115 research outputs found
Game saturation of intersecting families
We consider the following combinatorial game: two players, Fast and Slow,
claim -element subsets of alternately, one at each turn,
such that both players are allowed to pick sets that intersect all previously
claimed subsets. The game ends when there does not exist any unclaimed
-subset that meets all already claimed sets. The score of the game is the
number of sets claimed by the two players, the aim of Fast is to keep the score
as low as possible, while the aim of Slow is to postpone the game's end as long
as possible. The game saturation number is the score of the game when both
players play according to an optimal strategy. To be precise we have to
distinguish two cases depending on which player takes the first move. Let
and denote the score of
the saturation game when both players play according to an optimal strategy and
the game starts with Fast's or Slow's move, respectively. We prove that
holds
Instabilities in complex mixtures with a large number of components
Inside living cells are complex mixtures of thousands of components. It is
hopeless to try to characterise all the individual interactions in these
mixtures. Thus, we develop a statistical approach to approximating them, and
examine the conditions under which the mixtures phase separate. The approach
approximates the matrix of second virial coefficients of the mixture by a
random matrix, and determines the stability of the mixture from the spectrum of
such random matrices.Comment: 4 pages, uses RevTeX 4.
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Tur\'an numbers for -free graphs: topological obstructions and algebraic constructions
We show that every hypersurface in contains a large grid,
i.e., the set of the form , with . We use this to
deduce that the known constructions of extremal -free and
-free graphs cannot be generalized to a similar construction of
-free graphs for any . We also give new constructions of
extremal -free graphs for large .Comment: Fixed a small mistake in the application of Proposition
On high moments of strongly diluted large Wigner random matrices
We consider a dilute version of the Wigner ensemble of nxn random matrices
and study the asymptotic behavior of their moments in the limit of
infinite , and , where is the dilution parameter. We show
that in the asymptotic regime of the strong dilution, the moments with
depend on the second and the fourth moments of the random entries
and do not depend on other even moments of . This fact can be
regarded as an evidence of a new type of the universal behavior of the local
eigenvalue distribution of strongly dilute random matrices at the border of the
limiting spectrum. As a by-product of the proof, we describe a new kind of
Catalan-type numbers related with the tree-type walks.Comment: 43 pages (version four: misprints corrected, discussion added, other
minor modifications
Noise-Resilient Group Testing: Limitations and Constructions
We study combinatorial group testing schemes for learning -sparse Boolean
vectors using highly unreliable disjunctive measurements. We consider an
adversarial noise model that only limits the number of false observations, and
show that any noise-resilient scheme in this model can only approximately
reconstruct the sparse vector. On the positive side, we take this barrier to
our advantage and show that approximate reconstruction (within a satisfactory
degree of approximation) allows us to break the information theoretic lower
bound of that is known for exact reconstruction of
-sparse vectors of length via non-adaptive measurements, by a
multiplicative factor .
Specifically, we give simple randomized constructions of non-adaptive
measurement schemes, with measurements, that allow efficient
reconstruction of -sparse vectors up to false positives even in the
presence of false positives and false negatives within the
measurement outcomes, for any constant . We show that, information
theoretically, none of these parameters can be substantially improved without
dramatically affecting the others. Furthermore, we obtain several explicit
constructions, in particular one matching the randomized trade-off but using measurements. We also obtain explicit constructions
that allow fast reconstruction in time \poly(m), which would be sublinear in
for sufficiently sparse vectors. The main tool used in our construction is
the list-decoding view of randomness condensers and extractors.Comment: Full version. A preliminary summary of this work appears (under the
same title) in proceedings of the 17th International Symposium on
Fundamentals of Computation Theory (FCT 2009
Guaranteed clustering and biclustering via semidefinite programming
Identifying clusters of similar objects in data plays a significant role in a
wide range of applications. As a model problem for clustering, we consider the
densest k-disjoint-clique problem, whose goal is to identify the collection of
k disjoint cliques of a given weighted complete graph maximizing the sum of the
densities of the complete subgraphs induced by these cliques. In this paper, we
establish conditions ensuring exact recovery of the densest k cliques of a
given graph from the optimal solution of a particular semidefinite program. In
particular, the semidefinite relaxation is exact for input graphs corresponding
to data consisting of k large, distinct clusters and a smaller number of
outliers. This approach also yields a semidefinite relaxation for the
biclustering problem with similar recovery guarantees. Given a set of objects
and a set of features exhibited by these objects, biclustering seeks to
simultaneously group the objects and features according to their expression
levels. This problem may be posed as partitioning the nodes of a weighted
bipartite complete graph such that the sum of the densities of the resulting
bipartite complete subgraphs is maximized. As in our analysis of the densest
k-disjoint-clique problem, we show that the correct partition of the objects
and features can be recovered from the optimal solution of a semidefinite
program in the case that the given data consists of several disjoint sets of
objects exhibiting similar features. Empirical evidence from numerical
experiments supporting these theoretical guarantees is also provided
Age and nutritional state influence the effects of cholecystokinin on energy balance
AbstractCholecystokinin (CCK) is anorexic, irrespective whether it is applied intraperitoneally (IP) or intracerebroventricularly (ICV) in male Wistar rats. The metabolic effects depend on the route of administration: by the IP route it elicits hypothermia (presumably by type-1 receptors, CCK1R-s), while ICV administration is followed by fever-like hypermetabolism and hyperthermia via activation of CCK2R-s, which latter response seems to be most important in the postprandial (compensatory) hypermetabolism. The efficacy of the IP injected CCK varies with age: it causes strong anorexia in young adult 4 and 6-months old and again in old rats (aged 18–24months), but the middle-aged (12-month old) ones seem to be resistant to this effect. Such pattern of effects may contribute to the explanation of age-related obesity observed in middle-aged animals as well as to the aging anorexia and loss of body weight in old ones. Diet-induced obesity accelerates the appearance of CCK-resistance as well as the return of high sensitivity to CCK in further aging, while chronic calorie-restriction prevents the development of resistance, as if the speed of the age-related regulatory changes was altered by the nutritional state. The effects of ICV applied CCK also change with age: the characteristic anorexic and hypermetabolic/hyperthermic effects can be observed in young adult rats, but the effects gradually and monotonically decline with age and disappear by the old age of 24months. These disparate age-related patterns of CCK efficacy upon peripheral or central administration routes may indicate that although both peripheral and central CCKR-s exert anorexic effects, they may have dissimilar roles in the regulation of overall energy balance
On Connected Diagrams and Cumulants of Erdos-Renyi Matrix Models
Regarding the adjacency matrices of n-vertex graphs and related graph
Laplacian, we introduce two families of discrete matrix models constructed both
with the help of the Erdos-Renyi ensemble of random graphs. Corresponding
matrix sums represent the characteristic functions of the average number of
walks and closed walks over the random graph. These sums can be considered as
discrete analogs of the matrix integrals of random matrix theory.
We study the diagram structure of the cumulant expansions of logarithms of
these matrix sums and analyze the limiting expressions in the cases of constant
and vanishing edge probabilities as n tends to infinity.Comment: 34 pages, 8 figure
Lines, Circles, Planes and Spheres
Let be a set of points in , no three collinear and not
all coplanar. If at most are coplanar and is sufficiently large, the
total number of planes determined is at least . For similar conditions and
sufficiently large , (inspired by the work of P. D. T. A. Elliott in
\cite{Ell67}) we also show that the number of spheres determined by points
is at least , and this bound is best
possible under its hypothesis. (By , we are denoting the
maximum number of three-point lines attainable by a configuration of
points, no four collinear, in the plane, i.e., the classic Orchard Problem.)
New lower bounds are also given for both lines and circles.Comment: 37 page
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