A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and basis form in terms of Malliavin derivative on a projective Fock scale, and their uniform continuity and QS differentiability with respect to the inductive limit convergence is proved. A new form of QS calculus based on an inductive ⋆–algebraic structure in an indefinite spac...

Definition 1.1. Given a ring R, a genus with values in R is a ring homomorphism, Ω ⊗Q→ R, where Ω is the G-bordism ring. For example, the Â-genus and L-genus are ring maps from Ω⊗Q→ Q. The Atiyah-Singer theorem shows that  can be refined to a genus Ω⊗Q→ Z. The L-genus (or signature) is defined on Ω⊗Q, and the Todd genus is defined on the complex cobordism category. We can define genera via mul...

Graph homomorphism has been studied intensively. Given an m ×m symmetric matrix A, the graph homomorphism function is defined as

We consider the question of the existence of homomorphisms between Gn,p and odd cycles when p = c/n, 1 < c ≤ 4. We show that for any positive integer l, there exists ε = ε(l) such that if c = 1 + ε then w.h.p. Gn,p has a homomorphism from Gn,p to C2l+1 so long as its odd-girth is at least 2l+1. On the other hand, we show that if c = 4 then w.h.p. there is no homomorphism from Gn,p to C5. Note t...

Let (M,ω) be a closed symplectic manifold and Ham(M,ω) the group of Hamiltonian diffeomorphisms of (M,ω). Then the Seidel homomorphism is a map from the fundamental group of Ham(M,ω) to the quantum homology ring QH∗(M ; Λ). Using this homomorphism we give a sufficient condition for when a nontrivial loop ψ in Ham(M,ω) determines a nontrivial loop ψ × idN in Ham(M ×N, ω ⊕ η), where (N, η) is a c...

There are many instances of the principle that if A is an algebra of functions on X, then every ring homomorphism A → C is given by evaluation at a particular x ∈ X. Examples are the nullstellensatz in algebraic geometry and the result of Gelfand when X is a compact Hausdorff space. In these cases one can regard X as being included in Hom(A, C) as the set of those f : A → C which satisfy the se...

We develop an obstruction theory for homotopy of homomorphisms f, g : M → N between minimal differential graded algebras. We assume that M = ΛV has an obstruction decomposition given by V = V0⊕V1 and that f and g are homotopic on ΛV0. An obstruction is then obtained as a vector space homomorphism V1 → H(N ). We investigate the relationship between the condition that f and g are homotopic and th...

We investigate linear operators between C?-algebras which approximately preserve involution and orthogonality, the latter meaning that for some ?>0 we have ??(x)?(y)????x??y? all positive x,y with xy=0. establish structural properties of such maps concerning approximate Jordan-like equations almost commutation relations. In situations (e.g. when codomain is finite-dimensional), show ? can be ap...

The perturbation semigroup was first defined in the case of $*$-algebras by Chamseddine, Connes and van Suijlekom. In this paper, we take $\mathcal{E}$ as a concrete operator system with unit. We give definition gauge group $\mathcal{G}(\mathcal{E})$ $\mathcal{E}$, after that closed respect to Haagerup tensor norm. also show there is continuous homomorphism from collection unital completely bou...

Let R be an oriented compact surface without boundary of genus exceeding one, and let θ : π1(R;O) → Γ ⊂ PSL(2,C) be a homomorphism of its fundamental group onto a nonelementary group Γ of Möbius transformations. We present a complete, self-contained proof of the following facts: (i) θ is induced by a complex projective structure for some complex structure on R if and only if θ lifts to a homomo...