An ergodic theorem of a parabolic Anderson model driven by L��vy noise

In this paper, we study an ergodic theorem of a parabolic Andersen model driven by Lévy noise. Under the assumption that A = (a(i, j)) _i,j∈S is symmetric with respect to a σ-finite measure gp, we obtain the long-time convergence to an invariant probability measure ν _h starting from a bounded nonnegative A-harmonic function h based on self-duality property. Furthermore, under some mild conditions, we obtain the one to one correspondence between the bounded nonnegative A-harmonic functions and the extremal invariant probability measures with finite second moment of the nonnegative solution of the parabolic Anderson model driven by Lévy noise, which is an extension of the result of Y. Liu and F. X. Yang.     

Small deviations of series of independent positive random variables with weights close to exponential

Let ξ, ξ₀, ξ₁, ... be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article [1] in which the asymptotics of probabilities of small deviations of series S = Σ _j=0 ^∞ a(j)ξ _j was studied under different assumptions on the rate of decrease of the probability ?(ξ < x) as x → 0, as well as of the coefficients a(j) ≥ 0 as j → ∞. We study the asymptotics of ?(S < x) as x → 0 under the condition that the coefficients a(j) are close to exponential. In the case when the coefficients a(j) are exponential and ?(ξ < x) ～ bx ^α as x → 0, b > 0, a > 0, the asymptotics ?(S < x) is obtained in an explicit form up to the factor x ^o(1). Originality of the approach of the present paper consists in employing the theory of delayed differential equations. This approach differs significantly from that in [1].     

Graphs and ranks of monoids

For a finite monoid S, let ν(S) (ν_d(S)) denote the least number n such that there exists a graph (directed graph) Γ of order n with End(Γ)?S. Also let rank(S) be the smallest number of elements required to generate S. In this paper, we use Cayley digraphs of monoids, to connect lower bounds of ν(S) (ν_d(S)) to the lower bounds of rank(S). On the other hand, we connect upper bounds of rank(S) to upper bounds of ν(S) (ν_d(S)).     

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Let K⊂ℝ ^d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K ^Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX _i (t) = ∑ _j a(j−i) (X _j (t) −X _i (t))dt + σ (X _i (t))dB _i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a _S (i) = a(i) + a(−i) is recurrent, then each component X _i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a _S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a _S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000     

A remark about topological and measure entropy for a class of maps of the interval [0,1]

In this paper we consider the family of dynamical systems on [0,1] in which the evolution is described by the class of mapsS _a(x)=ax(1?x),x∈[0,1], for the countable setA={a _p}_p≥2 of values ofa∈[0,4] satisfying the conditions of Pianigiani. For such values ofa we evaluate the topological entropyh _t(S_a) and show a lower bound for the measure entropyh(S _a,μ_a), where μ_a is the a.c.S _a-invariant measure. Moreover, we show that the sequence {h _t(S_{a p}} is increasing withp, and that for a subsequence{p _k}?{p}, asa _{p k}→4, one has: \(h_t (S_{a_n } ) \to h_t (S_4 ,\mu _4 )\) .     

Self-adjusting stochastic processes

Let A_t(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that A_t+1(h, k) ? A_t(h, k). If the self-adjustment (due to transition h → kx) is A_{t + 1}(h, j) = λA_t(h, j) + (1 ? λ)δ_jk (0 < λ < 1), then with probability 1 the process is eventually periodic. If A₀(i, j) < 1 for all i, j and if the self-adjustment satisfies A_{t + 1}(h, k) = ?(A_t(h, k)) with ?(x) twice differentiable and increasing, x < ?(x) < 1 for 0 ? x < 1,?(1) = ?′(1) = 1, then, with probability 1, lim A_t does not exist.     

Richardson's iteration for nonsymmetric matrices

To solve the linear N×N system (1) Ax=a for any nonsingular matrix A, Richardson's iteration (2) x_j+1=x_j-α_j(Ax_j-a), j=1,2,…,n, which is applied in a cyclic manner with cycle length n is investigated, where the α_j are free parameters. The objective is to minimize the error |x_n+1-x|, where x is the solution of (1). If the spectrum of A is known to lie in a compact set S, one is led to the Chebyshev-type approximation problem (3) min_p-1∈Vnmax_z∈S|p(z)|, where V_n is the linear span of z,z²,…,zⁿ. If p solves (3), then the reciprocals of the zeros of p are optimal iteration parameters α_j. It is shown that for a real problem (1) the iteration (2) can be carried out with real arithmetic alone, even when there are complex α_j. The stationary case n=1 is solved completely, i.e., for all compact sets S the problem (3) is solved explicitly. As a consequence, the converging stationary iteration processes can be characterized. For arbitrary closed disks S the problem (3) can be solved for all n∈$\text{N}$, and a simple proof is provided. The lemniscates associated with S are introduced. They appear as an important tool for studying the stability of the iteration (2).     

On the h-triangles of sequentially (S r ) simplicial complexes via algebraic shifting

Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (S _r ) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition (S _r ). Let Δ be a (d?1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (h _i,j (Δ))_{0≤j≤i≤d} be the h-triangle of Δ and (h _i,j (Γ(Δ)))_{0≤j≤i≤d} be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (S _r ) and for every i and j with 0≤j≤i≤r?1, the equality h _i,j (Δ)=h _i,j (Γ(Δ)) holds true.     

Multiresolution expansion, approximation order and quasiasymptotic behavior of tempered distributions

Multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces Sr(R), r∈N₀. For a function h, which is smooth enough and of appropriate decay, it is shown that the derivatives of its projections to the corresponding spaces Vj, j∈Z, in a regular multiresolution analysis of L²(R), denoted by hj, multiplied by a polynomial weight converge in sup norm, i.e., hj→h in Sr(R) as j→∞. Analogous result for tempered distributions is obtained by duality arguments. The analysis of the approximation order of the projection operator within the framework of the theory of shift-invariant spaces gives a further refinement of the results. The order of approximation is measured with respect to the corresponding space of test functions. As an application, we give Abelian and Tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity.     

An analysis of (h,k, 1)-Shellsort

One classical sorting algorithm, whose performance in many cases remains unanalyzed, is Shellsort. Let h be a t-component vector of positive integers. An h-Shellsort will sort any given n elements in t passes, by means of comparisons and exchanges of elements. Let S>_j(h; n) denote the average number of element exchanges in the jth pass, assuming that all the n! initial orderings are equally likely. In this paper we derive asymptotic formulas of S_j(h; n) for any fixed h = (h, k, l), making use of a new combinatorial interpretation of S₃. For the special case h = (3, 2, 1), the analysis is further sharpened to yield exact expressions.     

On the number of antipodal bicolored necklaces

Let ν(2n) be the number of antipodal bicolored necklaces with 2n pearls. In this note, we find the first two terms of the asymptotic expansion of ν(2n). As a byproduct of this result, we also show that the sequence (ν(2n)) _n≥1 is non-holonomic, i.e., it satisfies no linear recurrence of a fixed finite order k with polynomial coefficients.     

Berezin kernels and analysis on Makarevich spaces

Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels.We consider the conformal group Conf(V) of a simple real Jordan algebra V. The maximal degenerate representations π_s (s ε ℂ) we shall study are induced by a character of a maximal parabolic subgroup P¯ of Conf(V). These representations π_s can be realized on a space I_s of smooth functions on V. There is an invariant bilinear form ℬ_s on the space Is. The problem we consider is to diagonalize this bilinear form ℬ_s, with respect to the action of a symmetric subgroup G of the conformal group Conf(V). This bilinear form can be written as an integral involving the Berezin kernel B_v an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of B_v. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity: D(_ν)B_ν=b(ν)B_ν+1, where D(_ν) is an invariant differential operator on G/K and b(ν) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I_−s to I_s. Furthermore, we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf(V).     

An improvement of the Nevanlinna-Gundersen theorem

A well-known result of Nevanlinna states that for two nonconstant meromorphic functions f and g on the complex plane C and for four distinct values aj∈C∪{∞}, if νf−aj=νg−aj for all 1?j?4, then g is a Möbius transformation of f. In 1983, Gundersen generalized the result of Nevanlinna to the case where the above condition is replaced by: min{νf−aj,1}=min{νg−aj,1} for j=1,2 and νf−aj=νg−aj for j=3,4. In this paper, we prove that the theorem of Gundersen remains valid to the case where min{νf−aj,1}=min{νg−aj,1} for j=1,2, and min{νf−aj,2}=min{νg−aj,2} for j=3,4. Furthermore, we work on the case where {aj} are small functions.     

Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains

Let h be a positive integer and S?=?{x ₁,?…?,?x _h } be a set of h distinct positive integers. We say that the set S is a divisor chain if x _σ(1) ∣?…?∣ x _σ(h) for a permutation σ of {1,?…?,?h}. If the set S can be partitioned as S?=?S ₁?∪?S ₂?∪?S ₃, where S ₁, S ₂ and S ₃ are divisor chains and each element of S _i is coprime to each element of S _j for all 1?≤?i?<?j?≤?3, then we say that the set S consists of three coprime divisor chains. The matrix having the ath power (x _i , x _j ) ^a of the greatest common divisor of x _i and x _j as its i, j-entry is called the ath power greatest common divison (GCD) matrix on S, denoted by (S ^?a ). The ath power least common multiple (LCM) matrix [S ^?a ] can be defined similarly. In this article, let a and b be positive integers and let S consist of three coprime divisor chains with 1?∈?S. We show that if a?∣?b, then the ath power GCD matrix (S ^?a ) (resp., the ath power LCM matrix [S ^?a ]) divides the bth power GCD matrix (S ^?b ) (resp., the bth power LCM matrix [S ^?b ]) in the ring M _h (Z) of h?×?h matrices over integers. We also show that the ath power GCD matrix (S ^?a ) divides the bth power LCM matrix [S ^?b ] in the ring M _h (Z) if a?∣?b. However, if a???b, then such factorizations are not true. Our results extend Hong's and Tan's theorems and also provide further evidences to the conjectures of Hong raised in 2008.     

Measure preserving composition operators

We consider composition operators T induced on functional Hilbert spaces H = L₂(S, ∑, μ) byTf(·) = f(h(·)) where h: S → S is a nonsingular transformation. For these mappings T: H → H we give conditions under which they accept invariant Borel probability measures, and we relate the two structures of T, i.e., that of a bounded linear operator to that of a measure preserving transformation.     

Möbius Isoparametric Hypersurfaces in S n+1 with Two Distinct Principal Curvatures

A hypersurface x : M → S ⁿ⁺¹ without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ^?2∑ _i (e _i (H) + ∑ _j (h _ij ?Hδ _ij )e _j (log ρ))θ _i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S ^{n +1} without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ⁻²∑ _i (e _i (H) + ∑ _j (h _ij −Hδ _ij )e _j (log ρ))θ _i vanishes and its M?bius shape operator ? = ρ⁻¹(S−Hid) has constant eigenvalues. Here {e _i } is a local orthonormal basis for I = dx·dx with dual basis {θ _i }, II = ∑ _ij h _ij θ _i ⊗θ _i is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S ^{n +1} is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric hypersurfaces in S ^{n +1} with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S ^{n +1} can take only the values 2, 3, 4, 6. Received September 7, 2001, Accepted January 30, 2002     

Sequence entropy and mixing

The relationship between sequence entropy and mixing is examined. Let T be an automorphism of a Lebesgue space X, $\text{L}$₀ denote the set of all partitions of X possessing finite entropy, and $\text{S}$ denote the set of all increasing sequences of positive integers. It is shown that: (1) T is mixing /a2 sup_{A ? B}h_A(T, α) = H(α) for all B∈I and α∈Z₀. (2) T is weakly mixing /a2 sup_Ah_A(T, α) = H(α) for all α∈Z₀. (3) If T is partially mixing with constant $\text{c (1 ?}\text{1}\text{e}\text{< c < 1)}$, then sup_{A ? B}h_A(T, α) > cH(α) for all B∈I and nontrivial α∈Z₀. (4) If sup_{A ? B}h_A(T, α) > 0 for all B∈I and nontrivial α∈Z₀, then T is weakly mixing.     

Ascending Nets of Prüfer V-Multiplication Domains

Abstract

Let D be an integral domain, S ? D a multiplicative set such that aD _S  ∩ D is a principal ideal for each a ∈ D and let D ^(S) = ? _s∈S D[X/s]. It is known that if D is a Prüfer v-multiplication domain (resp., generalized GCD domain, GCD domain), then so is D ^(S) respectively. When D is a Noetherian domain, we obtain a similar result for the power series analog D ^((S)) = ? _s∈S D[[X/s]] of D ^(S). Our approach takes care simultaneously of both cases D ^(S) and D ^((S)).