Pairs of B-Splines with Small Support on the Four-Directional Mesh Generating a Partition of Unity

Let τ be the four-directional mesh of the plane and let Σ₁ (respectively Λ₁) be the unit square (respectively the lozenge) formed by four (respectively eight) triangles of τ. We study spaces of piecewise polynomial functions in C ^k (R ²) with supports Σ₁ or Λ₁ having sufficiently high degree n, which are invariant with respect to the group of symmetries of Σ₁ or Λ₁ and whose integer translates form a partition of unity. Such splines are called complete Σ₁ and Λ₁-splines. We first give a general study of spaces of linearly independent complete Σ₁ and Λ₁-splines of class C ^k and degree n. Then, for any fixed k≥0, we prove the existence of complete Σ₁ and Λ₁-splines of class C ^k and minimal degree, but they are not unique. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of these splines.     

Distribution of the likelihood ratio criterion for testing Σ = Σ0, μ = μ0

The exact null distribution of the likelihood ratio criterion for testing H₀: Σ = Σ₀ and μ = μ₀ against alternatives H₁: Σ ≠ Σ₀ or μ ≠ μ₀ in N_p(μ, Σ) has been obtained as (a) a chi-square series and (b) a beta series. Percentage points have been tabulated for p = 2(1) 6, α = .005, .01, .025, .05, .1, and .25 and various values of sample size N.     

Similarity invariants of operators on a class of Gowers-Maurey spaces

This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces, Σ _dc spaces, where an infinite dimensional Banach space X is called a Σ _dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton. It shows that two strongly irreducible operators T ₁ and T ₂ on a Σ _dc space are similar if and only if the K ₀-group of the commutant algebra of the direct sum T ₁⊕T ₂ is isomorphic to the group of integers ?. On a Σ _dc space X, it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in (ΣSI)(X), it further gives a necessary and sufficient condition that two operators in (ΣSI)(X) are similar by using the ordered K ₀-groups. It also proves that every operator in (ΣSI)(X) has a unique (SI) decomposition up to similarity on a Σ _dc space X, where (ΣSI)(X) denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.     

Euler-Summierbarkeit konform-äquivalenter Reihen

Let Φ be a conformal one to one mapping of the unit disc with Φ (1)=1. Given the convergent series Σa _n putf(z):=Σa _n zn, |z|<1, and leta _n (Φ) be defined by the relation $$f(\Phi (w)) = \sum\limits_{n = 0}^\infty {a_n } (\Phi )w^n ,\left| w \right|< 1.$$ The series Σa _n (Φ) is called the conformal equivalent of Σa _n with respect to Φ and need not converge as was discovered byTurán. We prove that it is nevertheless summable (E;q),q>0.     

Rigidity of Area-Minimizing Hyperbolic Surfaces in Three-Manifolds

We prove that if M is a three-manifold with scalar curvature greater than or equal to ?2 and Σ?M is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of Σ is greater than or equal to 4π(g(Σ)?1), where g(Σ) denotes the genus of Σ. In the equality case, we prove that the induced metric on Σ has constant Gauss curvature equal to ?1 and locally M splits along Σ. We also obtain a rigidity result for cylinders (I×Σ,dt ²+g _Σ), where I=[a,b]?? and g _Σ is a Riemannian metric on Σ with constant Gauss curvature equal to ?1.     

Complexity of some natural problems on the class of computable I-algebras

We study computable Boolean algebras with distinguished ideals (I-algebras for short). We prove that the isomorphism problem for computable I-algebras is Σ ₁ ¹ -complete and show that the computable isomorphism problem and the computable categoricity problem for computable I-algebras are Σ ₃ ⁰ -complete.     

Δ1 Ultrapowers are totally rigid

Hirschfeld and Wheeler proved in 1975 that ∑₁ ultrapowers (= “simple models”) are rigid; i.e., they admit no non-trivial automorphisms. We later noted, essentially mimicking their technique, that the same is true of Δ₁ ultrapowers (= “Nerode semirings”), a class of models of Π₂ Arithmetic that overlaps, but is mutually non-inclusive with, the class of Σ₁ ultrapowers. Hirschfeld and Wheeler left as open the question whether some Σ₁ ultrapowers might admit proper isomorphic self-injections. We do not answer that question; but we do answer the corresponding question, in the negative, for the Δ₁ case.     

Convergence in the pth-mean and some Weak Laws of Large Numbers for weighted sums of random elements in separable normed linear spaces

Theorem. Let X_n, n ≥ 1, be a sequence of tight random elements taking values in a separable Banach space B such that |X_n|, n ≥ 1, is uniformly integrable. Let a_nk, n ≥ 1, k ≥ 1, be a double array of real numbers satisfying Σ_{k ≥ 1} |a_nk| ≤ Γ for every n ≥ 1 for some positive constant Γ. Then Σ_{k ≥ 1}a_nkX_k, n ≥ 1, converges to 0 in probability if and only if Σ_{k ≥ 1}a_nkf(X_k), n ≥ 1, converges to 0 in probability for every f in the dual space B¹.     

An inequality for 3-polytopes

We prove the inequality Σ_n≥7 (n−6) p_n≤v−12 for any 3-dimensional polytope with v vertices and p_n n-sided faces, such that Σ_n≥6 p_n≥3. The polytopes satisfying Σ_n≥7 (n−6) p_n=v−12 are described and an interpretation of our results is given in terms of density of n-sided faces in planar graphs.     

Characterizations of locally testable events

Let Σ be a finite alphabet, Σ^* the free monoid generated by Σ and χ the length of χ ∈ Σ^*. For any integer k0, f_k(χ) (t_k(χ)) is χ if χ < k + 1, and it is the prefix (suffix) of χ of length k, othewise. Also let m_k+1(χ) = {νχ = uνw and ν = k+1}. For χ, y ε Σ^* define χ ～ _k+1y iff f_k(χ) = f_k(y), t_k(χ) = t_k(y) and m_k+1(χ) = m_k+1(y). The relation ～_k+1 is a congruence of finite index over Σ^*. An event E ? Σ^* is (k+1)-testable iff it is a union of congruence classes of ～_k+1. E is locally testable (LT) if it is k+1-testable for some k. (This definition differs from that of [6] but is equivalent.)We show that the family of LT events is a proper sub-family of star-free events of dot-depth 1. LT events and k-testable events are characterized in terms of (a) restricted star-free expressions based on finite and cofinite events; (b) finite automata accepting these events; (c) semigroups; and (d) structural decomposition of such automata. Algorithms are given for deciding whether a regular event is (a) LT and (b) k+1-testable. Generalized definite events are also characterized.     

The shift on the inverse limit of a covering projection

A group endomorphismα : G → G is said to beweakly shift equivalent to the group endomorphismβ : H → H if there existsh ∈ H such thatα is shift equivalent to Ad[h] °β. Given covering projectionsa : X → X, b : Y → Y of compact, connected, locally path connected, semilocally simply connected metric spaces with fixed pointsx ₀ ∈X,y ₀ ∈Y respectively, the inverse limits $$\begin{array}{l} \sum\nolimits_a { = \lim } (X,a) = \{ (x_i )_{i \in Z^ + } ax_{i + 1} = x_1 ,i \in Z^ + \} , \\ \sum\nolimits_a { = \lim } (Y,b) = \{ (y_i )_{i \in Z^ + } by_{i + 1} = y_1 ,i \in Z^ + \} , \\ \end{array}$$ and the “shift” mapsσ _a : Σ _a → Σ _a ,σ _b : Σ _b → Σ _b defined byσ _a((x _i)i∈Z ⁺)=(x _i+1)i∈Z ⁺ ∈ Σ _a ,σ _b((y _i)i∈Z ⁺)=(y i + 1)i∈Z ⁺ ∈ Σ _b are considered. It is proven that ifσ _a andσ _b are topologically conjugate thena _# :π ₁(X, x ₀) →π ₁(X, x ₀) is weakly shift equivalent tob _# :π ₁(Y, y ₀) →π ₁(Y, y ₀). Furthermore, ifa : X → X andb : Y → Y are expanding endomorphisms of compact differentiable manifolds, weak shift equivalence is a complete invariant of topological conjugacy. The use of this invariant is demonstrated by giving a complete classification of the shifts of expanding maps on the klein bottle. The reader is referred to Section 4 of this work for a detailed statement of results.     

Simple continued fractions for some irrational numbers,II

In this paper we prove a theorem allowing us to determine the continued fraction expansion for Σ_k=0^∞u^?c(k), where c(k) is any sequence of positive integers that grows sufficiently quickly. As an application, we determine the continued fraction expansion for Liouville's famous transcendental number Σ_k=0^∞m^{?(k + 1)!}.     

The distance between two random vectors with given dispersion matrices

For two p-dimensional random vectors X and Y with dispersion matrices Σ₁₁ and Σ₂₂, respectively, we determine that covariance matrix Ψ₀ of X and Y that minimizes the L₂-distance between X and Y. There is a dual to this problem that is of interest in another context.     

Stereographische Projektion und quadratisches Spurenprinzip imR 4

In this article a new method of representing the projective spaceR ₄ in the three-dimensional space Σ is described. This method is based on the stereographical projection of a hypersphere and transfers the straight lines, planes and hyperplanes ofR ₄ into the couples of points, circles and spheres of Σ; the points of theR ₄ are represented by the inversions on the spheres of Σ.     

Power Majorization and Majorization of Sequences

Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x₁ ≥ x₂ ≥ … ≥ x_n and y₁ ≥ y₂ >- … ≥ y_n and Σx_i = Σy_i. We say that \(\bar x\) is power majorized by \(\bar y\) if Σx_i ^p ≤ Σy_i ^p for all real p ? [0, 1] and Σx_i ^p ≥ Σy_i ^p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) .     

Trattazione rapida dei problemi al contorno relativi alle equazioni differenziali lineari a derivate parziali

Let Σ andS be two real Hilbert spaces and Σ₀ a subspace of Σ. Moreover, supposeT:S→Σ be a bounded linear operator whose rangeT (S) is contained in Σ₀, andE:S→Σ be a linear operator such that the productE T:S→S is a bounded operator with a closed range. In this framework we present an artifice from which the alternative theorem for the equationE u=q(u?Σ₀,q ?S) follows. It is worthwhile to note thatEu=q may represent a boundary value problem for elliptic equations.     

A Tauberian theorem related to Weyl's criterion

Let {x_n} be a sequence of real numbers and let a(n) be a sequence of positive real numbers, with A(N) = Σ_n=1^Na(n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t₀ for which A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞. In case f(s) = Σ_n=1^∞a(n)e^?sx_n, s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f(σ)^?1f(σ + it₀) → 0 as σ → 0 + if and only if A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞.     

Envelopes,indicators and conservativeness

A well known theorem proved (independently) by J. Paris and H. Friedman states that B Σ_{n +1} (the fragment of Arithmetic given by the collection scheme restricted to Σ_{n +1}‐formulas) is a Π_{n +2}‐conservative extension of I Σ_n (the fragment given by the induction scheme restricted to Σ_n ‐formulas). In this paper, as a continuation of our previous work on collection schemes for Δ_{n +1}(T )‐formulas (see [4]), we study a general version of this theorem and characterize theories T such that T + B Σ_{n +1} is a Π_{n +2}‐conservative extension of T . We prove that this conservativeness property is equivalent to a model‐theoretic property relating Π_n ‐envelopes and Π_n ‐indicators for T . The analysis of Σ_{n +1}‐collection we develop here is also applied to Σ_{n +1}‐induction using Parsons' conservativeness theorem instead of Friedman‐Paris' theorem. As a corollary, our work provides new model‐theoretic proofs of two theorems of R. Kaye, J. Paris and C. Dimitracopoulos (see [8]): B Σ_{n +1} and I Σ_{n +1} are Σ_{n +3}‐conservative extensions of their parameter free versions, B Σ^–_{n +1} and I Σ^–_{n +1}. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reflection and forcing in E-recursion theory

E-recursive enumerability is compared via forcing to Σ₁ definability. It is shown that for every countable E-closed ordinal κ there is a set of reals, X, so that L_κ[X] is the least E-closed structure over X.     

On the index sets of Σ-subsets of the real numbers

We compute the levels of complexity in analytical and arithmetical hierarchies for the sets of the Σ-formulas defining in the hereditarily finite superstructure over the ordered field of the reals the classes of open, closed, clopen, nowhere dense, dense subsets of ? ⁿ , first category subsets in ? ⁿ as well as the sets of pairs of Σ-formulas corresponding to the relations of set equality and inclusion which are defined by them. It is also shown that the complexity of the set of the Σ-formulas defining connected sets is at least Π ₁ ¹ .