Crossed Product

In this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity. The above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products (SP) matrix exist.

Abstract. We study Rickard equivalences between p-blocks of twisted group algebras and their local structure, in connection with Dade's conjectures. We prove that an extended form of Broué's conjecture implies Dade's Inductive Conjecture in the Abelian defect group case; this is a ...

We develop a general space–time filter applied to panel data models in order to control for heterogeneity as well as both time and spatial dependence. Treatment of initial period observations is analyzed when the number of time periods is small. A second issue relates to a restriction implied by the filter specification on the space–time cross-product term that can greatly

Vector algebra is a powerful and needful tool for Physics but unfortunately, due to lack of mathematical skills, it becomes misleading for first undergraduate courses of science and engineering studies. Standard vector identities are usually proved using Cartesian components or geometrical arguments, accordingly. Instead, this work presents a new teaching strategy in order to derive symbolically vector identities without analytical expansions in components, either explicitly or using indicial notation. This strategy is mainly based on the correspondence between three-dimensional vectors and skew-symmetric second-rank tensors. Hence, the derivations are performed from skew tensors and dyadic products, rather than cross products. Some examples of skew-symmetric tensors in Physics are illustrated.

The recently developed theory of partial actions of discrete groups on $C^*$-algebras is extended. A related concept of actions of inverse semigroups on $C^*$-algebras is defined, including covariant representations and crossed products. The main result is that every partial crossed product is a crossed product by a semigroup action.

This conjecture deals with the prime count function, n(p) and p, prime. Let a^(n(p) – 1) == p + x^2 y^2 Richard Crandal and Carl Pomerance. Prime Numbers: A Computational Perspective, c. 2005 (p. 321) Definition of elliptical curve y^2 = x^3 + ax + b, let a=b=c y^2 = x^3 Then Let p^(n(p) – 1) == p + x^2 y^2 p^(n(p) – 1) == p + x^2 x^3 {2,3,.. p} 2^(1 – 1) == 2 + x^5 0 == 1 + x^5, x =-1 3^(2 – 1) == 3 + x^5 3^1 == 3 + x^5, x = 0 Hence, there is a solution for the first two primes, 2 and 3. The conjecture is that there is a solution for each and every prime. III Cross Products Let Y = y-ybar and X = x-xbar Then Y^2 = (y^2 + ybar^2-2 y ybar); X^2 = (x^2 + xbar^2-2 x xbar); and YX = (yx + ybar xbar-2 ybar xbar) Substitute for (2 y ybar); (2 x xbar); and (2 ybar xbar) (2 root[y]^2 root[ybar]^2); (2 root[x]^2 root[xbar]^2) ; and (2 root[ybar]^2 root[xbar]^2 such that p^(n(p) – 1) == p + x^2 y^2 or x^2 y^2 == p^(n(p)-1)-p, so that root[y]^2 root[ybar]^2 == y ybar = 2^(1-1)-2 =-1, etc. Then Y^2 = (y^3 + y^2 + 2) ; X^2 = (x^3 + x^2 + 2); YX = (yx^3 + yx^2 + 2).

We construct a gauge fixed action for topological membranes on $G_2$-manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds. Moreover on $M\times S^1$ the theory naturally reduces to the standard A-model on Calabi-Yau manifold and to a membrane theory localized

Let (A,H,D) be a spectral triple, namely: A is a C^*-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H,D) with additional properties which guaranty that the Connes metric induces the weak^*-topology on the state space of A. A ^*-automorphism respecting the metric defined a dynamical system. This article gives various answers to the question: is there a canonical spectral triple based upon the crossed product algebra Ax_αZ, characterizing the metric properties of the dynamical system ? If α is the noncommutative analog of an isometry the answer is yes. Otherwise, the metric bundle construction of Connes and Moscovici is used to replace (A,α) by an equivalent dynamical system acting isometrically. The difficulties relating to the non compactness of this...

An integer matrix $A\in M_d(\Z)$ induces a covering $\sigma_A$ of $\T^d$ and an endomorphism $\alpha_A:f\mapsto f\circ \sigma_A$ of $C(\T^d)$ for which there is a natural transfer operator $L$. In this paper, we compute the KMS states on the Exel crossed product $C(\T^d)\rtimes_{\alpha_A,L}\N$ and its Toeplitz extension. We find that $C(\T^d)\rtimes_{\alpha_A,L}\N$ has a unique KMS state, which has inverse temperature $\beta=\log|\det A|$. Its Toeplitz extension, on the other hand, exhibits a phase transition at $\beta=\log|\det A|$, and for larger $\beta$ the simplex of KMS$_\beta$ states is isomorphic to the simplex of probability measures on $\T^d$.

We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy, respectively. Furthermore, we introduce a group homomorphism, the socalled mod map, from the normalizers of the various full groups to the automorphism groups of the (ordered)K 0-groups, which are associated to the Cantor minimal systems. We show how this in turn is related to the automorphisms of the associatedC *-crossed products. Our results are analogues in the topological dynamical setting of results obtained by Dye, Connes-Krieger and Hamachi-Osikawa in measurable dynamics.

We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost--Connes naturally arises by extension of scalars from the "field with one element" to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that "field", while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra.

Improvement of vanilla from wilt disease through in vitro culture Vanilla (Vanilla planifolia A.) is one of the important industrial plants as a source of farmers' income and state earnings. Product of the plant is commonly used as a flavor in food and beverage industry. However, there is no variety of this plant which is resistant to stem base and