Optimal sequences of continuous functions converging to a Baire-1 function

A proof of Rosenthal's c_o- index conjecture is given. Our approach uses optimal sequences of continuous functions converging to a Baire-1 function. Their existence is obtained by the ”optimal sequences theorem” stated and proved here. For a sequence of functions and a countable ordinal, the variation is also introduced. If pointwise converges to f the relation between and is completely clarified. Finally, optimal sequences associated to a Baire-1 function are defined and their existence for every Baire-1 function is provided. Received: 10 November 2000 / Revised version: 12 March 2002 / Published online: 5 September 2002     

A classification of graph capacity functions

Given two graphs G = (V(G), E(G)) and H = (V(H), E(H)), the sum of G and H, G + H, is the disjoint union of G and H. The product of G and H, G × H, is the graph with the vertex set V(G × H) that is the Cartesian product of V(G) and V(H), and two vertices (g₁, h₁), (g₂, h₂) are adjacent if and only if [g₁, g₂] (ELEMENT) E(G) and [h₁, h₂] (ELEMENT) E(H). Let G denote the set of all graphs. Given a graph G, the G-matching function, γ_G, assigns any graph H (ELEMENT) G to the maximum integer k such that kG is a subgraph of H. The graph capacity function for G, P_G: G → (RFRAKTUR), is defined as P_G(H) = lim_n→z[γ_G(Hⁿ)]^1/n, where Hⁿ denotes the n-fold product of H × H × … × H. Different graphs G may have different graph capacity functions, all of which are increasing. In this paper, we classify all graphs whose capacity functions are additive, multiplicative, and increasing; all graphs whose capacity functions are pseudo-additive, pseudo-multiplicative, and increasing; and all graphs whose capacity functions fall under neither of the above cases. © 1996 John Wiley & Sons, Inc.     

A classification of rapidly growing Ramsey functions

Let be a number-theoretic function. A finite set of natural numbers is called -large if . Let be the Paris Harrington statement where we replace the largeness condition by a corresponding -largeness condition. We classify those functions for which the statement is independent of first order (Peano) arithmetic . If is a fixed iteration of the binary length function, then is independent. On the other hand is provable in . More precisely let where denotes the -times iterated binary length of and denotes the inverse function of the -th member of the Hardy hierarchy. Then is independent of (for ) iff .

     

FE classification of functions of many-valued logic

On the set of functions of many-valued logic, we consider the closure operator defined on the basis of systems of functional equations (the operator of FE closure). This operator generates an FE classification of the functions of many-valued logic, whose kernel consists of classes of S _G -type defined by groups of permutations of G. A number of results are obtained to guarantee FE precompleteness of classes of only this type in the classes of S _G type. The results obtained are illustrated by examples of functions of 2-, 3-, and 4-valued logics.     

A classification of bisymmetric polynomial functions over integral domains of characteristic zero

We describe the class of n-variable polynomial functions that satisfy Aczél’s bisymmetry property over an arbitrary integral domain of characteristic zero with identity.     

Complete classification of shape functions of self-similar solutions

In this paper we study some asymptotic profiles of shape functions of self-similar solutions to the initial-boundary value problem with Neumann boundary condition for the generalized KPZ equation: u_t=u_xx−|u_x|^q, where q is positive number. The shapes of solutions of the corresponding nonlinear ordinary differential equation are of very different nature. The properties depend on the critical value and initial data as usual.     

A regularity classification of boundary points for p-harmonic functions and quasiminimizers

In this paper it is shown that irregular boundary points for p-harmonic functions as well as for quasiminimizers can be divided into semiregular and strongly irregular points with vastly different boundary behaviour. This division is emphasized by a large number of characterizations of semiregular points. The results hold in complete metric spaces equipped with a doubling measure supporting a Poincaré inequality. They also apply to Cheeger p-harmonic functions and in the Euclidean setting to A-harmonic functions, with the usual assumptions on A.     

Topological classification of continuous selections of five linear functions

Continuous selections of linear functions play an important role in Morse theory for piecewise C ²-functions. In this article, the topological properties of continuous selections of linear functions are investigated in detail. These are then utilized to provide a complete classification of all continuous selections of five linear functions. This is done by showing that the first four Betti numbers of a simplicial complex induced by such a function fully determine that function up to topological equivalence. The number of different topological types of continuous selections of linear functions has been known only in the case of four or less selection functions so far. The main result of this article now states that there are exactly 26 different topological types of continuous selections of five linear functions.     

On the classification of geometric codes by polynomial functions

It is shown how to represent algebraically all functions that have a zero sum on all -dimensional subspaces ofPG(n,q) or ofAG(n,q). In this way one can calculate the dimensions of related codes, or one can represent interesting sets of points by functions.     

A note on the classification of consumer demand functions with respect to retailer pass-through rates

Tyagi (1999) derived conditions on the curvature of consumer demand functions which make it optimal for a profit-maximizing retailer to pass-through greater (less) than 100% of a manufacturer trade deal amount. Since the pass-through is customarily evaluated at the optimal wholesale price, then additional sufficient conditions are needed to ensure the existence of an optimal wholesale price. The purpose of this note is to derive the additional required conditions on the curvature of the consumer demand functions for the existence of a greater (less) than 100% retailer pass-through rate at the optimal wholesale price.     

Active classification using belief functions and information gain maximization

Obtaining reliable estimates of the parameters of a probabilistic classification model is often a challenging problem because the amount of available training data is limited. In this paper, we present a classification approach based on belief functions that makes the uncertainty resulting from limited amounts of training data explicit and thereby improves classification performance. In addition, we model classification as an active information acquisition problem where features are sequentially selected by maximizing the expected information gain with respect to the current belief distribution, thus reducing uncertainty as quickly as possible. For this, we consider different measures of uncertainty for belief functions and provide efficient algorithms for computing them. As a result, only a small subset of features need to be extracted without negatively impacting the recognition rate. We evaluate our approach on an object recognition task where we compare different evidential and Bayesian methods for obtaining likelihoods from training data and we investigate the influence of different uncertainty measures on the feature selection process.     

On the classification of APN functions up to dimension five

We classify the almost perfect nonlinear (APN) functions in dimensions 4 and 5 up to affine and CCZ equivalence using backtrack programming and give a partial model for the complexity of such a search. In particular, we demonstrate that up to dimension 5 any APN function is CCZ equivalent to a power function, while it is well known that in dimensions 4 and 5 there exist APN functions which are not extended affine (EA) equivalent to any power function. We further calculate the total number of APN functions up to dimension 5 and present a new CCZ equivalence class of APN functions in dimension 6.     

Towards the classification of self-dual bent functions in eight variables

In this paper, we classify quadratic and cubic self-dual bent functions in eight variables with the help of computers. There are exactly four and 45 non-equivalent self-dual bent functions of degree two and three, respectively. This result is achieved by enumerating all eigenvectors with ± 1 entries of the Sylvester Hadamard matrix with an integer programming algorithm based on lattice basis reduction. The search space has been reduced by breaking the symmetry of the problem with the help of additional constraints. The final number of non-isomorphic self-dual bent functions has been determined by exploiting that EA-equivalence of Boolean functions is related to the equivalence of linear codes.     

Combinatorial aspects of the topological classification of functions on a circle

We prove a necessary and sufficient condition of the topological equivalence of smooth functions with finitely many local extrema defined on a circle. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 829–836, June, 2008.