Wellfoundedness proofs by means of non-monotonic inductive definitions II: First order operators

In this paper, we give two proofs of the wellfoundedness of a recursive notation system for Π_N-reflecting ordinals. One is based on distinguished classes, and the other is based on -inductive definitions.     

Register computations on ordinals

We generalize ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. Ordinal register machines are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the Zermelo-Fraenkel axioms ZFC. This allows the following characterization of computable sets: a set of ordinals is ordinal register computable if and only if it is an element of Gödel’s constructible universe L.     

Inadmissible forcing

A structure is E-closed if it is closed under all partial E-recursive functions from V into V, a set theoretic extension of Kleene's partial recursive functions of finite type in the normal case. Let L(κ) be E-closed and ∑₁ inadmissible. Then L(κ) has reflection properties useful in the study of generic extensions of L(κ). Every set generic extension of L(κ) via countably closed forcing conditions is E-closed. A class generic construction shows: if L(κ) is countable, and inside L(κ) the greatest cardinal gc(κ), has uncountable cofinality, then there exists a T ⊆ gc(κ) such that L(κ, T) = E(T), the least E-closed set with T as a member. A partial converse is obtained via a selection theorem that implies E(X) is ∑₁ admissible when X is a set of ordinals and the greatest cardinal in the sense of E(X) has countable cofinality in E(X).     

Recognition by Spectrum of Some Linear Groups over the Binary Field

We focus our attention on the linear groups L _n (2) and obtain some general properties of these groups. We will show then that the linear groups L _p (2), where 2 is a primitive root mod p (p odd prime), are recognizable by spectrum. For example, the linear groups L ₃(2), L ₅(2), L ₁₁(2), L ₁₃(2), L ₁₉(2), L ₂₉(2), L ₃₇(2), L ₅₃(2), etc. are recognizable by spectrum.     

Almost Primitive Elements of Free lie Algebras of Small Ranks

Let K be a field, X = {x1, . . . , xn}, and let L(X) be the free Lie algebra over K with the set X of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free, A. I. Shirshov proved that subalgebras of free Lie algebras are free. A subset M of nonzero elements of the free Lie algebra L(X) is said to be primitive if there is a set Y of free generators of L(X), L(X) = L(Y ), such that M ? Y (in this case we have |Y | = |X| = n). Matrix criteria for a subset of elements of free Lie algebras to be primitive and algorithms to construct complements of primitive subsets of elements with respect to sets of free generators have been constructed. A nonzero element u of the free Lie algebra L(X) is said to be almost primitive if u is not a primitive element of the algebra L(X), but u is a primitive element of any proper subalgebra of L(X) that contains it. A series of almost primitive elements of free Lie algebras has been constructed. In this paper, for free Lie algebras of rank 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed.     

Weighted L p-Approximation of Derivatives by the Method of Gammaoperators

We consider Gammaoperators G _n on suitable Sobolev type subspaces of L _p(0, ∞) and characterize the global rate of approximation of derivatives f ^(r) through corresponding derivatives (G _n f)^(r) in an appropriate weighted L _p — metric by the rate of Ditzian and Totik’s second order weighted modulus of smoothness.     

A note on Linnik’s approach to the Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions

Abstract—Let χ (mod q), q > 1, be a primitive Dirichlet character. We first present a detailed account of Linnik’s deduction of the functional equation of L(s, χ) from the functional equation of ζ(s). Then we show that the opposite deduction can be obtained by a suitable modification of the method, involving finer arithmetic arguments.     

The Choquet theorem and stochastic equations

где,у- произвольное положительное число,В —абсолютная положительная постоянная, — произв ольная последовател ьность знаков ±1. Из этого неравенства следует, что если 1<P<∞ иg∈L _p (0, 1), то. Отсюда сразу вытекае т, что система Франкли на является безусловным базисом в пространствахL _p (0, 1) для 1 <р<∞.     

On leastp-th power methods in multiple regressions and location estimations

The solution of the matricial systemZ=HX+V is derived. The observations are given by the matrixZ(n×m) and arem-components vectors. The parameter matrix isH(n×g), whileX(g×m) is the unknown andV(n×m) the residual matrix. The solutionX is estimated in order to minimize anL _p -norm ofV defined with the help of anL _r -norm on theV rows. The minimization algorithm is a steepest descent method with a specially chosen step size to “avoid” the singularities. The general problem shrinks in simple regression or location problems whilem org is reduced to the unit. Possibly more important is the recursive solution of this same general problem; a solutionX _(n) being obtained, we require the solutionX _(n+1) of the same system plus one row. In this context the productHX can be seen as a filtered version of the originalZ. The inversion of a system ofg linear equations is required at each step of the recursive solution.     

On a class of lattices associated with n-cubes

A Lattice L(X) is defined starting from a cubical lattice L and an increasing diagonally closed subset X of L (Section 1). The lattice L(X) are proved to be—up to isomorphism—precisely those of signed simplexes of a simplical complex (Section 2); furthermore, an algebraic combinatorial characterization of the lattices L(X) is given (Section 3).     

On approximation of a class by another class and extremal subspaces inL 1

Получена оценка (в опр еделенном смысле неу лучшаемая) наилучшего приближе ния в метрикеL ₁=L₁(0,2π) 2π-перио дических функций кла сса W^rH^ω = {f:f∈C^r,ω(f ^(r),δ) ≦ ω(δ)}, r = 0, 1, ..., (ω(δ) — выпуклый вверх мо дуль непрерывности) ф ункциями класса W ₁ ^r+v N = {?: ?^r+v?1)(t) —локально абсолютн о непрерывна, ∥?(^r+v∥L₁≦N}, v≧2. Доказано, что каждое п одпространство нече тной размерности, реализу ющее поперечник (по Колмогорову) класс а W ₁ ^r+v в L₁, обладает аналог ичным свойством относител ьно класса W^rH^ω при любом выпуклом вверх ω(δ).     

On positive linear interpolation operators

В этой работе мы даем о бобщение понятия нор мальной системы точек, введен ного Фейером [3]. Наше определ ение включает и случа й бесконечного интерв ала (0, ∞). Доказано, в частности, что систе ма точек 0<x ₁ ⁽ⁿ⁾ /(n)<... _n ⁽ⁿ⁾ <∞ является нормальной в смысле нашего определения тогда и т олько тогда, когда вып олняются оценки — фиксированное чис ло, 0≦?<1. Мы доказываем, что есл и точкиx _k ⁽ⁿ⁾ /(n) являются ну лями многочлена ЛагерраL _n ^(α) (x), то они образуют норма льную систему в том и т олько том случае, когда ?1<α≦0. Мы получаем, таким обр азом, положительный интерполяционный пр оцесс для каждой нормальной системы т очек и устанавливаем теорему сходимости для того с лучая, когда эти точки являются ну лямиL _n ^(α) (x) при — 1相似文献   

Minkowski Length of 3D Lattice Polytopes

We study the Minkowski length L(P) of a lattice polytope P, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing L(P) where P is a 3D lattice polytope. We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if Q, a subpolytope of P, is the Minkowski sum of L=L(P) lattice polytopes Q _i , each of Minkowski length 1, then the total number of interior lattice points of the polytopes Q ₁,??,Q _L is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case.     

On graphsG for which all large trees areG-good

LetG be a graph satisfying x(G) = k. The following problem is considered: WhichG have the property that, if n is large enough, the Ramsey numberr(G, T) has the value (k — 1)(n — 1) + 1 for all treesT onn vertices? It is shown thatG has this property if and only if for somem, G is a subgraph of bothL _k,m andM _K.m , whereL _k,m andM _k,m are two particulark-chromatic graphs. Indeed, it is shown thatr(L _k,m ,M _k,m ,T _n ) = (k — 1)(n — 1) + 1 whenn is large.     

Systems of functions which are complete in measure

In this note it is proved that if a complete orthonormal system {? _n} in L₂[0, 1] contains a subsystem {? _nk} of a lacunary order p>2, then for some bounded measurable function h(x) the system {h(x)? _n(x)}_n≠_nk is complete in L₂[0, 1].     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains

Given a complete Riemannian manifoldM (or a regionU inR ^N ) and two second-order elliptic operators L₁, L₂ in M (resp.U, conditions, mainly in terms of proximity near infinity (resp. near ?U) between these operators, are found which imply that their Green’s functions are equivalent in size. For the case of a complete manifold with a given reference pointO the conditions are as follows:L ₁ andL ₂ are weakly coercive and locally well-behaved, there is an integrable and nonincreasing positive function Ф on [0, ∞[ such that the “distance” (to be defined) betweenL ₁ andL ₂ in each ballB(x, 1 ) ?M is less than Ф(d(x, O)). At the same time a continuity property of the bottom of the spectrum of such elliptic operators is proved. Generalizations are discussed. Applications to the domain case lead to Dini-type criteria for Lipschitz domains (or, more generally, Hölder-type domains).     

Properties of derivations on some convolution algebras

For all convolution algebras L ¹[0, 1); L _loc ¹ and A(ω) = ∩ _n L ¹(ω _n ), the derivations are of the form D _μ f = Xf * μ for suitable measures μ, where (Xf)(t) = tf(t). We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure μ. Moreover, for all these algebras we show that the extension of D _μ to a natural dual space is weak-star continuous.     

Reflection Theorems and the Tame Kernel of a Number Field

Let L be a number field containing the pth primitive root of unity ζ _p . We investigate the p-rank of the ideal class groups of some subfields of L by using reflection theorems and establish relations between the p-rank of the ideal class groups and that of groups of units of some subfields of L.

Let F be a number field and 𝒪 _F the ring of integers in F. We also study the p-rank of tame kernels of F and establish relations between the p-rank of K ₂𝒪 _F and that of some direct summands of the ideal class group of F(ζ _p ).     

On spectral characterizations and embeddings of graphs

The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G. In this paper all graphs with λ(G) greater than ?2 are characterized. Such a graph is a generalized line graph of the form L(T;1,0,…,0), L(T), L(H), where T is a tree and H is unicyclic with an odd cycle, or is one of 573 graphs that arise from the root system E₈. If G is regular with λ(G)>?2, then Gis a clique or an odd circuit. These characterizations are used for embedding problems; λ_R(H) = sup{λ(G)z.sfnc;HinG; Gregular}. H is an odd circuit, a path, or a complete graph iff λ_R(H)> ?2. For any other line graph H, λ_R(H) = ?2. A similar result holds for complete multipartite graphs.