n-bathycenters

Does there exist a polygon with the property that for a suitable point p in the plane every ray with endpoint p intersects the polygon in exactly n connected components? Does there exist a polygon with the property that there are two such points, or three, or a segment of such points? For polygon P call a point p with the property that every ray from p intersects P in exactly n connected components n-isobathic with respect to P. Define the n-bathycenter of a polygon P as the set of all points p that are n-isobathic with respect to P. Further define a set S to be an n-bathycenter if there exists a polygon P of which S is the n-bathycenter. This paper deals with the characterization of 2- and 3-bathycenters, together with some results on the general case.     

Multiscale Young measures in homogenization of continuous stationary processes in compact spaces and applications

We introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces. The latter are functions f○T:Rn×Q→Q with f○T(x,ω)=f(T(x)ω) where Q is a compact (Hausdorff topological) space, f∈C(Q) and T(x):Q→Q, x∈Rn, is an n-dimensional continuous dynamical system endowed with an invariant Radon probability measure μ. It can be easily shown that for almost all ω∈Q the realization f(T(x)ω) belongs to an algebra with mean value, that is, an algebra of functions in BUC(Rn) containing all translates of its elements and such that each of its elements possesses a mean value. This notion was introduced by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators, Mat. Zametki 33 (1983) 571-582, English transl. in Math. Notes 33 (1983) 294-300]. We then establish the existence of multiscale Young measures in the setting of algebras with mean value, where the compactifications of Rn provided by such algebras plays an important role. These parametrized measures are useful in connection with the existence of correctors in homogenization problems. We apply this framework to the homogenization of a porous medium type equation in Rn with a stationary continuous process as a stiff oscillatory external source. This application seems to be new even in the classical context of periodic homogenization.     

Path-path Ramsey-type numbers for the complete bipartite graph

For a fixed pair of integers r, s ≥ 2, all positive integers m and n are determined which have the property that if the edges of K_m,n (a complete bipartite graph with parts n and m) are colored with two colors, then there will always exist a path with r vertices in the first color or a path with s vertices in the second color.     

Pointwise bornological spaces

With each metric space (X,d) we can associate a bornological space (X,Bd) where Bd is the set of all subsets of X with finite diameter. Equivalently, Bd is the set of all subsets of X that are contained in a ball with finite radius. If the metric d can attain the value infinite, then the set of all subsets with finite diameter is no longer a bornology. Moreover, if d is no longer symmetric, then the set of subsets with finite diameter does not coincide with the set of subsets that are contained in a ball with finite radius. In this text we will introduce two structures that capture the concept of boundedness in both symmetric and non-symmetric extended metric spaces.     

Deterministic symmetric rendezvous with tokens in a synchronous torus

In the rendezvous problem, the goal for two mobile agents is to meet whenever this is possible. In the rendezvous with detection problem, an additional goal for the agents is to detect the impossibility of a rendezvous (e.g., due to symmetrical initial positions of the agents) and stop. We consider the rendezvous problem with and without detection for identical anonymous mobile agents (i.e., running the same deterministic algorithm) with tokens in an anonymous synchronous torus with a sense of direction, and show that there is a striking computational difference between one and more tokens. Specifically, we show that (1) two agents with a constant number of unmovable tokens, or with one movable token each, cannot rendezvous in an n×n torus if they have o(logn) memory, while they can solve the rendezvous with detection problem in an n×m torus as long as they have one unmovable token and O(logn+logm) memory; in contrast, (2) when two agents have two movable tokens each then the rendezvous problem (respectively, rendezvous with detection problem) is solvable with constant memory in an arbitrary n×m (respectively, n×n) torus; and finally, (3) two agents with three movable tokens each and constant memory can solve the rendezvous with detection problem in an n×m torus. This is the first publication in the literature that studies tradeoffs between the number of tokens, memory and knowledge the agents need in order to meet in a torus.     

“Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth

We obtain a representation of Feller’s branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion H with a drift that is affine linear in the local time accumulated by H at its current level. As in the classical Ray–Knight representation, the excursions of H are the exploration paths of the trees of descendants of the ancestors at time t = 0, and the local time of H at height t measures the population size at time t. We cope with the dependence in the reproduction by introducing a pecking order of individuals: an individual explored at time s and living at time t = H _s is prone to be killed by any of its contemporaneans that have been explored so far. The proof of our main result relies on approximating H with a sequence of Harris paths H ^N which figure in a Ray–Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of H ^N together with its local times and with the Girsanov densities that introduce the dependence in the reproduction.     

Enumeration of rectangular arrays of zeros and ones

For r=1,2 the rectangular arrays of zeros and ones with r rows and n columns, with m_i zeros and r_i changes in the ith row, and with s_i vertical changes between the ith and (i+1)st rows, i=1,…,r-1, are enumerated. The number of arrays of zeros and ones with 3 rows and n columns, with r_i changes in the ith row, i=1,2,3, and with s_i vertical changes between the ith and (i+1)st rows, i=1,2, is also determined.     

On diffusivity of a tagged particle in asymmetric zero-range dynamics

Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑jp(j)≠0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d?1, and at most order t in d=1 and d?3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g(k) increases, and g(k)/k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.     

Characterizations of derivations of Banach space nest algebras: All-derivable points

Let N be a nest on a complex Banach space X with N∈N complemented in X whenever N_-=N, and let AlgN be the associated nest algebra. We say that an operator Z∈AlgN is an all-derivable point of AlgN if every linear map δ from AlgN into itself derivable at Z (i.e. δ(A)B+Aδ(B)=δ(Z) for any A,B∈A with AB=Z) is a derivation. In this paper, it is shown that if Z∈AlgN is an injective operator or an operator with dense range, or an idempotent operator with ran(Z)∈N, then Z is an all-derivable point of AlgN. Particularly, if N is a nest on a complex Hilbert space, then every idempotent operator with range in N, every injective operator as well as every operator with dense range in AlgN is an all-derivable point of the nest algebra AlgN.     

Mourre's inequality and embedded bound states

A self-adjoint operator H with an eigenvalue λ embedded in the continuum spectrum is considered. Boundedness of all operators of the form AnP is proved, where P is the eigenprojection associated with λ and A is any self-adjoint operator satisfying Mourre's inequality in a neighborhood of λ and such that the higher commutators of H with A up to order n+2 are relatively bounded with respect to H.     

On permuteral subgroups in finite groups

The permutizer of a subgroup H in a group G is defined as the subgroup generated by all cyclic subgroups of G that permute with H. Call H permuteral in G if the permutizer of H in G coincides with G; H is called strongly permuteral in G if the permutizer of H in U coincides with U for every subgroup U of G containing H. We study the finite groups with given systems of permuteral and strongly permuteral subgroups and find some new characterizations of w-supersoluble and supersoluble groups.     

Modulability and duality of certain cones in pluripotential theory

Let p>0, and let Ep denote the cone of negative plurisubharmonic functions with finite pluricomplex p-energy. We prove that the vector space δEp=Ep−Ep, with the vector ordering induced by the cone Ep is σ-Dedekind complete, and equipped with a suitable quasi-norm it is a non-separable quasi-Banach space with a decomposition property with control of the quasi-norm. Furthermore, we explicitly characterize its topological dual. The cone Ep in the quasi-normed space δEp is closed, generating, and has empty interior.     

Vietoris topology on partial maps with compact domains

The space PK of partial maps with compact domains (identified with their graphs) forms a subspace of the hyperspace of nonempty compact subsets of a product space endowed with the Vietoris topology. Various completeness properties of PK, including ?ech-completeness, sieve completeness, strong Choquetness, and (hereditary) Baireness, are investigated. Some new results on the hyperspace K(X) of compact subsets of a Hausdorff X with the Vietoris topology are obtained; in particular, it is shown that there is a strongly Choquet X, with 1st category K(X).     

The complex of non-crossing diagonals of a polygon

Given a convex n-gon P in R² with vertices in general position, it is well known that the simplicial complex θ(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n−3. We prove that for any non-convex polygonal region P with n vertices and h+1 boundary components, θ(P) is a ball of dimension n+3h−4. We also provide a new proof that θ(P) is a sphere when P is convex with vertices in general position.     

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH ₂(M; ?) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.     

Color-bounded hypergraphs, IV: Stable colorings of hypertrees

We consider vertex colorings of hypergraphs in which lower and upper bounds are prescribed for the largest cardinality of a monochromatic subset and/or of a polychromatic subset in each edge. One of the results states that for any integers s≥2 and a≥2 there exists an integer f(s,a) with the following property. If an interval hypergraph admits some coloring such that in each edge E_i at least a prescribed number s_i≤s of colors occur and also each E_i contains a monochromatic subset with a prescribed number a_i≤a of vertices, then a coloring with these properties exists with at most f(s,a) colors. Further results deal with estimates on the minimum and maximum possible numbers of colors and the time complexity of determining those numbers or testing colorability, for various combinations of the four color bounds prescribed. Many interesting problems remain open.     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

The number of designs with geometric parameters grows exponentially

It is well-known that the number of 2-designs with the parameters of a classical point-hyperplane design PG _n-1(n, q) grows exponentially. Here we extend this result to the number of 2-designs with the parameters of PG _d (n, q), where 2 ≤ d ≤ n ? 1. We also establish a characterization of the classical geometric designs in terms of hyperplanes and, in the special case d = 2, also in terms of lines. Finally, we shall discuss some interesting configurations of hyperplanes arising in designs with geometric parameters.     

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x∈V(G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(mlogn) time whether G is a median graph with geodetic number 2.     

Towards a spectral theory of graphs based on the signless Laplacian, II

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular to those based on the adjacency matrix A and the Laplacian L. As demonstrated in the first part, the Q-theory can be constructed in part using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, common features with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. In this part, we introduce notions of enriched and restricted spectral theories and present results on integral graphs, enumeration of spanning trees, characterizations by eigenvalues, cospectral graphs and graph angles.