Jordan curves in the level sets of additive Brownian motion

This paper studies the topological and connectivity properties of the level sets of additive Brownian motion. More precisely, for each excursion set of this process from a fixed level, we give an explicit construction of a closed Jordan curve contained in the boundary of this excursion set, and in particular, in the level set of this process.

     

Normality and boundary behaviour of arbitrary and meromorphic functions along simple curves and applications

In this paper, we establish some theorems giving necessary and sufficient conditions for an arbitrary function defined in the unit disc of the complex plane to have boundary values along classes of an equivalence relation over simple curves. Our results generalize the well-known theorems on asymptotic and angular boundary behaviours of meromorphic functions (Lindelölf-, Lehto–Virtanen- and Seidel–Walsh-type theorems). The obtained results are applied to the study of boundary behaviour of meromorphic functions along curves using P-sequences, as well as in the proof of the uniqueness theorem similar to ?aginjan’s one. The constructed examples of functions show that the results cannot be improved.     

Maximal Sidon sets and matroids

A subset X of an abelian group Γ, written additively, is a Sidon set of orderh if whenever {(a_i,m_i):i∈I} and {(b_j,n_j):j∈J} are multisets of size h with elements in X and ∑_i∈Im_ia_i=∑_j∈Jn_jb_j, then {(a_i,m_i):i∈I}={(b_j,n_j):j∈J}. The set X is a generalized Sidon set of order(h,k) if whenever two such multisets have the same sum, then their multiset intersection has size at least k. It is proved that if X is a generalized Sidon set of order (2h−1,h−1), then the maximal Sidon sets of order h contained in X have the same cardinality. Moreover, X is a matroid where the independent subsets of X are the Sidon sets of order h.     

On the structure of WDC sets

WDC sets in were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial generalization of sets with positive reach and still admit the definition of curvature measures. Using results on singularities of convex functions, we obtain regularity results on the boundaries of WDC sets. In particular, the boundary of a compact WDC set can be covered by finitely many DC surfaces. More generally, we prove that any compact WDC set M of topological dimension can be decomposed into the union of two sets, one of them being a k‐dimensional DC manifold open in M, and the other can be covered by finitely many DC surfaces of dimension . We also characterize locally WDC sets among closed Lipschitz domains and among lower‐dimensional Lipschitz manifolds. Finally, we find a full characterization of locally WDC sets in the plane.     

Linear interval equations with symmetric solution sets

Criteria for symmetry and boundedness are found for the combined solution set of a system of linear algebraic equations with interval coefficients. It is shown that the problem of the best inner interval estimation of a symmetric solution set can be exactly solved by linear programming methods.     

Maximal independent sets in minimum colorings

Every graph G contains a minimum vertex-coloring with the property that at least one color class of the coloring is a maximal independent set (equivalently, a dominating set) in G. Among all such minimum vertex-colorings of the vertices of G, a coloring with the maximum number of color classes that are dominating sets in G is called a dominating-χ-coloring of G. The number of color classes that are dominating sets in a dominating-χ-coloring of G is defined to be the dominating-χ-color number of G. In this paper, we continue to investigate the dominating-χ-color number of a graph first defined and studied in [1].     

On mean approximation by polynomials with gaps on Caratheodory sets

The paper presents some new results on the possibility of approximation by polynomials with gaps. The approximations are done in the norm of the space L _p , 1 ≤ p < + ∞, on the Caratheodory sets in the complex plane. The lacunary versions of some results by Farell—Markushevich, S. Sinanian, A. L. Shahinian are obtained (Theorems 1, 3, 5). Similar approximations by the real parts of lacunary polynomials are given (Theorems 2, 4, 6). Dedicated to the memory of academician S. N. Mergelyan     

Functions on discrete sets holomorphic in the sense of Ferrand, or monodiffric functions of the second kind

We study the class of functions called monodiffric of the second kind by Isaac.They are discrete analogues of holomorphic functions of one or two complex variables.Discrete analogues of the Cauchy-Riemann operator,of domains of holomorphy in one discrete variable,and of the Hartogs phenomenon in two discrete variables are investigated.Two fundamental solutions to the discrete Cauchy-Riemann equation are studied:one with support in a quadrant,the other with decay at infinity.The first is easy to construct by induction;the second is accessed via its Fourier transform.     

Lineability of sets of nowhere analytic functions

Although the set of nowhere analytic functions on [0,1] is clearly not a linear space, we show that the family of such functions in the space of C^∞-smooth functions contains, except for zero, a dense linear submanifold. The result is even obtained for the smaller class of functions having Pringsheim singularities everywhere. Moreover, in spite of the fact that the space of differentiable functions on [0,1] contains no closed infinite-dimensional manifold in C([0,1]), we prove that the space of real C^∞-smooth functions on (0,1) does contain such a manifold in C((0,1)).     

Constructions of generalized Sidon sets

We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s₁+s₂,s_i∈S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g?3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Kolountzakis’ idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo, Ruzsa and Trujillo, Jia, and Habsieger and Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases.     

Operation properties and -equalities of complex fuzzy sets

A complex fuzzy set is a fuzzy set whose membership function takes values in the unit circle in the complex plane. This paper investigates various operation properties and proposes a distance measure for complex fuzzy sets. The distance of two complex fuzzy sets measures the difference between the grades of two complex fuzzy sets as well as that between the phases of the two complex fuzzy sets. This distance measure is then used to define δ-equalities of complex fuzzy sets which coincide with those of fuzzy sets already defined in the literature if complex fuzzy sets reduce to real-valued fuzzy sets. Two complex fuzzy sets are said to be δ-equal if the distance between them is less than 1-δ. This paper shows how various operations between complex fuzzy sets affect given δ-equalities of complex fuzzy sets. An example application of signal detection demonstrates the utility of the concept of δ-equalities of complex fuzzy sets in practice.     

Existence of good large sets of Steiner triple systems

The concept of good large set of Steiner triple systems (or GLS in short) was introduced by Lu in his paper “on large sets of disjoint Steiner triple systems”, [J. Lu, On large sets of disjoint Steiner triple systems, I-III, J. Combin. Theory (A) 34 (1983) 140-182]. In this paper a doubling construction for GLSs is displayed and some existence results are obtained.     

A one-to-one correspondence between colorings and stable sets

Given a graph G, we construct an auxiliary graph with vertices such that the set of all stable sets of is in one-to-one correspondence with the set of all colorings of G. Then, we show that the Max-Coloring problem in G reduces to the Maximum Weighted Stable set problem in .     

Relative difference sets fixed by inversion (ii)—character theoretical approach

In this paper, we continue our investigation of relative difference sets fixed by inversion. We exclusively focus our attention on abelian groups. New necessary conditions are obtained and a new family of such relative difference sets with forbidden subgroup Z/4Z is constructed. The methods we use are character theory of abelian groups and Galois rings over Z/4Z.     

Differentiability of the volume of a region enclosed by level sets

The level of a function f on Rn encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the function f is smooth, the volume function is again smooth for regular values of f. For critical values of f the volume function is only finitely differentiable. The initial motivation for this study comes from Radiotherapy, where such volume functions are used in an optimization process. Thus their differentiability properties become important.     

Fixed point theorems for better admissible multimaps on almost convex sets

We obtain new fixed point theorems on multimaps in the class Bp defined on almost convex subsets of topological vector spaces. Our main results are applied to deduce various fixed point theorems, coincidence theorems, almost fixed point theorems, intersection theorems, and minimax theorems. Consequently, our new results generalize well-known works of Kakutani, Fan, Browder, Himmelberg, Lassonde, and others.     

Asplund sets, differentiability and subdifferentiability of functions in Banach spaces

We show that Asplund sets are effective tools to study differentiability of Lipschitz functions, and ε-subdifferentiability of lower semicontinuous functions on general Banach spaces. If a locally Lipschitz function defined on an Asplund generated space has a minimal Clarke subdifferential mapping, then it is TBY-uniformly strictly differentiable on a dense Gδ subset of X. Examples are given of locally Lipschitz functions that are TBY-uniformly strictly differentiable everywhere, but nowhere Fréchet differentiable.     

Relative difference sets fixed by inversion (III)—Cocycle theoretical approach

In this paper, we give a characterization of a group G which contains a semiregular relative difference set R relative to a central subgroup N containing the commutator subgroup [G,G] of G such that 1∈R and rRr=R for all r∈R. In particular, these relative difference sets are fixed by inversion and inner automorphisms of the group are multipliers. We also present a construction of such relative difference sets.     

Large sets of resolvable idempotent Latin squares

An idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the v(v−1) off-diagonal cells can be resolved into v−1 disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of v−2 RILS(v)s pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs.