On the role of supercompact and extendible cardinals in logic

It is proved that the existence of supercompact cardinal is equivalent to a certain Skolem-Löwenheim Theorem for second order logic, whereas the existence of extendible cardinal is equivalent to a certain compactness theorem for that logic. It is also proved that a certain axiom schema related to model theory implies the existence of many extendible cardinals.     

On extensions of supercompactness

We show that, in terms of both implication and consistency strength, an extendible with a larger strong cardinal is stronger than an enhanced supercompact, which is itself stronger than a hypercompact, which is itself weaker than an extendible. All of these are easily seen to be stronger than a supercompact. We also study Cⁿ‐supercompactness.     

Continual version of the Perron effect of change of values of the characteristic exponents

We prove the existence of a perturbed two-dimensional system of ordinary differential equations such that its linear approximation has arbitrarily prescribed negative characteristic exponents, the perturbation is of arbitrarily prescribed higher order of smallness in a neighborhood of the origin, all of its nontrivial solutions are infinitely extendible to the right, and the whole set of their Lyapunov exponents is contained in the positive half-line, is bounded, and has positive Lebesgue measure. In the general case, we also obtain explicit representations of the exponents of these solutions via their initial values.     

Embedding (r, 1)-designs in finite projective planes

It is known that any non-trivial (r,1)-design on υ varieties (υ ? (r? 1)² ? 1) is extendible; this fact implies the existence of a projective plane of order r ? 1. In this paper it is shown that any non-trivial (r, 1)-design on (r ? 1)² ? α varieties, where r and α are appropriately bounded, is extendible; hence this fact implies the existence of a projective plane of order r ? 1. We also show that, for υ ? (r ? 1)² ? 2, any non-trivial (r, 1)-design on υ varieties is extendible.     

On search by address computation

We study the effect of data distribution on address computation data structures for searching, as typified by the priority queue problem. We compare several techniques showing that, in contrast to sorting, neither one nor multilevel bucket methods are uniformly efficient for this task. However, an enhancement of order preserving extendible hashing is shown to behave asymptotically independently of the amount of data and its distribution. Also conclusions regarding multiattribute file structures are presented.     

Extendible elements of the alternating groups

An element σ of A_n, the Alternating group of degree n, is extendible in S_n, the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

On the Samarskii-Andreev transmission conditions in the theory of elastic beams

It is proved that the transmission conditions for elastic beams in the case of a nonideal joint are limiting in the construction of asymptotics for the transmission problem for two thin elastic bodies if the boundary between the bodies is filled by slightly extendible material.     

Extendible elements of the alternating groups

An element?σ?of A_n , the Alternating group of degree n, is extendible in S_n , the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

On the sheaf of extendible ideals of topological algebras

In this paper a new sheaf for topological algebras, called the sheaf of extendible ideals, is introduced. It is shown that the sheaf space of this sheaf is uniformizable but not complete in general. The research of both of the authors was supported by Estonian Science Foundation grant 6205.     

Stably extendible tangent bundles over lens spaces

The purpose of this paper is to study the stable extendibility of the tangent bundle τn(p) over the (2n+1)-dimensional standard lens space Ln(p) for odd prime p. We investigate for which m the tangent bundle τn(p) is stably extendible to Lm(p) but is not stably extendible to Lm+1(p), where we consider m=∞ if τn(p) is stably extendible to Lk(p) for any k?n, and determine m in the case n?p−3.     

Lower bounds for the distribution function of a k-dimensionaln-extensible exchangeable process

A lower bound for the distribution function of a k-dimensional, n-extensible exchangeable process is provided when the marginals are uniform on the unit segment. The result is obtained by means of standard linear programming techniques. The lower bound for infinitely extendible exchangeable processes is the distribution of independent random variables.     

Endpoint extendable paths in tournaments

Let s(n) be the threshold for which each directed path of order smaller than s(n) is extendible from one of its endpoints in some tournament T_n It is shown that s(n) is asymptotic to 3n/4, with an error term at most 3 for infinitely many n. There are six tournaments with s(n) = n. © 1996 John Wiley & Sons, Inc.     

Ideal points of semisimple Jordan algebras

It is well known that I. L. Kantor has presented a local construction which assigns to a semisimple Jordan algebra a compact symmetric space with an extendible group of motions. For semisimple Jordan algebras of classical type we present a method of completion of the algebra space to a symmetric space in the large that generalizes the completion of a Euclidean space to a projective space and a conformai sphere.     

To what extent does the dual Banach spaceE′ determine the polynomials overE?

We show that under conditions of regularity, ifE′ is isomorphic toF′, then the spaces of homogeneous polynomials overE andF are isomorphic. Some subspaces of polynomials more closely related to the structure of dual spaces (weakly continuous, integral, extendible) are shown to be isomorphic in full generality.     

The pluripolar hull of a graph and fine analytic continuation

We show that if the graph of an analytic function in the unit disk D is not complete pluripolar in C² then the projection of its pluripolar hull contains a fine neighborhood of a point . Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function f in D extends to a function ℱ which is defined on a fine neighborhood of a point and is finely analytic at p then the pluripolar hull of the graph of f contains the graph of ℱ over a smaller fine neighborhood of p. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes of analytic functions in the disk which have non-trivial pluripolar hulls, among them C^∞ functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls. Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation.     

Stable extendibility of normal bundles over lens spaces

We study the stable extendibility of R-vector bundles over the (2n+1)-dimensional standard lens space Ln(p) with odd prime p, focusing on the normal bundle to an immersion of Ln(p) in the Euclidean space R2n+1+t. We show several concrete cases in which is stably extendible to Lk(p) for any k with k?n, and in several cases we determine the exact value m for which is stably extendible to Lm(p) but not stably extendible to Lm+1(p).     

Barycentric systems and stretchability

Using a general resolution of barycentric systems we give a generalization of Tutte's theorem on convex drawing of planar graphs. We deduce a characterization of the edge coverings into pairwise non-crossing paths which are stretchable: such a system is stretchable if and only if each subsystem of at least two paths has at least three free vertices (vertices of the outer face of the induced subgraph which are internal to none of the paths of the subsystem). We also deduce that a contact system of pseudo-segments is stretchable if and only if it is extendible.     

On extendible cardinals and the GCH

We give a characterization of extendibility in terms of embeddings between the structures H _λ . By that means, we show that the GCH can be forced (by a class forcing) while preserving extendible cardinals. As a corollary, we argue that such cardinals cannot in general be made indestructible by (set) forcing, under a wide variety of forcing notions.     

Sequential Arbitrage Measurements and Interest Rate Envelopes

This paper proposes new measures that provide us with the level of sequential arbitrage in bond markets. All the measures vanish in an arbitrage-free market and all of them are positive otherwise. Each measure is generated by a dual pair of optimization problems. Primal problems permit us to compute optimal sequential arbitrage strategies, if available. Each dual problem generates a concrete proxy for the term structure of interest rates. The set of proxies allows us to obtain the exact market price of any bond and may measure several effects. For instance, the credit risk spread of nondefault free bonds, or the embedded option price of callable or extendible bonds. The developed theory has been tested empirically.     

Relation Algebras andt-vertex Condition Graphs

The scheme associated with a graph is an association scheme iff the graph is strongly regular. Consider the problem of extending such an association scheme to a superscheme. The obstacles can be expressed in terms oft-vertex conditions. If a graph does not satisfy thet-vertex condition, a presuperscheme associated with it cannot be erected beyond the (t − 3)rd level. We give an example of an association scheme which is not extendible to a superscheme: it cannot be extended beyond the bottom level of a presuperscheme.