The ({alpha}+2{beta})-Inequality on a Torus

A theorem of Macbeath asserts that µ(A+B)min(1, µ(A)+µ(B))for any subsets A and B of a finite-dimensional torus. We conjecturethat, when the obvious exceptions are excluded, a stronger inequality holds, and we prove this conjecture under some technical restrictions.     

Measurability in C({2^\kappa }) } and Kunen cardinals

A cardinal κ is called a Kunen cardinal if the σ-algebra on κ × κ generated by all products A×B, where A,B ? κ, coincides with the power set of κ×κ. For any cardinal κ, let $C({2^\kappa })$ be the Banach space of all continuous real-valued functions on the Cantor cube $C({2^\kappa })$ . We prove that κ is a Kunen cardinal if and only if the Baire σ-algebra on $C({2^\kappa })$ for the pointwise convergence topology coincides with the Borel σ-algebra on $C({2^\kappa })$ for the norm topology. Some other links between Kunen cardinals and measurability in Banach spaces are also given.     

On correspondences between certain automorphic forms on Sp_{4n} (\mathbb{A}) and \widetilde{Sp}_{2n} (\mathbb{A})

In 2005, Ginzburg, Rallis and Soudry constructed, in terms of residues of certain Eisenstein series, and by use of the descent method, families of nontempered automorphic representations of $Sp_{4nm} (\mathbb{A})$ and $\widetilde{Sp}_{2n(2m - 1)} (\mathbb{A})$ , which generalized the classical work of Piatetski-Shapiro on Saito-Kurokawa liftings. In this paper, we introduce a new framework (Diagrams of Constructions) in order to establish explicit relations among the representations introduced in [GRS05]. In particular, we prove that these constructions yield bijections between a certain set of cuspidal automorphic forms on $\widetilde{Sp}_{2n} (\mathbb{A})$ and a certain set of square-integrable automorphic forms of $Sp_{4n} (\mathbb{A})$ . The proofs use new interpretations of composition of two consecutive descents with explicit identities, which we expect to be very useful to further investigation of the automorphic discrete spectrum of classical groups.     

Navigating in the Cayley Graphs of {\rm SL}_{N}(\mathbb{Z}) and {\rm SL}_{N}(\mathbb{F}_{p})

We give a nondeterministic algorithm that expresses elements of , for N ≥ 3, as words in a finite set of generators, with the length of these words at most a constant times the word metric. We show that the nondeterministic time-complexity of the subtractive version of Euclid’s algorithm for finding the greatest common divisor of N ≥ 3 integers a₁, ..., a_N is at most a constant times . This leads to an elementary proof that for N ≥ 3 the word metric in is biLipschitz equivalent to the logarithm of the matrix norm – an instance of a theorem of Mozes, Lubotzky and Raghunathan. And we show constructively that there exists K>0 such that for all N ≥ 3 and primes p, the diameter of the Cayley graph of with respect to the generating set is at most .Mathematics Subject Classification: 20F05     

Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities

Let be a space of quasipolynomials of dimension N=N ₁+⋅⋅⋅+N _n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ _i=0 ^N A _N−i (x)∂ _x ⁱ annihilating V and such that its leading coefficient A ₀ is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ _i=0 ^N u ⁱ A _N−i (∂ _u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).     

Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})

We prove real Paley-Wiener type theorems for the Dunkl transform ℱ _D on the space of tempered distributions. Let T∈S′(ℝ ^d ) and Δ _κ the Dunkl Laplacian operator. First, we establish that the support of ℱ _D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  _n→+∞ R ⁻²ⁿ Δ _κ ⁿ T=0 in S′(ℝ ^d ). Second, we prove that the support of ℱ _D (T) is included in ℝ ^d ∖B(0,M), M>0, if and only if for all R<M, we have lim  _n→+∞ R ²ⁿ  ℱ _D ⁻¹(‖y‖⁻²ⁿ ℱ _D (T))=0 in S′(ℝ ^d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.      

Operator-Like Wavelet Bases of L_{2}(\mathbb{R}^{d})

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.     

Symmetric Sequence Subspaces of C({alpha})

Two main results are shown: (1) a characterisation of the symmetricsequence spaces which embed into C(), and (2) a characterisation,in terms of the Orlicz function M, of the Orlicz sequence spacesh_M which embed into C(K) for some countable compact Hausdorffspace K. The proofs make use of the Schreier families of sets,which have found wide applications in Banach space theory.     

Signature Characters of Highest-Weight Representations of $U_{q}(\mathfrak {gl}_{n})$

We consider \(U_{q}(\mathfrak {gl}_{n})\), the quantum group of type A for |q| = 1, q generic. We provide formulas for signature characters of irreducible finite-dimensional highest weight modules and Verma modules. In both cases, the technique involves combinatorics of the Gelfand-Tsetlin bases. As an application, we obtain information about unitarity of finite-dimensional irreducible representations for arbitrary q: we classify the continuous spectrum of the unitarity locus. We also recover some known results in the classical limit \(q \rightarrow 1\) that were obtained by different means. Finally, we provide several explicit examples of signature characters.     

Multiplicity one theorem for ({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))

Let F be either or . Consider the standard embedding and the action of GL_n(F) on GL_n+1(F) by conjugation. We show that any GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Fréchet representations π of GL_n+1(F) and of GL_n(F),

An Atomic Decomposition Characterization of Flag Hardy Spaces H^{p}_{F}(\mathbb{R}^{n}\times\mathbb{R}^{m}) with Applications

In this paper, we give an atomic decomposition characterization of flag Hardy spaces $H^{p}_{F}(\mathbb{R}^{n}\times\mathbb{R}^{m})$ for 0<p≤1, which were introduced in (Han and Lu in arXiv:0801.1701). A remarkable feature of atoms of such flag Hardy spaces is that these atoms have only partial cancellation conditions. As an application, we prove a boundedness criterion for operators on flag Hardy spaces.     

On sets with variational mean curvature in L^{n}(\mathbf{R}^{n})

In this paper we give an example of a singular set with finite perimeter in the plane which is a subgraph with variational curvature in $L^2(\mathbf R ^2)$ . This is a extension to the class of graphs of a well celebrated example suggested by De Giorgi, where the boundary of the set was the union of two spiral curves.     

The reducible Specht modules for the Hecke algebra \mathcal{H}_{\mathbb{C},{-}1}(\mathfrak{S}_{n})

The reducible Specht modules for the Hecke algebra $\mathcal {H}_{\mathbb{F},q}(\mathfrak{S}_{n})$ have been classified except when q=?1. We prove one half of a conjecture which we believe classifies the reducible Specht modules when q=?1.     

Renewal reward process for T-related fuzzy random variables on (\mathbb {R}^{p}, \mathbb {R}^{q})

In this paper, following our previous studies, we investigate the renewal rewards process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random rewards are characterized as weighted fuzzy numbers under \(t\) -norm-based fuzzy operations on \(\mathbb {R}^{p}\) and \(\mathbb {R}^{q}\,\,p,\,q \ge 1,\) respectively. Many versions of \(T\) -related fuzzy renewal rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on \(\mathbb {R}^{p}\) . An application example is provided to illustrate the utility of the results.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .