Latin trades in groups defined on planar triangulations

For a finite triangulation of the plane with faces properly coloured white and black, let A_W\mathcal{A}_{W} be the abelian group constructed by labelling the vertices with commuting indeterminates and adding relations which say that the labels around each white triangle add to the identity. We show that A_W\mathcal{A}_{W} has free rank exactly two. Let A_W^*\mathcal{A}_{W}^{*} be the torsion subgroup of  A_W\mathcal{A}_{W} , and A_B^*\mathcal{A}_{B}^{*} the corresponding group for the black triangles. We show that A_W^*\mathcal{A}_{W}^{*} and A_B^*\mathcal{A}_{B}^{*} have the same order, and conjecture that they are isomorphic. For each spherical latin trade W, we show there is a unique disjoint mate B such that (W,B) is a connected and separated bitrade. The bitrade (W,B) is associated with a two-colourable planar triangulation and we show that W can be embedded in  A_W^*\mathcal{A}_{W}^{*} , thereby proving a conjecture due to Cavenagh and Drápal. The proof involves constructing a (0,1) presentation matrix whose permanent and determinant agree up to sign. The Smith normal form of this matrix determines A_W^*\mathcal{A}_{W}^{*} , so there is an efficient algorithm to construct the embedding. Contrasting with the spherical case, for each genus g≥1 we construct a latin trade which is not embeddable in any group and another that is embeddable in a cyclic group.     

Orlicz norm estimates for eigenvalues of matrices

Let φ be a supermultiplicative Orlicz function such that the function $t \mapsto \varphi \left( {\sqrt t } \right)$ is equivalent to a convex function. Then each complexn×n matrixT=(τ _ij ) _{i, j} satisfies the following eigenvalue estimate: $\left\| {\left( {\lambda _i \left( T \right)} \right)_{i = 1}^n } \right\|_{\ell _\varphi } \leqslant C\left\| ( \right\|\left( {\tau _{ij} } \right)_{i = 1}^n \left\| {_{_{\ell _{\varphi *} } } )_{j = 1}^n } \right\|\ell _{\bar \varphi } $ . Here, ?_* stands for Young’s conjugate function of φ, ?, $\bar \varphi $ is the minimal submultiplicative function dominating φ andC>0 a constant depending only on φ. For the power function φ(t)=t ^p ,p≥2 this is a celebrated result of Johnson, König, Maurey and Retherford from 1979. In this paper we prove the above result within a more general theory of related estimates.     

Trivectors yielding spreads in PG(5, 2)

Given an alternating trilinear form ${T\in {\rm Alt}(\times^{3}V_{6})}$ on V ₆ = V(6, 2) let ${\mathcal{L}_{T}}$ denote the set of those lines ${\langle a, b \rangle}$ in ${{\rm PG}(5,2)=\mathbb{P}V_{6}}$ which are T-singular, satisfying, that is, T(a, b, x) = 0 for all ${x\in {\rm PG}(5, 2).}$ If ${\mathcal{L}_{21}}$ is a Desarguesian line-spread in PG(5, 2) it is shown that ${\mathcal{L}_{T}=\mathcal{L}_{21}}$ for precisely three choices T ₁,T ₂,T ₃ of T, which moreover satisfy T ₁ + T ₂ + T ₃ = 0. For ${T\in\mathcal{T}:=\{T_{1},T_{2},T_{3}\}}$ the ${\mathcal{G}_{T}}$ -orbits of flats in PG(5, 2) are determined, where ${\mathcal{G}_{T}\cong {\rm SL}(3,4).2}$ denotes the stabilizer of T under the action of GL(6, 2). Further, for a representative U of each ${\mathcal{G}_{T}}$ -orbit, the T-associate U ^# is also determined, where by definition $$U^{\#}=\{v\in {\rm PG}(5,2)\, |\, T(u_{1},u_{2},v) = 0\, \,{\rm for\,all }\, \, u_{1},u_{2}\in U\}$$ .     

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.     

Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities

In this paper, we provide the Euler?CMaclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f??C ^??(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms where $\widehat{P}(y),P_{s}(y)$ and $\widehat{Q}(y),Q_{s}(y)$ are polynomials in y. The ?? _s and ?? _s are distinct, complex in general, and different from ?1. They also satisfy The results we obtain in this work extend the results of a recent paper [A.?Sidi, Numer. Math. 98:371?C387, 2004], which pertain to the cases in which $\widehat{P}(y)\equiv0$ and $\widehat{Q}(y)\equiv0$ . They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x??a+ and x??b?. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let $D_{\omega}=\frac{d}{d\omega}$ , h=(b?a)/n, where n is a positive integer, and define $\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)$ . Then with $\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}$ and $\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}$ , one of these results reads where ??(z) is the Riemann Zeta function and ?? _i are Stieltjes constants defined via $\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]$ , i=0,1,???.     

Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let $\{Y^{(a)}_{n}:n\ge1\}In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let {Y^(a)_n:n 3 1}\{Y^{(a)}_{n}:n\ge1\} be a sequence of independent and identically distributed random variables and {X^(a)_t:t 3 0}\{X^{(a)}_{t}:t\ge0\} be a Lévy process such that X₁^(a)=^dY₁^(a)X_{1}^{(a)}\stackrel{d}{=}Y_{1}^{(a)}, \mathbbEX₁^(a) < 0\mathbb{E}X_{1}^{(a)}<0 and \mathbbEX₁^(a)-0\mathbb{E}X_{1}^{(a)}\uparrow0 as a↓0. Let S^(a)_n=?_k=1ⁿ Y^(a)_kS^{(a)}_{n}=\sum _{k=1}^{n} Y^{(a)}_{k}. Then, under some mild assumptions, , for some random variable and some function Δ(⋅). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.     

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

A new class of integrable systems

This paper concerns the integrability of Hamiltonian systems with two degrees of freedom whose Hamiltonian has the form¶ H=¹/₂(x₁²+x₂²) +V(y₁,y₂) H={1\over2}(x_{1}^{2}+x_{2}^{2}) +V(y_{1},y_{2}) where¶¶ V(y₁,y₂)=¹/₂(a₁y₁²+a₂y₂²) + ¹/₄b₁y₁⁴ + ¹/₄b₂y₂⁴ + ¹/₂b₃y₁²y₂² + ?_k=1³g_k(y₁²+y₂²) ^k+2 V(y_{1},y_{2})={1\over2}\big(\alpha _{1}y_{1}^{2}+\alpha_{2}y_{2}^{2}\big) + {1\over4}\beta _{1}y_{1}^{4} + {1\over4}\beta_{2}y_{2}^{4} + {1\over2}\beta _{3}y_{1}^{2}y_{2}^{2} + \sum_{k=1}^{3}\gamma_{k}\big(y_{1}^{2}+y_{2}^{2}\big) ^{k+2} ¶¶ which, constitues a generalization of some well-known integrable systems. We give new values of the vector (a₁,a₂,b₁,b₂,b₃,g₁,g₂,g₃) (\alpha _{1},\alpha_{2},\beta _{1},\beta _{2},\beta _{3},\gamma _{1},\gamma _{2},\gamma _{3}) for which this system is completely integrable and we show that the system is linearized in the Jacobian variety Jac(G \Gamma ) of a smooth genus 2 hyperelliptic Riemann surface G \Gamma .     

Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u₀.For u₀$\in$H¹, local existence in time of solutions on an interval [0, T) is known, and there exist finite time blow-up solutions, that is u₀ such that $\textrm{lim} _{t\uparrow T <+\infty}|\nabla u(t)|_{L^{2}}=+\infty$. This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense.The question we address is to control the blow-up rate from above for small (in a certain sense) blow-up solutions with negative energy. In a previous paper [MeR], we established some blow-up properties of (NLS) in the energy space which implied a control $|\nabla u(t)|_{L^{2}} \leq C \frac{|\ln(T-t)|^{N/4}}{\sqrt{T-t}}$ and removed the rate of the known explicit blow-up solutions which is $\frac{C}{T-t}$.In this paper, we prove the sharp upper bound expected from numerics as$|\nabla u(t)|_{L^{2}} \leq C \left(\frac{\ln|\ln(T-t)|}{T-t} \right)^{1/2}$by exhibiting the exact geometrical structure of dispersion for the problem.     

Integration and Lipschitz functions

The aim of the paper is to prove that every f ∈ L ¹([0,1]) is of the form f = , where j _n,k is the characteristic function of the interval [k- 1 / 2 ⁿ , k / 2 ⁿ ) and Σ _n=0^∞Σ _k=1²ⁿ |a _n,k | is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b _n,k ) _n≧0 ^k=1,...,2n of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).      

<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Subalgebras Generated by a Single Operator in <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">H</Emphasis>)

In this paper, we characterize a C ^*-subalgebra C ^*(x) of B(H), generated by a single operator x. We show that if x is polar-decomposed by aq, where a is the partial isometry part and q is the positive operator part of x, then C ^*(x) is *-isomorphic to the groupoid crossed product algebra A_q×_a\mathbbG_a\mathcal{A}_{q}\times_{\alpha }\mathbb{G}_{a} , where A_q=C^*(q)\mathcal{A}_{q}=C^{*}(q) and \mathbbG_a\mathbb{G}_{a} is the graph groupoid induced by a partial isometry part a of x.     

Partitions involving D.H. Lehmer numbers

Let n ≥ 2 be a fixed integer, let q and c be two integers with q > n and (n, q) = (c, q) = 1. For every positive integer a which is coprime with q we denote by [`(a)]<sub>c</sub>{\overline{a}_{c}} the unique integer satisfying 1 ￡ [`(a)]_c ￡ q{1\leq\overline{a}_{c} \leq{q}} and a[`(a)]_c o c(mod q){a\overline{a}_{c} \equiv{c}({\rm mod}\, q)}. Put

On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials

This paper continues recent investigations started in Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) into the structure of the set H_q,2n ³ {\mathcal{H}_{q,2n}^{\ge}} of all Hankel nonnegative definite sequences, (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}}, of complex q × q matrices and its important subclasses H_q,2n ^{3 ,e}{\mathcal{H}_{q,2n}^{\ge,{\rm e}}} and ${\mathcal{H}_{q,2n}^>}${\mathcal{H}_{q,2n}^>} of all Hankel nonnegative definite extendable sequences and of all Hankel positive definite sequences, respectively. These classes of sequences arise quite naturally in the framework of matrix versions of the truncated Hamburger moment problem. In Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) a canonical Hankel parametrization [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} consisting of two sequences of complex q × q matrices was associated with an arbitrary sequence (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} of complex q × q matrices. The sequences belonging to each of the classes H_q,2n ³ , H_q,2n ^{3 ,e}{\mathcal{H}_{q,2n}^{\ge}, \mathcal{H}_{q,2n}^{\ge,{\rm e}}}, and ${\mathcal{H}_{q,2n}^>}${\mathcal{H}_{q,2n}^>} were characterized in terms of their canonical Hankel parametrization (see, Dyukarev et al. in Complex anal oper theory 3(4):759–834, 2009; Proposition 2.30). In this paper, we will study further aspects of the canonical Hankel parametrization. Using the canonical Hankel parametrization [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} of a sequence (s_j)_j=0²ⁿ ? H_q,2n ³ {(s_{j})_{j=0}^{2n} \in \mathcal{H}_{q,2n}^{\ge}}, we give a recursive construction of a monic right (resp. left) orthogonal system of matrix polynomials with respect to (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} (see Theorem 5.5). The matrices [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be expressed in terms of an arbitrary monic right (resp. left) orthogonal system with respect to (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} (see Theorem 5.11). This result will be reformulated in terms of nonnegative Hermitian Borel measures on \mathbbR{\mathbb{R}}. In this way, integral representations for the matrices [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be obtained (see Theorem 6.9). Starting from the monic orthogonal polynomials with respect to some classical probability distributions on \mathbbR{\mathbb{R}}, Theorem 6.9 is used to compute the canonical Hankel parametrization of their moment sequences. Moreover, we discuss important number sequences from enumerative combinatorics using the canonical Hankel parametrization.     

Triebel-Lizorkin Spaces of Para-Accretive Type and a Tb Theorem

In this article, we use a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type $\dot{F}^{\alpha,q}_{b,p}In this article, we use a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type , which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a necessary and sufficient condition for the boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with M _b TM _b ∈WBP is bounded from to if and only if and T ^* b=0 for , where ε is the regularity exponent of the kernel of T. Chin-Cheng Lin supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-008-021-MY3. Kunchuan Wang supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-259-009 and NCU Center for Mathematics and Theoretic Physics.     

Gromov-Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations

We compute, with symplectic field theory (SFT) techniques, the Gromov-Witten theory of ${\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}}We compute, with symplectic field theory (SFT) techniques, the Gromov-Witten theory of \mathbbP¹_a1,?,aa{\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}}, i.e., the complex projective line with a orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy (Carlet, The extended bigraded toda hierarchy. arXiv preprint arXiv:math-ph/0604024). We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x, y, z) = −xyz + P ₁(x) + P ₂(y) + P ₃(z) which contains as special cases the ones constructed on the space of Laurent polynomials (Dubrovin, Geometry of 2D topologica field theories. Integrable systems and quantum groups, Springer Lecture Notes in Mathematics 1620:120–348, 1996; Milanov and Tseng, The space of Laurent polynomials, \mathbbP¹{\mathbb{P}^1}-orbifolds, and integrable hierarchies. preprint arXiv:math/0607012v3 [math.AG]). We prove a mirror theorem stating that these Frobenius structures are isomorphic to the ones found before for polynomial \mathbbP¹{\mathbb{P}^1}-orbifolds. Finally we link rational SFT of Seifert fibrations over \mathbbP¹_a,b,c{\mathbb{P}^1_{a,b,c}} with orbifold Gromov-Witten invariants of the base, extending a known result (Bourgeois, A Morse-Bott approach to contact homology. Ph.D. dissertation, Stanford University, 2002) valid in the smooth case.     

Hölder Regularity for Solutions of Ultraparabolic Equations in Divergence Form

We consider the second order differential equation , where (x,t) ^N+1, 0<m ₀N, the coefficients a _i,j belong to a suitable space of vanishing mean oscillation functions VMO _L and B=(b _i,j ) is a constant real matrix. The aim of this paper is to study interior regularity for weak solutions to the above equation assuming that F _j belong to a function space of Morrey type.     

On the classification of simple inductive limit C<Superscript>*</Superscript>-algebras,II: The isomorphism theorem

In this article, it is proved that the invariant consisting of the scaled ordered K-group and the space of tracial states, together with the natural pairing between them, is a complete invariant for the class of unital simple C ^*-algebras which can be expressed as the inductive limit of a sequence

Epicompletion in Frames with Skeletal Maps,I: Compact Regular Frames

A frame homomorphism h : A ⟶ B is skeletal if x ^⊥⊥ = 1 in A implies that h(x)^⊥⊥ = 1 in B. It is shown that, in , the category of compact regular frames with skeletal maps, the subcategory , consisting of the frames in which every polar is complemented, coincides with the epicomplete objects in . Further, is the least epireflective subcategory, and, indeed, the target of the monoreflection which assigns to a compact regular frame A, the ideal frame ε A of , the boolean algebra of polars of A.      

A Generalized Radon Transform on the Plane

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ _α, β , and the Jacobi polynomials P_k^(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R _α, β and its dual R_a, b^*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in L_a, b^p(\mathbbR²₊)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations Lu:=?_j=1^ma_jD_a, b^ju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ _α, β . Another application is that, by an invariant property of R _α, β , a new product formula for the Jacobi polynomials of the type P_k^(b, a)(s)C_2k^a+b+1(t)=còòP_k^(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.