Explicit Exponents Research Articles

Inertial Proximal Block Coordinate Method for a Class of Nonsmooth Sum-of-Ratios Optimization Problems

In this paper, we consider a class of nonsmooth sum-of-ratios fractional optimization problems with block structure. This model class is ubiquitous and encompasses several important nonsmooth optimization problems in the literature. We first propose an inertial proximal block coordinate method for solving this class of problems by exploiting the underlying structure. The global convergence of our method is guaranteed under the Kurdyka--Lojasiewicz (KL) property and some mild assumptions. We then identify the explicit exponents of the KL property for three important structured fractional optimization problems. In particular, for the sparse generalized eigenvalue problem with either cardinality regularization or sparsity constraint, we show that the KL exponents are 1/2, and so, the proposed method exhibits linear convergence rate. Finally, we illustrate our theoretical results with both analytic and simulated numerical examples.

On the role of continuous symmetries in the solution of the three-dimensional Euler fluid equations and related models.

We review and apply the continuous symmetry approach to find the solution of the three-dimensional Euler fluid equations in several instances of interest, via the construction of constants of motion and infinitesimal symmetries, without recourse to Noether’s theorem. We show that the vorticity field is a symmetry of the flow, so if the flow admits another symmetry then a Lie algebra of new symmetries can be constructed. For steady Euler flows this leads directly to the distinction of (non-)Beltrami flows: an example is given where the topology of the spatial manifold determines whether extra symmetries can be constructed. Next, we study the stagnation-point-type exact solution of the three-dimensional Euler fluid equations introduced by Gibbon et al. (Gibbon et al. 1999 Physica D 132, 497–510. (doi:10.1016/S0167-2789(99)00067-6)) along with a one-parameter generalization of it introduced by Mulungye et al. (Mulungye et al. 2015 J. Fluid Mech. 771, 468–502. (doi:10.1017/jfm.2015.194)). Applying the symmetry approach to these models allows for the explicit integration of the fields along pathlines, revealing a fine structure of blowup for the vorticity, its stretching rate and the back-to-labels map, depending on the value of the free parameter and on the initial conditions. Finally, we produce explicit blowup exponents and prefactors for a generic type of initial conditions.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

Double marking of the past in Early Modern Greek

Abstract This paper examines double marking of the past feature in verbal formations with preverbs in Early Modern Greek (16th c.). This period is characterized by great linguistic variation reflected in every type of text. We compiled a dataset of 268 formations from a corpus which includes 250 excerpts of prose texts (155,717 words in total). Our analysis is based on quantitative and qualitative criteria. We argue that double marking of the past feature in this period can take two forms, that is, it can be realized as two distinct prefixes (explicit multiple exponence), or it can be realized by one verbal augment (prefix) and as part of the information carried by the verbal allomorph (implicit multiple exponence). The double exponence of this inflectional feature hints at a process of externalization of inflection, while pleonastic exponence of the past feature related to stem allomorphy can be explained as a tendency for explicit exponents of inflectional features.

BOUNDS FOR VOLUMES OF SUB-LEVEL SETS OF POLYNOMIALS AND APPLICATIONS

In this paper, we present some explicit exponents in the estimates for the volumes of sub-level sets of polynomials on bounded sets and applications to the decay of oscillatory integrals and the convergence of singular integrals.

On a basic mean value theorem with explicit exponents

We follow a paper by Sedunova regarding Vaughan’s basic mean value Theorem to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by Cojocaru and Murty. As an application we derive a basic mean value theorem for the von Mangoldt generalized functions.

Continuity of solutions to a nonlinear fractional diffusion equation

We study a parabolic equation for the fractional p-Laplacian of order s, for pge 2 and 0<s<1. We provide space-time Hölder estimates for weak solutions, with explicit exponents. The proofs are based on iterated discrete differentiation of the equation in the spirit of Moser’s technique.

Complementary Problems with Polynomial Data

Given polynomial maps \(f, g \colon \mathbb {R}^{n} \to \mathbb {R}^{n}\), we consider the polynomial complementary problem of finding a vector \(x \in \mathbb {R}^{n}\) such that $$ f(x) \ge 0, \quad g(x) \ge 0, \quad \text{ and } \quad \langle f(x), g(x) \rangle = 0. $$ In this paper, we present various properties on the solution set of the problem, including genericity, nonemptiness, compactness, uniqueness as well as error bounds with explicit exponents. These strengthen and generalize some previously known results.

Error bounds for the solution sets of generalized polynomial complementarity problems

ABSTRACT In this paper, several error bounds for the solution sets of the generalized polynomial complementarity problems (GPCPs) with explicit exponents are given. As the solution set of a GPCP is the solution set of a system of polynomial equalities and inequalities, the state-of-art results in error bounds for polynomial systems can be applied directly. Starting from this, a much better error bound result for the solution set of a GPCP based on exploring the intrinsic sparsity via tensor decomposition is established.

Exponents for the equidistribution of shears and applications

In [KK18], the authors introduced “soft” methods to prove the effective (i.e. with power savings error) equidistribution of “shears” in cusped hyperbolic surfaces. In this paper, we study the same problem but now allow full use of the spectral theory of automorphic forms to produce explicit exponents, and uniformity in parameters. We give applications to counting square values of quadratic forms.

Cramér’s estimate for stable processes with power drift

We investigate the upper tail probabilities of the all-time maximum of a stable Levy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Levy process. We also study the lower tail probabilities of the integrated stable Levy process in the presence of a power positive drift.

Error Bounds for the Solution Sets of Quadratic Complementarity Problems

In this article, two types of fractional local error bounds for quadratic complementarity problems are established, one is based on the natural residual function and the other on the standard violation measure of the polynomial equalities and inequalities. These fractional local error bounds are given with explicit exponents. A fractional local error bound with an explicit exponent via the natural residual function is new in the tensor/polynomial complementarity problems literature. The other fractional local error bounds take into account the sparsity structures, from both the algebraic and the geometric perspectives, of the third-order tensor in a quadratic complementarity problem. They also have explicit exponents, which improve the literature significantly.

On tangent cones at infinity of algebraic varieties

In this paper, we define the geometric and algebraic tangent cones at infinity of algebraic varieties and establish the following version at infinity of Whitney’s theorem [Local properties of analytic varieties, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton University Press, Princeton, N. J., 1965), pp. 205–244; Tangents to an analytic variety, Ann. of Math. 81 (1965) 496–549]: The geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of the geometric tangent cone at infinity using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. Moreover, we show that the tangent cone at infinity of a complex algebraic variety is actually the part at infinity of this variety [G.-M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, 2nd extended edn. (Springer, Berlin, 2008)]. We also show that the tangent cone at infinity of a complex algebraic variety can be computed using Gröbner bases.

The global weak sharp minima with explicit exponents in polynomial vector optimization problems

In this paper we discuss the global weak sharp minima property for vector optimization problems with polynomial data. Exploiting the imposed polynomial structure together with tools of variational analysis and a quantitative version of Łojasiewicz’s gradient inequality due to D’Acunto and Kurdyka, we establish the Holder type global weak sharp minima with explicitly calculated exponents.

Łojasiewicz inequalities with explicit exponents for smallest singular value functions

Let F(x):=(fij(x))i=1,…,p;j=1,…,q, be a (p×q)-real polynomial matrix and let f(x) be the smallest singular value function of F(x). In this paper, we first give the following nonsmooth version of Łojasiewicz gradient inequality for the function f with an explicit exponent: For anyx̄∈Rn, there existc>0andϵ>0such that we have for all‖x−x̄‖<ϵ,inf{‖w‖:w∈∂f(x)}≥c|f(x)−f(x̄)|1−τ,where∂f(x)is the limiting subdifferential offatx,d:=maxi=1,…,p;j=1,…,qdegfij,ℛ(n,d):=d(3d−3)n−1ifd≥2andℛ(n,d):=1ifd=1, andτ:=1ℛ(n+p,2d+2). Then we establish some versions of Łojasiewicz inequality for the distance function with explicit exponents, locally and globally, for the smallest singular value function f(x) of the matrix F(x).

On the least prime ideal and Siegel zeros

Let [Formula: see text] be a number field, [Formula: see text] be an integral ideal, and [Formula: see text] be the associated narrow ray class group. Suppose [Formula: see text] possesses a real exceptional character [Formula: see text], possibly principal, with a Siegel zero [Formula: see text]. For [Formula: see text] satisfying [Formula: see text] [Formula: see text], we establish an effective [Formula: see text]-uniform Linnik-type bound with explicit exponents for the least norm of a prime ideal [Formula: see text]. A special case of this result is a bound for the least rational prime represented by certain binary quadratic forms.