On schur rings over cyclic groups

In this paper, we introduce the notion of wedge product of Schur rings. We show that for any nontrivial Schur ringS over a cyclic groupG, if there is a subgroupH such that Σ _g ε _H g Σ _g∈H g ∉S, thenS is either a dot product or wedge product for some Schur rings over smaller cyclic groups.     

Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in

In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the black-box polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form f([`(x)]) = ?<sub>i = 1</sub><sup>k</sup> h<sub>i</sub> ([`(x)]) ·g_i ([`(x)])f(\bar x) = \sum\nolimits_{i = 1}^k {h_i (\bar x) \cdot g_i (\bar x)} , where each h _i is a polynomial that depends on only ρ linear functions, and each g _i is a product of linear functions (when h _i = 1, for each i, then we get the class of depth-3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When max _i (deg(h _i · g _i )) = d we say that f is computable by a ΣΠΣ(k; d;ρ) circuit. We obtain the following results.

A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

We consider Kerr spacetimes with parameters a and M such that |a|≪M, Kerr-Newman spacetimes with parameters |Q|≪M, |a|≪M, and more generally, stationary axisymmetric black hole exterior spacetimes (M,g)(\mathcal{M},g) which are sufficiently close to a Schwarzschild metric with parameter M>0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □ _g ψ=0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Σ _τ , then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ, a suitable energy flux through Σ _τ is bounded by C times the initial energy flux through Σ₀. This energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ, in terms of a higher order initial energy on Σ₀. Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.     

Complete growth functions of hyperbolic groups

. We study a generalization of the growth functions of finitely generated groups, namely the growth functions Σ _{g ∈ G} gz ^{| g |} with coefficients in the group ring ℤ[G]. Rationality and methods of computation of such functions are discussed, in particular for hyperbolic groups. The complete growth functions of surface groups are explicitly computed. The operator and geodesic growth functions are also studied. Oblatum 20-IX-1996 & 13-I-1997     

Necessary conditions for absolute double matrix summability factors

We obtain necessary conditions for a doubly triangular matrix A to have the property that a double series ΣΣ λ _mn b _mn is summable |A| _k whenever the series ΣΣb _mn is bounded |A| _k .     

Extending analyticK-subanalytic functions

Letg:U→ℝ (U open in ℝⁿ) be an analytic and K-subanalytic (i. e. definable in ℝ _an ^K , whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū\∑. Partially supported by the European RTN Network RAAG (contract no. HPRN-CT-00271)     

A Nichtnegativstellensatz for polynomials in noncommuting variables

Let S⋃{f} be a set of symmetric polynomials in noncommuting variables. If f satisfies a polynomial identity Σ _i h _i ^* fh _i = 1 + Σ _i g _i ^* s _i g _i for some s _i ∈ S ⋃ {1}, then f is obviously nowhere negative semidefinite on the class of tuples of nonzero operators defined by the system of inequalities s ≥ 0 (s ∈ S). We prove the converse under the additional assumption that the quadratic module generated by S is Archimedean. Supported by the European network RAAG (EC contract HPRN-CT-2001-00271). Igor Klep was supported from the state budget by the Slovenian Research Agency (project No. Z1-9570-0101-06).     

A generalization of the Morita-Mumford classes to extended mapping class groups for surfaces

Let Σ _g,1 be an oriented compact surface of genus g with 1 boundary component, and Γ _g,1 the mapping class group of Σ _g,1 . We define a bigraded series of cohomology classes m _i,j ∈H ^2i+j−2 (Γ _g,1 ;⋀ ^j H ₁(Σ _g,1 ;ℤ)), 2i+j−2≥1,i,j≥0. When j=0, the class m _i+1,0 is the i-th Morita- Mumford class [Mo][Mu]. It is proved that H ^r (Γ _g,1 ;⋀ ^s H ₁(Σ _g,1 ;ℚ)) is generated by m _i,j 's for the case r+s=2 and the case g≥5 and (r,s)=(1,3). Especially the Johnson homomorphism extended to the whole mapping class group by Morita [Mo3] has an implicit representation by the classes m _0,3 and m _0,2 m _1,1 over ℚ. Oblatum 28-IV-1995 & 8-II-1997     

Coboundaries of irreducible markov operators onC(K)

LetK be a compact Hausdorff space, and letT be an irreducible Markov operator onC(K). We show that ifgεC(K) satisfies sup _N ‖Σ _j ^=0N T ^j g‖<∞, then (and only then) there existsfεC(K) with (I − T)f=g. Generalizing the result to irreducible Markov operator representations of certain semi-groups, we obtain that bounded cocycles are (continuous) coboundaries. For minimal semi-group actions inC(K), no restriction on the semi-group is needed.     

The Ground State and the Long-Time Evolution in the CMC Einstein Flow

Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)³Vol_g(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by V_inf=(\frac-16Y(S))^\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g _i , K _i )} satisfying: (i) k _i  = −3, (ii) V_iˉ V_inf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q ₀((g _i , K _i )) ≤ Λ, where Q ₀ is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)²g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E₁)\tilde]=E₁(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E₁=Q₀+Q₁{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ V_inf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.     

Constrained energy problems with applications to orthogonal polynomials of a discrete variable

Given a positive measure Σ with gs > 1, we write Με ℳ^Σ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight functionw, we prove existence and uniqueness of a measure λ^Σ _w that minimizes the weighted logarithmic energy over the class ℳ^Σ. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure λ^Σ _w As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials. The research done by this author is in partial fulfillment of the Ph.D. requirements at the University of South Florida. The research done by this author was supported, in part, by U.S. National Science Foundation under grant DMS-9501130.     

The integral cohomology of toric manifolds

We prove that the integral cohomology of a smooth, not necessarily compact, toric variety X _Σ is determined by the Stanley-Reisner ring of Σ. This follows from a formality result for singular cochains on the Borel construction of X _Σ. As a consequence, we show that the cycle map from Chow groups to Borel-Moore homology is split injective.     

Birkhoff strata of the Grassmannian Gr<Superscript>(2)</Superscript>: Algebraic curves

We study algebraic varieties and curves arising in the Birkhoff strata of the Sato Grassmannian Gr ⁽²⁾ . We show that the big cell Σ ₀ contains the tower of families of the normal rational curves of all odd orders. The strata Σ _2n, n = 1, 2, 3,..., contain hyperelliptic curves of genus n and their coordinate rings. The strata Σ _2n+1 , n = 0, 1, 2, 3,..., contain (2m + 1, 2m + 3) plane curves for n = 2m, 2m − 1 (m ≥ 2) and also (3, 4) and (3, 5) curves in Σ ₃ and Σ ₅ . Curves in the strata Σ _2n+1 have zero genus.     

Existence of Symplectic Structures on Torus Bundles Over Surfaces

Let E be the total space of a locally trivial torus bundle over a surface Σ_g of genus g > 1. Using Seiberg–Witten theory and spectral sequences, we prove that E carries a symplectic structure if and only if the homology class of the fiber [T²] is nonzero in H₂(E, ℝ). Mathematics Subject Classifications (2000): 53D05, 57R57, 55R20.     

Spacelike Graphs with Parallel Mean Curvature in Pseudo-Riemannian Product Manifolds

The author introduces the w-function defined on the considered spacelike graph M.Under the growth conditions w = o(log z) and w = o(r),two Bernstein type theorems for M in Rmn+ mare got,where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively.As the ambient space is a curved pseudoRiemannian product of two Riemannian manifolds(Σ1,g1) and(Σ2,g2) of dimensions n and m,a Bernstein type result for n = 2 under some curvature conditions on Σ1 and Σ2 and the growth condition w = o(r) is also got.As more general cases,under some curvature conditions on the ambient space and the growth condition w = o(r) or w = o(√r),the author concludes that if M has parallel mean curvature,then M is maximal.     

Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution

The Fourier method is used to obtain a classical solution of an initial-boundary value problem for a first-order partial differential equation with involution in the function and its derivative. The series Σ produced by the Fourier method as a formal solution of the problem is represented as Σ = S ₀ + (Σ − Σ₀₎, where Σ₀ is the formal solution of a special reference problem for which the sum S ₀ can be explicitly calculated. Refined asymptotic formulas for the solution of the Dirac system are used to show that the series Σ − Σ₀ and the series obtained from it by termwise differentiation converge uniformly. Minimal smoothness assumptions are imposed on the initial data of the problem.     

Marginally Trapped Surfaces in L<Superscript>4</Superscript> and an Extended Weierstrass–Bryant Representation

Let E be the total space of a locally trivial torus bundle over a surface Σ_g of genus g > 1. Using Seiberg–Witten theory and spectral sequences, we prove that E carries a symplectic structure if and only if the homology class of the fiber [T²] is nonzero in H₂(E, R). Mathematics Subject Classifications (2000): 53C42, 53A10.     

Adjacency in subposets of the lattice of T 1-topologies on a set

Summary We study subposets of the lattice L_₁(X) of all T1-topologies on a set X, namely Σ_t(X), Σ₃(X) and Σ_lc(X), being respectively the collections of all Tychonoff, all T₃ and all locally compact Hausdorff topologies on X, with a view to deciding which elements of these partially ordered sets have and which do not have covers, that is to say immediate successors, in the respective posets. In the final section we discuss the subposet Σ _G of all Hausdorff group topologies on a group G.