A S(3, {4, 6}, 18) with a subdesign S(3, 4, 8) does not exist

For , a S(t,K,v) design is a pair, , with |V| = v and a set of subsets of V such that each t‐subset of V is contained in a unique and for all . If , , , and is a S(t,K,u) design, then we say has a subdesign on U. We show that a S(3,{4,6},18) design with a subdesign S(3,4,8) does not exist. © 2007 Wiley Periodicals, Inc. J Combin Designs 17: 36–38, 2009     

Bruno de Finetti and imprecision: Imprecise probability does not exist!

We review several of de Finetti’s fundamental contributions where these have played and continue to play an important role in the development of imprecise probability research. Also, we discuss de Finetti’s few, but mostly critical remarks about the prospects for a theory of imprecise probabilities, given the limited development of imprecise probability theory as that was known to him.     

(81, 16, 3) Abelian difference sets do not exist

Of the five abelian groups of order 81, three are known not to contain a (81, 16, 3) difference set. We prove that the remaining two groups behave similarly. Consequently, two of the missing entries in Lander's table can be filled with a negative response.     

Pseudogeometric graph for pG 4 (8, 175)

Doklady Mathematics -     

Nonaffine differential-algebraic curves do not exist

The paper outlines why the spectrum of maximal ideals Spec _? A of a countable-dimensional differential ?-algebra A of transcendence degree 1 without zero divisors is locally analytic, which means that for any ?-homomorphism ψ _M: A → ? (M ∈ Spec _? A) and any a ∈ A the Taylor series \(\widetilde {{\psi _M}}{\left( a \right)^{\underline{\underline {def}} }}\sum\limits_{m = 0}^\infty {\psi M\left( {{a^{\left( m \right)}}} \right)} \frac{{{z^m}}}{{m!}}\) has nonzero radius of convergence depending on the element a ∈ A.     

There does not exist an enumerable family of context-free grammars that generates the class of single-valued languages

Translated from Matematicheskie Zametki, Vol. 50, No. 1, pp. 34–40, July, 1991.     

To exist or not to exist: example generation in Real Analysis

This article deals with the activity of example generation as a special case of problem solving. We asked university students in the scientific-technological area to produce examples (which may exist or not) of mathematical objects fulfilling given requirements. For the analysis of students’ performances, we have developed a model that attempts to grasp the nature of the different stages in the solving process. The discussion of the findings allows us to outline specificities and educational potentialities of example generation activities.     

On averaging of differential inclusions in the case where the average of the right-hand side does not exist

We consider the problem of application of the averaging method to the asymptotic approximation of solutions of differential inclusions of standard form in the case where the average of the right-hand side does not exist.     

Directed graph representation of half-rate additive codes over GF(4)

We show that (n, 2 ⁿ ) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n, 2 ⁿ ) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.     

Recognition by noncommuting graph of finite simple groups L 4(q)

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the set of non-central elements of G, and two different vertices x and y are joined by an edge if and only if x and y do not commute as elements of G, i.e., [x, y] ≠ 1. We prove that if L ∈ {L ₄(7), L ₄(11), L ₄(13), L ₄(16), L ₄(17)} and G is a finite group such that ∇(G) ≅ ∇(L), then G ≅ L.     

The weakest m-convex topology stronger than an A-convex topology need not exist

It is known that every locally A-convex algebra (E, τ) can be endowed with, at least, one locally m-convex topology stronger thanτ. In this note, we answer several questions concerning such locally m-convex topologies. In particular, we show that, unlike what is asserted in several previous papers, the collection of all such topologies does not have any weakest element in general.     

The diagonal hypergraph of a matrix (bipartite graph)

Let A be an n x n matrix of 0's and 1's (a bipartite graph). The diagonal hypergraph of A is the hypergraph whose vertices correspond to the 1's (edges) of A and whose edges correspond to the positive diagonals (1-factors) of A. The numerical invariants of this hypergraph are investigated.     

A geometric realization without self-intersections does exist for Dyck's regular map

Even in our decade there is still an extensive search for analogues of the Platonic solids. In a recent paper Schulte and Wills [13] discussed properties of Dyck's regular map of genus 3 and gave polyhedral realizations for it allowing self-intersections. This paper disproves their conjecture in showing that there is a geometric polyhedral realization (without self-intersections) of Dyck's regular map {3, 8}₆ already in Euclidean 3-space. We describe the shape of this new regular polyhedron.     

The girth of a 4-homogeneous bipartite graph

In this paper, it is proved that the girth of a 4-homogeneous bipartite graph with valency greater than 2 is at most 12.