scholarly journals Simple Upper and Lower Bounds on the Ultimate Success Probability for Discriminating Arbitrary Finite-Dimensional Quantum Processes

2021 ◽  
Vol 126 (20) ◽  
Author(s):  
Kenji Nakahira ◽  
Kentaro Kato
Keyword(s):  
Lower Bounds ◽  
Success Probability ◽  
Upper And Lower Bounds ◽  
Quantum Processes ◽  
Finite Dimensional

scholarly journals Computing the exponent of a Lebesgue space

2020 ◽  
Vol 12 ◽  
Author(s):  
Timothy McNicholl
Keyword(s):  
Lower Bounds ◽  
Lebesgue Space ◽  
Random Variables ◽  
Upper And Lower Bounds ◽  
Effective Theory ◽  
Finite Dimensional ◽  
Stable Random Variables ◽  
Uniform Solution

We consider the question as to whether the exponent of a computably presentable Lebesgue space whose dimension is at least 2 must be computable.  We show this very natural conjecture is true when the exponent is at least 2 or when the space is finite-dimensional.  However, we also show there is no uniform solution even when given upper and lower bounds on the exponent.  The proof of this result leads to some basic results on the effective theory of stable random variables.  

Runtime Analysis of the (μ + 1) EA on Simple Pseudo-Boolean Functions

2006 ◽  
Vol 14 (1) ◽  
pp. 65-86 ◽  
Author(s):  
Carsten Witt
Keyword(s):  
Lower Bounds ◽  
Boolean Functions ◽  
Success Probability ◽  
Population Based ◽  
Upper And Lower Bounds ◽  
Rigorous Analysis ◽  
Search Spaces ◽  
Discrete Search ◽  
Family Trees ◽  
Theoretical Developments

Although Evolutionary Algorithms (EAs) have been successfully applied to optimization in discrete search spaces, theoretical developments remain weak, in particular for population-based EAs. This paper presents a first rigorous analysis of the (μ + 1) EA on pseudo-Boolean functions. Using three well-known example functions fromthe analysis of the (1 + 1) EA, we derive bounds on the expected runtime and success probability. For two of these functions, upper and lower bounds on the expected runtime are tight, and on all three functions, the (μ + 1) EA is never more efficient than the (1 + 1) EA. Moreover, all lower bounds growwith μ. On a more complicated function, however, a small increase of μ provably decreases the expected runtime drastically. This paper develops a newproof technique that bounds the runtime of the (μ + 1) EA. It investigates the stochastic process for creating family trees of individuals; the depth of these trees is bounded. Thereby, the progress of the population towards the optimum is captured. This new technique is general enough to be applied to other population-based EAs.

The Ranks of the Homotopy Groups of a Finite Dimensional Complex

2013 ◽  
Vol 65 (1) ◽  
pp. 82-119
Author(s):  
Yves Félix ◽  
Steve Halperin ◽  
Jean-Claude Thomas
Keyword(s):  
Error Bound ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Homotopy Groups ◽  
Finite Dimensional ◽  
Simply Connected

AbstractLet X be an n-dimensional, finite, simply connected CWcomplex and setWhen 0 < αX < ∞, we give upper and lower bounds for for k sufficiently large. We also show for any r that αX can be estimated from the integers rk πi (X), i ≤ nr with an error bound depending explicitly on r.

scholarly journals Programmability of covariant quantum channels

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 488
Author(s):  
Martina Gschwendtner ◽  
Andreas Bluhm ◽  
Andreas Winter
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Group Action ◽  
Space Dimension ◽  
Upper And Lower Bounds ◽  
Quantum Channels ◽  
Covariant Quantum ◽  
Finite Dimensional ◽  
Channel Input ◽  
Quantum Processor

A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.

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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