WA Contractions

The problem of unitary -dilation can be generalized by Langer [9, p.55] as follows: Let A be a positive linear operator on a Hilbert space H, 0 < mI A MI, and C_A = {T : QTⁿQ = P_HUⁿ|_H(n = 1,2,3,...) where Q = A^-1/2 and U is a unitary on some Hilbert space H₁ H}. Then T C_A if and only if T satisfies the condition: A + 2Re z(I - A)T + |z|²T*(A - 2I)T 0. Using the above generalization, we have a block-matrix criterion for an element in C_A as follows: T C_A if and only if P(A,z,T,n) 0(n = 1,2,3,...) [Theorem 2.5]. We define the operator radii w_A(.) by w_A(T) = inf;{r>0 : T/r C_A}. Applying the block-matrix criterion, we give some fundamental properties for w_A(.) and extend some earlier results involving operator radii w(.)( > 0) in Fong and Holbrook (1983), Haagerup and de la Harpe (1992), Holbrook (1968), Holbrook (1969) and Holbrook (1971) to the case of w_A(.). We have the equalities and . Inequalities involving completely bounded linear maps on unital C*-algebras are also provided [Theorem 4.5].     

G-covering systems of subgroups for the class of supersoluble groups

Let ? be a class of groups. Given a group G, assign to G some set of its subgroups Σ = Σ(G). We say that Σ is a G-covering system of subgroups for ? (or, in other words, an ?-covering system of subgroups in G) if G ∈ ? wherever either Σ = ? or Σ ≠ ? and every subgroup in Σ belongs to ?. In this paper, we provide some nontrivial sets of subgroups of a finite group G which are G-covering subgroup systems for the class of supersoluble groups. These are the generalizations of some recent results, such as in [1–3].     

On Weak Contractions

A new, elementary proof that weak contractions are decomposableis given. This result is used to characterize the weak contractionsamong all contractions.     

Contractions and minimalk-colorability

Coloring the vertex set of a graphG with positive integers, thechromatic sum (G) ofG is the minimum sum of colors in a proper coloring. Thestrength ofG is the largest integer that occurs in every coloring whose total is(G). Proving a conjecture of Kubicka and Schwenk, we show that every tree of strengths has at least ((2 + ) ^s–1 – (2 – ) ^s–1)/ vertices (s 2). Surprisingly, this extremal result follows from a topological property of trees. Namely, for everys 3 there exist precisely two treesT _s andR _s such that every tree of strength at leasts is edge-contractible toT _s orR _s .     

Compressions of Stable Contractions

The stability of compressions of stable contractions is studied and a sufficient orbit condition is given. On the other hand, it is shown that there are non-stable compressions of the 1-dimensional backward shift and a complete characterization of weighted unilateral shifts with this property is provided. Dilations of bilateral weighted shifts to backward shifts are also considered.     

Fuzzy subgroups of n-ary subgroups

We introduce a notion of t-fuzzy n-ary subgroup in an n-ary group (G, f) and have studied their related properties.     

Reflection subgroups and sub-root systems of the imprimitive complex reflection groups

Based on a graph-theoretic analysis, we determine all the irreducible reflection subgroups of the imprimitive complex reflection groups G(m, p, n), and describe the irreducible subsystems of all possible types in the root system R(m, p, n) of G(m, p, n).     

Contractions to k8

It is proved that the maximal number of edges in a graph with n ≧ 8 vertices that is not contractible to K₈ is 6n ? 21, unless 5 divides n, and the only graphs with n = 5m vertices and more than 6n ? 21 edges that are not contractible to K₈ are the K₅(2)-cockades that have exactly 6n ? 20 edges.     

Contractions with spectral boundary

Address beginning September 1986: Department of Mathematics University of Michigan Ann Arbor, Michigan 48109     

Fixed Points of Multivalued Contractions

A fixed-point theorem is proved for a broad class of closed-valued -contractions with for any positive t and with .     

On Contractions of Lie Algebras

It is well-known that application of invariant functions in various fields of pure or applied mathematics, physics and computer calculations is certainly of interest. In this paper, we study application of the invariant functions of Lie algebras, named \(\psi \) and \(\varphi \) function, respectively, to contractions of some lower-dimensional Lie algebras, with the goal of contributing to progress in any of these fields, particularly in physics and engineering.