Discrete semi-stable distributions

The purpose of this paper is to introduce and study the concepts of discrete semi-stability and geometric semi-stability for distributions with support inZ ₊. We offer several properties, including characterizations, of discrete semi-stable distributions. We establish that these distributions posses the property of infinite divisibility and that their probability generating functions admit canonical representations that are analogous to those of their continuous counterparts. Properties of discrete geometric semi-stable distributions are deduced from the results obtained for discrete semi-stability. Several limit theorems are established and some examples are constructed.     

On Geometric Infinite Divisibility and Stability

The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in Z ₊ and R ₊. Several new characterizations are obtained. We prove in particular that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on Z ₊ (resp. R ₊). These distributions are shown to arise as the only solutions to a stability equation. We also establish that the Mittag-Leffler distributions characterize geometric stability. Related stationary autoregressive processes of order one (AR(1)) are constructed. Importantly, we will use Poisson mixtures to deduce results for distributions on R ₊ from those for their Z ₊-counterparts.     

Free <Emphasis Type="Bold">C</Emphasis><Subscript>+</Subscript>-actions on <Emphasis Type="Bold">C</Emphasis><Superscript>3</Superscript> are translations

Let X be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C₊-action on it such that S=X//C₊ is smooth. We prove that X is isomorphic to S×C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C₊-action on C³ is a translation in a suitable coordinate system.     

Sturmian morphisms,the braid group <Emphasis Type="Italic">B</Emphasis><Subscript>4</Subscript>, Christoffel words and bases of <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F ₂) of automorphisms of the rank two free group F ₂ and show that it can be realized as a monoid in the group B ₄ of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F ₂ lifting any given basis of the free abelian group Z ². We further give an algorithm allowing to decide whether two elements of F ₂ form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes. Mathematics Subject Classification (2000) 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15     

A Notion of α-Monotonicity with Generalized Multiplications

The multiplications of van Harn et al. (1982, Z. Wahrsch. Verw. Gebiete, 61, 97–118) are used to generalize the definition of -monotonicity of Olshen and Savage (1970, J. Appl. Probab., 7, 21–34) and Steutel (1988, Statist. Neerlandica, 42, 137–140) for distributions with support in Z ₊ and R ₊. Several characterizations are offered and a convolution property is established. Some relevant stability equations are solved and a relationship with the important concept of self-decomposability is noted. Poisson mixtures are used to deduce results for the R ₊-case from those for the Z ₊-case.     

Discrete isoperimetric and Poincaré-type inequalities

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ ⁿ on the discrete cube {0, 1} ⁿ and on the lattice Z ⁿ . In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z ⁿ . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. Received: 30 April 1997 / Revised version: 5 June 1998     

Singular integrals on the discrete Besov space B1 0,1 (Z)

Summary We show the boundedness of singular integral operators on the discrete Besov space B₁^0,1 (Z). For this purpose, we introduce discrete special molecules on B₁^0,1 (Z).     

Constrained Approximation in Banach Spaces

   Abstract. We consider the problem of approximating vectors from a complemented subspace Z ₊ of a Banach space X by the projections onto Z ₊ of vectors from a subspace Y ₊ with a norm constraint on their projections onto the complementary subspace. Sufficient conditions are found for the existence of a unique best approximant and a characterization via a critical point equation is provided, thus extending known results on Hilbert spaces. These results are then applied in the case that X is L ^p (T), where T denotes the unit circle, Z ₊ consists of functions supported on a subset of the circle, and Y ₊ is the corresponding Hardy space.     

A Differential Calculus on the Z<Subscript>3</Subscript>-graded Quantum Group GL<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2)

In this work, we introduce Z₃-graded quantum group GL\({_q(2,{\mathbb C})}\) with the help of Z₃-graded quantum plane and a Z₃-graded bicovariant differential calculus on the Z₃-graded quantum group GL _q (2). The corresponding Z₃-graded quantum Lie superalgebra is obtained.     

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.     

Difference Families in Z2d+1⊕Z2d+1 and Infinite Translation Designs in Z⊕Z

We analyse 3-subset difference families of Z_2d+1⊕Z_2d+1 arising as reductions (mod 2d+1) of particular families of 3-subsets of Z⊕Z. The latter structures, namely perfect d-families, can be viewed as 2-dimensional analogues of difference triangle sets having the least scope. Indeed, every perfect d-family is a set of base blocks which, under the natural action of the translation group Z⊕Z, cover all edges {(x,y),(x′,y′)} such that |x−x′|, |y−y′|≤d. In particular, such a family realises a translation invariant (G,K₃)-design, where V(G)=Z⊕Z and the edges satisfy the above constraint. For that reason, we regard perfect families as part of the hereby defined translation designs, which comprise and slightly generalise many structures already existing in the literature. The geometric context allows some suggestive additional definitions. The main result of the paper is the construction of two infinite classes of d-families. Furthermore, we provide two sporadic examples and show that a d-family may exist only if d≡0,3,8,11 (mod 12).     

Asymptotics of Plancherel measures for the infinite-dimensional unitary group

We study a two-dimensional family of probability measures on infinite Gelfand-Tsetlin schemes induced by a distinguished family of extreme characters of the infinite-dimensional unitary group. These measures are unitary group analogs of the well-known Plancherel measures for symmetric groups.We show that any measure from our family defines a determinantal point process on Z₊×Z, and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes.     

Syzygies and Diagonal Resolutions for Dihedral Groups

Let G be a finite group with integral group ring Λ =Z[G]. The syzygies Ω_r(Z) are the stable classes of the intermediate modules in a free Λ-resolution of the trivial module. They are of significance in the cohomology theory of G via the “co-represention theorem” H^r(G, N) = Hom_𝒟er(Ω_r(Z), N). We describe the Ω_r(Z) explicitly for the dihedral groups D_4n+2, so allowing the construction of free resolutions whose differentials are diagonal matrices over Λ.     

Boundaries of random walks on graphs and groups with infinitely many ends

Consider an irreducible random walk {Z _n} on a locally finite graphG with infinitely many ends, and assume that its transition probabilities are invariant under a closed group Γ of automorphisms ofG which acts transitively on the vertex set. We study the limiting behaviour of {Z _n} on the spaceΩ of ends ofG. With the exception of a degenerate case,Ω always constitutes a boundary of Γ in the sense of Furstenberg, and {Z _n} converges a.s. to a random end. In this case, the Dirichlet problem for harmonic functions is solvable with respect toΩ. The degenerate case may arise when Γ is amenable; it then fixes a unique end, and it may happen that {Z _n} converges to this end. If {Z _n} is symmetric and has finite range, this may be excluded. A decomposition theorem forΩ, which may also be of some purely graph-theoretical interest, is derived and applied to show thatΩ can be identified with the Poisson boundary, if the random walk has finite range. Under this assumption, the ends with finite diameter constitute a dense subset in the minimal Martin boundary. These results are then applied to random walks on discrete groups with infinitely many ends.     

The Latent Scale Covariogram: A Tool for Exploring the Spatial Dependence Structure of Nonnormal Responses

When modeling spatially distributed normal responses Y_i in terms of vectors x_i of explanatory variables, one may fit a linear model assuming independence, and then use the empirical variogram of the residuals to determine an appropriate parametric form for the autocorrelation function. Suppose, however, that the responses are not normally distributed—for example, Poisson or Bernoulli. One may model spatial dependence using a hierarchical generalized linear model in which, conditional on a latent Gaussian field Z = {Z_i}, the Y_i have independent distributions from the exponential family, with an appropriate link function connecting their conditional means with the linear predictors x^t_iβ + Z_i. The question then is how to determine an appropriate model for the autocorrelation function of Z. The empirical variogram of the Y_i is no longer appropriate, since (unless the link function is the identity) it is on the wrong scale. We propose here an alternative, the latent scale covariogram, whose graph reflects the autocorrelation structure of the underlying normal field. We illustrate its use on several real datasets, together with a simulated dataset, and obtain results quite different from those obtained using the variogram. Supplementary materials for this article are available online.     

Complete study on a bi-center problem for the <Emphasis Type="Italic">Z</Emphasis><Subscript>2</Subscript>-equivariant cubic vector fields

For the planar Z ₂-equivariant cubic systems having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Liapunov constants are completely discussed. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z ₂-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme 〈6 ∐ 6〉 is proved.     

A Note on Polynomially Growing C<Subscript>0</Subscript>-Semigroups

We characterize polynomial growth of a C₀-semigroup in terms of the first power of the resolvent of its generator. We do this for a class of semigroups which includes C₀-semigroups on Hilbert spaces and analytic semigroups on Banach spaces. Furthermore, we characterize polynomial growth for discrete semigroups.     

Limit behavior of additive functionals of semistable processes and processes attracted to semistable processes

The limit behavior of distributions of functionals of the form $$v_T = \frac{1}{T}\int\limits_o^T {h\left( {x_t } \right)dt,} $$ is studied in the paper; here x_t is a process with values in R_m which is semistable or is attracted to a semistable process, and h, h:R_m→-R₁, is bounded and measurable. Sufficient conditions are obtained for the existence of a limit distribution. It is shown that in the one-dimensional case these conditions are actually also necessary. Results are also presented on the joint behavior of several functionals of similar type.     

Vacuum effects for a one-dimensional “hydrogen atom” with <Emphasis Type="Italic">Z</Emphasis> &gt; <Emphasis Type="Italic">Z</Emphasis><Subscript>cr</Subscript>

For a supercritical Coulomb source with a chargeZ > Z_cr in 1+1 dimensions, we study the nonperturbative properties of the vacuum density ρ_VP(x) and the energy ?_VP. We show that for corresponding problem parameters, nonlinear effects in the supercritical region can lead to behavior of the vacuum energy differing significantly from the perturbative quadratic growth, to the extent of an (almost) quadratic decrease of the form ?|η|Z² into the negative region. We also show that although approaches for calculating vacuum expectations values and the behavior of ρ_VP(x) in the supercritical region for various numbers of spatial dimensions indeed have many common features, ?_VP for 1+1 dimensions in the supercritical region nevertheless has several specific features determined by the one-dimensionality of the problem.     

On the modularity of the GL<Subscript>2</Subscript>-twisted spinor L-function

In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke’s converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat’s Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL₂-twisted spinor L-function Z _{G ⊗ h} (s) related to automorphic forms G,h on the symplectic group GSp₂ and GL₂. This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insights into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan’s work on the modularity of the Rankin-Selberg L-series.