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Abstract: Abstract The Ambarzumyan theorem is studied for third order differential equation with discontinuity conditions inside a finite interval. To deduce this theorem, the estimates of solutions, the counting lemma and estimates of eigenvalues are discussed. PubDate: 2023-01-16
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Abstract: Abstract We consider a Bernstein–Schnabl operator \(L_n\) which commutes with the derivative and consequently can be represented as a differential operator with constant coefficients. The inverse of \(L_n\) restricted to polynomials is investigated. We obtain Voronovskaja type results for \(L_n^{-1}\) and compare them with the corresponding results for \(L_n\) . A conjecture in this sense is presented. PubDate: 2023-01-09
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Abstract: Abstract In this article we will explore Dirichlet Laplace eigenvalues of balls with spherically symmetric metrics. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radii. As a special case we shall show that the Dirichlet Laplace eigenvalues of balls with small radii on the sphere are smaller than the corresponding eigenvalues of the Euclidean balls with the same radii. The opposite correspondence is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces. PubDate: 2023-01-06
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Abstract: Abstract The goal of this paper is to show the existence of a bounded variation solution, which is based on the Anzellotti pairing to an evolution problem associated with the minimal surface equations. A key ingredient in the proof is to approximate the parabolic minimal surface problem by a quasilinear parabolic problem involving a parameter \(p>1\) , and then by establishing some energy estimates independent of p, we take the limit as \(p\rightarrow 1^{+}\) to obtain the desired result. PubDate: 2023-01-03
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Abstract: Abstract We present a simple example for the failure of the Calderón–Zygmund estimate for the \(\overline{\partial }\) -operator when the Sobolev (k, p)-norms are replaced by the \(C^k\) -norms. This example is discussed in the context of elliptic bootstrapping, Fredholm theory, and the regularity of J-holomorphic curves. PubDate: 2023-01-02
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Abstract: Abstract By virtue of a Reilly type integral formula associated with affine connections, we obtain a Colesanti type inequality. As an application, the Minkowski type and Heintze–Karcher type inequalities are provided. On the other hand, we also consider eigenvalue problem for Wentzell boundary conditions and achieve some relationships between the first nonzero eigenvalue of Wentzell boundary conditions and the first nonzero eigenvalue of Laplacian acting on functions on the boundary. PubDate: 2023-01-02
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Abstract: Abstract In this paper, we present a relation between Jacobi–Reeb dynamics and the dynamics associated with a mechanical Hamiltonian system with respect to a linear Poisson structure on a vector bundle. For this purpose, we will use the so-called Jacobi bundle metrics induced by the mechanical Hamiltonian system. These constructions extend classical results on the relation between standard mechanical Hamiltonian systems on cotangent bundles and Reeb dynamics. PubDate: 2022-12-20
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Abstract: Abstract Given a nondecreasing sequence \(\Lambda =\{\lambda _n>0\}\) such that \(\displaystyle \lim _{n\rightarrow \infty } \lambda _n=\infty ,\) we consider the sequence \(\mathcal {N}_\Lambda :=\left\{ \lambda _ne^{i\theta _n},n\in \,\mathbb {N}\right\} \) , where \(\theta _n\) are independent random variables uniformly distributed on \([0,2\pi ].\) We discuss the conditions on the sequence \(\Lambda \) under which \(\mathcal N_\Lambda \) is a zero set (a uniqness set) of a given weighted Fock space almost surely. The critical density of the sequence \(\Lambda \) with respect to the weight is found. PubDate: 2022-12-15
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Abstract: Abstract We consider the perturbed Stark operator \(H_q\varphi = -\varphi '' + x\varphi + q(x)\varphi \) , \(\varphi (0)=0\) , in \(L^2({\mathbb R}_+)\) , where q is a real function that belongs to \({\mathfrak {A}}_r =\left\{ q\in \mathcal {A}_r\cap \text {AC}[0,\infty ): q'\in \mathcal {A}_r\right\} \) , where \(\mathcal {A}_r = L^2_{\mathbb R}({\mathbb R}_+,(1+x)^r dx)\) and \(r>1\) is arbitrary but fixed. Let \(\left\{ \lambda _n(q)\right\} _{n=1}^\infty \) and \(\left\{ \kappa _n(q)\right\} _{n=1}^ \infty \) be the spectrum and associated set of norming constants of \(H_q\) . Let \(\{a_n\}_{n=1}^\infty \) be the zeros of the Airy function of the first kind, and let \(\omega _r:{\mathbb N}\rightarrow {\mathbb R}\) be defined by the rule \(\omega _r(n) = n^{-1/3}\log ^{1/2}n\) if \(r\in (1,2)\) and \(\omega _r(n) = n^{-1/3}\) if \(r\in [2,\infty )\) . We prove that \(\lambda _n(q) = -a_n + \pi (-a_n)^{-1/2}\int _0^\infty {{\,\textrm{Ai}\,}}^2(x+a_n)q(x)dx + O(n^{-1/3}\omega _r^2(n))\) and \(\kappa _n(q) = - 2\pi (-a_n)^{-1/2}\int _0^\infty {{\,\textrm{Ai}\,}}(x+a_n){{\,\textrm{Ai}\,}}'(x+a_n)q(x)dx + O(\omega _r^3(n))\) , uniformly on bounded subsets of \({\mathfrak {A}}_r\) . In order to obtain these asymptotic formulas, we first show that \(\lambda _n:\mathcal {A}_r\rightarrow {\mathbb R}\) and \(\kappa _n:\mathcal {A}_r\rightarrow {\mathbb R}\) are real analytic maps. PubDate: 2022-12-03
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Abstract: Abstract Well-posedness issues for the initial value problem (IVP) associated to the three-wave mixing Schrödinger system with quadratic nonlinearities posed on d-dimensional Zoll manifold M with initial data in \(\mathbf {H}^s(M)\) are investigated. New bilinear Strichartz’s type estimates for the interacting evolution groups are obtained considering that the system parameter \(\eta \) be a rational number. As a consequence, bilinear estimates for the interacting nonlinearities are obtained so as to prove the local well-posedness results with initial data in \(\mathbf {H}^s(M)\) whenever \(s>\frac{1}{4}\) if \(d = 2\) and \(s > \frac{d - 2}{2}\) if \(d \ge 3\) . Using some conserved quantities combined with the sharp Gagliardo-Nirenberg inequality the global well-posedness results are also obtained in dimensions \(d = 2, 3\) for initial data with Sobolev regularity \(s \ge 1\) . The present work improves the earlier results for the \(2\times 2\) Schrödinger system in Nogueira and Panthee (J Math Anal Appl 494:124574, 2021) relaxing the restrictions on the system parameter and consequently the local and the global well-posedness results there. PubDate: 2022-12-01
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Abstract: Abstract We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions, and antilinear eigenfunction expansions. The study is motivated by physical symmetries in quantum mechanics with non-self-adjoint operators. PubDate: 2022-11-29
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Abstract: Abstract We consider a general second order linear elliptic equation in a finely perforated domain. The shapes of cavities and their distribution in the domain are arbitrary and non-periodic; they are supposed to satisfy minimal natural geometric conditions. On the boundaries of the cavities we impose either the Dirichlet or a nonlinear Robin condition; the choice of the type of the boundary condition for each cavity is arbitrary. Then we suppose that for some cavities the nonlinear Robin condition is sign-definite in certain sense. Provided such cavities and ones with the Dirichlet condition are distributed rather densely in the domain and the characteristic sizes of the cavities and the minimal distances between the cavities satisfy certain simple condition, we show that a solution to our problem tends to zero as the perforation becomes finer. Our main result are order sharp estimates for the \(L_2\) - and \(W_2^1\) -norms of the solution uniform in the \(L_2\) -norm of the right hand side in the equation. PubDate: 2022-11-28
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Abstract: Abstract The objects of our study are webs in the geometry of volume-preserving diffeomorphisms. We introduce two local invariants of divergence-free webs: a differential one, directly related to the curvature of the natural connection of a divergence-free 2-web introduced by Tabachnikov (Diff Geom Appl 3:265-284, 1993), and a geometric one, inspired by the classical notion of planar 3-web holonomy defined by Blaschke and Bol (Geometrie der Gewebe. Grundlehren der mathematischen Wissenschaften, vol. 49. Springer, Berlin, 1938). We show that triviality of either of these invariants characterizes trivial divergence-free web-germs up to equivalence. We also establish some preliminary results regarding the full classification problem, which jointly generalize the theorem of Tabachnikov on normal forms of divergence-free 2-webs. They are used to provide a canonical form and a complete set of invariants of a generic divergence-free web in the planar case. Lastly, the relevance of local triviality conditions and their potential applications in numerical relativity are discussed. PubDate: 2022-11-23
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Abstract: Abstract In this paper we study a kind of Riemann problem for the Lamé–Navier system in the plane on a smooth as well as on a fractal closed contour. By using the Kolosov–Muskhelisvili formula, we reduce this problem to a pair of Riemann boundary value problems for analytic functions, and after that we get the necessary and sufficient conditions for the solvability of the problem and obtain explicit formulas for its solution. PubDate: 2022-11-21
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Abstract: Abstract We study a Lagrangian extension of the 5d Martínez Alonso–Shabat equation \(\mathcal {E}\) $$\begin{aligned} u_{yz}=u_{tx}+u_y\,u_{xs}-u_x\,u_{ys} \end{aligned}$$ that coincides with the cotangent equation \(\mathcal {T}^{*}\mathcal {E}\) to the latter. We describe the Lie algebra structure of its symmetries (which happens to be quite nontrivial and is described in terms of deformations) and construct two families of recursion operators for symmetries. Each family depends on two parameters. We prove that all the operators from the first family are hereditary, but not compatible in the sense of the Nijenhuis bracket. We also construct two new parametric Lax pairs that depend on higher-order derivatives of the unknown functions. PubDate: 2022-11-21
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Abstract: Abstract We implement a version of conformal field theory in a doubly connected domain with numerous conformal types to connect it to the theory of annulus SLE of various types, including the standard annulus SLE, the reversible annulus SLE, and the annulus SLE with several force points. This implementation considers the statistical fields generated under the OPE multiplication by the Gaussian free field and its central/background charge modifications with a weighted combination of Dirichlet and excursion-reflected boundary conditions. We derive the Eguchi–Ooguri version of Ward’s equations and Belavin–Polyakov–Zamolodchikov equations for those statistical fields and use them to show that the correlations of fields in the OPE family under the insertion of the one-leg operators are martingale-observables for various annulus SLEs. We find Coulomb gas (Dotsenko–Fateev integral) solutions to the parabolic partial differential equations for partition functions of conformal field theory for the reversible annulus SLE. PubDate: 2022-11-19
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Abstract: Abstract Stable harmonic mappings were first introduced and studied by Hernández and Martín (Math Proc Camb Philos Soc 155:343–359, 2013). Continuing research in this direction, we first prove a result on stability of the integral transform of harmonic mappings. Afterwards, we proved that the harmonic convolution \((\varphi +\alpha \overline{\varphi })*f\) is a stable harmonic convex mapping for \( \alpha \le 1\) , where \(\varphi \) is a convex univalent map and f is a stable harmonic convex map. Furthermore, we consider the meromorphic analogs of stable harmonic mappings and establish various interesting results for these new classes of mappings. PubDate: 2022-11-14 DOI: 10.1007/s13324-022-00760-z
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Abstract: Abstract Let \(P_{2k}\) be a homogeneous polynomial of degree 2k and assume that there exist \(C>0\) , \(D>0\) and \(\alpha \ge 0\) such that $$\begin{aligned} \left\langle P_{2k}f_{m},f_{m}\right\rangle _{L^2(\mathbb {S}^{d-1})}\ge \frac{1}{C\left( m+D\right) ^{\alpha }}\left\langle f_{m},f_{m}\right\rangle _{\mathbb {S}^{d-1}} \end{aligned}$$ for all homogeneous polynomials \(f_{m}\) of degree m. Assume that \(P_{j}\) for \(j=0, \dots ,\beta <2k\) are homogeneous polynomials of degree j. The main result of the paper states that for any entire function f of order \( \rho <\left( 2k-\beta \right) /\alpha \) there exist entire functions q and h of order bounded by \(\rho \) such that $$\begin{aligned} f=\left( P_{2k}-P_{\beta }- \dots -P_{0}\right) q+h\text { and }\Delta ^k r=0. \end{aligned}$$ This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem for parabola-shaped domains on the plane, with data given by entire functions of order smaller than \(\frac{1}{2}\) . PubDate: 2022-11-11 DOI: 10.1007/s13324-022-00758-7
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Abstract: Abstract We study the mixed Dirichlet–Neumann problem for the Laplace equation in the complement of a bounded convex polygonal quadrilateral in the extended complex plane. The Dirichlet / Neumann conditions at opposite pairs of sides are \(\{0,1\}\) and \(\{0,0\},\) resp. The solution to this problem is a harmonic function in the unbounded complement of the polygon known as the potential function of the quadrilateral. We compute the values of the potential function u including its value at infinity. The main result of this paper is Theorem 4.3 which yields a formula for \(u(\infty )\) expressed in terms of the angles of the polygonal given quadrilateral and the well-known special functions. We use two independent numerical methods to illustrate our result. The first method is a Mathematica program and the second one is based on using the MATLAB toolbox PlgCirMap. The case of a quadrilateral, which is the exterior of the unit disc with four fixed points on its boundary, is considered as well. PubDate: 2022-11-10 DOI: 10.1007/s13324-022-00732-3
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Abstract: Abstract The partial Taylor sums \(S_n\) , \(n \ge 0\) , are finite rank operators on any Banach space of analytic functions on the open unit disc. In the classical setting of disc algebra \({\mathcal {A}}\) , the precise value of \(\Vert S_n\Vert _{{\mathcal {A}} \rightarrow {\mathcal {A}}}\) is not known. These numbers are referred as the Lebesgue constants and they grow like \(\log n\) , modulo a multiplicative constant, when n tends to infinity. In this note, we study \(\Vert S_n\Vert \) when it acts on the local Dirichlet space \({\mathcal {D}}_\zeta \) . There are several distinguished ways to put a norm on \({\mathcal {D}}_\zeta \) and each choice naturally leads to a different operator norm for \(S_n\) , as an operator on \({\mathcal {D}}_\zeta \) . We consider three different norms on \({\mathcal {D}}_\zeta \) and, in each case, evaluate the precise value of \(\Vert S_n\Vert _{{\mathcal {D}}_\zeta \rightarrow {\mathcal {D}}_\zeta }\) . In all cases, we also show that the maximizing function is unique. These formulas indicate that \(\Vert S_n\Vert _{{\mathcal {D}}_\zeta \rightarrow {\mathcal {D}}_\zeta } \asymp \sqrt{n}\) as n grows. Hence, in the light of uniform boundedness principle, there is a function \(f \in {\mathcal {D}}_\zeta \) such that the local sequence \(\Vert S_nf\Vert _{{\mathcal {D}}_{\zeta }}\) , \(n \ge 1\) , is unbounded. We provide two explicit constructions. PubDate: 2022-11-07 DOI: 10.1007/s13324-022-00756-9
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