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Introduction

## Publications

Publications (60)

The Turán number of hypergraphs has been studied extensively. Here we deal with a recent direction, the linear Turán number, and restrict ourselves to linear triple systems, a collection of triples on a set of points in which any two triples intersect in at most one point. For a fixed linear triple system F, the linear Turán number exL(n,F) is the...

This work studies the evolution of cortical networks during the transition from escape strategy to avoidance strategy in auditory discrimination learning in Mongolian gerbils trained by the well-established two-way active avoidance learning paradigm. The animals were implanted with electrode arrays centered on the surface of the primary auditory co...

In this paper a random graph model [Formula presented] is introduced, which is a combination of fixed torus grid edges in (ℤ∕Nℤ)² and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices u,v∈(ℤ∕Nℤ)² with graph distance d on the torus grid is pd = c∕Nd, where c is some constant. We...

Spiking neural networks are motivated from principles of neural systems and may possess unexplored advantages in the context of machine learning. A class of \textit{convolutional spiking neural networks} is introduced, trained to detect image features with an unsupervised, competitive learning mechanism. Image features can be shared within subpopul...

It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{n,n}$ there is a monochromatic connected component with at least ${2n\over r}$ vertices. It would be interesting to know whether we can additionally require that this large component be balanced; that is, is it true that in every $r$-coloring of $K_{n,n}$ t...

Ongoing fluctuations of neuronal activity have long been considered intrinsic noise that introduces unavoidable and unwanted variability into neuronal processing, which the brain eliminates by averaging across population activity (Georgopoulos et al., 1986; Lee et al., 1988; Shadlen and Newsome, 1994; Maynard et al., 1999). It is now understood, th...

The brain generates oscillatory neuronal activity at a broad range of frequencies and the presence and amplitude of certain oscillations at specific times and in specific brain regions are highly correlated with states of arousal, sleep, and with a wide range of cognitive processes. The neuronal mechanisms underlying the generation of brain rhythms...

In this paper a random graph model $G_{\mathbb{Z}^2_N,p_d}$ is introduced,
which is a combination of fixed torus grid edges in $(\mathbb{Z}/N
\mathbb{Z})^2$ and some additional random ones. The random edges are called
long, and the probability of having a long edge between vertices
$u,v\in(\mathbb{Z}/N \mathbb{Z})^2$ with graph distance $d$ on the...

Power efficiency became an important factor in High Performance Computing (HPC). FPGA-based dataflow machines are the best candidates for power efficient computing, because of the maximized memory bandwidth utilization, and user-defined optimal caching. However, input data streams are required with optimized data locality. This paper focuses on the...

In the paper, an field-programmable gate array (FPGA)-based framework is described to efficiently accelerate unstructured finite volume computations where the same mathematical expression has to be evaluated at every point of the mesh. The irregular memory access patterns caused by the unstructured spatial discretization are eliminated by a novel m...

Combinatorial optimization problems play important role in many applications such as vehicle routing, robot action planning, or scheduling hard disk requests. In most cases the solution is a plan that contains series of actions. If an applicable partial plan is given, the actors can start their job while an optimization process works on the rest of...

Many real-life applications of processor-arrays suffer from memory bandwidth limitations. In many cases an unstructured mesh is given (computation on sensor data, simulations of physical systems - PDEs), where the vertices represent computations with dependencies represented by the edges. Utilization of processing elements (PEs) during these comput...

Reflecting on problems posed by Gyárfás [Ramsey Theory Yesterday, Today and Tomorrow, Birkhäuser, Basel, 2010, pp. 77–96] and Mubayi [Electron J Combin 9 (2002), #R42], we show in this note that every r-edge-coloring of Kn contains a monochromatic component of diameter at most five on at least n/(r−1) vertices. Copyright © 2011 Wiley Periodicals, I...

We show that any properly edge-colored Kn contains a rainbow cycle with at least (4/7 - o(1))n edges. This improves the lower bound of n/2 - 1 proved in [Akbari, Etesami, Manini and Mahmoody, Australas. J. Combin. 37 (2007), 33-42]. © 2011 Combinatorial Mathematics Society of Australasia (Inc.).

We show in this paper that in every 3-coloring of the edges of K n all but o(n) of its vertices can be partitioned into three monochromatic cycles. From this, using our earlier results, actually it follows that we can partition all the vertices into at most 17 monochromatic cycles, improving the best known bounds. If the colors of the three monochr...

A hypergraph is called an r by r grid if it is isomorphic to a pattern of r
horizontal and r vertical lines. Three sets form a triangle if they pairwise
intersect in three distinct singletons. A hypergraph is linear if every pair of
edges meet in at most one vertex. In this paper we construct large linear
r-hypergraphs which contain no grids. Moreo...

In this article, we study the tripartite Ramsey numbers of paths. We show that in any two-coloring of the edges of the complete tripartite graph K(n, n, n) there is a monochromatic path of length (1 − o(1))2n. Since R(P2n+1,P2n+1)=3n, this means that the length of the longest monochromatic path is about the same when two-colorings of K3n and K(n, n...

In this paper we deal with codes identifying sets of vertices in random networks; that is, (1,⩽ℓ)-identifying codes. These codes enable us to detect sets of faulty processors in a multiprocessor system, assuming that the maximum number of faulty processors is bounded by a fixed constant ℓ. The (1,⩽1)-identifying codes are of special interest. For r...

Let c be a positive constant. Suppose that r = o(n5/12) and the members of are chosen sequentially at random to form an intersecting hypergraph . We show that whp (A sequence of events is said to occur with high probability (whp) .) consists of a simple hypergraph of size Θ(r/n1/3), a distinguished vertex v and all r-sets that contain v and meet ev...

We prove—for sufficiently large n—the following conjecture of Faudree and Schelp: $$
R{\left( {P_{n} ,P_{n} ,P_{n} } \right)} = \left\{ {\begin{array}{*{20}c}
{{2n - 1{\kern 1pt} \;{\text{for}}\;{\text{odd}}\;n,}} \\
{{{\text{2n - 2}}\;{\text{for}}\;{\text{even}}\;n,}} \\
\end{array} } \right.
$$ , for the three-color Ramsey numbers of paths on n v...

Let c be a positive constant. Suppose that r = o(n 5/12) and the members of [n] r are chosen sequentially at random to form an intersecting hypergraph H. We show that whp 1 H consists of a simple hypergraph S of size Θ(r/n 1/3), a distinguished vertex v and all r-sets which contain v and meet every edge of S. This is a continuation of the study of...

Improving a result of Erdős, Gyárfás and Pyber for large n we show that for every integer r⩾2 there exists a constant n0=n0(r) such that if n⩾n0 and the edges of the complete graph Kn are colored with r colors then the vertex set of Kn can be partitioned into at most 100rlogr vertex disjoint monochromatic cycles.

For integers 2 les k les r, a set of binary codewords is a k-out-of-r multiple user tracing code, if from the bitwise "or" of at most r codewords one can determine at least k of them. Single user tracing codes (k = 1) were introduced by Csuros and Ruszinko (2005) and the order of magnitude of their rate was determined to be 1/r by Alon and Asodi. H...

The most important communication channels were investigated in the situation of multiple users. These models do not assume any central intelligence there is no coordination between the transmitters, so fit to the usual conditions of an ad-hoc or sensor network. Better bounds on the minimum code word length and efficient code constructions were achi...

Assume that the edges of a complete bipartite graph K(A, B) are colored with r colors. In this paper we study coverings of B by vertex disjoint monochromatic cycles, connected matchings, and connected subgraphs. These problems occur in several applications.

In this paper we deal with codes identifying sets of vertices in random graphs, that is l-identifying codes. These codes enable us to detect sets of faulty processors in a multiprocessor system, assuming that the maximum number of faulty processors is bounded by a fixed constant l. The l-identifying codes or simply identifying codes are of special...

A family of subsets of [n] is positive linear combination free if the characteristic vector of neither member is the positive linear combination of the characteristic vectors of some other ones. We construct a positive linear combination free family which contains (1-o(1))2n subsets of [n] and we give tight bounds on the o(1)2n term. The problem wa...

The zero-error capacity region of r-out-of-T user multiple-access OR channel is investigated. A family F of subsets of [n] = {1, ..., n} is an r-single-user-tracing superimposed code (r-SUT) if there exists such a single-user-tracing function φ:2<sup>[n]</sup> → F that for all F' ⊆ F with 1 ≤ |F'| ≤ r, φ(∪<sub>A∈F</sub>'A) ∈ F'. In this corresponde...

We investigate the zero-error capacity region of r-out of T user multiple access OR channel. A family ℱ subsets of [n]={1, . . . ,n} is an r-single-user-tracing superimposed code (r-SUT) if there exists such a single-user-tracing function φ: 2<sup>[n]</sup>→ℱ that for all ℱ⊆ℱ with |ℱ'|≤r, φ(∪<sub>A∈</sub>ℱ<sub>'</sub>A...

Let c be a positive constant. We show that if r = #cn and the members of are chosen sequentially at random to form an intersecting hypergraph then with limiting probability (1 + c ) -1 ,asn ##, the resulting family will be of maximum .

Let p be a positive integer and let Q be a subset of {0, 1,...,p}. Call p sets A(1),A(2),...,A(p) of a ground set X a (p,Q)-system if the number of sets A, containing x is in Q for every x E X. In hypergraph terminology, a (p,Q)-system is a hypergraph with p edges such that each vertex x has degree d(x) E Q. A family of sets F with ground set X is...

A family of subsets of (n) is positive linear combination free if the characteristic vector of neither edge is the positive linear combination of the characteristic vectors of some other ones. We construct a positive linear combination free family which contains (1 o(1))2n subsets of (n) and we give tight bounds on the o(1)2n term. The problem was...

Let p be a positive integer and let Q be a subset of {0, 1,…,p}. Call p sets A1, A2,…, Ap of a ground set X a (p, Q)-system if the number of sets Ai containing a; is in Q for every x ∈ X. In hypergraph terminology, a (p, Q)-system is a hypergraph with p edges such that each vertex x has degree d(x) ∈ Q. A family of sets T with ground set X is calle...

Let G be a graph on vertex set [.], and for X C_ [.] let N(X) be the union of X and it's neighborhood in G. A family of sets ' C- 2 ["] is G-intersectin# if N(X) Y 0 for all X, Y '. In this paper we study the cardinality and structure of the largest k-uniform G-intersecting families on a fixed graph G.

A t-round χ-coloring is defined as a sequence ψ1, …, ψt of t (not necessarily distinct) edge colorings of a complete graph, using at most χ colors in each of the colorings. For positive integers k⩽n and t let χt(k, n) denote the minimum number χ of colors for which there exists a t-round χ-coloring of Kn such that all (k2) edges of each Kk⊆Kn get d...

Let H be a simple graph having no isolated vertices. An (H,k)-vertex-cover of a simple graph G = (V,E) is a collection of subgraphs of G satisfying
1. , for all i = 1, ..., r,
2. ,
3. , for all , and
4. each is in at most k of the . We consider the existence of such vertex covers when H is a complete graph, , in the context of extremal and...

Let fd (G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n > n0 (D) vertices, f2(G) = n - D - 1 and f3(G) ≥ n - O(D3). For d ≥ 4, fd (G) depends strongly on the actual structure of G, not only on the maximum degree of...

Consider a connected r-regular n-vertex graph G with random independent edge lengths, each uniformly distributed on (0; 1). Let mst(G) be the expected length of a minimum spanning tree. We show in this paper that if G is suciently highly edge connected then the expected length of a minimum spanning tree is nr(3). If we omit the edge connectivity co...

It is shown in this note that with high probability it is enough to destroy all triangles in order to get a cover graph from a random graph G n,p with p # # log n/n for any constant #<2/3. On the other hand, this is not true for somewhat higher densities: If p # #(log n) 3 /(n log log n)with#>1/8then with high probability we need to delete more edg...

It is known that lim
inf<sub>n→∞</sub>((2n+1)/2-Θ(C<sub>2n+1</sub>))=0 and
that the lim sup of this difference is at most 1/4, where Θ(G) is
the Shannon (1956) capacity of the graph G. We prove that the above lim
sup is at most 1/6 and conjecture that the limit itself exists and
equals 0. We show that the lim sup is small by constructing large
inde...

A family of n-dimensional unit norm vectors is an Euclidean
superimposed code if the sums of any two distinct at most m-tuples of
vectors are separated by a certain minimum Euclidean distance d. Ericson
and Gyorfi (1988) proved that the rate of such a code is between (log
m)/4m and (log m)/m for m large enough. In this paper-improving the
above lon...

Assume that G is a triangle-free graph. Let be the minimum number of edges one has to add to G to get a graph of diameter at most d which is still triangle-free. It is shown that for connected graphs of order n and of fixed maximum degree. The proof is based on relations of and the clique-cover number of edges of graphs. It is also shown that the m...

An isomorphism certiflcate of a labeled tournament T is a labeled subdigraph of T which to- gether with an unlabeled copy of T allows the errorless reconstruction of T. It is shown that any tournament on n vertices contains an isomorphism certiflcate with at most n log2n edges. This answers a question of Fishburn, Kim and Tetali. A score certiflcat...

A family of n-dimensional unit norm vectors is a Euclidean superimposed code, if the sums of any two distinct at most m-tuples of vectors are separated by a certain minimum Euclidean distance d. Ericson and Gyorfi (see IEEE Trans. Inform. Theory, vol.34, no.4, p.877-80, 1988) proved that the rate of such a code is between (log m)/4m and (logm)/m fo...

An intersectingsystemoftype ([exists at least one exists], [for all], k, n) is a collection [open face F]={[script F]1, ..., [script F]m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair [script F]i and [script F]j of distinct members of [open face F] there exists an A[set mem...

A search problem of G. O. H. Katona was solved in [3] where an unknownpoint x in a 2-dimensional grid has to be located using queries of type \isx = (x 1 ; x 2 ) a = (a 1 ; a 2 )?". Here a is an arbitrary lattice point andx a means that a i b i (i = 1; 2). In the recent paper we consider thegeneralization of this problem for arbitrary dimension d.1...

Pippenger (1981) showed in a probabilistic way that the capacity
of a collision channel with multiplicity feedback is one. In this
correspondence, using an Erdos-Renyi type search strategy, we settle a
long-standing open problem by giving a constructive proof of this
result. Moreover, we prove that two different capacity definitions are
equivalent,...

The apparentRF value of a compound after successive development steps in incremental multiple development (IMD) is determined. The model
developed enables prediction of the migration and the zone width of investigated compounds in IMD with linearly and quadratically
increasing development distances. For the experimental investigations we examined n...

Let T(r,n) denote the maximum number of subsets of an n-set
satisfying the condition that no set is covered by the union of r
others, while let T*(ε,n) be the maximum size of a <ε
part intersecting family, i.e. of a family where the size of the
intersection of any two sets is < than the ε<sup>th</sup> part
of the smaller one. By partially answering...

Various different definitions were investigated in random multiple access theory for capacity of the multiple-access collision channel. However, as it was pointed out by Tsybakov (1985), almost nothing about the relations between the various definitions is known. In this paper we try to fullfil this gap showing that some widely used capacity defini...

This presentation deals with the prediction of planar chromatographic retention data in incremental multiple development. The value R'(Fn), the retention factor after the nth development step (i.e. the ratio of the total migration distance of the component during consecutive development steps to the chromatographic distance of the last development...

Let T (r; n) denote the maximum number of subsets of an n-set satisfying the condition in the title. It is proved in a purely combinatorial way, that for n sufficiently large log 2 T (r; n) n 8 \Delta log 2 r r 2 holds. 1. Introduction The notion of the r-cover-free families was introduced by Kautz and Singleton in 1964 [17]. They initiated investi...

## Projects

Projects (2)

Olfactory bulb neurones oscillate reliably in phase with the rhythm of breathing, even in the absence of any detectable odour (Adrian, 1950). This respiration-locked neuronal rhythm propagates to olfactory cortical areas and closely connected downstream structures such as the hippocampus (Fontanini and Bower, 2003 & 2005). In 2014 we showed that respiration-locked neuronal oscillations also occur in the whisker barrel cortex in mice (Ito et al., 2014), a primary somatosensory cortex that has no connections with olfactory or olfactory-related areas. We demonstrated that the olfactory bulb was the main driving force behind these oscillations, showing for the first time, that respiration-locked olfactory bulb output activity propagated through the cortico-thalamic network to non-olfactory areas. Even more surprising was our finding that the high-frequency gamma oscillations (30-100 Hz) that are widely implicated in cognitive processing, are power-modulated in phase with respiration (Ito et al., 2014). This finding suggested that respiratory modulation of cognition-relevant gamma oscillations might explain the long-known influence of controlled (e.g. Yogic) respiration on cognitive and emotional states. In 2016 we showed that the same influence of respiration on cortical slow oscillations and gamma activity is also observed in Humans (Heck et al., 2016) and can be reproduced in a graph theory model of a cortical network (Heck et al., 2017). Our findings in humans were soon after confirmed by Zeloano et al. (2016), who were also able to show that respiration does indeed modulate memory and emotional processing on a cycle by cycle basis. We are now working on a more detailed understanding of the mechnisms underlying respiratory modulation of brain activity and how it influences brain function.