Nilfactors of R^m-actions and configurations in sets of positive upper density in R^m

We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of R^m, with positive upper density. Let V={0,v_1,...,v_k} be a subset of R^m. We show that for r large enough, we can find an isometric copy of rV arbitrarily close to E. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss showing a similar property for m=k=2

Biregular cages of girth five

Let $2 \le r < m$ and $g$ be positive integers. An $({r,m};g)$--graph} (or biregular graph) is a graph with degree set ${r,m}$ and girth $g$, and an $({r,m};g)$-cage (or biregular cage) is an $({r,m};g)$-graph of minimum order $n({r,m};g)$. If $m=r+1$, an $({r,m};g)$-cage is said to be a semiregular cage. In this paper we generalize the reduction and graph amalgam operations from M. Abreu, G. Araujo-Pardo, C. Balbuena, D. Labbate (2011) on the incidence graphs of an affine and a biaffine plane obtaining two new infinite families of biregular cages and two new semiregular cages. The constructed new families are $({r,2r-3};5)$-cages for all $r=q+1$ with $q$ a prime power, and $({r,2r-5};5)$-cages for all $r=q+1$ with $q$ a prime. The new semiregular cages are constructed for r=5 and 6 with 31 and 43 vertices respectively

Asymptotic behavior of Tor over complete intersections and applications

Let $R$ be a local complete intersection and $M,N$ are $R$-modules such that \ell(\Tor_i^R(M,N))<\infty for $i\gg 0$. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of \ell(\Tor_i^R(M,N)) using Eisenbud operators and show that they have well-behaved growth. We define and study a function $\eta^R(M,N)$ which generalizes Serre's intersection multiplicity $\chi^R(M,N)$ over regular local rings and Hochster's function $\theta^R(M,N)$ over local hypersurfaces. We use good properties of $\eta^R(M,N)$ to obtain various results on complexities of \Tor and \Ext, vanishing of \Tor, depth of tensor products, and dimensions of intersecting modules over local complete intersections

Orthogonal basis for spherical monogenics by step two branching

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space R^m. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on R^m. Fix the direct sum R^m = R^p x R^q. In this paper we will study the decomposition of the space M_n(R^m;C_m) of spherical monogenics of order n under the action of Spin(p) x Spin(q). As a result we obtain a Spin(p) x Spin(q)-invariant orthonormal basis for M_n(R^m;C_m). In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space M_n(R^m;C_m).Comment: submitte

On Z4-linear Reed-Muller like codes

For each r, 0 <= r <= m, it is presented the class of quaternary linear codes LRM(r,m) whose images under the Gray map are binary codes with parameters of Reed-Muller RM(r,m) code of order r

Coherent State Transforms and the Weyl Equation in Clifford Analysis

We study a transform, inspired by coherent state transforms, from the Hilbert space of Clifford algebra valued square integrable functions $L^2({\mathbb R}^m,dx)\otimes {\mathbb C}_{m}$ to a Hilbert space of solutions of the Weyl equation on ${\mathbb R}^{m+1}= {\mathbb R} \times {\mathbb R}^m$, namely to the Hilbert space ${\mathcal M}L^2({\mathbb R}^{m+1},d\mu)$ of ${\mathbb C}_m$-valued monogenic functions on ${\mathbb R}^{m+1}$ which are $L^2$ with respect to an appropriate measure $d\mu$. We prove that this transform is a unitary isomorphism of Hilbert spaces and that it is therefore an analog of the Segal-Bargmann transform for Clifford analysis. As a corollary we obtain an orthonormal basis of monogenic functions on ${\mathbb R}^{m+1}$. We also study the case when ${\mathbb R}^m$ is replaced by the $m$-torus ${\mathbb T}^m.$ Quantum mechanically, this extension establishes the unitary equivalence of the Schr\"odinger representation on $M$, for $M={\mathbb R}^m$ and $M={\mathbb T}^m$, with a representation on the Hilbert space ${\mathcal M}L^2({\mathbb R} \times M,d\mu)$ of solutions of the Weyl equation on the space-time ${\mathbb R}\times M$

Beilinson's Hodge conjecture for smooth varieties

Consider the cycle class map cl_{r,m} : CH^r(U,m;\Q) \to \Gamma H^{2r-m}(U,\Q(r)), where CH^r(U,m;\Q) is Bloch's higher Chow group (tensored with \Q) of a smooth complex quasi-projective variety U, and H^{2r-m}(U,\Q(r)) is singular cohomology. We study the image of cl_{r,m} in terms of kernels of Abel-Jacobi maps. When r=m, we deduce from the Bloch-Kato theorem that the cokernel of cl_{r,m} at the generic point of U is the same for integral or rational coefficients.Comment: 33 page

Decomposition of direct product at an arbitrary Brillouin zone point: D(★R)(m)D^{(\bigstar{R})(m)} ⊗\otimes D(★−R)(m)D^{(\bigstar{-R})(m)}

A general rule is presented for the decomposition of the direct product of irreducible representation at arbitrary Brillouin zone point $\bf{R}$ with its negative: the number of the appearences of the zone center representation equals the dimensionality of the representation. This rule is applicable for all space groups. Although in most situations the interesting physics takes place at high symmetry points in the Brillouin zone, this general rule is useful for situations where double excitations are considered. It is shown that double excitations from arbitrary Brillouin point $\bf{R}$ have the right symmetry to participate in all optical experiments regardless of polarization directions.Comment: 7 pages, 2 figure

An extension of harmonic functions along fixed direction

Let a function $u(x,y)$ be harmonic in the domain $$D\times V_r=D\times \{y\in \mathbb{R}^m: |y|<r\}\subset \mathbb{R}^n\times \mathbb{R}^m$$ and for each fixed point $x^0$ from some a set $E\subset D$, %which is not embedded in countable association of $N$-sets of $Lh_0(D)$, the function $u(x^0,y)$, as a function of variable $y$, can be extended to a harmonic function on the whole $\mathbb{R}^m$. Then $u(x,y)$ harmonically extends to the domain $D\times \mathbb{R}^m$ as a function of variables $x$ and $y$.Comment: 4 page

Characterizations of compact sets in fuzzy sets spaces with LpL_p metric

In this paper, we present characterizations of totally bounded sets, relatively compact sets and compact sets in the fuzzy sets spaces $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ equipped with $L_p$ metric, where $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ are two kinds of general fuzzy sets on $\mathbb{R}^m$ which do not have any assumptions of convexity or star-shapedness. Subsets of $F_B(\mathbb{R}^m)^p$ include common fuzzy sets such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and the general fuzzy star-shaped numbers introduced by Qiu et al. The existed compactness criteria are stated for three kinds of fuzzy sets spaces endowed with $L_p$ metric whose universe sets are the former three kinds of common fuzzy sets respectively. Constructing completions of fuzzy sets spaces with respect to $L_p$ metric is a problem which is closely dependent on characterizing totally bounded sets. Based on preceding characterizations of totally boundedness and relatively compactness and some discussions on convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy sets spaces mentioned in this paper can be obtained by using the $L_p$-extension. We also clarify relation among all the ten fuzzy sets spaces discussed in this paper, which consist of five pairs of original spaces and the corresponding completions. Then, we show that the subspaces of $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ mentioned in this paper have parallel characterizations of totally bounded sets, relatively compact sets and compact sets. At last, as applications of our results, we discuss properties of $L_p$ metric on fuzzy sets space and relook compactness criteria proposed in previous work.Comment: This paper is submitted to Fuzzy Sets and Systems at 29/08/201