Index of expansions

2022

In this paper, we study the notion of an index of sub-expansions in an expansion. We prove the index inequality as an application.

THE DOMINATING NUMBER OF EXPANSIONS, 2022

In this paper, we study the notion of dominating number of expansions.

Algebra and Logic, 1982

Journal of Informetrics, 2019

Starting from the notion of h-type indices for infinite sequences we investigate if these indices satisfy natural inequalities related to the arithmetic, the geometric and the harmonic mean. If f denotes an h-type index, such as the h-or the g-index, then we investigate inequalities such as min(f(X),f(Y)) ≤ f((X + Y)/2) ≤ max(f(X), f(Y)). We further investigate if: f(min(X,Y)) = min(f(X),f(Y)) and if f(max(X,Y)) = max(f(X),f(Y)). It is shown that the h-index satisfies all the equalities and inequalities we investigate but the g-index does not always, while it is always possible to find a counterexample involving the R-index. This shows that the h-index enjoys a number of interesting mathematical properties as an operator in the partially ordered positive cone (R +) ∞ of all infinite sequences with non-negative real values. In a second part we consider decreasing vectors X and Y with components at most at distance d. Denoting by D the constant sequence (d,d,d, …) and by Y-D the vector (max(y r-d), 0) r , we prove that under certain natural conditions, the double inequality h(Y-D) ≤ h(X) ≤ h(Y+D) holds.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1980

SOME REMARKS ON THE STRUCTURE OF EXPANSIONS by ROMAN MURAWSKI in Poznali (Poland) l) 0. Peano arithmetic P, as it was shown by K. GODEL, is not complete. Hence it is natural to investigate its extensions. We shall be interested in extensions "generated" by certain second order theories, namely by second order arithmetics T called' simply analysis. The question arises : what can be proved in T about natural number, i.e. what is the theory P" = {cp E L (P) : T t p]. The semantical counterpart of this problem is the following: Call a model 93 I. P T-expandable iff i t is a number-theoretic part of a model of T. It turns out (cf. [I21 part 11, Theorem 37) that YJl t= PT iff there is a model !J? such that !J? 3 XV and % is T-expandable. Hence T-expandable models are good representatives of models of P".

Southeast Asian Bulletin of Mathematics, 2000

We study an index set function which is related to the well-known Levinson inequality and to an inequality from .

Journal of the London Mathematical Society, 1998

This paper is mainly devoted to the study of the index of a map at a zero, and the index of a polynomial map over n. For semi-quasi-homogeneous maps we prove that the index at a zero coincides with the index at this zero of its quasi-homogeneous part. For a class of polynomial maps with finite zero set we provide a method which makes easier the computation of its index over n. Finally we relate the index and the multiplicity.

Journal of Number Theory, 2009

Let q ∈ (1, 2); it is known that each x ∈ [0, 1/(q − 1)] has an expansion of the form x = ∞ n=1 a n q −n with a n ∈ {0, 1}. It was shown in [4] that if q < ( √ 5 + 1)/2, then each x ∈ (0, 1/(q − 1)) has a continuum of such expansions; however, if q > ( √ 5 + 1)/2, then there exist infinitely many x having a unique expansion .

2015

These are the notes for the talk Splittings, comfortably embedded subvarieties and index theorems I gave at the RIMS Symposium on Topological and geometrical methods of complex differential equations in Kyoto, 19-23 January 2004. I wish to sincerely thank prof. Shishikura and prof. Ito for the invitation. These notes contain some well known facts (maybe with a new interpretation) and statements of new results which will be proved in a forthcoming paper. 1. WHAT IS AN INDEX THEOREM? Let X be a n-dimensional complex variety and let ϕ ∈ H•(X) be a (nonzero) element of its cohomology. Often it is not possible to “calculate ” such an element directly. It is then important when one can calculate such element using tools like differential geometry or complex analysis. For instance the Chern classes of a vector bundle on X can be calculated using the Chern-Weil theory of connections, provided X is nonsingular. In applications however it is important to know the image of P (ϕ) ∈ H2n− • where...

2007

We give a prescription for how to compute the Callias index, using as regulator an exponential function. We find agreement with old results in all odd dimensions. We show that the problem of computing the dimension of the moduli space of self-dual strings can be formulated as an index problem in even-dimensional (loop-)space. We think that the regulator used in this Letter can be applied to this index problem.