Search

In this paper, we focus on the nonlinear least squares problem: \min _{{\boldsymbol {x}}\in \mathbb {H}^d}\|\lvert A{\boldsymbol {x}}\rvert -{\boldsymbol {b}}\| where A\in \mathbb {H}^{m\times d}, {\boldsymbol {b}}\in \mathbb {R}^m with \mathbb {H}\in \left \{\mathbb {R},\mathbb {C}\right \} and consider the uniqueness and stability of solutions. This problem arises in applications such as phase retrieval and absolute value rectification neural networks. While several results have been developed to characterize the uniqueness and stability of solutions when {\boldsymbol {b}}=\lvert A{\boldsymbol {x}}_0\rvert for some {\boldsymbol {x}}_0\in \mathbb {H}^d, no existing results address the case where {\boldsymbol {b}} is arbitrary. In this paper, we investigate the uniqueness and stability of solutions for the more general case where {\boldsymbol {b}} is not necessarily equal to \lvert A{\boldsymbol {x}}_0\rvert for any {\boldsymbol {x}}_0\in \mathbb {H}^d. We prove that for any matrix A\in \mathbb {H}^{m\times d}, there is always a vector {\boldsymbol {b}}\in \mathbb {R}^m for which the solution to the nonlinear least squares problem is not unique. However, we show that such "bad" vectors {\boldsymbol {b}} are negligible in practice; specifically, if {\boldsymbol {b}}\in \mathbb {R}_{ }^m does not lie in some measure zero set, then the solution is unique. Furthermore, we establish certain conditions under which the solution is guaranteed to be unique. Regarding the stability of solutions, we prove that the solution is not uniformly stable. However, if we restrict the vectors {\boldsymbol {b}} to a convex set where the solution to the least squares problem is unique, then the solution becomes stable. To the best of our knowledge, our results represent the first theoretical results of the uniqueness and stability of solutions for the nonlinear least squares problem. [ABSTRACT FROM AUTHOR]

In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

An exchange matrix B dominates an exchange matrix B' if the signs of corresponding entries weakly agree, with the entry of B always having weakly greater absolute value. When B dominates B', interesting things happen in many cases (but not always): the identity map between the associated mutation-linear structures is often mutation-linear; the mutation fan for B often refines the mutation fan for B'; the scattering (diagram) fan for B often refines the scattering fan for B'; and there is often an injective homomorphism from the principal-coefficients cluster algebra for B' to the principal-coefficients cluster algebra for B, preserving \mathbf {g}-vectors and sending the set of cluster variables for B' (or an analogous larger set) into the set of cluster variables for B (or an analogous larger set). The scope of the description "often" is not the same in all four contexts and is not settled in any of them. In this paper, we prove theorems that provide examples of these dominance phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the study of the Bohr phenomenon for a quasisubordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for K-quasiconformal harmonic mappings by replacing the constant term by the absolute value of the analytic part of the given function. We also obtain the Bohr radius for the subordination family of odd analytic functions. [ABSTRACT FROM AUTHOR]

We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities. For instance, if the matrix \mathbb {A} is partitioned into three blocks A,B,C, then \begin{gather*} |\mathbb {A}|^3 \ge U|A|^3U^* + V|B|^3V^*+ W|C|^3W^*,\\ \sqrt {3} |\mathbb {A}| \ge U|A|U^* + V|B|V^*+ W|C|W^*, \end{gather*} for some isometries U,V,W, and \begin{equation*} \mu _4^2(\mathbb {A}) \le \mu _3^2(A) +\mu _2^2(B) + \mu _1^2(C) \end{equation*} where \mu _j stands for the j-th singular value. Our theorem may be used to extend a result by Bhatia and Kittaneh for the Schatten p-norms and to give a singular value version of Cauchy's Interlacing Theorem. [ABSTRACT FROM AUTHOR]

We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multiwavelet frame coefficients \[ \{\lvert \mathcal {W}_{\phi _i} f(\alpha ^{m}\beta n,\alpha ^{m}) \rvert : i\in \{1,2,3\}, m,n\in \mathbb {Z}\} \] for every \alpha >1,\beta >0 with \beta \ln (\alpha)\leq 4\pi /(1+4p), p>0, when the three wavelets \phi _i are suitable linear combinations of the Poisson wavelet P_p of order p and its Hilbert transform \mathscr {H}P_p. For complex-valued signals we find that this is not possible for any choice of the parameters \alpha >1,\beta >0, and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients. [ABSTRACT FROM AUTHOR]

We consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a function defined on a domain equal to the complement in Rn of the union of a finite number of bounded Lipschitz domains. The mean curvature H = H(x,t) is assumed to have absolute value controlled from above by a locally bounded, Lp-function, p ∈ [1,2n/(n+2)], n ≥ 3. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value. We establish an approximation of the Lyapunov exponent of a rational map ƒ of P1 of degree d > 1 defined over K in terms of the multipliers of periodic points of f having the formally exact period n, with an explicit error estimate in terms of ƒ, n, and d. As an immediate consequence, we obtain an estimate on the blow-up of the Lyapunov exponent function near a pole in one-dimensional parameter families of rational maps over K. Combined with an improvement of our former archimedean counterpart, this non-archimedean quantitative approximation of Lyapunov exponents allows us to establish [label = −] a quantification of Silverman's and Ingram's recent comparison between the critical height and any ample height on the dynamical moduli space Md(Q) except for the flexible Lattès locus, an improvement of McMullen's finiteness of the multiplier maps in two aspects: reduction to multipliers of cycles having a given formally exact period, and an explicit computation on the magnitude of the formally exact period of cycles, and a characterization of non-affine isotrivial rational maps defined over a function field C(X) of a complex normal projective variety X in terms of the growth of the degree of the multipliers of cycles. [ABSTRACT FROM AUTHOR]

In this article we prove that suitable positive powers of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix converge in law, as the size of the matrix tends to infinity, to a Gaussian multiplicative chaos measure once correctly normalized. We prove this in the whole L1- or subcritical phase of the chaos measure. [ABSTRACT FROM AUTHOR]

This article answers a question posed by Draisma and Horobet, who asked for a closed formula for the expected number of real eigenvalues of a random real symmetric tensor drawn from the Gaussian distribution relative to the Bombieri norm. This expected value is equal to the expected number of real critical points on the unit sphere of a Kostlan polynomial. We also derive an exact formula for the expected absolute value of the determinant of a matrix from the Gaussian Orthogonal Ensemble. [ABSTRACT FROM AUTHOR]

Let X1, . . ., Xn be a standard normal sample in Rd. We compute exactly the expected volume of the Gaussian polytope conv [X1, . . ., Xn], the symmetric Gaussian polytope conv [±X1, . . .,±Xn], and the Gaussian zonotope [0,X1] + · · · + [0,Xn] by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson’s formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including conv [l1X1, . . ., lnXn] and conv[±l1X1, . . .,±lnXn], where l1, . . ., ln ≥ 0 are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the kth intrinsic volume of the regular simplex Δn−1 to the expected maximum of independent standard Gaussian random variables ξ1, . . ., ξn given that the maximum has multiplicity k. Namely, we show that Vk(Δn−1) = (2π)k/2/k! · lim ε↓0 ε1−kE[max{ξ1, . . ., ξn}1{ξ(n) −ξ(n−k+1) ≤ε}], where ξ(1) ≤ ·· · ≤ ξ(n) denote the order statistics. A similar result holds for the cross-polytope if we replace ξ1, . . ., ξn with their absolute values. [ABSTRACT FROM AUTHOR]

The canonical height associated to a polarized endomorphism of a projective variety, constructed by Call and Silverman and generalizing the Néron-Tate height on a polarized abelian variety, plays an important role in the arithmetic theory of dynamical systems. We generalize this construction to polarized correspondences, prove various fundamental properties, and show how the global canonical height decomposes as an integral of a local height over the space of absolute values on the algebraic closure of the field of definition. [ABSTRACT FROM AUTHOR]

We prove some new continuity results for the Julia sets J and J+ of the complex Hénon map Hc,a (x, y) = (x2 + c + ay, ax), where a and c are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue (1 + t)λ, where λ is a root of unity and t is real and small in absolute value. These maps have a semi-parabolic fixed point when t is 0, and we use the techniques that we have developed in a prior work for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero |t|, the Hénon map is hyperbolic and has connected Julia set. We prove that the Julia sets J and J+ depend continuously on the parameters as t → 0, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of Hénon maps is stable on J and J+ when t is non-negative. [ABSTRACT FROM AUTHOR]

Let K/Q be an abelian extension and let D be the absolute value of the discriminant of K. We show that for each e > 0, the smallest rational prime that splits completely in K is O(D1/4+e). Here the implied constant depends only on e and the degree of K. This generalizes a theorem of Elliott, who treated the case when K/Q has prime conductor. [ABSTRACT FROM AUTHOR]