Second-Order Optimality Conditions for Constrained Domain Optimization

This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ∫ _Ω g(x) dx=1, a domain is sought which maximizes either , fixed x ₀∈Ω, or ℱ(Ω)=∫ _Ω F(x,u(x)) dx, where u solves the Dirichlet problem Δu(x)=−f(x), x∈Ω, u(x)=0, x∈∂Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the cost functionals and ℱ. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Singularity Spectrum of Generic ��-H?lder Regular Functions After Time Subordination

A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by B₁^a{\mathcal {B}}_{1}^{\alpha} the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function g ? B₁^ag\in {\mathcal {B}}_{1}^{\alpha} one obtains a time subordination of g simply by considering the composite function Z=g○f, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space E^a=M×B₁^a\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha} equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of g○f are investigated for a generic (typical) element (f,g)∈ℰ ^α . In particular we determine the generic H?lder singularity spectrum of g○f.     

Deformations of circle-valued Morse functions on surfaces

Let M be a smooth connected orientable compact surface and let F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) be a space of all Morse functions f : M → S ¹ without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S ¹ is either a constant map or a covering map. The space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) is endowed with the C ^∞-topology. We present the classification of connected components of the space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) . This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on ∂M.     

Optimal interpolation of convergent algebraic series

Let , –1<x ₁<...<x _n <1. Denote , t∈(–1,1). Given a function f∈W we try to recover f(ζ) at fixed point ζ∈(–1,1) by an algorithm A on the basis of the information f(x ₁),...,f(x _n ). We find the intrinsic error of recovery . This work is supported by RFBR (grant 07-01-00167-a and grant 06-01-00003).     

Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

Characterization of $ {\mathbb{A}_{16}} $ by a noncommuting graph

Let G be a finite non-Abelian group. We define a graph Γ _G ; called the noncommuting graph of G; with a vertex set G − Z(G) such that two vertices x and y are adjacent if and only if xy ≠ yx: Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite non-Abelian simple group and G is a group such that Γ _S ≅ Γ _G ; then S ≅ G: It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except \mathbbA₁₀ {\mathbb{A}_{10}} , L ₄(8), L ₄(4), and U ₄(4). In this paper, we prove that if \mathbbA₁₆ {\mathbb{A}_{16}} denotes the alternating group of degree 16; then, for any finite group G; the graph isomorphism G_\mathbbA16 @ G_G {\Gamma_{{\mathbb{A}_{16}}}} \cong {\Gamma_G} implies that \mathbbA₁₆ @ G {\mathbb{A}_{16}} \cong G .     

Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n

In this paper, we introduce the subfamilies H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.     

On invariant notions of Segre varieties in binary projective spaces

Invariant notions of a class of Segre varieties S_(m)(2){\mathcal{S}_{(m)}(2)} of PG(2 ^m − 1, 2) that are direct products of m copies of PG(1, 2), m being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains S_(m)(2){\mathcal{S}_{(m)}(2)} and is invariant under its projective stabiliser group G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} . By embedding PG(2 ^m − 1, 2) into PG(2 ^m − 1, 4), a basis of the latter space is constructed that is invariant under G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according as m is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} -invariant geometric spread of lines of PG(2 ^m − 1, 2). This spread is also related with a G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} -invariant non-singular Hermitian variety. The case m = 3 is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} , while the points of PG(7, 2) form five orbits.     

On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D\{ b } ? \mathbbR³ f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR³ D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR³ b \in {\mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(\mathbbR³)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRⁿ g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} such that A ⊂ U.     

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

Multilevel frames for sparse tensor product spaces

For Au = f with an elliptic differential operator and stochastic data f, the m-point correlation function of the random solution u satisfies a deterministic equation with the m-fold tensor product operator A ^(m) of A. Sparse tensor products of hierarchic FE-spaces in are known to allow for approximations to which converge at essentially the rate as in the case m = 1, i.e. for the deterministic problem. They can be realized by wavelet-type FE bases (von Petersdorff and Schwab in Appl Math 51(2):145–180, 2006; Schwab and Todor in Computing 71:43–63, 2003). If wavelet bases are not available, we show here how to achieve the fast computation of sparse approximations of for Galerkin discretizations of A by multilevel frames such as BPX or other multilevel preconditioners of any standard FEM approximation for A. Numerical examples illustrate feasibility and scope of the method.     

The Asymptotic Schottky Problem

Let ${\mathcal{M}_g}Let M_g{\mathcal{M}_g} denote the moduli space of compact Riemann surfaces of genus g and let A_g{\mathcal{A}_g} be the moduli space of principally polarized abelian varieties of dimension g. Let J : M_g ? A_g{J : \mathcal{M}_g \rightarrow \mathcal{A}_g} be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image J_g : = J(M_g){\mathcal{J}_g := J(\mathcal{M}_g)} is contained in a proper subvariety of A_g{\mathcal{A}_g} when g ≥  4. The classical and long-studied Schottky problem is to characterize the Jacobian locus J_g{\mathcal{J}_g} in A_g{\mathcal{A}_g}. In this paper we address a large scale version of this problem posed by Farb and called the coarse Schottky problem: What does J_g{\mathcal{J}_g} look like “from far away”, or how “dense” is J_g{\mathcal{J}_g} in the sense of coarse geometry? The large scale geometry of A_g{\mathcal{A}_g} is encoded in its asymptotic cone, Cone_￥(A_g){{\rm Cone}_\infty(\mathcal{A}_g)}, which is a Euclidean simplicial cone of real dimension g. Our main result asserts that the Jacobian locus J_g{\mathcal{J}_g} is “coarsely dense” in A_g{\mathcal{A}_g}, which implies that the subset of Cone_￥(A_g){{\rm Cone}_\infty(\mathcal{A}_g)} determined by J_g{\mathcal{J}_g} actually coincides with this cone. The proof shows that the Jacobian locus of hyperelliptic curves is coarsely dense in A_g{\mathcal{A}_g} as well. We also study the boundary points of the Jacobian locus J_g{\mathcal{J}_g} in A_g{\mathcal{A}_g} and in the Baily–Borel and the Borel–Serre compactification. We show that for large genus g the set of boundary points of J_g{\mathcal{J}_g} in these compactifications is “small”.     

On Lie derivations of Lie ideals of prime algebras

LetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureA _c and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let be the factor Lie algebra and let δ: be a Lie derivation. Suppose that char(A) ≠ 2 andA does not satisfySt ₁₄, the standard identity of degree 14. We show thatR ΩC =Z(R) and there exists a derivation of algebrasD:A →A _c such that for allx∈R. Our result solves an old problem of Herstein.     

Homogenization with corrector for a multidimensional periodic elliptic operator near an edge of an inner gap

For a second order differential operator A \mathcal{A} _ε  = −div g(x/ε)∇ + ε ⁻²p(x/ε) in L ₂(ℝ ^d ) with periodic coefficients and small parameter ε > 0 we study an approximation of the resolvent of A \mathcal{A} _ε at a point close to an edge of an inner gap in the spectrum of A \mathcal{A} _ε . Under certain regularity conditions, we construct an approximation (with a first order corrector taken into account) for the resolvent with error estimate of order O(ε ²). We show that a proper effective operator and a proper corrector are associated to each (regular) edge of the gap. Bibliography: 14 titles.     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.     

Bounds on the degree of APN polynomials: the case of x ?1?+?g(x)

In this paper we consider APN functions ${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$ of the form f(x) = x ^?1 + g(x) where g is any non ${\mathcal{F}_{2}}$ -affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields ${\mathcal{F}_{2^m}}$ . Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ?? 3 where these functions are equivalent to x ³.     

Homotopy Categories for Simply Connected Torsion Spaces

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups with s ∈ S, or in the abelian group where is the group of fractions of the form with s ∈ S. In the first case, for n > 1 the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations , obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For instance, the map is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces , are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If S is generated by a set P of primes and S ^p is generated by a prime p ∈ P one has that for n > 1 the category is equivalent to the product category . If the multiplicative system S is generated by a finite set of primes, then localized category is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for .     

Regularity of Weak Solutions of Nonlinear Equations with Discontinuous Coefficients

In this paper, we prove that the weak solutions u∈Wloc^1, p （Ω） （1 〈p〈∞） of the following equation with vanishing mean oscillation coefficients A（x）： -div[（A（x）△↓u·△↓u）p-2/2 A（x）△↓u＋│F（x）│^p-2 F（x）]=B（x, u, △↓u）, belong to Wloc^1, q （Ω）（A↓q∈（p, ∞）, provided F ∈ Lloc^q（Ω） and B（x, u, h） satisfies proper growth conditions where Ω ∪→R^N（N≥2） is a bounded open set, A（x）=（A^ij（x）） N×N is a symmetric matrix function.     

Arithmetic of semigroups of series in multiplicative systems

We study the arithmetic of a semigroup M_P\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?_{n = 0}^￥ a_nc_n(x)    ( a_n 3 0,?_{n = 0}^￥ a_n = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { c_n }_{n = 0}^￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M_P\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R ⁿ are true. We describe the class I₀(M_P)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.