Asynchronously parallel optimization solver for finding multiple minima

We propose and analyze an asynchronously parallel optimization algorithm for finding multiple, high-quality minima of nonlinear optimization problems. Our multistart algorithm considers all previously evaluated points when determining where to start or continue a local optimization run. Theoretical results show that when there are finitely many minima, the algorithm almost surely starts a finite number of local optimization runs and identifies every minimum. The algorithm is applicable to general optimization settings, but our numerical results focus on the case when derivatives are unavailable. In numerical tests, a Python implementation of the algorithm is shown to yield good approximations of many minima (including a global minimum), and this ability is shown to scale well with additional resources. Our implementation’s time to solution is shown also to scale well even when the time to perform the function evaluation is highly variable. An implementation of the algorithm is available in the libEnsemble library at https://github.com/Libensemble/libensemble.     

Invariant and stationary measures for the action on Moduli space

We prove some ergodic-theoretic rigidity properties of the action of Open image in new window on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of Open image in new window is supported on an invariant affine submanifold.The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.     

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let Open image in new window denote a weight in Open image in new window which belongs to the Muckenhoupt class Open image in new window and let Open image in new window denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure Open image in new window . The sharp Tauberian constant of Open image in new window with respect to Open image in new window , denoted by Open image in new window , is defined by

Open image in new window

In this paper, we show that the Solyanik estimate

$$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$

holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator Open image in new window and a weight Open image in new window :

Open image in new window

We show that we have Open image in new window if and only if Open image in new window . As a corollary of our methods we obtain a quantitative embedding of Open image in new window into Open image in new window .

     

Positive definite and Gram tensor complementarity problems

Given a continuous function Open image in new window and Open image in new window , the non-linear complementarity problem \(\text{ NCP }(g,q)\) is to find a vector Open image in new window such that

$$\begin{aligned} x \ge 0,~~y:=g(x) +q\ge 0~~\text{ and }~~x^Ty=0. \end{aligned}$$

We say that g has the Globally Uniquely Solvable (\(\text{ GUS }\))-property if \(\text{ NCP }(g,q)\) has a unique solution for all Open image in new window and C-property if \(\mathrm{NCP}(g,q)\) has a convex solution set for all Open image in new window . In this paper, we find a class of non-linear functions that have the \(\text{ GUS }\)-property and C-property. These functions are constructed by some special tensors which are positive semidefinite. We call these tensors as Gram tensors.

     

Approximation by local <Emphasis Type="Italic">L</Emphasis>-splines corresponding to a linear differential operator of the second order

For the class of functions

Open image in new window

, where \(\mathcal{L}_2 (D)\) is a linear differential operator of the second order whose characteristic polynomial has only real roots, we construct a noninterpolating linear positive method of exponential spline approximation possessing extremal and smoothing properties and locally inheriting the monotonicity of the initial data (the values of a function

Open image in new window

at the points of a uniform grid). The approximation error is calculated exactly for this class of functions in the uniform metric.

     

Preserving preference rankings under non-financial background risk

We investigate the impact of a non-financial background risk ??? on thepreference rankings between two independent financial risks Open image in new window ₁ and Open image in new window ₂ for anexpected-utility maximizer. More precisely, we provide necessary and sufficientconditions for the alternative (x₀+ Open image in new window ₁,y₀+ ???) to be preferred to(x₀+ Open image in new window ₂,y₀+ ???)whenever (x₀+ Open image in new window ₁,y₀) ispreferred to (x₀+ Open image in new window ₂,y₀). Utilityfunctions that preserve the preference rankings are fully characterized. Theirpractical relevance is discussed in light of recent results on the constraintsfor the modelling of the preference for the disaggregation of harms.     

Asymptotic Normality of Some Graph Sequences

For a simple finite graph G denote by Open image in new window the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If \(E_n\) is the graph on n vertices with no edges then Open image in new window coincides with Open image in new window , the ordinary Stirling number of the second kind, and so we refer to Open image in new window as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of \(E_n\), is asymptotically normal—essentially, as n grows, the histogram of Open image in new window , suitably normalized, approaches the density function of the standard normal distribution. In light of Harper’s result, it is natural to ask for which sequences \((G_n)_{n \ge 0}\) of graphs is there asymptotic normality of Open image in new window . Thanh and Galvin conjectured that if for each n, \(G_n\) is acyclic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that \(G_n\) has no more than \(o(\sqrt{n/\log n})\) components. Here we settle Thanh and Galvin’s conjecture in the affirmative, and significantly extend it, replacing “acyclic” in their conjecture with “co-chromatic with a quasi-threshold graph, and with negligible chromatic number”. Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in the Weyl algebra, and work of Kahn on the matching polynomial of a graph.     

Power boundedness in the Fourier and Fourier–Stieltjes algebras on homogeneous spaces

We study power boundedness in the Fourier and Fourier–Stieltjes algebras, Open image in new window and Open image in new window of a homogeneous space Open image in new window The main results characterizes when all elements with spectral radius at most one, in any of these algebras, are power bounded.     

A weird relation between two cardinals

For a set M, let \({\text {seq}}(M)\) denote the set of all finite sequences which can be formed with elements of M, and let \([M]^2\) denote the set of all 2-element subsets of M. Furthermore, for a set A, let Open image in new window denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory \(\textsf {ZF}\): There exists a set M such that Open image in new window and no function Open image in new window is finite-to-one.     

Monitoring the process mean based on attribute inspection when a small sample is available

This paper proposes a new control chart, denoted by Open image in new window to evaluate the stability of a process mean when a small sample is available. This chart is based on attribute inspection rather than the physical measurements (taken with an instrument, such as a caliper or precision balance) of the quality characteristics of interest of the sampled items. The main goal is to recover measurements on a continuous scale by generating random measurements using the frequencies observed for the sample as inputs. The average sample obtained using these recovery measures ( Open image in new window ) is calculated and used to draw the standard Open image in new window chart. The average sample Open image in new window can be shown to be a mixture of normal distributions. The values of the lower control limit (LCL) and the upper control limit (UCL) are chosen to minimize the average run length (ARL).     

<Emphasis Type="Italic">C</Emphasis>*-Homomorphisms and duality of <Emphasis Type="Italic">C</Emphasis>*-discrete quantum groups

Let

Open image in new window

be a C*-discrete quantum group and let

Open image in new window

be the discrete quantum group associated with

Open image in new window

. Suppose that there exists a continuous action of

Open image in new window

on a unital C*-algebra

Open image in new window

so that

Open image in new window

becomes a

Open image in new window

-algebra. If there is a faithful irreducible vacuum representation π of

Open image in new window

on a Hilbert space H =

Open image in new window

with a vacuum vector Ω, which gives rise to a

Open image in new window

-invariant state, then there is a unique C*-representation (θ, H) of

Open image in new window

supplemented by the action. The fixed point subspace of

Open image in new window

under the action of

Open image in new window

is exactly the commutant of θ(

Open image in new window

).

     

The Parseval equality for multiple Fourier-Stieltjes series with respect to the Haar system

Let a function f : \(\Pi ^{ * ^m } \) → ? be Lebesgue integrable on \(\Pi ^{ * ^m } \) and Riemann-Stieltjes integrable with respect to a function G : \(\Pi ^{ * ^m } \) → ? on \(\Pi ^{ * ^m } \). Then the Parseval equality

Open image in new window

holds, where

Open image in new window

(k) = (f, χk) = (L)

Open image in new window

f(x)χk(x) dx and \(\widehat{dG}\)(k) =

Open image in new window

χk(x) dG(x) are Fourier coefficients of the function f and Fourier-Stieltjes coefficients of the function G with respect to the Haar system, respectively; the integrals in the equality and in the definition of the coefficients of the function G are the Riemann-Stieltjes integrals; the series in the right-hand side of the equality converges in the sense of rectangular partial sums; and the overline indicates the complex conjugation. If f : Π ^m → ? is a complex-valued Lebesgue integrable function, G is a complex-valued function of bounded variation on Π ^m ,

Open image in new window

are Fourier-Lebesgue coefficients of the function f _x(t) = f(x ⊕ t), where ⊕ is the group addition, then the Parseval equality

Open image in new window

holds for almost all x ∈ \(\Pi ^{ * ^m } \) in the sense of any summation method with respect to which the Fourier series of Lebesgue integrable functions are summable to these functions almost everywhere (the integral here is interpreted in the sense of Lebesgue-Stieltjes).

     

Geodesic mapping onto Kählerian spaces of the first kind

In the present paper a generalized Kählerian space Open image in new window of the first kind is considered as a generalized Riemannian space \(\mathbb{G}\mathbb{R}_N \) with almost complex structure F _i ^h that is covariantly constant with respect to the first kind of covariant derivative.Using a non-symmetric metric tensor we find necessary and sufficient conditions for geodesic mappings f: Open image in new window with respect to the four kinds of covariant derivatives. These conditions have the form of a closed system of partial differential equations in covariant derivatives with respect to unknown components of the metric tensor and the complex structure of the Kählerian space Open image in new window .     

Strongly Gauduchon Spaces

The authors define strongly Gauduchon spaces and the class■■, which are generalization of strongly Gauduchon manifolds in complex spaces. Comparing with the case of Kahlerian, the strongly Gauduchon space and the class■are similar to the Kahler space and the Fujiki class■■ respectively. Some properties about these complex spaces are obtained. Moreover, the relations between the strongly Gauduchon spaces and the class■■ are studied.     

A WIN-WIN approach to multiple objective linear programming problems

Interactive decision making arose as a means to overcome the uncertainty concerning the decision maker's (DM) value function. So far the search is confined to nondominated alternatives, which assumes that a win–lose strategy is adopted. The purpose of this paper is to suggest a complementary interactive algorithm that uses an interior point method to solve multiple objective linear programming problems. As the algorithm proceeds, the DM has access to intermediate solutions. The sequence of intermediate solutions has a very interesting characteristic: all of the criteria are improved, that is, a solution Open image in new window , that follows another solution Open image in new window , has the values of all objectives greater than those of Open image in new window . This WIN-WIN feature allows the DM to reach a nondominated solution without making any trade-off among the objective functions. However, there is no impediment in proceeding with traditional multiobjective methods.     

Commuting variety of Witt algebra

Let \(\mathfrak{g} = W_1 \) be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let Open image in new window be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety Open image in new window is reducible, and not equidimensional. Irreducible components of Open image in new window and their dimensions are precisely given. As a consequence, the variety Open image in new window is not normal.     

AFFINE EXTENSIONS OF PRINCIPAL ADDITIVE BUNDLES OVER A PUNCTURED SURFACE

The aim of this article is to make a first step towards the classification of complex normal affine

Open image in new window

_α -threefolds X. We consider the case where the restriction of the quotient morphism π: X → S to π^?1 (S _* ), where S _* denotes the complement of some regular closed point in S, is a principal

Open image in new window

_α -bundle. The variety SL₂ will be of special interest and a source of many examples. It has a natural right

Open image in new window

_α -action such that the quotient morphism SL₂ →

Open image in new window

² restricts to a principal

Open image in new window

_α -bundle over the punctured plane

Open image in new window

.

     

Nonseparable closed vector subspaces of separable topological vector spaces

In 1983 P. Domański investigated the question: For which separable topological vector spaces E, does the separable space Open image in new window have a nonseparable closed vector subspace, where \(\hbox {c}\) is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space Open image in new window has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line \(\mathbb M\) the space of all continuous real-valued functions on \(\mathbb M\) endowed with the pointwise convergence topology, \(C_p(\mathbb M)\) contains a nonseparable closed vector subspace while \(C_p(\mathbb M)\) is separable.     

Statistical simulation of one type of pairs of random variables with the use of fictitious jumps

Statistical simulation based on the sampling technique for a pair of random variables (T,

Open image in new window

), where T ∈ [0, +∞) and

Open image in new window

∈ ? ^d (d ≥ 1) is considered. The simultaneous distribution of the pair is specified in the form that is common for analogous problems in various fields. It has the form

Open image in new window

, where f is a function and m is a measure. The first variable T is the well-known random waiting time. A simulation method for the pair (T,

Open image in new window

) is constructed using a realization of an auxiliary Markov sequence of trial pairs. Applications of this method in particle transport theory and in kinetics of rarefied gases are discussed.

     

Weakly infinite-dimensional spaces modulo simplicial complexes

The classes of spaces

Open image in new window

-wid and ?-wid are introduced for the class

Open image in new window

of finite simplicial complexes and the class ? of compact polyhedra. If

Open image in new window

, then

Open image in new window

-wid = wid, ?-wid = S-wid. It is proved that S-wid ? ?-wid and ?-wid = S-? _τ -wid for any triangulation τ of the class ?.