We treat equivariant completions of toric contraction morphisms as an application of the toric Mori theory. For this purpose, we generalize the toric Mori theory for non-Q-factorial toric varieties. So, our theory seems to be quite different from Reid’s original combinatorial toric Mori theory. We also explain various examples of non-Q-factorial contractions, which imply that the Q-factoriality...

In the paper we begin a description of functional methods of quantum field theory for systems of interacting q-particles. These particles obey exotic statistics and are the q-generalization of the colored particles which appear in many problems of condensed matter physics, magnetism and quantum optics. Motivated by the general ideas of standard field theory we prove the q-functional analogues o...

In the paper we begin a description of functional methods of quantum field theory for systems of interacting q-particles. These particles obey exotic statistics and are the q-generalization of the colored particles which appear in many problems of condensed matter physics, magnetism and quantum optics. Motivated by the general ideas of standard field theory we prove the q-functional analogues o...

We develop a finite KK-theory of C∗-algebras following ArlettazH.Inassaridze’s approach to finite algebraic K-theory [1] . The BrowderKaroubi-Lambre’s theorem on the orders of the elements for finite algebraic K-theory [ , ] is extended to finite KK-theory. A new bivariant theory, called torsion KK-theory is defined as the direct limit of finite KK-theories. Such bivariant K-theory has almost a...

We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and find many theorems of classical number theory arise as particular cases of extremely general combinatorial structure laws. We then see that the relatively recently-defined q-bracket operator 〈f〉q, studied by Bloch– Okounkov, Zagier, and others for its quasimodular properties...

Matrices over a finite field having fixed rank and restricted support are a natural qanalogue of rook placements on a board. We develop this q-rook theory by defining a corresponding analogue of the hit numbers. Using tools from coding theory, we show that these q-hit and q-rook numbers obey a variety of identities analogous to the classical case. We also explore connections to earlier q-analog...

In this lecture we work exclusively with first-order logic with equality. Fix a signature σ. A theory T is a set of sentences (closed formulas) that is closed under semantic entailment, i.e., if T |= F then F ∈ T . Given a σ-structure A it is clear that the set of sentences that hold in A is a theory. We denote this theory by Th(A) and call it the theory of A. We say that a theory is complete i...

let $g$ be a non-abelian group and let $z(g)$ be the center of $g$. associate with $g$ there is agraph $gamma_g$ as follows: take $gsetminus z(g)$ as vertices of$gamma_g$ and joint two distinct vertices $x$ and $y$ whenever$yxneq yx$. $gamma_g$ is called the non-commuting graph of $g$. in recent years many interesting works have been done in non-commutative graph of groups. computing the clique...

We show that if Q is a closed, reduced, complex orbifold of dimension n such that every local group acts as a subgroup of SU(2) < SU(n), then the K-theory of the unique crepant resolution of Q is isomorphic to the orbifold K-theory of Q.

We obtain connection coefficients between q-binomial and q-trinomial coefficients. Using these, one can transform q-binomial identities into a q-trinomial identities and back again. To demonstrate the usefulness of this procedure we rederive some known trinomial identities related to partition theory and prove many of the conjectures of Berkovich, McCoy and Pearce, which have recently arisen in...