scholarly journals Bounds on the sum of (log(p))2 Terms

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Lower Bounds ◽  
Estimation Error ◽  
Research Paper ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Log P ◽  
Fast Computation

In this research paper, we implement the theory of the primorial function, to develop the Supremum and Infimum bounds for the sum of (log(p))2. There are, however, considerable computational difficulties related to these bounds. Therefore, from a pragmatic point of view, a set of Upper and Lower bounds had been developed to bypass this issue. Despite the increased estimation error, the Upper and Lower bounds are still considered sufficiently accurate, while facilitating an easy and fast computation of the estimate of the sum.

scholarly journals Discrepancy of Symmetric Products of Hypergraphs

10.37236/1066 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Benjamin Doerr ◽  
Michael Gnewuch ◽  
Nils Hebbinghaus
Keyword(s):  
Direct Product ◽  
Lower Bounds ◽  
Research Paper ◽  
Arithmetic Progressions ◽  
Symmetric Product ◽  
Upper And Lower Bounds ◽  
Symmetric Products ◽  
Cartesian Products ◽  
Better Than

For a hypergraph ${\cal H} = (V,{\cal E})$, its $d$–fold symmetric product is defined to be $\Delta^d {\cal H} = (V^d,\{E^d |E \in {\cal E}\})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound ${\rm disc}(\Delta^d {\cal H},2) \le {\rm disc}({\cal H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and ${\rm disc}(\Delta^d {\cal H},c) = \Omega_d({\rm disc}({\cal H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product ${\cal H}^d$, which satisfies ${\rm disc}({\cal H}^d,c) = O_{c,d}({\rm disc}({\cal H},c)^d)$.

scholarly journals Maximal prime gaps bounds

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Estimation Error ◽  
The Other ◽  
Upper And Lower Bounds ◽  
Close Approximation ◽  
Implicit Constant ◽  
Error Cost ◽  
Prime Gaps

This paper presents research results, pertinent to the maximal prime gaps bounds. Four distinct bounds are presented: Upper bound, Infimum, Supremum and finally the Lower bound. Although the Upper and Lower bounds incur a relatively high estimation error cost, the functions representing them are quite simple. This ensures, that the computation of those bounds will be straightforward and efficient. The Lower bound is essential, to address the issue of the value of the lower bound implicit constant C, in the work of Ford et al (Ford, 2016). The concluding Corollary in this paper shows, that the value of the constant C does diverge, although very slowly. The constant C, will eventually take any arbitrary value, providing that a large enough N (for p <= N) is considered. The Infimum/Supremum bounds on the other hand are computationally very demanding. Their evaluation entails computations at an extreme level of precision. In return however, we obtain bounds, which provide an extremely close approximation of the maximal prime gaps. The Infimum/Supremum estimation error gradually increases over the range of p and attains at p = 18361375334787046697 approximately the value of 0.03.

scholarly journals Discrepancy of Products of Hypergraphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Benjamin Doerr ◽  
Michael Gnewuch ◽  
Nils Hebbinghaus
Keyword(s):  
Direct Product ◽  
Lower Bounds ◽  
Research Paper ◽  
Arithmetic Progressions ◽  
Symmetric Product ◽  
Upper And Lower Bounds ◽  
Cartesian Products ◽  
International Audience ◽  
Better Than

International audience For a hypergraph $\mathcal{H} = (V,\mathcal{E})$, its $d$―fold symmetric product is $\Delta^d \mathcal{H} = (V^d,\{ E^d | E \in \mathcal{E} \})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound $\textrm{disc}(\Delta^d \mathcal{H},2) \leq \textrm{disc}(\mathcal{H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and $\textrm{disc}(\Delta^d \mathcal{H},c) = \Omega_d(\textrm{disc}(\mathcal{H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product $\mathcal{H}^d$, which satisfies $\textrm{disc}(\mathcal{H}^d,c) = O_{c,d}(\textrm{disc}(\mathcal{H},c)^d)$.

LARGE SORTING AND ROUTING PROBLEMS ON THE HYPERCUBE AND RELATED NETWORKS

1991 ◽  
Vol 01 (02) ◽  
pp. 113-124
Author(s):  
GIOVANNI MANZINI
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Log P ◽  
Routing Problems ◽  
Cube Connected Cycles

In this paper we present upper and lower bounds for the problems of sorting and routing n items using p processors on the hypercube, the shuffle, the butterfly and the Cube-Connected-Cycles. We consider problems where n is large; more precisely we assume that n = Ω (p2). The main results of this paper are an algorithm for routing n packets in O((n/p) log p) time, and an algorithm for sorting n integers in the range 0, 1, …, R – 1 in [Formula: see text] time.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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