1 almost complete intersection. We then apply the obtained results to the case of Jacobian ideals of hyperplane arrangements.
We construct the L-1 contractive solutions to the Cauchy problem for a scalar conservation law with a discontinuous flux function. For this end, we give a new method for constructing the L-1 contractive solutions to the Cauchy problem for the scalar conservation law.
Let (Omega,Sigma,lambda) be a finite measure space and r(L-infinity(lambda)) be a natural Mackey topology on L-infinity(lambda). Let T : L-infinity(lambda)-> L-infinity(lambda) be a Bochner representable operator, that is, there exists g is an element of L-1 (lambda,L-infinity(lambda)) so that T(u) = integral(Omega)u(omega)g(omega)d lambda(omega) . It is shown that T is a nuclear operator between the locally convex space (L-infinity (lambda), tau(L-infinity (lambda), L-1(lambda))) and the Banach space L-infinity(lambda) and T has a well-defined trace: tr T = integral(Omega)g(omega)(omega) d lambda (omega).
It is well known that the quotient of the derived subgroup of the Shephard-Todd complex reflection group G32 (which has rank 4) by its center is isomorphic to the derived subgroup of the Weyl group of type E6. We show that this isomorphism can be realized through the second exterior power, and take the opportunity to propose an alternative construction of the
