A conditional dichotomy theorem for stochastic processes with independent increments

Let X(t) and Y(t) be two stochastically continuous processes with independent increments over [0, T] and Lévy spectral measures M_t and N_t, respectively, and let the “time-jump” measures M and N be defined over [0, T] × $\text{R}$?{0} by M((t₁, t₂] × A) = M_t2(A) ? M_t1(A) and N((T₁, t₂] × A) = N_t2(A) ? N_t1(A). Under the assumption that M is equivalent to N, it is shown that the measures induced on function space by X(t) and Y(t) are either equivalent or orthogonal, and necessary and sufficient conditions for equivalence are given. As a corollary a complete characterization of the set of admissible translates of such processes is obtained: a function f is an admissible translate for X(t) if and only if it is an admissible translate for the Gaussian component of X(t). In particular, if X(t) has no Gaussian component, then every nontrivial translate of X(t) is orthogonal to it.     

A limit theorem for perturbed operator semigroups with applications to random evolutions

Let U(t) and S(t) be strongly continuous contraction semigroups on a Banach space L with infinitesimal operators A and B, respectively. Suppose the closure of A + αB generates a semigroup T_α(t). The behavior of T_α(t) as α goes to infinity is examined. In particular, suppose S(t) converges strongly to P. If the closure of PA generates a semigroup T(t) on $\text{R}$(P), then T_α(t) goes to T(t) on $\text{R}$(P). If PA = 0 and if BVf = ?f for fε$\text{N}$(P), conditions are given that imply T_α(αt) converges on $\text{R}$(P) to a semigroup generated by the closure of PAVA.The results are used to obtain new and known limit theorems for random evolutions, which in turn give approximation theorems for diffusion processes.     

Algebraic properties of codimension series of PI-algebras

Let c _n (R), n = 0, 1, 2, …, be the codimension sequence of the PI-algebra R over a field of characteristic 0 with T-ideal T(R) and let c(R, t) = c ₀(R) + c ₁(R)t + c ₂(R)t ² + … be the codimension series of R (i.e., the generating function of the codimension sequence of R). Let R ₁,R ₂ and R be PI-algebras such that T(R) = T(R1)T(R ₂). We show that if c(R ₁, t) and c(R ₂, t) are rational functions, then c(R, t) is also rational. If c(R ₁, t) is rational and c(R ₂, t) is algebraic, then c(R, t) is also algebraic. The proof is based on the fact that the product of two exponential generating functions behaves as the exponential generating function of the sequence of the degrees of the outer tensor products of two sequences of representations of the symmetric groups S _n .     

Strong relative property (T) and spectral gap of random walks

We consider strong relative property (T) for pairs (Γ, G) where Γ acts on G. If N is a connected nilpotent Lie group and Γ is a group of automorphisms of N, we choose a finite index subgroup Γ ⁰ of Γ and obtain that (Γ , [Γ ⁰, N]) has strong relative property (T) provided Zariski-closure of Γ has no compact factor of positive dimension. We apply this to obtain the following: Let G be a connected Lie group with solvable radical R and a semisimple Levi subgroup S. If S _nc denotes the product of noncompact simple factors of S and S _T denotes the product of simple factors in S that have property (T), then we show that (Γ , R) or ${(\Gamma S_{T}, \overline{S_{T}R})}$ has strong relative property (T) for a ’Zariski-dense’ closed subgroup Γ of S _nc if and only if R = [S _nc , R]. We also provide some applications to the spectral gap of π (μ) =  ∫ π (g) d μ (g) where π is a certain unitary representation and μ is a probability measure.     

RU and (U,N)-implications satisfying Modus Ponens

In this paper it is investigated when some kinds of fuzzy implication functions derived from uninorms satisfy the Modus Ponens with respect to a continuous t-norm T, or equivalently, when they are T-conditionals. The study is done for RU-implications and $(U,N)$-implications with N a continuous fuzzy negation leading to a lot of solutions in both cases. For RU-implications T-conditionality only depends on the underlying t-norm of the uninorm used to derive the residual implication. On the contrary, for $(U,N)$-implications the underlying t-norm is never relevant and only the region out of the t-norm is so. Even the t-conorm can be not relevant also in some cases.     

Derivations and right ideals of algebras

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X₁,…,X_t) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X₁,…,X_t). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x₁,…,x_t))?m for all x₁,…,x_t∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(V_D) of finite rank. In addition, if f(X₁,…,X_t) is multilinear, then b can be chosen such that rank(b)?2(6t+13)m+2.     

Constant tolerance intersection graphs of subtrees of a tree

A chordal graph is the intersection graph of a family of subtrees of a host tree. In this paper we generalize this. A graph G=(V,E) has an (h,s,t)-representation if there exists a host tree T of maximum degree at most h, and a family of subtrees {S_v}_v∈V of T, all of maximum degree at most s, such that uv∈E if and only if |S_u∩S_v|?t. For given h,s, and t, there exist infinitely many forbidden induced subgraphs for the class of (h,s,t)-graphs. On the other hand, for fixed h?s?3, every graph is an (h,s,t)-graph provided that we take t large enough. Under certain conditions representations of larger graphs can be obtained from those of smaller ones by amalgamation procedures. Other representability and non-representability results are presented as well.     

The prime dicompletion of a di-uniformity on a plain texture

Working within a plain texture (S,S), the authors construct a completion of a dicovering uniformity υ on (S,S) in terms of prime S-filters. In case υ is separated, a separated completion is then obtained using the T₀-quotient, and it is shown that this construction produces a reflector. For a totally bounded di-uniformity it is verified that these constructions lead to dicompactifications of the uniform ditopology. A condition is given under which complementation is preserved on passing to these completions, and an example on the real texture (R,R,ρ) is presented.     

Stochastic integral representation of some martingales

Let {X_t} be a continuous square integrable martingale. Denote its increasing (natural) process by {A_t}. Let S_t, T_t be the left and right inverses of A_t, respectively. Then for any square integrable martingale {Y_t} defined on {X_t}, Y_t = ∝₀^tψ_sdX_s, R₀ < t < S_∞ where S_∞ = lim_t→∞S_t, R₀ = inf {t: X_t ≠ 0} provided that Y(T(t)) is σ(X(T(s)): s ? t)-measurable. All martingales are assumed to be zero at t = 0. Brownian motion and Poisson processes are considered also.     

The genus of a module

Let R be a Dedekind domain satisfying the Jordan-Zassenhaus theorem (e.g., the ring of integers in a number field) and Λ a module finite R-algebra. We extend classical results of Jacobinski, Roiter, and Drozd on orders and lattices. In particular, it is shown that the genus of a finitely generated Λ-module M is finite. Moreover, given M, there exist a positive integer t and a finite extension S of R such that a Λ-module N is the genus of M if and only if M^(t) ? N^(t) if and only if M ? S ? N ? S.     

Hyperamarts: Conditions for regularity of continuous parameter processes

The regularity of trajectories of continuous parameter process (X_t)_t∈R+ in terms of the convergence of sequence E(X_Tn) for monotone sequences (T_n) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (X_n)_1≤n≤∞ of integrable random variables, lim X_n exists and is equal to X_∞ and (X_T) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(X_Tn) = E(X_∞) for every increasing sequence (T_n) of extended simple stopping times converging to ∞. By applying these discrete parameter theorems, convergence theorems about continuous parameter processes are obtained. For example, it is shown that a progressive, optionally separable process (X_t)_t∈R+ with E{X_T} < ∞ for every bounded stopping time T is right continuous if lim E(X_Tn) = E(X_T) for every bounded stopping time T and every descending sequence (T_n) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained.     

On universally catenarian pairs

If 1≤n<∞ and R⊆S are integral domains, then (R,S) is called an n-catenarian pair if for each intermediate ring T (that is each ring T such that R⊆T⊆S) the polynomial ring in n indeterminates, T[n] is catenarian. This implies that (R,S) is m-catenarian for all m<n. The main purpose of this paper is to prove that 1-catenarian and universally catenarian pairs are equivalent in several cases. An example of a 1-catenarian pair which is not 2-catenarian is given.     

Construction of vector lists and isomorph rejection

It is observed that if Δ is a system of orbit representatives for the action of a finite group G on a G-stable subset L of a finite set S and if F is a field of characteristic zero, then F^S is an algebra of dimension |S|. Furthermore, if S = R^D, the set of functions from a finite set D to a finite set R, then F^S has a multilinear structure. A general problem is stated: Given linear operators T₁ and T₂, construct vectors v₁, … v_t?F^S such that T₂I_Δ = T₁(v₁ + … + v_t) where l_Δ is the indicator or characteristic function of Δ. (Note that this construction gives, in some sense, a solution to the problem of isomorph rejection.) For appropriate choices of T₁ and T₂, two approaches to this construction problem are considered. These are the principle of inclusion-exclusion and backtrack computer programming. In particular, these approaches are discussed when S = R^D and the vectors to be constructed are pure or homogeneous tensors.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

A semialgebraic closure for commutative algebra

Let A⊂R be rings containing the rationals. In R let S be a multiplicatively closed subset such that 1∈S and 0∉S, T a preorder of R (a proper subsemiring containing the squares) such that S⊂T and I an A-submodule of R. Define ρ(I) (or ρ_S,T(I)) to be

ρ(I)={a∈R|sa^2m+t∈I^2m for some m∈N,s∈S and t∈T}.     

On the direct image of intersections in exact homological categories

Given a regular epimorphism f:X?Y in an exact homological category C, and a pair (U,V) of kernel subobjects of X, we show that the quotient (f(U)∩f(V))/f(U∩V) is always abelian. When C is nonpointed, i.e. only exact protomodular, the translation of the previous result is that, given any pair (R,S) of equivalence relations on X, the difference mappingδ:Y/f(R∩S)?Y/(f(R)∩f(S)) has an abelian kernel relation. This last result actually holds true in any exact Mal'cev category. Setting Y=X/T, this result says that the difference mapping determined by the inclusion T∪(R∩S)?(T∪R)∩(T∪S) has an abelian kernel relation, which casts a new light on the congruence distributive property.     

Intersection of Yager’s implications with QL and D-implications

The intersection of the different classes of implications is one of the most popular topics nowadays due to the large number of construction methods of these operators. In this paper, we deal with the characterization of the intersection of Yager’s implications with QL and D-implications. Some initial steps have already been made with the intersection of Yager’s implications with (S, N), R and QL-implications, however some questions remain unanswered. In particular, we solve an open problem related to the characterization of those implications that are both QL-implications and f-generated implications with f(0) < ∞, fully determining the expression of the QL-implications generated by a continuous t-conorm that belong to the considered intersection. Furthermore, we perform a similar study for D-implications and finally, we study the intersection of Yager’s implications with their φ-conjugates.     

HLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATIONS

We show the existence of dissipative H¨older continuous solutions of the Boussinesq equations. More precise, for any β∈(0,1/5), a time interval [0, T ] and any given smooth energy profile e : [0, T ] →(0, ∞), there exist a weak solution(v, θ) of the 3 d Boussinesq equations such that(v, θ) ∈ Cβ(T~3× [0, T ]) with e(t) =′his T~3|v(x, t)|~2 dx for all t ∈ [0, T ]. Textend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30].     

The normalizer group of an ergodic automorphism of type III and the commutant of an ergodic flow

It was proved by W. Krieger that for an ergodic automorphism T of type III there is an ergodic flow (F_t)_tR, whose isomorphism class uniquely determines the weak equivalence class of T. It will be shown that if N[T] is the normalizer group of the full group [T] and [ ] is the closure of [T], then the quotient group is topologically isomorphic to the commutant, c((F_t)_tR), of the flow (F_t)_tR. For various examples of flows (F_t)_tR the commutant c((F_t)_tR) is studied.