The Geometry of the Lie Algebra of the Orthogonal Group O(ℝ4)

In the six-dimensional space of bivectors, a Lie product similar to the standard vector product in is introduced. The Lie algebra constructed is proved to be isomorphic to the Lie algebra of the orthogonal group , and the isomorphism is a canonical isometry between and the space of antisymmetric operators in . Bibliography: 2 titles.     

Chromatic Uniqueness of the Complement of T_m(l, 4, m - 6)

In the paper, we prove that the complement of Tm(1,4,m - 6)(m > 10) is chromatically unique if and only if m ≠5k(k ≥ 2).     

Asymptotic Behavior of Renormalization Constants in Higher Orders of the Perturbation Expansion for the (4−∈)-Dimensionally Regularized O(n)-Symmetric φ4 Theory

Higher-order asymptotic expansions for renormalization constants and critical exponents of the O(n)-symmetric ⁴ theory are found based on the instanton approach in the minimal subtraction scheme for the (4–)-dimensional regularization. The exactly known expansion terms differ substantially from their asymptotic values. We find expressions that improve the asymptotic expansions for unknown expansion terms of the renormalization constants.     

The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let

The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let

Estimation of the Separation of Two Matrices (Ⅱ)

In this paper we give a lower bound of the separation $sep_F(A,B)$ of two diagonalizable matrices A and B. The key to finding the lower bound of $sep_F(A,B)$ is to find an upper bound for the condition number of a transformation matrix Q which transforms a diagonalizable matrix A to a diagonal form. The obtained lower bound of $sep_F(A,B)$ involves the eigenvalues of A and B as well as the departures form the normality $\delta_F(A)$ and $\delta_F(B)$.     

On the (in)finiteness of the image of Reshetikhin–Turaev representations

We state a simple criterion to prove the infiniteness of the image of Reshetikhin–Turaev irreducible representations of the mapping class groups of surfaces. We use it to study some of the Reshetikhin–Turaev representations associated to the tori with one and two punctures and derive an alternative proof of the results of Funar (Pac J Math 188:251–274, 1999).     

Irreducible generating sets of complete semigroups of unions B x (D) defined by semilattices of the class Σ2(X, 4)

In complete semigroups of unions B _x (D) defined by semilattices of class ??₂(X, 4), we selecte subsets of certain type on which equivalent binary relations are defined and by means of these relations, irreducible generating sets of the considered semigroups are described.     

An elementary proof of the irrationality of ζ(3)

An elementary proof of the irrationality of ζ(3) is presented. The proof is based on a two times more dense sequence of Diophantine approximations to this number than the sequence in the original proof of Apery.     

The existence spectrum of Meta(K 4 (3) >K 4 (3) − e)

Let H and J1 be both t-uniform hypergraphs. Let J2 be a sub-hypergraph of J1. In this paper, the metamorphosis of a hypergraph decomposition is introduced, denoted by (H, J1 J2)-design, which is a generalization of the concept of metamorphosis of a graph decomposition. Let Meta(J1J2) denote the set of all integers v such that there exists a (Kv((3)), J1J2)-design. We completely determine the set Meta(K4((3))K4((3))-e).     

On the Integral Curvature of Curve-(III)

In this paper we prove the following theorem.It is a generalization of Tenchel's theorem on theintegral curvature of curve.Theorem.If 1 is the length of a curve C=AB and φ is the angle between the tangent vectors ofC at A,B,then the integral curvature of C     

On the (n − 2)-transversals of n convex subsets of the plane

In this paper we consider families of distinct ovals in the plane, with the property that certain subfamilies have stabbing lines (transversals). Our main result says that if any k member of the family can be stabbed by a line avoiding all the other ovals and k is large enough, then the family consists of at most k+1 ovals. For any n4 we show a family of n ovals, whose n–2 element subfamilies have, but the n–1 element subfamilies do not have, transversals.     

On the Various Realizations of the Basic Representation of An−1(1) and the Combinatorics of Partitions

The infinite dimensional Lie algebra l _n = A _n–1 ⁽¹⁾ can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of l _n corresponding to all partitions of n can be described very simply by using combinatorial constructions.     

Operators on the orthogonal complement of the Dirichlet space(Ⅱ)

In this paper we first prove that a dual Hankel operator R φ on the orthogonal complement of the Dirichlet space is compact for φ∈ W 1,∞(D),and then that a semicommutator of two Toeplitz operators on the Dirichlet space or two dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space is compact.We also prove that a dual Hankel operator R φ with φ∈ W 1,∞(D) is of finite rank if and only if B φ is orthogonal to the Dirichlet space for some finite Blaschke product B,and give ...     

On the size of the set A(A + 1)

Let F _p be the field of a prime order p. For a subset A ì F_p{A \subset F_p} we consider the product set A(A + 1). This set is an image of A ×  A under the polynomial mapping f(x, y) = xy +  x : F _p ×  F _p → F _p . In the present note we show that if |A| <  p ^1/2, then

Investigating the temporal variations of the time-clustering behavior of the Koyna–Warna (India) reservoir-triggered seismicity

The time-clustering behavior of the 1996–2005 seismicity of Koyna–Warna region (India), a unique site where reservoir-triggered earthquakes have been continuously occurring over the last about 50 year, has been analyzed. The scaling exponent α, estimated by using the Allan Factor method, a powerful tool to investigate clusterization in point processes, shows co-seismic and pre-seismic enhancements associated with the occurrence of the major events.