Reflexive relations on algebras with Boolean lattice reducts

For any algebra A, let Ref(A) be the algebra of compatible reflexive binary relations on A under intersection, composition, and converse with the universal and identity relations as constants. We characterize all Ref(A) where A is a finite algebra with a Boolean lattice reduct.     

Monadic MV-algebras I: a study of subvarieties

In this paper, we study and classify some important subvarieties of the variety of monadic MV-algebras. We introduce the notion of width of a monadic MV-algebra and we prove that the equational class of monadic MV-algebras of finite width k is generated by the monadic MV-algebra [0, 1] ^k . We describe completely the lattice of subvarieties of the subvariety ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ generated by [0, 1] ^k . We prove that the subvariety generated by a subdirectly irreducible monadic MV-algebra of finite width depends on the order and rank of ?A, the partition associated to A of the set of coatoms of the boolean subalgebra B(A) of its complemented elements, and the width of the algebra. We also give an equational basis for each proper subvariety in ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ . Finally, we give some results about subvarieties of infinite width.     

Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL ²(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleH _b (g)=(I?P)bg. This paper continues earlier investigations of the authors and others by determining conditions under whichH _b is bounded, compact, or lies in the Schatten-von Neumann idealS _p , 1<p<∞     

Iterated vanishing cycles

If A ^? is a bounded, constructible complex of sheaves on a complex analytic space X, and ${f : X \rightarrow \mathbb{C}}$ and ${g : X \rightarrow \mathbb{C}}$ are complex analytic functions, then the iterated vanishing cycles φ _g [?1](φ _f [?1]A ^?) are important for a number of reasons. We give a formula for the stalk cohomology H*(φ _g [?1]φ _f [?1]A ^?) _x in terms of relative polar curves, algebra, and Morse modules of A ^?.     

Boolean topological graphs of semigroups: the lack of first-order axiomatization

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class ?? of algebras let G(??)={G(A)∣A∈??}. Assume that ?? is a class of semigroups possessing a nontrivial member with a neutral element and let ? be the universal Horn class generated by G(??). We prove that the Boolean core of ?, i.e., the topological prevariety generated by finite members of ? equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ?. We derive analogous results when ?? is a class of monoids or groups with a nontrivial member.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

MV-closures of Wajsberg hoops and applications

In this paper we construct, given a Wajsberg hoop A, an MV-algebra MV(A) such that the underlying set A of A is a maximal filter of MV(A) and the quotient MV(A)/A is the two element chain. As an application we provide a topological duality for locally finite Wajsberg hoops based on a previously known duality for locally finite MV-algebras. We also give another duality for k-valued Wajsberg hoops based on a different representation of k-valued MV-algebras and show the relation to the first duality. We also apply this construction to give a topological representation for free k-valued Wajsberg hoops.     

The Equivalence Between Seven Classes of Wavelet Multipliers and Arcwise Connectivity They Induce

Let A be a d×d expansive matrix with |detA|=2. An A-wavelet is a function $\psi\in L^{2}(\mathbb{R}^{d})$ such that $\{2^{\frac{j}{2}}\psi(A\cdot-k):\,j\in \mathbb{Z},\,k\in \mathbb{Z}^{d}\}$ is an orthonormal basis for $L^{2}(\mathbb{R}^{d})$ . A measurable function f is called an A-wavelet multiplier if the inverse Fourier transform of $f\hat{\psi}$ is an A-wavelet whenever ψ is an A-wavelet, where $\hat{\psi}$ denotes the Fourier transform of ψ. A-scaling function multiplier, A-PFW multiplier, semi-orthogonal A-PFW multiplier, MRA A-wavelet multiplier, MRA A-PFW multiplier and semi-orthogonal MRA A-PFW multiplier are defined similarly. In this paper, we prove that the above seven classes of multipliers are equivalent, and obtain a characterization of them. We then prove that if the set of all A-wavelet multipliers acts on some A-scaling function (A-wavelet, A-PFW, semi-orthogonal A-PFW, MRA A-wavelet, MRA A-PFW, semi-orthogonal MRA A-PFW), the orbit is arcwise connected in $L^{2}(\mathbb{R}^{d})$ , and that if the generator of an orbit is an MRA A-PFW, the orbit is equal to the set of all MRA A-PFWs whose Fourier transforms have same module, and is also equal to the set of all MRA A-PFWs with corresponding pseudo-scaling functions having the same module of their Fourier transforms.     

Weighted norm inequalities for the vectorvalued Hardy-Littlewood maximal functions of one parameter rectangles

Let N denote the Hardy-Littlewood maximal operator for the familyR of one parameter rectangles. In this paper, we obtain that for 1 _w ^p (l^r) to L _W ^P (l^r) if and only if w ∈ A_P(R); for 1≤p<∞, N is bounded from L _W ^P (l^r) to weak L _W ^P (l^r) if and only if W ∈ A_P(R). Here we say W∈A_p (1), if $$\begin{gathered} \mathop {sup}\limits_{R \in R} \left( {\tfrac{1}{{|R|}}\smallint _r wdx} \right)\left( {\tfrac{1}{{|R|}}\smallint _R w^{ - 1/(p - 1)} dx} \right)^{p - 1}< \infty ,1< p< \infty , \hfill \\ (Nw)(x) \leqslant Cw(x)a.e.,p = 1 \hfill \\ \end{gathered} $$ ,     

Sylow-metacyclic groups andQ-admissibility

A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2 ^a 3 ^b and to a list of known “almost simple” groups.     

Fully invariant and verbal congruence relations

A congruence relation θ on an algebra A is fully invariant if every endomorphism of A preserves θ. A congruence θ is verbal if there exists a variety ${\mathcal{V}}$ such that θ is the least congruence of A such that ${{\bf A}/\theta \in \mathcal{V}}$ . Every verbal congruence relation is known to be fully invariant. This paper investigates fully invariant congruence relations that are verbal, algebras whose fully invariant congruences are verbal, and varieties for which every fully invariant congruence in every algebra in the variety is verbal.     

On diversity and stability of unit bases for the Euclidean metric

We prove the existence of a family Ω(n) of 2 ^c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E ⁿ , 1) of the Euclidean space E ⁿ , n ≥ 2, such that (a) for each graph G ? Ω(n), any homomorphism of G to (E ⁿ , 1) is an isometry of E ⁿ ; moreover, for each subgraph G ₀ of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G ₀ to (E ⁿ , 1) is an isometry (of the set of the vertices of G ₀); (b) each graph G ? Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.     

Strongly dependent theories

We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ? ? and |I|≥?_|T|+(µ), then some J ? I of cardinality µ⁺ is an indiscernible sequence over A.     

Euler method vs. GESS method for dynamical systems

In this paper we introduce GESS method and show that dynamics of the systemy′ =A(s,t,y)y is more faithfully approximated by GESS method than by Euler method. Numerical experiments are given for the comparison of GESS method with Euler method.     

Characterization and Connectivity of Generalized Filters in L 2(ℝ d )

Let A be a d×d real expansive integer matrix (i.e., a matrix with real entries whose eigenvalues are all of modules greater than one) with |det?A|=2, and let m (which is called A-dilation generalized filter) be a 2π? ^d periodic function with the property that |m(s)|²+|m(s+2π h ₂)|²=1, where h ₂∈(A ^τ )^?1? ^d ?? ^d . In this paper, we characterize the set of all A-dilation generalized filters and show that this set is path-connected in $L^{2}({\mathbb{T}}^{d})$ -norm by using the technique of filter multipliers. We also obtain an equivalent condition for an A-dilation generalized filter to be an A-dilation low pass filter. These extend the results of Manos Papadakis et al. from one dimensional case to high dimensions and matrix dilations cases.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

L p -Norms, Log-Barriers and Cramer Transform in Optimization

We show that the Laplace approximation of a supremum by L ^p -norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem P and its dual P ^* appear naturally when using this simple approximation technique for the value function g of P or its Legendre–Fenchel conjugate g ^*. In addition, minimizing the LBF of the dual P ^* is just evaluating the Cramer transform of the Laplace approximation of g. Finally, this technique permits to sometimes define an explicit dual problem P ^* in cases when the Legendre–Fenchel conjugate g ^* cannot be derived explicitly from its definition.     

Time-Varying Fractionally Integrated Processes with Finite or Infinite Variance and Nonstationary Long Memory

Recently, Philippe et al. (C.R. Acad. Sci. Paris. Ser. I 342, 269–274, 2006; Theory Probab. Appl., 2007, to appear) introduced a new class of time-varying fractionally integrated filters A(d)x _t =∑ _j=0 ^∞ a _j (t)x _t?j , B(d)x _t =∑ _j=0 ^∞ b _j (t)x _t?j depending on arbitrary given sequence d=(d _t ,t∈?) of real numbers, such that A(d)^?1=B(?d), B(d)^?1=A(?d) and such that when d _t ≡d is a constant, A(d)=B(d)=(1?L) ^d is the usual fractional differencing operator. Philippe et al. studied partial sums limits of (nonstationary) filtered white noise processes X _t =B(d)ε _t and Y _t =A(d)ε _t in the case when (1) d is almost periodic having a mean value $\bar{d}\in (0,1/2)$ , or (2) d admits limits d _±=lim? _t→±∞ d _t ∈(0,1/2) at t=±∞. The present paper extends the above mentioned results of Philippe et al. into two directions. Firstly, we consider the class of time-varying processes with infinite variance, by assuming that ε _t ,t∈? are iid rv’s in the domain of attraction of α-stable law (1<α≤2). Secondly, we combine the classes (1) and (2) of sequences d=(d _t ,t∈?) into a single class of sequences d=(d _t ,t∈?) admitting possibly different Cesaro limits $\bar{d}_{\pm}\in(0,1-(1/\alpha))$ at ±∞. We show that partial sums of X _t and Y _t converge to some α-stable self-similar processes depending on the asymptotic parameters $\bar{d}_{\pm}$ and having asymptotically stationary or asymptotically vanishing increments.     

Selection theorems,based on generalized variation and oscillation

In this paper we prove selection theorems for everywhere and almost everywhere convergent subsequences, based on the notion of uniformly limited oscillation, ofA, Λ-oscillation and ofA, Λ-variation, whereA is a system of intervals and Λ a sequence of reals. By this, we generalize and strengthen the selection theorems of Schrader (for oscillation) and of Waterman (for Λ-variation).