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[en] In this paper we prove the exponential instability of the fractional Calderón problem and thus prove the optimality of the logarithmic stability estimate from Rüland and Salo (2017 arXiv:1708.06294). In order to infer this result, we follow the strategy introduced by Mandache in (2001 Inverse Problems 17 1435) for the standard Calderón problem. Here we exploit a close relation between the fractional Calderón problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in Rüland and Salo (2017 arXiv:1708.06294). Finally, in one dimension, we show a close relation between the fractional Calderón problem and the truncated Hilbert transform. (paper)
[en] Given two weighted graphs (X, bk, mk), k = 1,2 with b1 ∼ b2 and m1 ∼ m2, we prove a weighted L1-criterion for the existence and completeness of the wave operators W±(H2, H1, I1,2), where Hk denotes the natural Laplacian in #Script Small L#2(X, mk) w.r.t. (X, bk, mk) and I1,2 the trivial identification of #Script Small L#2(X, m1) with #Script Small L#2(X, m2). In particular, this entails a general criterion for the absolutely continuous spectra of H1 and H2 to be equal.
[en] The consensus problem of second-order multi-agent systems with quantized link is investigated in this Letter. Some conditions are derived for the quantized consensus of the second-order multi-agent systems by the stability theory. Moreover, a result characterizing the relationship between the eigenvalues of the Laplacians matrix and the quantized consensus is obtained. Examples are given to illustrate the theoretical analysis. -- Highlights: ► A second-order multi-agent model with quantized data is proposed. ► Two sufficient and necessary conditions are obtained. ► The relationship between the eigenvalues of the Laplacians matrix and the quantized consensus is discovered.
[en] In a celebrated paper ''Can one hear the shape of a drum?'' M. Kac [Am. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.
[en] We consider random Hamiltonians defined on long-range percolation graphs over . The Hamiltonian consists of a randomly weighted Laplacian plus a random potential. We prove uniform existence of the integrated density of states and express the IDS using a Pastur-Shubin trace formula.
[en] In the present article, we wish to discuss q-analogues of Laplace-type integrals on diverse types of q-special functions involving Fox’s -functions. Some of the discussed functions are the q-Bessel functions of the first kind, the q-Bessel functions of the second kind, the q-Bessel functions of the third kind, and the q-Struve functions as well. Also, we obtain some associated results related to q-analogues of the Laplace-type integral on hyperbolic sine (cosine) functions and some others of exponential order type as an application to the given theory.
[en] The author examines the Schroedinger operator H = -Δ + q(x) where Δ is the Laplacian and q(x)epsilon R/sup n/. He gives sufficient conditions for the spectrum of H to contain the interval of the form [a,infinity) and sufficient conditions for the essential spectrum of H to contain the interval of the form [b.infinity). The estimates for the lower bounds of a and b are positive numbers. q(x) is allowed to be negative in some region. The results are in R2 and in R/sup n/
[en] A function f, defined on the vertices of a graph G, induces nodal domains on the graph. Nodal domains of discrete and metric graphs are of growing interest among physicists and mathematicians. In this paper, several results regarding the nodal domain counts of discrete graphs are derived. One such result is a global upper bound for the number of nodal domains of G, in terms of its chromatic number. Another result is a criterion of resolution of (Laplacian) isospectral graphs via their nodal counts. Several additional results are also shown