Abstract: We consider how m-dimensional Bakry-Émery Ricci curvature affects the geometry of the boundary \(\partial M\) . By using the Reilly’s formula with respect to f-Laplacian, geometric inequalities involving f-mean curvature are obtained. Furthermore, we also achieve the relationship between f-mean curvature of the boundary submanifold and the mean curvature of submanifold \(x:\partial M\rightarrow \mathbb {R}^N(c)\) into space form \(\mathbb {R}^N(c)\) . PubDate: 2018-05-30 DOI: 10.1007/s13324-018-0237-5

Authors:Dmitrii Gerasimov; Igor Popov; Irina Blinova; Anton Popov Abstract: In this paper we investigate scattering problem for a quantum graph (a ring), connected with two semi-infinite leads via a Dirac delta function at boundary. We prove incompleteness of the system of resonance states in \(L_2\) on finite subgraph for the Kirchhoff coupling condition at the vertex and discuss a relation with the factorization of the characteristic function in Sz-Nagy functional model. The sensitivity of the incompleteness property to variation of the operator or the graph structure is considered. The cases of the Landau and the Dirac operators at the graph edges demonstrate the same completeness/incompleteness property as the Schrödinger case. At the same time, small variation of the graph structure restores the completeness property. PubDate: 2018-05-23 DOI: 10.1007/s13324-018-0233-9

Authors:César E. Torres Ledesma Abstract: In this work, we consider the existence of solution to the following fractional advection–dispersion equation 0.1 $$\begin{aligned} -\frac{d}{dt} \left( p {_{-\infty }}I_{t}^{\beta }(u'(t)) + q\; {_{t}}I_{\infty }^{\beta }(u'(t))\right) + b(t)u = f(t, u(t)),\;\;t\in \mathbb {R}\end{aligned}$$ where \(\beta \in (0,1)\) , \(_{-\infty }I_{t}^{\beta }\) and \(_{t}I_{\infty }^{\beta }\) denote left and right Liouville–Weyl fractional integrals of order \(\beta \) respectively, \(0<p=1-q<1\) , \(f:\mathbb {R}\times \mathbb {R} \rightarrow \mathbb {R}\) and \(b:\mathbb {R} \rightarrow \mathbb {R}^{+}\) are continuous functions. Due to the general assumption on the constant p and q, the problem (0.1) does not have a variational structure. Despite that, here we study it performing variational methods, combining with an iterative technique, and give an existence criteria of solution for the problem (0.1) under suitable assumptions. PubDate: 2018-05-22 DOI: 10.1007/s13324-018-0234-8

Authors:Thomas Bieske; Robert D. Freeman Abstract: We prove a \(\texttt {p}(\cdot )\) -Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. We then establish the existence and uniqueness (up to a set of zero \(\texttt {p}(\cdot )\) -capacity) of a minimizer to the Dirichlet energy integral for the variable exponent case. PubDate: 2018-05-22 DOI: 10.1007/s13324-018-0235-7

Authors:Vagif S. Guliyev; Arash Ghorbanalizadeh; Yoshihiro Sawano Abstract: We investigate the direct and inverse theorems for trigonometric polynomials in the Morrey space \({{\mathcal {M}}}^{p(\cdot ),\lambda (\cdot )}\) with variable exponents. For this space, we obtain estimates of the K-functional in terms of the modulus of smoothness and the Bernstein type inequality for trigonometric polynomials. PubDate: 2018-05-12 DOI: 10.1007/s13324-018-0231-y

Authors:Arun Kajla; Sheetal Deshwal; P. N. Agrawal Abstract: In the present paper we introduce a Durrmeyer variant of Jain operators based on a function \(\rho (x)\) where \(\rho \) is a continuously differentiable function on \([0,\infty ),~\rho (0)=0\) and \(\inf \rho ^{'}(x)\ge \) a, \(\text{ a }>0,~\text{ x }\in [0,\infty )\) . For these new operators, some indispensable auxiliary results are established first. Then, the degree of approximation with the aid of Ditzian–Totik modulus of smoothness and the rate of convergence for functions whose derivatives are of bounded variation, is obtained. Further, we focus on the study of a Voronovskaja type asymptotic theorem, quantitative Voronovskaya and Grüss-Voronovskaya type theorems. PubDate: 2018-05-05 DOI: 10.1007/s13324-018-0229-5

Authors:Vladimir G. Tkachev Abstract: We establish sharp inequalities for the Riesz potential and its gradient in \(\mathbb {R}^{n}\) and indicate their usefulness for potential analysis, moment theory and other applications. PubDate: 2018-04-28 DOI: 10.1007/s13324-018-0230-z

Authors:Li-Juan Cheng; Anton Thalmaier; James Thompson Abstract: We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using Ledoux’s argument, and uniform quantitative gradient estimates, firstly for \(C^2_b\) functions with boundary conditions and then for the unit spectral projection operators of Dirichlet and Neumann Laplacians. PubDate: 2018-04-24 DOI: 10.1007/s13324-018-0228-6

Authors:Martin Schlichenmaier Abstract: For compact quantizable Kähler manifolds the Berezin-Toeplitz quantization schemes, both operator and deformation quantization (star product) are reviewed. The treatment includes Berezin’s covariant symbols and the Berezin transform. The general compact quantizable case was done by Bordemann–Meinrenken–Schlichenmaier, Schlichenmaier, and Karabegov–Schlichenmaier. For star products on Kähler manifolds, separation of variables, or equivalently star product of (anti-) Wick type, is a crucial property. As canonically defined star products the Berezin-Toeplitz, Berezin, and the geometric quantization are treated. It turns out that all three are equivalent, but different. PubDate: 2018-04-12 DOI: 10.1007/s13324-018-0225-9

Authors:Nick Gravin; Dmitrii V. Pasechnik; Boris Shapiro; Michael Shapiro Abstract: We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope \(\mathcal {P}\subset \mathbb {R}^d\) is a rational function. Its denominator is the product of linear forms dual to the vertices of \(\mathcal {P}\) raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak non-degeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S. PubDate: 2018-04-12 DOI: 10.1007/s13324-018-0226-8

Authors:Ali Aral; Daniela Inoan; Ioan Raşa Abstract: In this paper we consider two different general linear positive operators defined on unbounded interval and obtain estimates for the differences of these operators in quantitative form. Our estimates involve an appropriate K-functional and a weighted modulus of smoothness. Similar estimates are obtained for Chebyshev functional of these operators as well. All considerations are based on rearrangement of the remainder in Taylor’s formula. The obtained results are applied for some well known linear positive operators. PubDate: 2018-04-12 DOI: 10.1007/s13324-018-0227-7

Authors:Yuri Kondratiev; Yuri Kozitsky Abstract: We correct an estimate stated in Lemma 4.1 of our work in this journal. The correction implies that Theorem 2.5 is proved except for the point-wise boundedness of the first two correlation functions. PubDate: 2018-04-10 DOI: 10.1007/s13324-018-0224-x

Authors:Fei Meng; Fang Liu Abstract: In this paper, we study the Fourier transforms for two equations arising in the kinetic theory. The first equation is the spatially homogeneous Boltzmann equation. The Fourier transform of the spatially homogeneous Boltzmann equation has been first addressed by Bobylev (Sov Sci Rev C Math Phys 7:111–233, 1988) in the Maxwellian case. Alexandre et al. (Arch Ration Mech Anal 152(4):327–355, 2000) investigated the Fourier transform of the gain operator for the Boltzmann operator in the cut-off case. Recently, the Fourier transform of the Boltzmann equation is extended to hard or soft potential with cut-off by Kirsch and Rjasanow (J Stat Phys 129:483–492, 2007). We shall first establish the relation between the results in Alexandre et al. (2000) and Kirsch and Rjasanow (2007) for the Fourier transform of the Boltzmann operator in the cut-off case. Then we give the Fourier transform of the spatially homogeneous Boltzmann equation in the non cut-off case. It is shown that our results cover previous works (Bobylev 1988; Kirsch and Rjasanow 2007). The second equation is the spatially homogeneous Landau equation, which can be obtained as a limit of the Boltzmann equation when grazing collisions prevail. Following the method in Kirsch and Rjasanow (2007), we can also derive the Fourier transform for Landau equation. PubDate: 2018-03-22 DOI: 10.1007/s13324-018-0223-y

Authors:Andrew Thomack; Zachariah Tyree Abstract: Li and Wei (Proc Am Math Soc 137:195–204, 2009) studied the density of zeros of Gaussian harmonic polynomials with independent Gaussian coefficients. They derived a formula for the expected number of zeros of random harmonic polynomials as well as asymptotics for the case that the polynomials are drawn from the Kostlan ensemble. In this paper we extend their work to cover the case that the polynomials are drawn from the Weyl ensemble by deriving asymptotics for this class of harmonic polynomials. PubDate: 2018-03-16 DOI: 10.1007/s13324-018-0220-1

Authors:Artem Hulko Abstract: In this paper we define a one-dimensional discrete Dirac operator on \({\mathbb {Z}}\) . We study the eigenvalues of the Dirac operator with a complex potential. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity. We also estimate the number of eigenvalues for the discrete Schrödinger operator with complex potential on \({\mathbb {Z}}\) . That is we extend the result obtained by Hulko (Bull Math Sci, to appear) to the whole \({\mathbb {Z}}\) . PubDate: 2018-03-06 DOI: 10.1007/s13324-018-0222-z

Authors:Sergey Buterin; Vjacheslav Yurko Abstract: Inverse spectral problems are studied for second order integral and integro-differential operators. Uniqueness results are obtained, and algorithms for the solutions are provided along with necessary and sufficient conditions for the solvability of these nonlinear inverse problems. PubDate: 2018-03-03 DOI: 10.1007/s13324-018-0217-9

Authors:Kaushal Verma Abstract: We highlight an intrinsic connection between classical quadrature domains and the well-studied theme of removable singularities of analytic sets in several complex variables. Exploiting this connection provides a new framework to recover several basic properties of such domains, namely the algebraicity of their boundary, a better understanding of the associated defining polynomial and the possible boundary singularities that can occur. PubDate: 2018-03-02 DOI: 10.1007/s13324-018-0221-0