scholarly journals Bit-Parallelism Score Computation with Multi Integer Weight

2019 ◽  
Vol 11 (1) ◽  
pp. 35-50
Author(s):  
Setyorini Setyorini ◽  
◽  
Kuspriyanto Kuspriyanto ◽  
Dwi Hendratmo Widyantoro ◽  
Adi Pancoro ◽  
...  
Keyword(s):  
Integer Weight

scholarly journals CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK

2010 ◽  
Vol 06 (02) ◽  
pp. 281-309 ◽  
Author(s):  
F. G. GARVAN
Keyword(s):  
Partition Function ◽  
Rank Function ◽  
Maass Forms ◽  
Hecke Eigenforms ◽  
Half Integer ◽  
Integer Weight

Let spt (n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt (n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt (n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

scholarly journals Parameterized Algorithms for Finding a Collective Set of Items

2020 ◽  
Vol 34 (02) ◽  
pp. 1838-1845
Author(s):  
Robert Bredereck ◽  
Piotr Faliszewski ◽  
Andrzej Kaczmarczyk ◽  
Dušan Knop ◽  
Rolf Niedermeier
Keyword(s):  
Parameterized Complexity ◽  
Weighted Average ◽  
Fixed Parameter Tractability ◽  
Parameterized Algorithms ◽  
Utility Value ◽  
Average Operator ◽  
Fixed Parameter ◽  
Ordered Weighted Average ◽  
Integer Weight

We extend the work of Skowron et al. (AIJ, 2016) by considering the parameterized complexity of the following problem. We are given a set of items and a set of agents, where each agent assigns an integer utility value to each item. The goal is to find a set of k items that these agents would collectively use. For each such collective set of items, each agent provides a score that can be described using an OWA (ordered weighted average) operator and we seek a set with the highest total score. We focus on the parameterization by the number of agents and we find numerous fixed-parameter tractability results (however, we also find some W[1]-hardness results). It turns out that most of our algorithms even apply to the setting where each agent has an integer weight.

On the Sizes of Gaps in the Fourier Expansion of Modular Forms

2005 ◽  
Vol 57 (3) ◽  
pp. 449-470 ◽  
Author(s):  
Emre Alkan
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Complex Multiplication ◽  
The Riemann Zeta Function ◽  
Classical Paper ◽  
Riemann Zeta ◽  
Image Position ◽  
Almost All ◽  
Integer Weight

AbstractLet be a cusp form with integer weight k ≥ 2 that is not a linear combination of forms with complex multiplication. For n ≥ 1, letConcerning bounded values of i f (n) we prove that for ∊ > 0 there exists M = M(∊, f ) such that Using results of Wu, we show that if f is a weight 2 cusp form for an elliptic curve without complex multiplication, then . Using a result of David and Pappalardi, we improve the exponent to for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate i f (n) on the average using well known bounds on the Riemann Zeta function.

scholarly journals NONVANISHING OF JACOBI POINCARÉ SERIES

2010 ◽  
Vol 89 (2) ◽  
pp. 165-179 ◽  
Author(s):  
SOUMYA DAS
Keyword(s):  
Exponential Type ◽  
Poincaré Series ◽  
Jacobi Forms ◽  
Poincare Series ◽  
Integer Weight

AbstractWe prove that, under suitable conditions, a Jacobi Poincaré series of exponential type of integer weight and matrix index does not vanish identically. For the classical Jacobi forms, we construct a basis consisting of the ‘first’ few Poincaré series, and also give conditions, both dependent on and independent of the weight, that ensure the nonvanishing of a classical Jacobi Poincaré series. We also obtain a result on the nonvanishing of a Jacobi Poincaré series when an odd prime divides the index.

POLYNOMIAL TIME SOLVABILITY OF THE WEIGHTED RING ARC-LOADING PROBLEM WITH INTEGER SPLITTING

2004 ◽  
Vol 05 (02) ◽  
pp. 193-200 ◽  
Author(s):  
JINJIANG YUAN ◽  
SANMING ZHOU
Keyword(s):  
Polynomial Time ◽  
Polynomial Algorithm ◽  
Maximum Load ◽  
Negative Integer ◽  
Routing Scheme ◽  
Discrete Algorithms ◽  
Loading Problem ◽  
Wavelength Translation ◽  
Integer Weight

In the Weighted Ring Arc-Loading Problem with Integer Splitting, we are given a set of communication requests each associated with a non-negative integer weight. The problem is to find a routing scheme such that the maximum load on arcs of the ring is minimized, subject to that the weight of each request may be split into two integral parts routed in two directions around the ring, where the load of an arc is the sum of parts routed through the arc. A pseudo-polynomial algorithm for this problem is implied by a result in [G. Wilfong and P. Winkler, Ring routing and wavelength translation, Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, San Fancisco, CA, 1998, 333-341]. By refining the rounding technique developed in the same paper, we prove that the problem can be solved in polynomial time.

scholarly journals Cosheaf representations of relations and Dowker complexes

2021 ◽  
Author(s):  
Michael Robinson
Keyword(s):  
Weight Function ◽  
Simplicial Complex ◽  
Binary Relation ◽  
Homotopy Equivalent ◽  
Covariant Functor ◽  
The Matrix ◽  
Abstract Simplicial Complex ◽  
Integer Weight

AbstractThe Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction—vertices for the complex are either the rows or the columns of the matrix representing the relation—the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes.

scholarly journals An O(n^3) time algorithm for the maximum weight b-matching problem on bipartite graphs

2020 ◽  
Author(s):  
Fatemeh Rajabi-Alni ◽  
Alireza Bagheri ◽  
Behrouz Minaei-Bidgoli
Keyword(s):  
Computational Biology ◽  
Bipartite Graph ◽  
Association Studies ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Maximum Weight ◽  
Positive Real ◽  
Matching Problem ◽  
Genome Wide ◽  
Integer Weight

Abstract Background: A matching between two sets A and B assigns some elements of A to some elements of B. Finding the similarity between two sets of elements by advantage of the matching is widely used in computational biology for example in the contexts of genome-wide and sequencing association studies. Frequently, the capacities of the elements are limited. That is, the number of the elements that can be matched to each element should not exceed a given number. Results: We use bipartite graphs to model relationships between pairs of objects. Given an undirected bipartite graph G = (A [ B;E), the b-matching of G matches each vertex v in A (resp. B) to at least 1 and at most b(v) vertices in B (resp. A), where b(v) denotes the capacity of v. We propose the rst O(n3) time algorithm for nding the maximum weight b-matching of G, where jAj + jBj = O(n). Conclusions: The b-matching has been studied widely for the bipartite graphs with integer weight edges. But our algorithm is the rst algorithm for the maximum (respectively minimum) b-matching problem with non positive real (respectively non negative real) edge weights.

scholarly journals The $t$-Pebbling Number is Eventually Linear in $t$

10.37236/640 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael Hoffmann ◽  
Jiří Matoušek ◽  
Yoshio Okamoto ◽  
Philipp Zumstein
Keyword(s):  
Initial Distribution ◽  
Adjacent Vertex ◽  
Weighted Graphs ◽  
Graph Pebbling ◽  
Pebbling Number ◽  
Integer Weight

In graph pebbling games, one considers a distribution of pebbles on the vertices of a graph, and a pebbling move consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The $t$-pebbling number $\pi_t(G)$ of a graph $G$ is the smallest $m$ such that for every initial distribution of $m$ pebbles on $V(G)$ and every target vertex $x$ there exists a sequence of pebbling moves leading to a distibution with at least $t$ pebbles at $x$. Answering a question of Sieben, we show that for every graph $G$, $\pi_t(G)$ is eventually linear in $t$; that is, there are numbers $a,b,t_0$ such that $\pi_t(G)=at+b$ for all $t\ge t_0$. Our result is also valid for weighted graphs, where every edge $e=\{u,v\}$ has some integer weight $\omega(e)\ge 2$, and a pebbling move from $u$ to $v$ removes $\omega(e)$ pebbles at $u$ and adds one pebble to $v$.

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