An adaptive radial basis algorithm (ARBF) for expensive black-box global optimization

Powerful response surface methods based on kriging and radial basis function (RBF) interpolation have been developed for expensive, i.e. computationally costly, global nonconvex optimization. We have implemented some of these methods in the solvers rbfSolve and EGO in the TOMLAB Optimization Environment (http://www.tomopt.com/tomlab/). In this paper we study algorithms based on RBF interpolation. The practical performance of the RBF algorithm is sensitive to the initial experimental design, and to the static choice of target values. A new adaptive radial basis interpolation (ARBF) algorithm, suitable for parallel implementation, is presented. The algorithm is described in detail and its efficiency is analyzed on the standard test problem set of Dixon–Szegö. Results show that it outperforms the published results of rbfSolve and several other solvers.     

The sum of digits of primes in $${\mathbb{Z}}$$[i]

We study the distribution of the complex sum-of-digits function s _q with basis q = –a±i, \({a \in \mathbb{Z}^+}\) for Gaussian primes p. Inspired by a recent result of Mauduit and Rivat (http://iml.univ-mrs.fr/~rivat/publications.html) for the real sum-of-digits function, we here get uniform distribution modulo 1 of the sequence (αs _q (p)) provided \({\alpha \in \mathbb{R} \setminus \mathbb{Q}}\) and q is prime with a ≥ 28. We also determine the order of magnitude of the number of Gaussian primes whose sum-of-digits evaluation lies in some fixed residue class mod m.     

<Emphasis Type="Italic">um</Emphasis>-Topology in multi-normed vector lattices

Let \({\mathcal {M}}=\{m_\lambda \}_{\lambda \in \Lambda }\) be a separating family of lattice seminorms on a vector lattice X, then \((X,{\mathcal {M}})\) is called a multi-normed vector lattice (or MNVL). We write \(x_\alpha \xrightarrow {\mathrm {m}} x\) if \(m_\lambda (x_\alpha -x)\rightarrow 0\) for all \(\lambda \in \Lambda \). A net \(x_\alpha \) in an MNVL \(X=(X,{\mathcal {M}})\) is said to be unbounded m-convergent (or um-convergent) to x if \(|x_\alpha -x |\wedge u \xrightarrow {\mathrm {m}} 0\) for all \(u\in X_+\). um-Convergence generalizes un-convergence (Deng et al. in Positivity 21:963–974, 2017; Kandi? et al. in J Math Anal Appl 451:259–279, 2017) and uaw-convergence (Zabeti in Positivity, 2017. doi: 10.1007/s11117-017-0524-7), and specializes up-convergence (Ayd?n et al. in Unbounded p-convergence in lattice-normed vector lattices. arXiv:1609.05301) and \(u\tau \)-convergence (Dabboorasad et al. in \(u\tau \)-Convergence in locally solid vector lattices. arXiv:1706.02006v3). um-Convergence is always topological, whose corresponding topology is called unbounded m-topology (or um-topology). We show that, for an m-complete metrizable MNVL \((X,{\mathcal {M}})\), the um-topology is metrizable iff X has a countable topological orthogonal system. In terms of um-completeness, we present a characterization of MNVLs possessing both Lebesgue’s and Levi’s properties. Then, we characterize MNVLs possessing simultaneously the \(\sigma \)-Lebesgue and \(\sigma \)-Levi properties in terms of sequential um-completeness. Finally, we prove that every m-bounded and um-closed set is um-compact iff the space is atomic and has Lebesgue’s and Levi’s properties.     

A Polynomial Bound for Untangling Geometric Planar Graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002) asked if every n-vertex geometric planar graph can be untangled while keeping at least n ^ε vertices fixed. We answer this question in the affirmative with ε=1/4. The previous best known bound was \(\Omega(\sqrt{\log n/\log\log n})\). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least \(\Omega(\sqrt{n})\) vertices fixed, while the best upper bound was \(\mathcal{O}((n\log n)^{2/3})\). We answer a question of Spillner and Wolff (http://arxiv.org/abs/0709.0170) by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than \(3(\sqrt{n}-1)\) vertices fixed.     

Domination problem for AM-compact abstract Uryson operators

The Boolean algebra of fragments of a positive abstract Uryson operator recently was described in M. Pliev (Positivity, doi:10.1007/s11117-016-0401-9, 2016). Using this result, we prove a theorem of domination for AM-compact positive abstract Uryson operators from a Dedekind complete vector lattice E to a Banach lattice F with an order continuous norm.     

Congruences between word length statistics for the finitary alternating and symmetric groups

Bacher and de la Harpe (arxiv:1603.07943, 2016) study conjugacy growth series of infinite permutation groups and their relationships with p(n), the partition function, and \(p(n)_\mathbf{e }\), a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory due to Bacher and de la Harpe (arxiv:1603.07943, 2016) also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function p(n) and Atkin’s generalization to the k-colored partition function \(p_{k}(n)\), we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes \(\ell \ge 5\).     

Implementing Brouwer’s database of strongly regular graphs

Andries Brouwer maintains a public database of existence results for strongly regular graphs on \(n\le 1300\) vertices. We have implemented most of the infinite families of graphs listed there in the open-source software Sagemath (The Sage Developers, http://www.sagemath.org), as well as provided constructions of the “sporadic” cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of n.     

Eigenvalues of the complex Laplacian on compact non-Kähler manifolds

We consider \(\lambda \) is the principle eigenvalue of the complex Laplacian on a compact Hermitian manifold M. We prove that \(\lambda \ge C\) where C depends only on the dimension n, the diameter d, the Ricci curvature of the Levi-Civita connection on M, and a norm, expressed in curvature, that determines how much M fails to be Kähler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng (on curvature tensors of Hermitian manifolds, 2016. arXiv:1602.01189). We combine these results to obtain the main estimate. We also discuss several special cases in which one can obtain a lower bound solely in terms of the Riemannian geometry.     

Quasi-periodic water waves

We present the result and the ideas of the recent paper (Berti and Montalto, Quasi-periodic standing wave solutions of gravity-capillary water waves, http://arxiv.org/abs/1602.02411, 2016) concerning the existence of Cantor families of small-amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean, with infinite depth, in irrotational regime, under the action of gravity and surface tension at the free boundary. These quasi-periodic solutions are linearly stable.     

Classification of affine symmetry groups of orbit polytopes

Let G be a finite group acting linearly on a vector space V. We consider the linear symmetry groups \({\text {GL}}(Gv)\) of orbits \(Gv\subseteq V\), where the linear symmetry group \({\text {GL}}(S)\) of a subset \(S\subseteq V\) is defined as the set of all linear maps of the linear span of S which permute S. We assume that V is the linear span of at least one orbit Gv. We define a set of generic points in V, which is Zariski open in V, and show that the groups \({\text {GL}}(Gv)\) for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group \({\text {GL}}(Gw)\) such that V is the linear span of Gw. If the underlying characteristic is zero, “isomorphic” can be replaced by “conjugate in \({\text {GL}}(V)\).” Moreover, in the characteristic zero case, we show how the character of G on V determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (Geom Dedicata 6(3):331–337, 1977.  https://doi.org/10.1007/BF02429904).     

A BKR Operation for Events Occurring for Disjoint Reasons with High Probability

Given events A and B on a product space \(S={\prod }_{i = 1}^{n} S_{i}\), the set \(A \Box B\) consists of all vectors x = (x₁,…,x_n) ∈ S for which there exist disjoint coordinate subsets K and L of {1,…,n} such that given the coordinates x_i,i ∈ K one has that x ∈ A regardless of the values of x on the remaining coordinates, and likewise that x ∈ B given the coordinates x_j,j ∈ L. For a finite product of discrete spaces endowed with a product measure, the BKR inequality

$$ P(A \Box B) \le P(A)P(B) $$

(1)

was conjectured by van den Berg and Kesten (J Appl Probab 22:556–569, 1985) and proved by Reimer (Combin Probab Comput 9:27–32, 2000). In Goldstein and Rinott (J Theor Probab 20:275–293, 2007) inequality Eq. 1 was extended to general product probability spaces, replacing \(A \Box B\) by the set Open image in new window consisting of those outcomes x for which one can only assure with probability one that x ∈ A and x ∈ B based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality Eq. 1 results, due to the fact that Open image in new window . In particular, it may be the case that \(A \Box B\) is empty, while Open image in new window is not. We propose the further extension Open image in new window depending on probability thresholds s and t, where Open image in new window is the special case where both s and t take the value one. The outcomes Open image in new window are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.

     

Character sums with smooth numbers

We use the large sieve inequality for smooth numbers due to Drappeau et al. (Smooth-supported multiplicative functions in arithmetic progressions beyond the \(x^{1/2}\)-barrier, Preprint, 2017. arXiv:1704.04831), together with some other arguments, to improve their bounds on the frequency of pairs \((q,\chi )\) of moduli q and primitive characters \(\chi \) modulo q, for which the corresponding character sums with smooth numbers are large.     

Super duality and crystal bases for quantum ortho-symplectic superalgebras II

Let \(\mathcal {O}^\mathrm{int}_q(m|n)\) be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type B, C, D introduced in Kwon (Int Math Res Not, 2015. doi: 10.1093/imrn/rnv076). It is a natural counterpart of the category of finitely dominated integrable modules over a quantum group of type B, C, D from a viewpoint of super duality. Continuing the previous work on type B and C (Kwon in Int Math Res Not, 2015. doi: 10.1093/imrn/rnv076), we classify the irreducible modules in \(\mathcal {O}^\mathrm{int}_q(m|n)\) and prove the existence and uniqueness of their crystal bases in case of type D. A new combinatorial model of classical crystals of type D is introduced, whose super analog gives a realization of crystals for the highest weight modules in \(\mathcal {O}^\mathrm{int}_q(m|n)\).     

Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety

Let \({\textnormal {R}}\) be a real closed field, \(\mathcal{P},\mathcal{Q} \subset {\textnormal {R}}[X_{1},\ldots,X_{k}]\) finite subsets of polynomials, with the degrees of the polynomials in \(\mathcal{P}\) (resp., \(\mathcal{Q}\)) bounded by d (resp., d ₀). Let \(V \subset {\textnormal {R}}^{k}\) be the real algebraic variety defined by the polynomials in \(\mathcal{Q}\) and suppose that the real dimension of V is bounded by k′. We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family \(\mathcal{P}\) on V is bounded by

$\sum_{j=0}^{k'}4^j{s +1\choose j}F_{d,d_0,k,k'}(j),$

where \(s = \operatorname {card}\mathcal{P}\), and

$F_{d,d_0,k,k'}(j)=\binom{k+1}{k-k'+j+1} (2d_0)^{k-k'}d^j \max\{2d_0,d \}^{k'-j}+2(k-j+1).$

In case 2d ₀≤d, the above bound can be written simply as

$\sum_{j = 0}^{k'} {s+1 \choose j}d^{k'} d_0^{k-k'} O(1)^{k}= (sd)^{k'} d_0^{k-k'} O(1)^k$

(in this form the bound was suggested by Matousek 2011). Our result improves in certain cases (when d ₀?d) the best known bound of

$\sum_{1 \leq j \leq k'}\binom{s}{j} 4^{j} d(2d-1)^{k-1}$

on the same number proved in Basu et al. (Proc. Am. Math. Soc. 133(4):965–974, 2005) in the case d=d ₀.

The distinction between the bound d ₀ on the degrees of the polynomials defining the variety V and the bound d on the degrees of the polynomials in \(\mathcal{P}\) that appears in the new bound is motivated by several applications in discrete geometry (Guth and Katz in arXiv:1011.4105v1 [math.CO], 2011; Kaplan et al. in arXiv:1107.1077v1 [math.CO], 2011; Solymosi and Tao in arXiv:1103.2926v2 [math.CO], 2011; Zahl in arXiv:1104.4987v3 [math.CO], 2011).     

A binomial Laurent phenomenon algebra associated with the complete graph

In this paper, we find the exchange graph of \(\mathcal {A}({\tau _n})\), the rank n binomial Laurent phenomenon algebra associated with the complete graph \(K_n\). More specifically, we prove that the exchange graph is isomorphic to that of \(\mathcal {A}(t_n)\), a rank n linear Laurent phenomenon algebra associated with the complete graph which is discussed in Lam and Pylyavskyy (Linear Laurent phenomenon algebras, arXiv:1206.2612v2, 2012).     

Heinz mean curvature estimates in warped product spaces $$M\times _{e^{\psi }}N$$

If a graph submanifold (x, f(x)) of a Riemannian warped product space \((M^m\times _{e^{\psi }}N^n,\tilde{g}=g+ e^{2\psi }h)\) is immersed with parallel mean curvature H, then we obtain a Heinz-type estimation of the mean curvature. Namely, on each compact domain D of M, \(m\Vert H\Vert \le \frac{A_{\psi }(\partial D)}{V_{\psi }(D)}\) holds, where \(A_{\psi }(\partial D)\) and \(V_{\psi }(D)\) are the \({\psi }\)-weighted area and volume, respectively. In particular, \(H=0\) if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.01257, 2016). This generalizes the known cases \(n=1\) or \(\psi =0\). We also conclude minimality using a closed calibration, assuming \((M,g_*)\) is complete where \(g_*=g+e^{2\psi }f^*h\), and for some constants \(\alpha \ge \delta \ge 0\), \(C_1>0\) and \(\beta \in [0,1)\), \(\Vert \nabla ^*\psi \Vert ^2_{g_*}\le \delta \), \(\mathrm {Ricci}_{\psi ,g_*}\ge \alpha \), and \({\mathrm{det}}_g(g_*)\le C_1 r^{2\beta }\) holds when \(r\rightarrow +\infty \), where r(x) is the distance function on \((M,g_*)\) from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.     

Continued fractions for square series generating functions

We consider new series expansions for variants of the so-termed ordinary geometric square series generating functions originally defined in the recent article titled “Square Series Generating Function Transformations” (arXiv:1609.02803). Whereas the original square series transformations article adapts known generating function transformations to construct integral representations for these square series functions enumerating the square powers of \(q^{n^2}\) for some fixed non-zero q with \(|q| < 1\), we study the expansions of these special series through power series generated by Jacobi-type continued fractions, or J-fractions. We prove new exact expansions of the hth convergents to these continued fraction series and show that the limiting case of these convergent generating functions exists as \(h \rightarrow \infty \). We also prove new infinite q-series representations of special square series expansions involving square-power terms of the series parameter q, the q-Pochhammer symbol, and double sums over the q-binomial coefficients. Applications of the new results we prove within the article include new q-series representations for the ordinary generating functions of the special sequences, \(r_p(n)\), and \(\sigma _1(n)\), as well as parallels to the examples of the new integral representations for theta functions, series expansions of infinite products and partition function generating functions, and related unilateral special function series cited in the first square series transformations article.     

Algebraic Hamiltonian actions

In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X. We study the quotient morphism \({\mu_{G,X}//G : X//G \rightarrow \mathfrak{g} //G}\) of the moment map \({\mu_{G,X} : X\rightarrow \mathfrak{g}}\) . We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μ _G,X //G have the same dimension. We also study the “Stein factorization” of μ _G,X //G. Namely, let C _G,X denote the spectrum of the integral closure of \({\mu_{G,X}^{*}(\mathbb{K}[\mathfrak{g}]^G)}\) in \({\mathbb{K}(X)^G}\) . We investigate the structure of the \({\mathfrak{g} //G}\) -scheme C _G,X . Our results partially generalize those obtained by F. Knop for the actions on cotangent bundles and symplectic vector spaces.     

The cohomology of orbit spaces of certain free circle group actions

Suppose that \(G =\mathbb{S}^1\) acts freely on a finitistic space X whose (mod p) cohomology ring is isomorphic to that of a lens space \(L^{2m-1}(p;q_1,\ldots,q_m)\) or \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). The mod p index of the action is defined to be the largest integer n such that α ⁿ ?≠?0, where \(\alpha \,\epsilon\, H^2(X/G;\mathbb{Z}_p)\) is the nonzero characteristic class of the \(\mathbb{S}^1\)-bundle \(\mathbb{S}^1\hookrightarrow X\rightarrow X/G\). We show that the mod p index of a free action of G on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\) is p???1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free G-action on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). It is note worthy that the mod p index for free G-actions on the cohomology lens space is not defined.     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).