Spectral invariant subalgebras of reduced groupoid <Emphasis Type="Italic">C</Emphasis>*-algebras

We introduce the notion of property (RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S ₂ ^l (G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C*-algebra C _r ^* (G) of G when G has property (RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G ⁰ of G, gives rise to a canonical map τ _c on the algebra C _c (G) of complex continuous functions with compact support on G. We show that the map τ _c can be extended continuously to S ₂ ^l (G) and plays the same role as an n-trace on C _r ^* (G) when G has property (RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on C _r ^* (G).     

The unramified computation of Rankin–Selberg integrals expressed in terms of Bessel models for split orthogonal groups: Part I

In this paper, we compute the local integrals, with normalized unramified data, over a p-adic field F, arising from general Rankin–Selberg integrals for SO _m × GL_r+k+1, where the orthogonal group is split over F, \(k \leqslant \left[ {\frac{{m - 1}}{2}} \right]\), and the irreducible representation of SO _m (F) has a Bessel model with respect to an irreducible representation of the split orthogonal group SO_m?2k?1(F). Our proof is by “analytic continuation from the unramified computation in the generic case”. We let the unramified parameters of the representations involved vary, and express the local integrals in terms of the Whittaker models of the representations, which exist at points in general position. Then we apply analytic continuation and the known unramified computation in the generic case. We discuss some applications to poles of partial L-functions and functorial lifting.     

A numerical algorithm for solving a nonstationary problem of optimal control

Let {φ _n ^(α,β) (z)} _n=0 ^∞ be a system of Jacobi polynomials orthonormal on the circle |z| = 1 with respect to the weight (1 ? cos τ)^α+1/2(1 + cos τ)^β+1/2 (α, β > ?1), and let \(\psi _n^{\left( {\alpha ,\beta } \right)*} \left( z \right): = z^n \overline {\psi _n^{\left( {\alpha ,\beta } \right)} \left( {{1 \mathord{\left/ {\vphantom {1 {\bar z}}} \right. \kern-\nulldelimiterspace} {\bar z}}} \right)}\)). We establish relations between the polynomial φ _n ^(α,?1/2) (z) and the nth (C, α ? 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?3/2 and also between the polynomial φ _n ^(α,?1/2)* (z) and the nth (C, α + 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?1/2. We use these relations to derive an asymptotic formula for φ _n ^(α,?1/2) (z); the formula is uniform inside the disk |z| < 1. It follows that φ _n ^(α,?1/2) (z) ≠ 0 in the disk |z| ≤ ρ for fixed φ ∈ (0, 1) and α > ?1 if n is sufficiently large.     

On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l _r,k ^α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product

$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$

, and generated by the classical orthogonal Laguerre polynomials L _k ^α (x) (k = 0, 1,...). The polynomials l _r,k ^α (x) are represented as expressions containing the Laguerre polynomials L _n ^α?r (x). An explicit form of the polynomials l _r,k+r ^α (x) is established as an expansion in the powers x ^r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l _r,k ^α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.

     

Improved bounds on the generalized acyclic chromatic number

An r-acyclic edge chromatic number of a graph G, denoted by a _r ^′ r(G), is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {|C|, r} colors. We prove that a _r ^′ r(G) ≤ (4r + 1)Δ(G), when the girth of the graph G equals to max{50, Δ(G)} and 4 ≤ r ≤ 7. If we relax the restriction of the girth to max {220, Δ(G)}, the upper bound of a _r ^′ r(G) is not larger than (2r + 5)Δ(G) with 4 ≤ r ≤ 10.     

Neighbor sum distinguishing total chromatic number of <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-minor free graph

A k-total coloring of a graph G is a mapping ?: V (G) ? E(G) → {1; 2,..., k} such that no two adjacent or incident elements in V (G) ? E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v: We say that ? is a k-neighbor sum distinguishing total coloring of G if f(u) 6 ≠ f(v) for each edge uv ∈ E(G): Denote χ _Σ ^″ (G) the smallest value k in such a coloring of G: Pil?niak and Wo?niak conjectured that for any simple graph with maximum degree Δ(G), χ _Σ ^″ ≤ Δ(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K ₄-minor free graph G with Δ(G) > 5; χ _Σ ^″ = Δ(G) + 1 if G contains no two adjacent Δ-vertices, otherwise, χ _Σ ^″ (G) = Δ(G) + 2.     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$

     

Integrable Crystals and Restriction to Levi Subgroups Via Generalized Slices in the Affine Grassmannian

Let G be a connected reductive algebraic group over ?, and let Λ _G ⁺ be the monoid of dominant weights of G. We construct integrable crystals B^G(λ), λ ∈ Λ _G ⁺ , using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group of G. We also construct tensor product maps \(P{\lambda _1},{\lambda _2}:{B^G}({\lambda _2}) \to {B^G}({\lambda _1} + {\lambda _2}) \cup \{ 0\} \) in terms of multiplication in generalized transversal slices. Let L ? G be a Levi subgroup of G. We describe the functor Res _L ^G : Rep(G) → Rep(L) of restriction to L in terms of the hyperbolic localization functors for generalized transversal slices.     

On finite alperin <Emphasis Type="Italic">p</Emphasis>-groups with homocyclic commutator subgroup

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C _n ² for p≠ 3 and d(G′) ≤ C _n ² + C _n ³ for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C _n ² . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C _n ² + C _n ³ , then G″ is an elementary abelian group.     

Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes

We study the operator-valued positive dyadic operator

$${T_\lambda }\left( {f\sigma } \right): = \sum\limits_{Q \in D} {{\lambda _Q}} \int_Q {fd\sigma 1Q}, $$

where the coefficients {λ _Q : C → D} _Q∈D are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy–Littlewood property. An example of a Banach lattice with the Hardy–Littlewood property is a Lebesgue space.

In the two-weight case, we prove that the L _C ^p (σ) → L _D ^q (ω) boundedness of the operator T _λ( · σ) is characterized by the direct and the dual L ^∞ testing conditions:

$$\left\| {{1_Q}{T_\lambda }} \right\|{\left( {{1_Q}f\sigma } \right)||_{L_D^q\left( \omega \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\sigma } \right)}}\sigma {\left( Q \right)^{1/p}}$$

,

$${\left\| {{1_Q}{T_\lambda }*\left( {{1_{Qg\omega }}} \right)} \right\|_{L_{C*}^{p'}\left( \sigma \right)}} \lesssim {\left\| g \right\|_{L_{D*}^\infty \left( {Q,\omega } \right)}}\omega {\left( Q \right)^{1/q'}}$$

.

Here L _C ^p (σ) and L _D ^q (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < p ≤ q < ∞, and locally finite Borel measures σ and ω.

In the unweighted case, we show that the L _C ^p (μ) → L _D ^p (μ) boundedness of the operator T _λ( · μ) is equivalent to the end-point direct L ^∞ testing condition:

$${\left\| {{1_Q}{T_\lambda }\left( {{1_Q}f\mu } \right)} \right\|_{L_D^1\left( \mu \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\mu } \right)}}\left( {Q,\mu } \right)\mu \left( Q \right)$$

.

This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.     

On criteria of estimation of the errors of cubature formulas

The paper considers cubature formulas for calculating integrals of functions f(X), X = (x ₁, …, x _n ) which are defined on the n-dimensional unit hypercube K ⁿ = [0, 1] ⁿ and have integrable mixed derivatives of the kind \(\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)\), 0 ≤ α _j ≤ 2. We estimate the errors R[f] = \(\smallint _{K^n } \) f(X)dX ? Σ _{k = 1} ^N c _k f(X(k)) of cubature formulas (c _k > 0) as functions of the weights c _k of nodes X(k) and properties of integrable functions. The error is estimated in terms of the integrals of the derivatives of f over r-dimensional faces (r≤n) of the hypercube K ⁿ : |R(f)| ≤ \(\sum _{\alpha _j } \) G(α _j )\(\int_{K^r } {\left| {\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)} \right|} \) dX _r , where coefficients G(α _j ) are criteria which depend only on parameters c _k and X(k). We present an algorithm to calculate these criteria in the two- and n-dimensional cases. Examples are given. A particular case of the criteria is the discrepancy, and the algorithm proposed is a generalization of those used to compute the discrepancy. The results obtained can be used for optimization of cubature formulas as functions of c _k and X(k).     

Removability of isolated singularities for anisotropic elliptic equations with gradient absorption

We study the well-posedness of the third-order degenerate differential equation \(\left( {{P_3}} \right):\alpha {\left( {Mu} \right)^{\prime \prime \prime }}\left( t \right) + {\left( {Mu} \right)^{\prime \prime }}\left( t \right) = \beta Au\left( t \right) + f\left( t \right)\), (t ∈ [0, 2p]) with periodic boundary conditions \(Mu\left( 0 \right) = Mu\left( {2\pi } \right),\;Mu'\left( 0 \right) = Mu'\left( {2\pi } \right),\;Mu''\left( 0 \right) = Mu''\left( {2\pi } \right)\), in periodic Lebesgue–Bochner spaces L^p(T,X), periodic Besov spaces B_p,q^s(T,X) and periodic Triebel–Lizorkin spaces F_p,q^s(T,X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) \( \cap \)D(B) ? D(M) and α, β, γ ∈ R. Using known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of (P₃) in the above three function spaces.     

On the towers of torsion Bertrandias and Payan modules

For an odd prime p, let K/k be a Galois p-extension and S be a set of primes of k containing the primes lying over p. For the p ^r th roots \({\mu _{{p^r}}}\left( K \right)\) of unity in K, we describe the so-called Sha group Sha _S (G(K/k), \({\mu _{{p^r}}}\left( K \right)\)) in terms of the Galois groups of certain subfields of K corresponding to S. As an application, we investigate a tower of extension fields \({\left\{ {{k_{{T^i}}}} \right\}_i} \geqslant 0\) where \({k_{{T^{i + 1}}}}\) is defined as the fixed field of a free part of the Galois group of the Bertrandias and Payan extension of \({k_{{T^i}}}\) over \({k_{{T^i}}}\). This is called a tower of torsion parts of the Bertrandias and Payan extensions over k. We find a relation between the degrees \({\left\{ {\left[ {{k_{{T^{i + 1}}}}:{k_{{T^i}}}} \right]} \right\}_{i \geqslant 0}}\) over the towers. Using this formula we investigate whether the towers are stationary or not.     

Exact distribution of MLE of covariance matrix in a GMANOVA-MANOVA model

For a GMANOVA-MANOVA model with normal error: Y = XB1Z1 T B2Z2 T E, E- Nq×n(0, In (?) ∑), the present paper is devoted to the study of distribution of MLE, ∑, of covariance matrix ∑. The main results obtained are stated as follows: (1) When rk(Z) -rk(Z2) ≥ q-rk(X), the exact distribution of ∑ is derived, where z = (Z1,Z2), rk(A) denotes the rank of matrix A. (2) The exact distribution of |∑| is gained. (3) It is proved that ntr{[S-1 - ∑-1XM(MTXT∑-1XM)-1MTXT∑-1]∑}has X2(q_rk(x))(n-rk(z2)) distribution, where M is the matrix whose columns are the standardized orthogonal eigenvectors corresponding to the nonzero eigenvalues of XT∑-1X.     

On the approximation of continuous functions of two variables

It is shown that if P _m ^α,β (x) (α, β > ?1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ _mn ^α,β (x, y)} _m,n=0 ^r = {P _m ^α,β (x)P _n ^α,β (y)} _{m, n=0} ^r (r = m + n ≤ N ? 1) is an orthogonal system on the set Ω _N×N = ?ub;(x _i , y _i ) _i,j=0 ^N , where x _i and y _i are the zeros of the Jacobi polynomial P _n ^α,β (x). Given an arbitrary continuous function f(x, y) on the square [?1, 1]², we construct the discrete partial Fourier-Jacobi sums of the rectangular type S _{m, n, N} ^α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S _{m, n, N} ^α,β ∥ of the discrete sums S _{m, n, N} ^α,β (f; x, y) for ?1/2 < α, β < 1/2, m + n ≤ N ? 1 is O((mn) ^{q + 1/2}), where q = max?ub;α,β?ub;. As a consequence of this result, several approximate properties of the discrete sums S _{m, n, N} ^α,β (f; x, y) are considered.     

Parity check matrices and product representations of squares

Let \(N_{\mathbb{F}} \)(n,k,r) denote the maximum number of columns in an n-row matrix with entries in a finite field \(\mathbb{F}\) in which each column has at most r nonzero entries and every k columns are linearly independent over \(\mathbb{F}\). We obtain near-optimal upper bounds for \(N_{\mathbb{F}} \)(n,k,r) in the case k > r. Namely, we show that \(N_\mathbb{F} (n,k,r) \ll n^{\frac{r}{2} + \frac{{cr}}{k}} \) where \(c \approx \frac{4}{3}\) for large k. Our method is based on a novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for finding short linear dependencies. We present additional applications of this method to a problem on hypergraphs and a problem in combinatorial number theory.     

Estimates of the Root Functions of a One-Dimensional Schrödinger Operator with a Strong Boundary Singularity

For any operator defined by the differential operation Lu = ?u″ + q(x)u on the interval G = (0, 1) with complex-valued potential q(x) locally integrable on G and satisfying the inequalities \(\int_{{x_1}}^{{x_2}} {\zeta |(q(\zeta ))|d\zeta \leqslant ln({x_1}/{x_2})} \) and \(\int_{{x_1}}^{{x_2}} {\zeta |(q(1 - \zeta ))|d\zeta \leqslant \gamma ln({x_1}/{x_2})} \) with some constant γ for all sufficiently small 0 < x₁ < x₂, we estimate the norms of root functions in the Lebesgue spaces L _p (G), 1 ≤ p < ∞. We show that for sufficiently small γ these norms satisfy the same estimates asymptotic in the spectral parameter as in the unperturbed case.     

Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =(x_1,x_2,x_3)∈R~(n_1)×R~(n_2)×R~(n_3) and ξ =(ξ_1,ξ_2,ξ_3)∈R~(n_1)×R~(n_2)×R~(n_3). One of our main results is the following:Assume that m(ξ) is a function on R~(n_1+n_2+n_3) satisfying ■ with s_i n_i(1/p-1/2) for 1≤i≤3. Then T_m is bounded from H~p(R~(n_1)×R~(n_2)×R~(n_3) to H~p(R~(n_1)×R~(n_2)×R~(n_3)for all 0 p≤1 and ■ Moreover, the smoothness assumption on s_i for 1≤i≤3 is optimal. Here we have used the notations m_(j,k,l)(ξ)=m(2~jξ_1,2~kξ_2,2~lξ_3)Ψ(ξ_1)Ψ(ξ_2)Ψ(ξ_3) and Ψ(ξ_i) is a suitable cut-off function on R~(n_i) for1≤i≤3, and W~(s_1,s_2,s_3) is a three-parameter Sobolev space on R~(n_1)×R~(n_2)× R~(n_3).Because the Fefferman criterion breaks down in three parameters or more, we consider the L~p boundedness of the Littlewood-Paley square function of T_mf to establish its boundedness on the multi-parameter Hardy spaces.     

The program iteration method in a game problem of guidance

Let a sequence of d-dimensional vectors n _k = (n _k ¹ , n _k ² ,..., n _k ^d ) with positive integer coordinates satisfy the condition n _k ^j = α _j m _k +O(1), k ∈ ?, 1 ≤ j ≤ d, where α ₁ > 0,..., α _d > 0 and {m _k } _k=1 ^∞ is an increasing sequence of positive integers. Under some conditions on a function φ: [0,+∞) → [0,+∞), it is proved that, if the sequence of Fourier sums \({S_{{m_k}}}\) (g, x) converges almost everywhere for any function g ∈ φ(L)([0, 2π)), then, for any d ∈ ? and f ∈ φ(L)(ln⁺ L) ^d?1([0, 2π) ^d ), the sequence \({S_{{n_k}}}\) (f, x) of rectangular partial sums of the multiple trigonometric Fourier series of the function f and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.     

On Frankl and Füredi’s conjecture for 3-uniform hypergraphs

Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N^(r) has the largest Lagrangian of all r-graphs with m edges. Denote this r-graph by C _r,m and the Lagrangian of a hypergraph by λ(G). In this paper, we first show that if \(\leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3 \end{array}} \right)\), G is a left-compressed 3-graph with m edges and on vertex set [t], the triple with minimum colex ordering in G ^c is (t ? 2 ? i)(t ? 2)t, then λ(G) ≤ λ(C _3,m ). As an implication, the conjecture of Frankl and Füredi is true for \(\left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)\).