Major Theorems Research Articles

List multicolorings of graphs with measurable sets

AbstractIn an ordinary list multicoloring of a graph, the vertices are “colored” with subsets of pre‐assigned finite sets (called “lists”) in such a way that adjacent vertices are colored with disjoint sets. Here we consider the analog of such colorings in which the lists are measurable sets from an arbitrary atomless, semifinite measure space, and the color sets are to have prescribed measures rather than prescribed cardinalities. We adapt a proof technique of Bollobás and Varopoulos to prove an analog of one of the major theorems about ordinary list multicolorings, a generalization and extension of Hall's marriage theorem, due to Cropper, Gyárfás, and Lehel. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 179–193, 2007

After Godel

This paper describes the enormous impact of Gödel's work on mathematical logic and recursion theory. After a brief description of the major theorems that Gödel proved, it focuses on subsequent work extending what he did, sometimes by quite different methods. The paper closes with a new result, applying Gödel's methods to show that if scientific epistemology (what Chomsky calls our “scientific competence”) could be completely represented by a particular Turing machine, then it would be impossible for us to know that fact.

Is Cultural-Historical Activity Theory Threatened to Fall Short of its Own Principles and Possibilities as a Dialectical Social Science?

In recent years, many researchers engaged in diverse areas and approaches of “cultural-historical activity theory” (CHAT) realized an increasing international interest in Lev S. Vygotsky’s, A. N. Leont’ev’s, and A. Luria’s work and its continuations. Not so long ago, Yrjö Engeström noted that the activity approach was still “the best-held secret of academia” (p. 64) and highlighted the “impressive dimension of theorizing behind” it. Certainly, this remark reflects a time when CHAT was off the beaten tracks. But if this situation begins to change today, in which direction will CHAT be heading? Will it continue to be one of those projects “unique for its practical, political, and civic engagement” committed “to ideals of social justice, equality, and social change” as it was in the beginning (Stetsenko & Arievitch, 2004, p. 58)? Although a positive future of CHAT seems to lie ahead, we consider in this article some of the problematics that may challenge all those who want to pass the “impressive dimensions of theorizing” from “insider” circles to a larger audience and from one generation to another as well as encourage newcomers to become part of this tradition through critical engagement in its theory and practice. A key to these engagements, we suggest, is not only the comprehensive empirical and philosophical basis, but also the role of dialectics as both topic and method. Therefore, the challenge for newcomers (as well as for “old-timers”) to take on the tradition of CHAT is not a small one indeed. We assume that a major reason for the increasing interest in CHAT lies in its potential to provide a non-reductionist approach to human development, which is due to its affinity to dialectics; however, the close interrelation to a tradition that reaches back to the theories of Georg W.F. Hegel and Karl Marx, among others, is not the easiest to master.In consideration of these difficulties, the purpose of this article is to investigate how contemporary approaches within CHAT can continue to provide a dialectical framework to preserve and renew the critical intention of this tradition, and how we run the risk of losing this sting. Thereby, we sensitize researchers to the problem of developing a cultural-historical approach within a historical situation that confronts us with new, unanswered questions. In this light, we also problematize the use of scientific language, for it may lead us to speak and argue un-dialectically when in fact we intend or ought to think dialectically.This article seeks to convey insights and arguments of how we can relate our theoretical approaches to a tradition of dialectical thinking and in what ways this is paramount for a critical engagement in theory and practice. In the first part, we therefore discuss not only some major theorems in Hegel’s and Marx’s work but also, and above all, Vygotsky’s way of developing the cultural-historical approach of psychology. Second, we argue that the contemporary, widely known version of CHAT, related to Yrjö Engeström’s theoretical and empirical work, neglects different aspects of dialectical thinking and consequently narrows its potential to a socio-critical approach to societal practice and human development. A crucial question of this scrutiny will be the notion of contradictions and how development is supposed to be achieved. In general, our intention is not only to clarify the role of dialectics as a method for activity theory but also to problematize the role of the subjects of research in CHAT and to confront ourselves with the problems of practicing and developing a critical science in face of a complex and challenging societal world.

Erratum to “Hyperfinite and standard unifications for physical theories”

This corrects the major theorem on product consequence operators.

The role of race track principles

The paper shows an alternative way of presenting differential calculus. It is shown that the Race Track Principles (RTP) (or any of the variants) is, in fact, equivalent to the Mean Value Theorem. Moreover, it is demonstrated how major theorems of differential calculus can be derived using RTP. The benefits of using RTP as a means to introduce theoretical part of differential calculus are analysed.

Carleson Type Theorems for Certain Convolution Operators

We prove mapping theorems for some convolution operators, acting from Sobolev type spaces in \(\mathbb{R}^{n}\) to Lorentz spaces defined on \(\mathbb{R}^{{n + 1}}_{ + }\) with a fractional-order Carleson measure. As an application of the major theorems, we give some a priori estimates for the solutions of certain elliptic equations.

Classical and approximate sampling theorems; studies in the [formula omitted] and the uniform norm

The approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker–Kotel’nikov–Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for f ∈ B π w p , 1 ⩽ p < ∞ , w > 0 , implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm. Turning now to L p ( R ) -space, it is shown that the classical sampling theorem for f ∈ B π w p , 1 < p < ∞ (here p = 1 must be excluded), implies the L p ( R ) -approximate sampling theorem with convergence in the L p ( R ) -norm, provided that f is locally Riemann integrable and belongs to a certain class Λ p . Basic in the proof is an intricate result on the representation of the integral ∫ R | f ( u ) | p du as the limit of an infinite Riemann sum of | f | p for a general family of partitions of R ; it is related to results of O. Shisha et al. (1973–1978) on simply integrable functions and functions of bounded coarse variation on R . These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago.

Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals

The Whittaker–Shannon–Kotel'nikov sampling theorem enables one to reconstruct signals f bandlimited to [ − π W , π W ] from its sampled values f ( k / W ) , k ∈ Z , in terms of ( S W f ) ( t ) ≡ ∑ k = − ∞ ∞ f ( k W ) sinc ( W t − k ) = f ( t ) ( t ∈ R ) . If f is continuous but not bandlimited, one normally considers lim W → ∞ ( S W f ) ( t ) in the supremum-norm, together with aliasing error estimates, expressed in terms of the modulus of continuity of f or its derivatives. Since in practice signals are however often discontinuous, this paper is concerned with the convergence of S W f to f in the L p ( R ) -norm for 1 < p < ∞ , the classical modulus of continuity being replaced by the averaged modulus of smoothness τ r ( f ; W −1 ; M ( R ) ) p . The major theorem enables one to sample any bounded signal f belonging to a certain subspace Λ p of L p ( R ) , the jump discontinuities of which may even form a set of measure zero on R . A corollary gives the counterpart of the approximate sampling theorem, now in the L p -norm. The averaged modulus, so far only studied for functions defined on a compact interval [ a , b ] , had first to be extended to functions defined on the whole real axis R . Basic tools are the de La Vallée Poussin means and a semi-discrete Hilbert transform.

Least Criminals for Some Given Conjectures in Group Theory

In the theory of finite groups, many major theorems have been proved by the minimum counterexample technique or sometimes called “least criminals” technique. It uses the fact that if there is a counterexample to a given theorem, then there is a counterexample of smallest possible order. In this paper, we will find the least criminals for five given conjectures in group theory. Keywords: Least Criminals; Group Theory; Conjectures

Matrix inequalities by means of embedding

In this expository study some basic matrix inequalities obtained by embedding bilinear formsAx, xandAx, yinto 2 × 2 matrices are investigated. Many classical inequalities are reproved or refined by the proposed unified approach. Some inequalities involving the matrix absolute value |A| are given. A new proof of Ky Fan's singular value majorization theorem is presented.

A Mathematical Excursion: From the Three—door Problem to a Cantor-Type Set

2. STARTING POINT: THE THREE-DOOR PROBLEM. Mathematics has always been enriched by a- diversity of games and intellectual curiosities. These have provided an endless supply of problems that have acquired a life of their own, far removed from the recreational aspect of their origins. For example, the first building blocks of probability owe their existence to the analysis of gambling games carried out by Fermat and Pascal in the beginning of the XVIIth century. Undoubtedly Fermat himself was much attracted to mathematics thanks to Bachet's Problemes plaisants et delectables of 1612 [1], which was an introduction to Bachet's most famous book: the Latin translation of Diophantus' Arithmetica, in whose margins Fermat wrote the note that made his major theorem famous. Another important instance, E. Lucas' Re'cre'ations Mathematiques [15], was a source of interesting problems at the beginning of the present century. Let us start our excursion by setting a simple problem in the form of a seemingly innocent game.

Harmonic Analysis on the Quantum Lorentz Group

This work begins with a review of complexification and realification of Hopf algebras. We emphasize the notion of multiplier Hopf algebras for the description of different classes of functions (compact supported, bounded, unbounded) on complex quantum groups and the construction of the associated left and right Haar measure. Using a continuation of $6j$ symbols of $SU_q (2)$ with complex spins, we give a new description of the unitary representations of $SL_q (2,\CC)_{\RR}$ and find explicit expressions for the characters of $SL_q (2,\CC)_{\RR}$. The major theorem of this article is the Plancherel theorem for the Quantum Lorentz Group.

Some remarks on Scheiblechner's treatment of isop models

Scheiblechner (1995) proposes a probabilistic axiomatization of measurement called ISOP (isotonic ordinal probabilistic models) that replaces Rasch's (1980) specific objectivity assumptions with two interesting ordinal assumptions. Special cases of Scheiblechner's model include standard unidimensional factor analysis models in which the loadings are held constant, and the Rasch model for binary item responses. Closely related are the doubly-monotone item response models of Mokken (1971; see also Mokken & Lewis, 1982; Sijtsma, 1988; Molenaar, 1991; Sijtsma & Junker, 1996; and Sijtsma & Hemker, in press). More generally, strictly unidimensional latent variable models have been considered in some detail by Holland and Rosenbaum (1986), Ellis and van den Wollenberg (1993), and Junker (1990, 1993). The purpose of this note is to provide connections with current research in foundations and nonparametric latent variable and item response modeling that are missing from Scheiblechner's (1995) paper, and to point out important related work by Hemker, Sijtsma, Molenaar, & Junker (1996, 1997), Ellis and Junker (in press) and Junker and Ellis (1997). We also discuss counterexamples to three major theorems in the paper. By carrying out these three tasks, we hope to provide researchers interested in the foundations of measurement and item response modeling the opportunity to give the ISOP approach the careful attention it deserves.

Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids

We use the support-operator method to derive new discrete approximations of the divergence, gradient, and curl using discrete analogs of the integral identities satisfied by the differential operators. These new discrete operators are adjoint to the previously derived natural discrete operators defined using ‘natural’ coordinate-invariant definitions, such as Gauss' theorem for the divergence. The natural operators cannot be combined to construct discrete analogs of the second-order operators div grad, grad div, and curl curl because of incompatibilities in domains and in the ranges of values for the operators. The same is true for the adjoint operators. However, the adjoint operators have complementary domains and ranges of values and the combined set of natural and adjoint operators allow a consistent formulation for all the compound discrete operators. We also prove that the operators satisfy discrete analogs of the major theorems of vector analysis relating the differential operators, including div A = 0 if and only if A = curl B ; curl A = 0 if and only if A = grad ϕ .

Sharpening Jensen's Inequality and a Majorization Theorem

Sharpening Jensen's Inequality and a Majorization Theorem

On New Majorization Theorems

On New Majorization Theorems

On solvability of inverse eigenvalue problems with Hermitian matrices

Taking advantages of Krylov, Bogoljubov, and Weinstein's and Temple's theories and Schur's majorization theorem, we present new sufficient conditions for the solvability of multiplicative inverse eigenvalue problems and so-called generalized inverse eigenvalue problems with Hermitian matrices.

Sparse parameterized problems

Sparse languages play an important role in classical structural complexity theory. In this paper we introduce a natural definition of sparse problems for parameterized complexity theory. We prove an analog of Mahaney's theorem: there is no sparse parameterized problem which is hard for the tth level of the W hierarchy, unless the W hierarchy itself collapses up to level t. The main result is proved for the most general form of parametric many:1 reducibility, where the parameter functions are not assumed to be recursive. This provides one of the few instances in parameterized complexity theory of a full analog of a major classical theorem. The proof involves not only the standard technique of left sets, but also substantial circuit combinatorics to deal with the problem of small weft, and a diagonalization to cope with potentially nonrecursive parameter functions. The latter techniques are potentially of interest for further explorations of parameterized complexity analogs of classical structural results.

Self-Monitoring Machines and anωωHierarchy of Loops

An ordinal hierarchy of recursive functions is developed based on the level to which a function requires a machine computing it to monitor and make decisions concerning itself. The major theorem states that the functions with self-monitoring level belowωωare precisely the class of loop functions (or primitive recursive functions).

Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above

Comparison and rigidity theorems are proved for curves of bounded geodesic curvature in singular spaes of curvature bounded above. Most of these estimates do not appear in the literature even for smooth curves in Riemannian manifolds. Geodesic curvature (which agrees with the usual one in the smooth case) is defined by comparison to curves of constant curvature in a model space. Two methods of comparison are used, preserving either sidelengths of inscribed triangles or arclength and chordlength. Using a majorization theorem of Reshetnyak, we obtain best possible global comparisons for arclength, chordlength, width and base angles in a CAT( K) space. A criterion for a metric ball to be a CAT( K) space is also given, in terms of the radius and the radial uniqueness property.