An ω-hierarchy of axiom systemZF

In this paper we develop a sequenceZ ₀,...,Z _n ,...(n) of axiom systems for set theory, such that (1) the consistency of any system within the sequence is provable in its succeeding systems, (2) the first system in the sequence is Zermelo 's systemZ and the union of all systems in the sequence is justZF.     

Stability of <Emphasis Type="Italic">n</Emphasis>-particle pseudorelativistic systems

For a system Z_n of n identical pseudorelativistic particles, we show that under some restrictions on the pair interaction potentials, there is an infinite sequence of numbers n_s, s = 1, 2,..., such that the system Z_n is stable for _n = n_s, and the inequality sup _sn_s+1n _s ⁻¹ < + ∞ holds. Furthermore, we show that if the system Z_n is stable, then the discrete spectrum of the energy operator for the relative motion of the system Z_n is nonempty for some values of the total momentum of the particles in the system. The stability of n-particle systems was previously studied only for nonrelativistic particles. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 528–537, September, 2007.     

The pauli principle,stability, and bound states in systems of identical pseudorelativistic particles

Based on analyzing the properties of the Hamiltonian of a pseudorelativistic system Z_n of n identical particles, we establish that for actual (short-range) interaction potentials, there exists an infinite sequence of integers n_s, s = 1, 2, …, such that the system is stable and that sup _s n_s+1 n_s⁻¹ < + ∞. For a stable system Z_n, we show that the Hamiltonian of relative motion of such a system has a nonempty discrete spectrum for certain fixed values of the total particle momentum. We obtain these results taking the permutation symmetry (Pauli exclusion principle) fully into account for both fermion and boson systems for any value of the particle spin. Similar results previously proved for pseudorelativistic systems did not take permutation symmetry into account and hence had no physical meaning. For nonrelativistic systems, these results (except the estimate for n_s+1 n_s⁻¹ ) were obtained taking permutation symmetry into account but under certain assumptions whose validity for actual systems has not yet been established. Our main theorem also holds for nonrelativistic systems, which is a substantial improvement of the existing result. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 116–129, October, 2008.     

Subsequences of normal sequences

In this paper, we characterize a set of indices τ={τ(0)<τ(1)<…} such that forany normal sequence (α(0), α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ(1)),…) is a normal sequence of the same type. Assume that_n→∞. Then, we prove that τ has this property if and only if the 0–1 sequence (θ _τ (0), whereθ _τ (i)=1 or 0 according asi∈{τ(j);j=0, 1,…} or not, iscompletely deterministic in the sense of B. Weiss.     

A unified characterization of reproducing systems generated by a finite family,II

By a “reproducing” method forH =L ²(ℝ ⁿ ) we mean the use of two countable families {e _α : α ∈A}, {f _α : α ∈A}, inH, so that the first “analyzes” a function h ∈H by forming the inner products {<h,e _α >: α ∈A} and the second “reconstructs” h from this information:h = Σα∈A <h,e _α >:f _α. A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations. A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that “unifies” all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on ℝ ⁿ . Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices.     

Limit theorems for a recursive maximum process with location-dependent periodic intensity-parameter

We investigate the recursive sequence Z _n : =  max {Z _n − 1,λ(Z _n − 1)X _n } where X _n is a sequence of iid random variables with exponential distributions and λ is a periodic positive bounded measurable function. We prove that the Césaro mean of the sequence λ(Z _n ) converges toward the essential minimum of λ. Subsequently we apply this result and obtain a limit theorem for the distributions of the sequence Z _n . The resulting limit is a Gumbel distribution.     

On linguistic dynamical systems,families of graphs of large girth,and cryptography

The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K, i.e., a family F of nonlinear polynomial maps f _α : K ⁿ → K ⁿ depending on “time” α ∈ {K − 0} such that f _α ⁻¹ = f _−αM, the relation f _α1 (x) = f _α2 (x) for some x ∈ Kⁿ implies α₁ = α₂, and each map f _α has no invariant points. The neighborhood {f _α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kⁿ. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α₁, υ, α₂), s ≤ d, where α_i + α_i+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ _a = f _α1 × ⋯ × f _αs (υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems L_n(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system L_n(q). If K = F_q, the graphs L(n, F_q) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops to 4. We introduce some other families of graphs of large girth related to the dynamical systems L_n(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 214–234.     

Mixing and sweeping out

A weakly mixing transformationT and a sequence (d _n) are constructed such thatT is uniformly mixing on (d _n),T is uniformly sweeping out on ([αd _n]) for allα∈(0, 1), and for all rationalα∈(0, 1)T is not mixing on ([αd _n]).     

Parameter estimation for a misspecified arma model with infinite variance innovations

We consider a general linear model , where the innovations Z_t belong to the domain of attraction of an α-stable law for α<2, so that neither Z_t nor X_t have a finite variance. We do not assume that (X_t) is a standardARMA process of the form φ(B)X_t=ϕ(B)Z_t, but we fit anARMA process of a given order to the data X₁,...,X_n by estimating the coefficients of φ and ϕ. Given that (X_t) is anARMA process, it has been proved that the Whittle estimator is a consistent estimator of the true coefficients of ϕ and φ. Moreover, it then has a heavytailed limit distribution and the rate of convergence is (n/logn)^1/α, which compares favorably with the L² situation with rate . In this note we study the limit properties of the Whittle estimator when the underlying model is not necessarily anARMA process. Under general conditions we show that the Whittle estimate converges in probability. It converges weakly to a distribution which does not have a finite moment of order a and the rate of convergence is again (n/logn)^1/α. We also give an analytic expression for the limit distribution. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

On invariant additive subgroups

Suppose thatR is a prime ring with the centerZ and the extended centroidC. An additive subgroupA ofR is said to be invariant under special automorphisms if (1+t)A(1+t)⁻¹ ⊆A for allt ∈R such thatt ²=0. Assume thatR possesses nontrivial idempotents. We prove: (1) If chR ≠ 2 or ifRC ≠C ₂, then any noncentral additive subgroup ofR invariant under special automorphisms contains a noncentral Lie ideal. (2) If chR=2,RC=C ₂ andC ≠ {0, 1}, then the following two conditions are equivalent: (i) any noncentral additive subgroup invariant under special automorphisms contains a noncentral Lie ideal; (ii) there isα ∈Z / {0} such thatα ² Z ⊆ {β ²:β ∈Z}.     

Approximation d’un nombre réel par des unités algébriques

 In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .     

Quadratically Hyponormal Recursively Generated Weighted Shifts Need Not Be Positively Quadratically Hyponormal

We study a class of weighted shifts W _α defined by a recursively generated sequence α ≡ α₀, … , α _m−2, (α _m−1, α _m , α _m+1)^∧ and characterize the difference between quadratic hyponormality and positive quadratic hyponormality. We show that a shift in this class is positively quadratically hyponormal if and only if it is quadratically hyponormal and satisfies a finite number of conditions. Using this characterization, we give a new proof of [12, Theorem 4.6], that is, for m = 2, W _α is quadratically hyponormal if and only if it is positively quadratically hyponormal. Also, we give some new conditions for quadratic hyponormality of recursively generated weighted shift W _α (m ≥ 2). Finally, we give an example to show that for m ≥ 3, a quadratically hyponormal recursively generated weighted shift W _α need not be positively quadratically hyponormal.     

Approximation d’un nombre réel par des unités algébriques

 In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .

(Received 14 March 2000; in revised form 16 November 2000)     

On the distribution of powers of a complex number

Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo ${\mathbb{Z}[i],}Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo \mathbbZ[i],{\mathbb{Z}[i],} where i=?{-1}{i=\sqrt{-1}} and \mathbbZ[i]=\mathbbZ+i\mathbbZ{\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}} is the ring of Gaussian integers. For any z ? \mathbbC,{z\in \mathbb{C},} one may naturally call the quantity z modulo \mathbbZ[i]{\mathbb{Z}[i]} the fractional part of z and write {z} for this, in general, complex number lying in the unit square S:={z ? \mathbbC:0 ￡ \mathfrakR(z),\mathfrakJ(z) < 1 }{S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) <1 \}}. We first show that if α is a complex non-real number which is algebraic over \mathbbQ{\mathbb{Q}} and satisfies |α| > 1 then there are two limit points of the sequence {ξ α ⁿ  +ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over \mathbbQ(i){\mathbb{Q}(i)} all lie in the unit disc |z| ≤  1 and x ? \mathbbQ(a,i).{\xi\in\mathbb{Q}(\alpha,i).} Then we prove a result in the opposite direction which implies that, for any fixed a ? \mathbbC{\alpha\in\mathbb{C}} of modulus greater than 1 and any sequence z_n ? \mathbbC,n=0,1,2,...,{z_n\in\mathbb{C},n=0,1,2,\dots,} there exists x ? \mathbbC{\xi \in \mathbb{C}} such that the numbers ξ α ⁿ −z _n , n = 0, 1, 2, . . . , all lie ‘far’ from the lattice \mathbbZ[i]{\mathbb{Z}[i]}. In particular, they all can be covered by a union of small discs with centers at (1+i)/2+\mathbbZ[i]{(1+i)/2+\mathbb{Z}[i]} if |α| is large.     

Potential theory of subordinate killed Brownian motion in a domain

 Subordination of a killed Brownian motion in a bounded domain D⊂ℝ ^d via an α/2-stable subordinator gives a process Z _t whose infinitesimal generator is −(−Δ| _D )^α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we study the properties of the process Z _t in a Lipschitz domain D by comparing the process with the rotationally invariant α-stable process killed upon exiting D. We show that these processes have comparable killing measures, prove the intrinsic ultracontractivity of the generator of Z _t , prove the intrinsic ultracontractivity of the semigroup of Z _t , and, in the case when D is a bounded C ^1,1 domain, obtain bounds on the Green function and the jumping kernel of Z _t . Received: 4 April 2002 / Revised version: 1 July 2002 / Published online: 19 December 2002 This work was completed while the authors were in the Research in Pairs program at the Mathematisches Forschungsinstitut Oberwolfach. We thank the Institute for the hospitality. The research of the first author is supported in part by NSF Grant DMS-9803240. The research of the second author is supported in part by MZT grant 037008 of the Republic of Croatia. Mathematics Subject Classification (2000): Primary 60J45; Secondary 60J75, 31C25 Key words or phrases: Killed Brownian motions – Stable processes – Subordination – Fractional Laplacian     

On driftless one-dimensional SDE’s with respect to stable Levy processes

The time-dependent SDE dX _t = b(t, X _t−)dZ _t with X ₀ = x ₀ ∈ ℝ, and a symmetric α-stable process Z, 1 < α ⩽ 2, is considered. We study the existence of nonexploding solutions of the given equation through the existence of solutions of the equation in class of time change processes, where is a symmetric stable process of the same index α as Z. The approach is based on using the time change method, Krylov’s estimates for stable integrals, and properties of monotone convergence. The main existence result extends the results of Pragarauskas and Zanzotto (2000) for 1 < α < 2 and those of T. Senf (1993) for α = 2. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 517–531, October–December, 2007.     

Factorization in dedekind domains with finite class group

LetD be a Dedekind domain. It is well known thatD is then an atomic integral domain (that is to say, a domain in which each nonzero nonunit has a factorization as a product of irreducible elements). We study factorization properties of elements in Dedekind domains with finite class group. IfD has the property that any factorization of an elementα into irreducibles has the same length, thenD is called a half factorial domain (HFD, see [41]). IfD has the property that any factorization of an elementα into irreducibles has the same length modulor (for somer>1), thenD is called a congruence half factorial domain of orderr. In Section I we consider some general factorization properties of atomic integral domains as well as the interrelationship of the HFD and CHFD property in the Dedekind setting. In Section II we extend many of the results of [41], [42] and [36] concerning HFDs when the class group ofD is cyclic. Finally, in Section III we consider the CHFD property in detail and determine some basic properties of Dedekind CHFDs. IfG is any Abelian group andS any subset ofG−[0], then {G, S} is called a realizable pair if there exists a Dedekind domainD with class groupG such thatS is the set of nonprincipal classes ofG which contain prime ideals. We prove that for a finite abelian groupG there exists a realizable pair {G, S} such that any Dedekind domain associated to {G, S} is CHFD for somer>1 but not HFD if and only ifG is not isomorphic toZ ₂,Z ₂,Z ₂ ⊕Z ₂, orZ ₃ ⊕Z ₃. The first author received support under the John M. Bennett Fellowship at Trinity University and also gratefully acknowledges the support of The University of North Carolina at Chapel Hill.     

Temperature inversion symmetry in gauge-Higgs unification models

We study the temperature inversion symmetry R → 1/T for the finite-temperature effective potential of the N=1, d=5 supersymmetric SU(3) _c ×SU(3) _w model on the orbifold S ¹ /Z ₂ . For the value of the Wilson line parameter α = 1 (SU(2) _L breaks to U′(1)), we show that the effective potential contains a symmetric part and an antisymmetric part under ξ → 1/ξ, ξ = RT. For α = 0 (SU(2) _L is preserved in this case), we find that the only contribution to the effective potential that breaks the temperature inversion symmetry comes from the fermions in the fundamental representation of the gauge group with the Z ₂ parities (+, +) or (−,−). This is interesting because it implies that the bulk effective potential corresponding to models with fundamental fermions localized at a fixed point in the orbifold (and models with no bulk fundamental fermions) has the temperature inversion symmetry. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 109–130, April, 2009.     

Division algebras with a projective basis

Letk be any field andG a finite group. Given a cohomology class α∈H ²(G,k ^*), whereG acts trivially onk ^*, one constructs the twisted group algebrak ^αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z _n×Z_n). This paper has two main results: First we prove that ifD=k ^α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k ^α G is a product of cyclic algebras. Furthermore, ifD _p is ap-primary factor ofD, thenD _p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD _p, p odd. Ifp=2 we show thatD ₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z ₂×Z₂n.     

Completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight

We study the problem on the completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight. Theorems of general form are obtained. In particular, the necessary and sufficient conditions on α, β, q ₁, and q ₂ for which the known orthogonal systems are everywhere dense in asymmetric spaces L _(α,β);q ([0, 1]) are found. Theorem. Let α, β, q ₁, q ₂ ∈ [1,+∞]. The following orthogonal systems are dense in asymmetric spaces L _(α,β);q ([0, 1]) if and only if either max{α, β, q ₁, q ₂} < + ∞ or max {α, β} < +∞, q ₁ = q ₂ = +∞: trigonometric, algebraic, Haar’s system, Meyer’s system of wavelets, Shannon-Kotel’nikov wavelets, Stromberg and Lemarie-Battle wavelets, the Walsh system, and the Franklin system. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.