Lines Avoiding Balls in Three Dimensions Revisited

Let $\mathcal{B}$ be a collection of n arbitrary balls in ?³. We establish an almost-tight upper bound of O(n ^3+?? ), for any ??>0, on the complexity of the space $\mathcal{F}(\mathcal{B})$ of all the lines that avoid all the members of $\mathcal{B}$ . In particular, we prove that the balls of $\mathcal{B}$ admit O(n ^3+?? ) free isolated tangents, for any ??>0. This generalizes the result of Agarwal et al.?(Discrete Comput. Geom. 34:231?C250, 2005), who established this bound only for congruent balls, and solves an open problem posed in that paper. Our bound almost meets the recent lower bound of ??(n ³) of Glisse and Lazard (Proc. 26th Annu. Symp. Comput. Geom., pp. 48?C57, 2010). Our approach is constructive and yields an algorithm that computes the discrete representation of the boundary of $\mathcal{F}(B)$ in O(n ^3+?? ) time, for any ??>0.     

Some Recent Developments on Hopf��s Holomorphic Form

Hopf??s theorem on surfaces in ${\mathbb{R}^3}$ with constant mean curvature (Hopf in Math Nach 4:232?C249, 1950-51) was a turning point in the study of such surfaces. In recent years, Hopf-type theorems appeared in various ambient spaces, (Abresch and Rosenberg in Acta Math 193:141?C174, 2004 and Abresch and Rosenberg in Mat Contemp Sociedade Bras Mat 28:283-298, 2005). The simplest case is the study of surfaces with parallel mean curvature vector in ${M_k^n \times \mathbb{R}, n \ge 2}$ , where ${M_k^n}$ is a complete, simply-connected Riemannian manifold with constant sectional curvature k ?? 0. The case n?=?2 was solved in Abresch and Rosenberg 2004. Here we describe some new results for arbitrary n.     

Generating sets of finite singular transformation semigroups

J.M. Howie proved that $\operatorname {Sing}_{n}$ , the semigroup of all singular mappings of {1,…,n} into itself, is generated by its idempotents of defect 1 (in J. London Math. Soc. 41, 707–716, 1966). He also proved that if n≥3 then a minimal generating set for $\operatorname {Sing}_{n}$ contains n(n?1)/2 transformations of defect 1 (in Gomes and Howie, Math. Proc. Camb. Philos. Soc. 101. 395–403, 1987). In this paper we find necessary and sufficient conditions for any set for transformations of defect 1 in $\operatorname {Sing}_{n}$ to be a (minimal) generating set for $\operatorname {Sing}_{n}$ .     

Infinite Families of Pellian Polynomials and Their Continued Fraction Expansions

We investigate families $ \lbrace D_k(X)\rbrace_{k\in{\rm N}} $ of quadratic integral polynomials and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of $ \sqrt {D_k(X)} $ is constant. Furthermore, we show that the period lengths of $ \sqrt {D_k(X)} $ go to infinity with k. For each member of the families involved, we show how to easily determine the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction expansion of $ \sqrt {D_k(X)} $ is related to that of $ \sqrt {C} $ . This continues work in [3]-[5].     

Quadratic functions with prescribed spectra

We study a class of quadratic p-ary functions ${{\mathcal{F}}_{p,n}}$ from ${\mathbb{F}_{p^n}}$ to ${\mathbb{F}_p, p \geq 2}$ , which are well-known to have plateaued Walsh spectrum; i.e., for each ${b \in \mathbb{F}_{p^n}}$ the Walsh transform ${\hat{f}(b)}$ satisfies ${|\hat{f}(b)|^2 \in \{ 0, p^{(n+s)}\}}$ for some integer 0 ≤ s ≤ n ? 1. For various types of integers n, we determine possible values of s, construct ${{\mathcal{F}}_{p,n}}$ with prescribed spectrum, and present enumeration results. Our work generalizes some of the earlier results, in characteristic two, of Khoo et. al. (Des Codes Cryptogr, 38, 279–295, 2006) and Charpin et al. (IEEE Trans Inf Theory 51, 4286–4298, 2005) on semi-bent functions, and of Fitzgerald (Finite Fields Appl 15, 69–81, 2009) on quadratic forms.     

Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.     

A proof of a theorem by Fried and MacRae and applications to the composition of polynomial functions

Fried and MacRae (Math. Ann. 180, 220?C226 (1969)) proved that for univariate polynomials ${p,q, f, g \in \mathbb{K}[t]}$ ( ${\mathbb{K}}$ a field) with p, q nonconstant, p(x) ? q(y) divides f(x) ? g(y) in ${\mathbb{K}[x,y]}$ if and only if there is ${h \in \mathbb{K}[t]}$ such that f?=?h(p(t)) and g?=?h(q(t)). Schicho (Arch. Math. 65, 239?C243 (1995)) proved this theorem from the viewpoint of category theory, thereby providing several generalizations to multivariate polynomials. In the present note, we give a new proof of one of these generalizations. The theorem by Fried and MacRae yields a way to prove the following fact for nonconstant functions f, g from ${\mathbb{C}}$ to ${\mathbb{C}}$ : if both the composition ${f \circ g}$ and g are polynomial functions, then f has to be a polynomial function as well. We give an algebraic proof of this fact and present a generalization to multivariate polynomials over algebraically closed fields. This provides a way to prove a generalization of a result by Carlitz (Acta Sci. Math. (Szeged) 24, 196?C203 (1963)) that describes those univariate polynomials over finite fields that induce bijective functions on all of their finite extensions.     

Modular Decomposition of the Orlik-Terao Algebra

Let ${\mathcal{A}}$ be a collection of n linear hyperplanes in ${\mathbb{k}^\ell}$ , where ${\mathbb{k}}$ is an algebraically closed field. The Orlik-Terao algebra of ${\mathcal{A}}$ is the subalgebra ${{\rm R}(\mathcal{A})}$ of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of ${\mathcal{A}}$ . It determines an irreducible subvariety ${Y (\mathcal{A})}$ of ${\mathbb{P}^{n-1}}$ . We show that a flat X of ${\mathcal{A}}$ is modular if and only if ${{\rm R}(\mathcal{A})}$ is a split extension of the Orlik-Terao algebra of the subarrangement ${\mathcal{A}_X}$ . This provides another refinement of Stanley’s modular factorization theorem [34] and a new characterization of modularity, similar in spirit to the fibration theorem of [27]. We deduce that if ${\mathcal{A}}$ is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.     

The second variational formula for Willmore submanifolds in Sn

In [17] the third author presented Moebius geometry for sub-manifolds in Sⁿ and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces $ \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace $ in S^m+1 (which can be obtained by exchanging radii in the Clifford tori $ S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})$ and show that they are stable Willmore hypersurfaces. In case of surfaces in S³, the stability of the Clifford torus $ S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} $ was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sⁿ.     

An example related to Gregory’s Theorem

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π ₁ ¹ set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model.     

Topological Aspects of Differential Chains

In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus??the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or? $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U?M, but not generally in the strong dual topology of? $\mathcal{B}'(U)$ .     

How Close is the Sample Covariance Matrix to?the?Actual Covariance Matrix?

Given a probability distribution in ? ⁿ with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N=N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose that the distribution is supported in a centered Euclidean ball of radius $O(\sqrt{n})$ . We conjecture that the optimal sample size is N=O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson (J. Funct. Anal. 164:60?C72, 1999), which states that N=O(nlog?n) for distributions with finite second moment, and a recent result of Adamczak et al. (J. Am. Math. Soc. 234:535?C561, 2010), which guarantees that N=O(n) for subexponential distributions.     

Vector partition functions and generalized dahmen and micchelli spaces

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory. Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space $ \mathcal{F}(X) $ contains the partition function $ {\mathcal{P}_{(X)}} $ . We prove a “localization formula” for any f in $ \mathcal{F}(X) $ , inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function $ {\mathcal{P}_{(X)}} $ is a quasi-polynomial on the Minkowski differences $ \mathfrak{c} - B(X) $ , where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.     

Blocks with Defect Group \boldsymbol{Q_{2^n}\times C_{2^m}} and \boldsymbol{SD_{2^n}\times C_{2^m}}

We determine the numerical invariants of blocks with defect group $Q_{2^n}\times C_{2^m}$ and $SD_{2^n}\times C_{2^m}$ , where $Q_{2^n}$ denotes a quaternion group of order 2 ⁿ , $C_{2^m}$ denotes a cyclic group of order 2 ^m , and $SD_{2^n}$ denotes a semidihedral group of order 2 ⁿ . This generalizes Olsson’s results for m?=?0. As a consequence, we prove Brauer’s k(B)-Conjecture, Olsson’s Conjecture, Brauer’s Height-Zero Conjecture, the Alperin–McKay Conjecture, Alperin’s Weight Conjecture and Robinson’s Ordinary Weight Conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper follows (and uses) (Sambale, J Pure Appl Algebra 216:119–125, 2012; Proc Amer Math Soc, 2012).     

Geometric Sufficient Conditions for Compactness of the Complex Green Operator

We establish compactness estimates for $\overline{\partial}_{M}$ on a compact pseudoconvex CR-submanifold M of ? ⁿ of hypersurface type that satisfies the (analogue of the) geometric sufficient conditions for compactness of the $\overline{\partial}$ -Neumann operator given in (Straube in Ann. Inst. Fourier, 54(3):699?C710, 2004; Munasinghe and Straube in Pac. J. Math., 232(2):343?C354 2007). These conditions are formulated in terms of certain short time flows in complex tangential directions.     

On locally bounded above solutions of an equation of the Goła̧b–Schinzel type

We characterize solutions ${f, g : \mathbb{R} \to \mathbb{R}}$ of the functional equation f(x + g(x)y) = f(x)f(y) under the assumption that f is locally bounded above at each point ${x \in \mathbb{R}}$ . Our result refers to Go?a?b and Schinzel (Publ Math Debr 6:113–125, 1959) and Wo?od?ko (Aequationes Math 2:12–29, 1968).     

Simplicial complexes with rigid depth

We extend a result of Minh and Trung (Adv. Math. 226:1285–1306, 2011) to get criteria for depth ${I = \rm {depth}\sqrt{I}}$ , where I is an unmixed monomial ideal of the polynomial ring S?=?K[x ₁, . . . , x _n ]. As an application we characterize all the pure simplicial complexes Δ which have rigid depth, that is, which satisfy the condition that for every unmixed monomial ideal ${I\subset S}$ with ${\sqrt{I}=I_\Delta}$ one has depth(I)?=?depth(I _Δ).     

Changing-sign bubble solutions for an anisotropic sinh-Poisson equation

We consider the following anisotropic sinh-Poisson equation $${\rm div} (a(x) \nabla u)+ 2\varepsilon^2 a(x) {\rm sinh}\,u=0\ \ {\rm in}\ \Omega, \quad u=0 \ \ {\rm on}\ \partial \Omega,$$ where ${\Omega \subset \mathbb{R}^2}$ is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient ${a(x)}$ on the existence of bubbling solutions. We show that there exists a family of solutions u _?? concentrating positively and negatively at ${\bar{x}}$ , a given local critical point of a(x), for ?? sufficiently small, for which with the property $$2\varepsilon^2a(x){\rm sinh} u_\varepsilon \rightharpoonup 8\pi\sum\limits_{j=1}^{m}b_j\delta_{\bar{x}},$$ where ${b_j=\pm 1}$ . This result shows a striking difference with the isotropic case (a(x) ?? Constant) in Bartolucci and Pistoia (IMA J Appl Math 72(6):706?C729, 2007), Jost et?al. (Calc Var Partial Differ Equ 31:263?C276, 2008) and Esposito and Wei (Calc Var Partial Differ Equ 34:341?C375, 2009).