An improved characterization of normal sets and some counter-examples

We strengthen G. Rauzy’s characterization of normal sets by observing that the so-called elementary sets are precisely the F_σδ sets. This answers in the affirmative Rauzy’s open question: Are finite unions of normal sets necessarily normal? We also generalize the notion and characterization of normal sets from subsets of ? to subsets of ? ^d . This allows us to answer a question of E. Lesigne and M. Wierdl with the following construction: There exist two sequences of real numbersU=(u _n ) _nε?,V=(v _n ) _nε? such thatαU+βV=(αu _n +βv _n ) _nε? is uniformly distributed mod 1 if and only if exactly one of the real numbers α, β vanishes. Additionally, we provide the ‘ultimate’ counter-example (stronger than that of H. G. Meijer and R. Sattler) to a conjecture of S. Uchiyama by constructing a sequence of integersU which is u.d. in ? (i.e. u.d. modk for allk ε ?), but such that for all realα, αU is not u.d. mod 1.     

Cone Dependence—A Basic Combinatorial Concept

We call A ? $ \mathbb{E} $ ⁿ cone independent of B ? $ \mathbb{E} $ ⁿ , the euclidean n-space, if no a = (a ₁,..., a _n ) ∈ A equals a linear combination of B \ {a} with non-negative coefficients. If A is cone independent of A we call A a cone independent set. We begin the analysis of this concept for the sets P(n) = {A ? {0, 1} ⁿ ? $ \mathbb{E} $ ⁿ : A is cone independent} and their maximal cardinalities c(n) ? max{|A| : A ∈ P(n)}. We show that lim _{n → ∞} $ \frac{{c\left( n \right)}}{{2^n }} $ > $\frac{1}{2}$ , but can't decide whether the limit equals 1. Furthermore, for integers 1 < k < ? ≤ n we prove first results about c _n (k, ?) ? max{|A| : A ∈ P _n (k, ?)}, where P _n (k, ?) = {A : A ? V ⁿ _k and V ⁿ _? is cone independent of A} and V ⁿ _k equals the set of binary sequences of length n and Hamming weight k. Finding c _n (k, ?) is in general a very hard problem with relations to finding Turan numbers.     

On the relation of generalized Valiron summability to Cesàro summability

A family (V _a ^k ) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (B_α,γ), (E _ρ, (T_α), (S _β) and (V _a) are members of this family. In §3 some properties of the (B _α,γ) and (V _a ^k ) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (V _a ^k ) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem.If s _n (n ≥ 0) isa real sequence satisfying $$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$ , and if s_n →s (V _a ^k ) thens _n → s (C, 2ρ).     

Recursive linear orders with recursive successivities

A successivity in a linear order is a pair of elements with no other elements between them. A recursive linear order with recursive successivities $\text{U}$ is recursively categorical if every recursive linear order with recursive successivities isomorphic to $\text{U}$ is in fact recursively isomorphic to $\text{U}$. We characterize those recursive linear orders with recursive successivities that are recursively categorical as precisely those with order type k₁+g₁+k₂+g₂+…+g_n-1+k_n where each k_n is a finite order type, non-empty for i?{2,…,n-1} and each g_i is an order type from among {ω,ω^*,ω+ω^*}∪{k·η:k<ω}.     

Limiting Spectral Distribution of Random k-Circulants

Consider random k-circulants A _k,n with n????,k=k(n) and whose input sequence {a _l } _l??0 is independent with mean zero and variance one and $\sup_{n}n^{-1}\sum_{l=1}^{n}\mathbb{E}|a_{l}|^{2+\delta}<\infty$ for some ??>0. Under suitable restrictions on the sequence {k(n)} _n??1, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists, and we identify the limits. In particular, we prove the following: Suppose g??1 is fixed and p ₁ is the smallest prime divisor of g. Suppose $P_{g}=\prod_{j=1}^{g}E_{j}$ where {E _j }_1??j??g are i.i.d. exponential random variables with mean one. (i) If k ^g =?1+sn where s=1 if g=1 and $s=o(n^{p_{1}-1})$ if g>1, then the empirical spectral distribution of n ^?1/2 A _k,n converges weakly in probability to $U_{1}P_{g}^{1/(2g)}$ where U ₁ is uniformly distributed over the (2g)th roots of unity, independent of P _g . (ii) If g??2 and k ^g =1+sn with $s=o(n^{p_{1}-1})$ , then the empirical spectral distribution of n ^?1/2 A _k,n converges weakly in probability to $U_{2}P_{g}^{1/(2g)}$ where U ₂ is uniformly distributed over the unit circle in ?², independent of P _g . On the other hand, if k??2, k=n ^o(1) with gcd?(n,k)=1, and the input is i.i.d. standard normal variables, then $F_{n^{-1/2}A_{k,n}}$ converges weakly in probability to the uniform distribution over the circle with center at (0,0) and radius $r=\exp(\mathbb{E}[\log\sqrt{E}_{1}])$ .     

Approximation by antiderivatives

Let Ω ?C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ ^(?n)} ofn-fold antiderivatives (i.e., we haveφ ⁽⁰⁾(z)∶=φ(z) andφ ^(?n)(z)=dφ ^(?n?1)(z)/dz for alln ∈ N₀ and z ∈ Ω) such that the following properties hold:

1. For any compact setB ?Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n _k} such that {φ^?nk} converges tof uniformly onB.
2. For any open setU ?Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m _k} such that {φ^?mk} converges tof compactly onU.
3. For any measurable setE ?Ω and any functionf that is measurable onE, there exists a sequence {p _k} such that {φ ^(-Pk)} converges tof almost everywhere onE.

     

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).     

Convergence of weighted sums of tight random elements

Convergence of weighted sums of tight random elements {V_n} (in a separable Banach space) which have zero expected values and uniformly bounded rth moments (r > 1) is obtained. In particular, if {a_nk} is a Toeplitz sequence of real numbers, then | Σ_k=1^∞a_nkf(V_k)| → 0 in probability for each continuous linear functional f if and only if 6Σ_k=1^∞a_nkV_k 6→ 0 in probability. When the random elements are independent and max_{1≤k≤n | ank | = $\text{O}$(n^?8)} for some $\text{0 <}\text{1}\text{s}\text{< r ? 1}$, then |Σ_k=1^∞a_nkV_k 6→ 0 with probability 1. These results yield laws of large numbers without assuming geometric conditions on the Banach space. Finally, these results can be extended to random elements in certain Fréchet spaces.     

Isothermic Submanifolds

We propose an answer to a question raised by F. Burstall: Is there any interesting theory of isothermic submanifolds of ? ⁿ of dimension greater than two? We call an n-immersion f(x) in ? ^m isothermic _k if the normal bundle of f is flat and x is a line of curvature coordinate system such that its induced metric is of the form $\sum_{i=1}^{n} g_{ii}\,\mathrm{d} x_{i}^{2}$ with $\sum_{i=1}^{n} \epsilon_{i} g_{ii}=0$ , where ?? _i =1 for 1??i??n?k and ?? _i =?1 for n?k<i??n. A smooth map (f ₁,??,f _n ) from an open subset ${\mathcal{O}}$ of ? ⁿ to the space of m×n matrices is called an n-tuple of isothermic _k n-submanifolds in ? ^m if each f _i is an isothermic _k immersion, $(f_{i})_{x_{j}}$ is parallel to $(f_{1})_{x_{j}}$ for all 1??i,j??n, and there exists an orthonormal frame (e ₁,??,e _n ) and a GL(n)-valued map (a _ij ) such that $\mathrm{d}f_{i}= \sum_{j=1}^{n} a_{ij} e_{j}\,\mathrm {d} x_{j}$ for 1??i??n. Isothermic₁ surfaces in ?³ are the classical isothermic surfaces in ?³. Isothermic _k submanifolds in ? ^m are invariant under conformal transformations. We show that the equation for n-tuples of isothermic _k n-submanifolds in ? ^m is the $\frac{O(m+n-k,k)}{O(m)\times O(n-k,k)}$ -system, which is an integrable system. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

A class of facet producing graphs for vertex packing polyhedra

We examine a family of graphs called webs. For integers n ? 2 and $\text{k, 1 ? k ?}\text{1}\text{2}\text{n}$, the web W(n, k) has vertices V_n = {1, …, n} and edges {(i, j): j = i+k, …, i+n ? k, for i?V_n (sums mod n)}. A characterization is given for the vertex packing polyhedron of W(n, k) to contain a facet, none of whose projections is a facet for the lower dimensional vertex packing polyhedra of proper induced subgraphs of W(n, k). Simple necessary and sufficient conditions are given for W(n, k) to contain W(n′, k′) as an induced subgraph; these conditions are used to show that webs satisfy the Strong Perfect Graph Conjecture. Complements of webs are also studied and it is shown that if both a graph and its complement are webs, then the graph is either an odd hole or its complement.     

Connected Baranyai’s theorem

Let K _n ^h = (V, ( _h ^V )) be the complete h-uniform hypergraph on vertex set V with ¦V¦ = n. Baranyai showed that K _n ^h can be expressed as the union of edge-disjoint r-regular factors if and only if h divides rn and r divides \((_{h - 1}^{n - 1} )\) . Using a new proof technique, in this paper we prove that λK _n ^h can be expressed as the union \(\mathcal{G}_1 \cup ... \cup \mathcal{G}_k \) of k edge-disjoint factors, where for 1≤i≤k, \(\mathcal{G}_i \) is r _i -regular, if and only if (i) h divides r _i n for 1≤i≤k, and (ii) \(\sum\nolimits_{i = 1}^k {r_i = \lambda (_{h - 1}^{n - 1} )} \) . Moreover, for any i (1≤i≤k) for which r _i ≥2, this new technique allows us to guarantee that \(\mathcal{G}_i \) is connected, generalizing Baranyai’s theorem, and answering a question by Katona.     

Extremal traces on some group-invariant C1-algebras

Let $\text{U}$ be a UHF-algebra of Glimm type n^∞, and {α_g: g?G} a strongly continuous group of ¹-automorphisms of product type on $\text{U}$, for G compact. Let $\text{U}$^α be the C¹-subalgebra of fixed elements of $\text{U}$. We show that any extremal normalized trace on $\text{U}$^α arises as the restriction of a symmetric product state ? on $\text{U}$ of the form ? = ?_k?1 ω. As an example we classify the extremal traces on $\text{U}$^α for the case G = SU(n), α_g = ?_{k ? 1} Ad(g).     

Interpretation and optimization of the k-means algorithm

The paper gives a new interpretation and a possible optimization of the wellknown k-means algorithm for searching for a locally optimal partition of the set A = {a _i ∈ ? ⁿ : i = 1, …, m} which consists of k disjoint nonempty subsets π₁, …, π _k , 1 ? k ? m. For this purpose, a new divided k-means algorithm was constructed as a limit case of the known smoothed k-means algorithm. It is shown that the algorithm constructed in this way coincides with the k-means algorithm if during the iterative procedure no data points appear in the Voronoi diagram. If in the partition obtained by applying the divided k-means algorithm there are data points lying in the Voronoi diagram, it is shown that the obtained result can be improved further.     

Phragmén-Lindelöf conditions for graph varieties

For P_m ∈ ?[z₁, …, z_n], homogeneous of degree m we investigate when the graph of P_m in ?ⁿ⁺¹ satisfies the Phragmén-Lindelöf condition PL(?ⁿ⁺¹, log), or equivalently, when the operator $i{\partial \over \partial_{x_{n+1}}}+P_{m}(D)$ admits a continuous solution operator on C^∞(?ⁿ⁺¹). This is shown to happen if the varieties V_+- ? {z ∈ ?ⁿ: P_m(z) = ±1} satisfy the following Phragmén-Lindelöf condition (SPL): There exists A ≥ 1 such that each plurisubharmonic function u on V_+- satisfying u(z) ≤ ¦z¦+ o(¦z¦) on V_+- and u(x) ≤ 0 on V_+- ∩ ?ⁿ also satisfies u(z) A¦ Im z¦ on V_+-. Necessary as well as sufficient conditions for V_+- to satisfy (SPL) are derived and several examples are given.     

The Paranormality of Products and Their Subsets

A topological space is called paranormal if any countable discrete system of closed sets {D_n:n = 1, 2, 3,...} can be expanded to a locally finite system of open sets {U_n:n = 1, 2, 3,...}, i.e., D_n is contained in U_n for all n, and D_m ∩ U_n≠ Ø if and only if D_m = D_n. It is proved that if X is a countably compact space whose cube is hereditarily paranormal, then X is metrizable.     

Grassmann and Weyl embeddings of orthogonal grassmannians

Given a non-singular quadratic form q of maximal Witt index on $V := V(2n+1,\mathbb{F})$ , let Δ be the building of type B _n formed by the subspaces of V totally singular for q and, for 1≤k≤n, let Δ _k be the k-grassmannian of Δ. Let ε _k be the embedding of Δ _k into PG(? ^k V) mapping every point 〈v ₁,v ₂,…,v _k 〉 of Δ _k to the point 〈v ₁∧v ₂∧?∧v _k 〉 of PG(? ^k V). It is known that if $\mathrm{char}(\mathbb{F})\neq2$ then $\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}$ . In this paper we give a new very easy proof of this fact. We also prove that if $\mathrm{char}(\mathbb{F}) = 2$ then $\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}-{{2n+1}\choose{k-2}}$ . As a consequence, when 1<k<n and $\mathrm{char}(\mathbb{F}) = 2$ the embedding ε _k is not universal. Finally, we prove that if $\mathbb{F}$ is a perfect field of characteristic p>2 or a number field, n>k and k=2 or 3, then ε _k is universal.     

Quasi-free derivations on the Canonical Anticommutation Relation algebra

An unbounded 1-derivation δ on a C¹-algebra $\text{U}$ is called approximately bounded if there is an increasing sequence of full matrix subalgebras {$\text{U}$_n} whose union is dense in the domain of $\text{U}$ and a sequence {h_n} of self-adjoint elements of $\text{U}$ such that h_n implements δ on $\text{U}$_n for every n, and {∥h_n ? Q_n(h_n)∥} is a bounded sequence where Q_n is the canonical conditional expectation of $\text{U}$ onto $\text{U}$_n. We prove that a quasi-free derivation on the Canonical Anticommutation Relation algebra is approximately bounded if the self-adjoint operator from which it arises is of finite multiplicity and bounded. We conjecture that all quasi-free derivations are approximately bounded. We also prove that a quasi-free derivation is bounded if and only if the self-adjoint operator from which it arises is of the trace class.     

Combinatorial models for the finite-dimensional Grassmannians

Let ? ⁿ be a linear hyperplane arrangement in ? ⁿ . We define two corresponding posetsG _k (? ⁿ andV _k (? ⁿ ) of oriented matroids, which approximate the GrassmannianG _k (? ⁿ ) and the Stiefel manifoldV _k (? ⁿ ). The basic conjectures are that the “OM-Grassmannian”G _k (? ⁿ ) has the homotopy type ofG _k (? ⁿ ), and that the “OM-Stiefel bundle” Δπ: ΔV _k (? ⁿ ) → ΔG _k (? ⁿ ) is a surjective map. These conjectures can be proved in some cases: we survey the known results and add some new ones. The conjectures fail if they are generalized to nonrealizable oriented matroids ? ⁿ .     

Hirzebruch Functional Equation: Classification of Solutions

The Hirzebruch functional equation is \(\sum\nolimits_{i = 1}^n {\prod\nolimits_{j \ne i} {(1/f({z_j} - {z_i})) = c} } \) with constant c and initial conditions f(0) = 0 and f'(0) = 1. In this paper we find all solutions of the Hirzebruch functional equation for n ≤ 6 in the class of meromorphic functions and in the class of series. Previously, such results have been known only for n ≤ 4. The Todd function is the function determining the two-parameter Todd genus (i.e., the χ_a,b-genus). It gives a solution to the Hirzebruch functional equation for any n. The elliptic function of level N is the function determining the elliptic genus of level N. It gives a solution to the Hirzebruch functional equation for n divisible by N. A series corresponding to a meromorphic function f with parameters in U ? ?^k is a series with parameters in the Zariski closure of U in ?^k, such that for the parameters in U it coincides with the series expansion at zero of f. The main results are as follows: (1) Any series solution of the Hirzebruch functional equation for n = 5 corresponds either to the Todd function or to the elliptic function of level 5. (2) Any series solution of the Hirzebruch functional equation for n = 6 corresponds either to the Todd function or to the elliptic function of level 2, 3, or 6. This gives a complete classification of complex genera that are fiber multiplicative with respect to ?P^n?1 for n ≤ 6. A topological application of this study is an effective calculation of the coefficients of elliptic genera of level N for N = 2,..., 6 in terms of solutions of a differential equation with parameters in an irreducible algebraic variety in ?⁴.