In this paper we prove a special case of the transversal conjecture of Tverberg and Vrećica. We consider the case when the numbers of parts ri in this conjecture are powers of the same prime. We also prove some results on common transversals that generalize the classical nonembeddability theorems. We also prove an analogue of colored Tverberg’s theorem by Živaljević and Vrećica. Instead of mult...

Let K/k be a Galois extension of number fields, with Galois group G = Gal(K/k), and suppose ρ : G → GLn(C) is a nontrivial irreducible representation of G. Stark’s conjectures [Tate 84] aim to unravel the arithmetic information encoded in the leading coefficient of the Taylor series for the Artin L-function L(s, ρ) of ρ at s = 0. When G is abelian and one modifies the Artin L-function by removi...

In this paper we prove the case \(dim(V_3)=3\) of a conjecture second author about exterior operad \({\Lambda }^{S^2}_{V_d}\). For introduce collection natural involutions on set homogeneous cycle-free d-partitions complete graph \(K_{2d}\), and show that these correspond to relations in }^{S^2}_{V_d}(2d+1)\). When \(d=3\) correspondence allows us give an explicit description determinant-like m...

Vasil'ev posed Problem 16.26 in [The Kourovka Notebook: Unsolved Problems in Group Theory, 16th ed.,Sobolev Inst. Math., Novosibirsk (2006).] as follows:Does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? Conjecture: $k = 5$.In [Zvezdina, On nonabelian simple groups having the same prime graph a...

In this paper‎, ‎using the discrete Fourier transform in the finite-dimensional Hilbert space C^n‎, ‎a class of nonRieszable equal norm tight frames is introduced ‎and‎ using this class, a bound for Fiechtinger Conjecture is presented. By the Fiechtinger Conjecture that has been proved recently, for given A,C>0 there exists a universal constant delta>0 independent of $n$ such that every C-equal...

In this paper, using a technique of Verona and Verona, we show that a result announced recently by Eberhard and Wenczel implies the truth of the Rockafellar conjecture. We then show that there is a gap in the logic of the Eberhard–Wenczel result, which we tried unsuccessfully to close. We also discuss briefly the connection with maximally monotone multifunctions of type (FPV). One of the fundam...

A special case, called the divergence-free case, of the Jacobian Conjecture in dimension two is proved. This note outlines an argument for a special case of the Jacobian conjecture in dimension two: Suppose F : C → C is a polynomial so that F (0) = 0, F ′(0) = I, det(F ′(z)) = 1, z ∈ C. (1) where I is the identity transformation on C. Write F (x, y) = ( r(x, y) + x s(x, y) + y ) , (x, y) ∈ C wh...

It was conjectured that the smallest minimal point sets of PG(2s, q), q a square, that meet every s-subspace and that generate the whole space are Baer subgeometries PG(2s, √ q). This was shown in 1971 by Bruen for s = 1, and by Metsch and Storme [5] for s = 2. Our main interest in this paper is to prepare a possible proof of this conjecture by proving a strong theorem on line-blocking sets in ...

Henning and Yeo [SIAM J. Discrete Math. 26 (2012) 687–694] conjectured that a 3-regular digraph D contains two vertex disjoint directed cycles of different length if either D is of sufficiently large order or D is bipartite. In this paper, we disprove the first conjecture. Further, we give support for the second conjecture by proving that every bipartite 3-regular digraph, which either possesse...

The well-known ABC-conjecture is generally formulated as follows: The ABC-conjecture. Consider the set S of triples (A, B, C) ∈ N 3 such that ABC = 0, gcd{A, B, C} = 1 and A + B = C Then for every ǫ > 0, there exists a constant K ǫ such that C ≤ K ǫ · R(ABC) 1+ǫ for all triples (A, B, C) ∈ S, where R(ABC) denotes the square-free part of the product ABC. The ABC-conjecture is studied in many pap...