820 research outputs found
Degeneration and orbits of tuples and subgroups in an Abelian group
A tuple (or subgroup) in a group is said to degenerate to another if the
latter is an endomorphic image of the former. In a countable reduced abelian
group, it is shown that if tuples (or finite subgroups) degenerate to each
other, then they lie in the same automorphism orbit. The proof is based on
techniques that were developed by Kaplansky and Mackey in order to give an
elegant proof of Ulm's theorem. Similar results hold for reduced countably
generated torsion modules over principal ideal domains. It is shown that the
depth and the description of atoms of the resulting poset of orbits of tuples
depend only on the Ulm invariants of the module in question (and not on the
underlying ring). A complete description of the poset of orbits of elements in
terms of the Ulm invariants of the module is given. The relationship between
this description of orbits and a very different-looking one obtained by Dutta
and Prasad for torsion modules of bounded order is explained.Comment: 13 pages, 1 figur
Counting independent sets in hypergraphs
Let be a triangle-free graph with vertices and average degree . We
show that contains at least independent sets. This improves a recent result of the
first and third authors \cite{countingind}. In particular, it implies that as
, every triangle-free graph on vertices has at least
independent sets, where . Further, we show that for all , there exists a triangle-free
graph with vertices which has at most
independent sets, where . This disproves a
conjecture from \cite{countingind}.
Let be a -uniform linear hypergraph with vertices and average
degree . We also show that there exists a constant such that the
number of independent sets in is at least This is tight apart from the constant
and generalizes a result of Duke, Lefmann, and R\"odl
\cite{uncrowdedrodl}, which guarantees the existence of an independent set of
size . Both of our lower bounds follow
from a more general statement, which applies to hereditary properties of
hypergraphs
New lower bounds for the independence number of sparse graphs and hypergraphs
We obtain new lower bounds for the independence number of -free graphs
and linear -uniform hypergraphs in terms of the degree sequence. This
answers some old questions raised by Caro and Tuza \cite{CT91}. Our proof
technique is an extension of a method of Caro and Wei \cite{CA79, WE79}, and we
also give a new short proof of the main result of \cite{CT91} using this
approach. As byproducts, we also obtain some non-trivial identities involving
binomial coefficients
Combinatorics of finite abelian groups and Weil representations
The Weil representation of the symplectic group associated to a finite
abelian group of odd order is shown to have a multiplicity-free decomposition.
When the abelian group is p-primary, the irreducible representations occurring
in the Weil representation are parametrized by a partially ordered set which is
independent of p. As p varies, the dimension of the irreducible representation
corresponding to each parameter is shown to be a polynomial in p which is
calculated explicitly. The commuting algebra of the Weil representation has a
basis indexed by another partially ordered set which is independent of p. The
expansions of the projection operators onto the irreducible invariant subspaces
in terms of this basis are calculated. The coefficients are again polynomials
in p. These results remain valid in the more general setting of finitely
generated torsion modules over a Dedekind domain.Comment: 26 pages, 3 figures Revised version, to appear in Pacific Journal of
Mathematic
Two Proofs for Shallow Packings
We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A delta-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u, v in W is greater than delta, where delta > 0 is an integer parameter. The delta-packing number is then defined as the cardinality of the largest delta-separated subcollection of V. Haussler showed an asymptotically tight bound of Theta((n / delta)^d) on the delta-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X\u27 of X of size m 0 and a real parameter 1 <= d_1 <= d (this generalizes the standard notion of bounded primal shatter dimension when d_1 = d). In this case when V is "k-shallow" (all vector lengths are at most k), we show that its delta-packing number is O(n^{d_1} k^{d-d_1} / delta^d), matching Haussler\u27s bound for the special cases where d_1=d or k=n. We present two proofs, the first is an extension of Haussler\u27s approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler\u27s proof
Life History and Population Dynamics of Tenualosa ilisha of Sundarban Estuary in Bay of Bengal, India for Sustainable Fishery Management
1870-1880The population structure of Hilsa Shad (Tenualosa ilisha), a choicest table fish, in the estuaries of India, Bangladesh and Myanmar of Bay of Bengal has been studied by different methods and each provided complementary data on population structure. Considering the present scenario of climate change, increasing pollutant load in Indo–Bangladesh and its effect on reproduction and maturation a thorough and detailed understanding of the life cycle of Hilsa is a pre requisite criteria. However, understanding of the life history of anadromous Hilsa Shad, considering the widespread climatic changes, would be immensely important. Further, in view of increasing pollutant loads in the Indo-Bangladesh estuary region, the important area of Hilsa reproduction and maturation, a detailed work on the life history strategies of Hilsa is also need of the hour. Results of such studies would be important for sustainable management of this highly economic biological resource. The present paper deals with the aspects of life history and population dynamics of Hilsa Shad in Sundarban estuaries. The data collection was done during the period of June 2011 to March 2012 at Frasergunje Fishing Harbour and offshore, northern Bay of Bengal. The length and weight of total 617 Hilsa fish were measured under this study. Monthly variations of length and weight, length frequency distribution, monthly variation of the allometry coefficient, movement pattern, and catch per unit effort were estimated. The exploitation rate of Hilsa species was found to be 0.78 and the maximum sustainable yield was 11700.18 tonnes whereas the annual catch was 18126.00 tonnes. Highest weight of adult Hilsa was recorded during the monsoon i.e. the months of June, July and August. The result of relative yield per recruitment indicated that the mortality due to current fishing period and pressure were high. Widespread fishing of juvenile and growing Hilsa (. In our study we have observed over harvest of Hilsa fish, especially Jatka (<500 gms and <230 mm) from West Bengal coastal areas
Uniform Brackets, Containers, and Combinatorial Macbeath Regions
We study the connections between three seemingly different combinatorial structures - uniform brackets in statistics and probability theory, containers in online and distributed learning theory, and combinatorial Macbeath regions, or Mnets in discrete and computational geometry. We show that these three concepts are manifestations of a single combinatorial property that can be expressed under a unified framework along the lines of Vapnik-Chervonenkis type theory for uniform convergence. These new connections help us to bring tools from discrete and computational geometry to prove improved bounds for these objects. Our improved bounds help to get an optimal algorithm for distributed learning of halfspaces, an improved algorithm for the distributed convex set disjointness problem, and improved regret bounds for online algorithms against ?-smoothed adversary for a large class of semi-algebraic threshold functions
Effect of Spin Orbit Coupling in non-centrosymmetric half-Heusler alloys
Spin-orbit coupled electronic structure of two representative non-polar
half-Heusler alloys, namely 18 electron compound CoZrBi and 8 electron compound
SiLiIn have been studied in details. An excursion through the Brillouin zone of
these alloys from one high symmetry point to the other revealed rich local
symmetry of the associated wave vectors resulting in non-trivial spin splitting
of the bands and consequent diverse spin textures in the presence of spin-orbit
coupling. Our first principles calculations supplemented with low energy
model Hamiltonian revealed the presence of linear
Dresselhaus effect at the X point having symmetry and Rashba effect
with both linear and non-linear terms at the L point with point group
symmetry. Interestingly we have also identified non-trivial Zeeman spin
splitting at the non-time reversal invariant W point and a pair of
non-degenerate bands along the path to L displaying vanishing spin
polarization due to the non-pseudo polar point group symmetry of the wave
vectors. Further a comparative study of CoZrBi and SiLiIn suggest, in addition,
to the local symmetry of the wave vectors, important role of the participating
orbitals in deciding the nature and strength of spin splitting. Our
calculations identify half-Heusler compounds with heavy elements displaying
diverse spin textures may be ideal candidate for spin valleytronics where spin
textures can be controlled by accessing different valleys around the high
symmetry k-points
-set problem in graphs
A subset of a graph is a -set if every
vertex is adjacent to at least but not more than
vertices in D. The cardinality of a minimum -set of , denoted as
, is called the -domination number of . Given a
graph and an integer , the decision version of the -set
problem is to decide whether has a -set of cardinality at most .
In this paper, we first obtain an upper bound on using
probabilistic methods, for bounded minimum and maximum degree graphs. Our bound
is constructive, by the randomized algorithm of Moser and Tardos [MT10], We
also show that the - set problem is NP-complete for chordal graphs.
Finally, we design two algorithms for finding of a tree
and a split graph, for any fixed , which answers an open question posed in
[CHHM13]
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