Some extensions of the Cauchy-Davenport theorem

The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s in $\mathbb{Z}/p\mathbb{Z}$, the cardinality of the sumset $A+B=\{a+b|a\in A,b\in B\}$ is bounded below by $\min (r+s-1,p)$; moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G and positive integers $r,s⩽|G|$, the analogous sharp lower bound, namely the function${\mu }_G(r,s)=\min \{|A+B||A,B\subset G,|A|=r,|B|=s\}.$ Important progress on this topic has been achieved in recent years, leading to the determination of ${\mu }_G$ for all abelian groups G. In this note we survey the history of earlier results and the current knowledge on this function.     

Finite rank Toeplitz operators: Some extensions of D. Luecking's theorem

The recent theorem by D. Luecking about finite rank Bergman-Toeplitz operators is extended to weights being distributions with compact support and to the spaces of harmonic functions.     

Some extensions of fan's best approximation theorem

A well-known Ky Fan's best approximation theorem which has been of great importance in nonlinear analysis, game theory, and minimax theorems is extended to a class of factorizable multifunctions.     

Some applications of Ball's extension theorem

We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice , equipped with the metric, in any -uniformly convex Banach space is of order .

     

Some generalizations of Chirka's extension theorem

In this paper, we generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of - where is the open unit disc in and is the graph of a continuous valued function on - to higher dimensions, for certain classes of graphs 1$">. In particular, we show that Chirka's extension theorem generalizes to configurations in 1$">, involving graphs of (non-holomorphic) polynomial maps with small coefficients.

     

Unitary extensions of two-parameter local semigroups of isometric operators and the Krein extension theorem

A notion of two-parameter local semigroups of isometric operators in Hilbert space is discussed. It is shown that under certain conditions such a semigroup can be extended to a strongly continuous two-parameter group of unitary operators in a larger Hilbert space. As an application a simple proof of the Eskin bidimensional version of the Krein extension theorem is given.     

The Mardesic factorization theorem for extension theory and c-separation

We shall prove a type of Mardesic factorization theorem for extension theory over an arbitrary stratum of CW-complexes in the class of arbitrary compact Hausdorff spaces. Our result provides that the space through which the factorization occurs will have the same strong countability property (e.g., strong countable dimension) as the original one had. Taking into consideration the class of compact Hausdorff spaces, this result extends all previous ones of its type. Our factorization theorem will simultaneously include factorization for weak infinite-dimensionality and for Property C, that is, for C-spaces.

A corollary to our result will be that for any weight and any finitely homotopy dominated CW-complex , there exists a Hausdorff compactum with weight and which is universal for the property and weight . The condition means that for every closed subset of and every map , there exists a map which is an extension of . The universality means that for every compact Hausdorff space whose weight is and for which is true, there is an embedding of into .

We shall show, on the other hand, that there exists a CW-complex which is not finitely homotopy dominated but which has the property that for each weight , there exists a Hausdorff compactum which is universal for the property and weight .

     

Some contributions to the theory of cyclic quartic extensions of the rationals

K is a cyclic quartic extension of Q iff $\text{K = Q((rd + p d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>)</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where d > 1, p and r are rational integers, d squarefree, for which p² + q² = r²d for some integer q. Following a paper of A. A. Albert we show that the absolute discriminant, $\text{d(}\text{K}\text{Q}\text{)}$, of the general cyclic quartic extension is given by $\text{d(}\text{K}\text{Q}\text{) = (W}{}^2\text{d}{}^2\text{)}$ for an explicitly computable rational integer W. We next find that the relative discriminant, $\text{d(}\text{K}\text{F}\text{)}$, is given by $\text{d(}\text{K}\text{F}\text{) = (W d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where $\text{F = Q (d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is K′s uniquely determined quadratic subfield. We use this last result in conjunction with Corollary 3, page 359, of Narkiewicz's “Elementary and Analytic Theory of Algebraic Numbers” (PWN-Polish Scientific Publishers, 1974) to establish the following Theorem 1: If the (wide) class number of$\text{F = Q(d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$is odd then every cyclic quartic extensionKofQcontainingFhas a relative integral basis overF. We give a second, more organic, proof of Theorem 1 which also allows us to prove the following converse result, namely Theorem 2: Suppose the quadratic fieldFis contained in some cyclic quartic extension ofQand suppose thatFhas even (wide) class number. There then is a cyclic quartic extensionKofQcontainingFsuch thatKhas no relative integral basis overF.     

Some remarks on the folk theorem in game theory

It is argued that although the pathological multiplicity of Nash equilibria of super games stated by the folk theorem can be removed by introducing limited observations into super games with a continuum of players, the consideration of super games in terms of the Nash equilibrium concept involves a more fundamental and conceptual difficulty.     

On extensions of the domain invariance theorem

If f maps continuously a compact subset X of Rⁿ into Rⁿ and x is a point whose distance from the boundary ∂X is greater than double diameter of the fibres of the points in f(∂X) then f(x) is in the interior of f(X). This theorem extends some results due to Borsuk and Sitnikov.     

Some remarks concerning the individual ergodic theorem of information theory

Let (X,μ, T) be an ergodic dynamic system and let ξ = (C₁, C₂, ...) be a discrete decomposition of X. Conditions are considered for the existence almost everywhere of $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left| {\log \mu (C_{\xi ^n } (x))} \right|,$$ whereC _ξn(x) is the element of the decomposition ξⁿ = ξ V T ξ V ... < T^n-1ξ containing x. It is proved that the condition H(ξ) < ∞ is close to being necessary. If T is a Markov automorphism and ξ is the decomposition into states, then the limit exists, even if H(ξ) = ∞, and is equal to the entropy of the chain.     

An extension of H. Cartan's theorem

In this article we prove that if , , is a bounded pseudoconvex domain with real analytic boundary, then for each , there exists a fixed open neighborhood of and an open neighborhood of in such that any can be extended holomorphically to , and that the action defined by

is real analytic in joint variables. This extends H. Cartan's theorem beyond the boundary. Some applications are also discussed here.

     

A stable theorem of the alternative: an extension of the gordan theorem

A theorem with a number of equivalent alternatives is proposed as an extension of the classical Gordan theorem of the alternative. The theorem can handle nonzero unrestricted variables which cannot be directly treated by ordinary theorems of the alternative. Like the Gordan theorem, the extended theorem has the stability feature that small perturbations in the data will not invalidate an alternative that is in force. The theorem has useful applications in establishing the boundedness and uniqueness of feasible points of polyhedral sets and of solutions to linear programming problems.     

Some extensions of the Lanczos-Ortiz theory of canonical polynomials in the Tau Method

Lanczos and Ortiz placed the canonical polynomials (c.p.'s) in a central position in the Tau Method. In addition, Ortiz devised a recursive process for determining c.p.'s consisting of a generating formula and a complementary algorithm coupled to the formula. In this paper a) We extend the theory so as to include in the formalism also the ordinary linear differential operators with polynomial coefficients with negative height

where denotes the degree of . b) We establish a basic classification of the c.p.'s and their orders , as primary or derived, depending, respectively, on whether or such does not exist; and we state a classification of the indices , as generic , singular , and indefinite . Then a formula which gives the set of primary orders is proved. c) In the rather frequent case in which all c.p.'s are primary, we establish, for differential operators with any height , a recurrency formula which generates bases of the polynomial space and their multiple c.p.'s arising from distinct , , so that no complementary algorithmic construction is needed; the (primary) c.p.'s so produced are classified as generic or singular, depending on the index . d) We establish the general properties of the multiplicity relations of the primary c.p.'s and of their associated indices. It becomes clear that Ortiz's formula generates, for , the generic c.p.'s in terms of the singular and derived c.p.'s, while singular and derived c.p.'s and the multiples of distinct indices are constructed by the algorithm.