scholarly journals On The Independence Number Of Some Strong Products Of Cycle-Powers

2015 ◽  
Vol 40 (2) ◽  
pp. 133-141 ◽  
Author(s):  
Marcin Jurkiewicz ◽  
Marek Kubale ◽  
Krzysztof Ocetkiewicz
Keyword(s):  
Lower Bounds ◽  
Independence Number ◽  
Computational Results ◽  
Shannon Capacity ◽  
Noisy Channels ◽  
Running Time ◽  
The Third ◽  
Independence Numbers

Abstract In the paper we give some theoretical and computational results on the third strong power of cycle-powers, for example, we have found the independence numbers α((C102)√3) = 30 and α((C144)√3) = 14. A number of optimizations have been introduced to improve the running time of our exhaustive algorithm used to establish the independence number of the third strong power of cycle-powers. Moreover, our results establish new exact values and/or lower bounds on the Shannon capacity of noisy channels.

scholarly journals Fundamental limitations on distillation of quantum channel resources

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Bartosz Regula ◽  
Ryuji Takagi
Keyword(s):  
Lower Bounds ◽  
Quantum Channel ◽  
Quantum Communication ◽  
Fault Tolerant ◽  
State Of The Art ◽  
Quantum Systems ◽  
Noisy Channels ◽  
General Transformation ◽  
Fundamental Limitations ◽  
Strong Converse

AbstractQuantum channels underlie the dynamics of quantum systems, but in many practical settings it is the channels themselves that require processing. We establish universal limitations on the processing of both quantum states and channels, expressed in the form of no-go theorems and quantitative bounds for the manipulation of general quantum channel resources under the most general transformation protocols. Focusing on the class of distillation tasks — which can be understood either as the purification of noisy channels into unitary ones, or the extraction of state-based resources from channels — we develop fundamental restrictions on the error incurred in such transformations, and comprehensive lower bounds for the overhead of any distillation protocol. In the asymptotic setting, our results yield broadly applicable bounds for rates of distillation. We demonstrate our results through applications to fault-tolerant quantum computation, where we obtain state-of-the-art lower bounds for the overhead cost of magic state distillation, as well as to quantum communication, where we recover a number of strong converse bounds for quantum channel capacity.

scholarly journals On the Interactive Capacity of Finite-State Protocols

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 17
Author(s):  
Assaf Ben-Yishai ◽  
Young-Han Kim ◽  
Rotem Oshman ◽  
Ofer Shayevitz
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Noisy Channel ◽  
Shannon Capacity ◽  
Finite State

The interactive capacity of a noisy channel is the highest possible rate at which arbitrary interactive protocols can be simulated reliably over the channel. Determining the interactive capacity is notoriously difficult, and the best known lower bounds are far below the associated Shannon capacity, which serves as a trivial (and also generally the best known) upper bound. This paper considers the more restricted setup of simulating finite-state protocols. It is shown that all two-state protocols, as well as rich families of arbitrary finite-state protocols, can be simulated at the Shannon capacity, establishing the interactive capacity for those families of protocols.

Reidemeister moves and parity polynomials of virtual knot diagrams

2017 ◽  
Vol 26 (10) ◽  
pp. 1750051
Author(s):  
Myeong-Ju Jeong
Keyword(s):  
Lower Bounds ◽  
Virtual Knot ◽  
The Third ◽  
Reidemeister Moves ◽  
Polynomial Formula ◽  
Knot Diagrams

When two virtual knot diagrams are virtually isotopic, there is a sequence of Reidemeister moves and virtual moves relating them. I introduced a polynomial [Formula: see text] of a virtual knot diagram [Formula: see text] and gave lower bounds for the number of Reidemeister moves in deformation of virtually isotopic knot diagrams by using [Formula: see text]. In this paper, I introduce bridge diagrams and polynomials of virtual knot diagrams based on parity of crossings, and show that the polynomials give lower bounds for the number of the third Reidemeister moves. I give an example which shows that the result is distinguished from that obtained from [Formula: see text].

LOWER BOUNDS FOR THE NORMALIZED HEIGHT AND NON-DENSE SUBSETS OF SUBVARIETIES OF ABELIAN VARIETIES

2010 ◽  
Vol 06 (03) ◽  
pp. 471-499 ◽  
Author(s):  
EVELINA VIADA
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Abelian Variety ◽  
Lower Bounds ◽  
Finite Rank ◽  
Height Function ◽  
Algebraic Numbers ◽  
Algebraic Subgroup ◽  
Rank Zero ◽  
The Third

This work is the third part of a series of papers. In the first two, we considered curves and varieties in a power of an elliptic curve. Here, we deal with subvarieties of an abelian variety in general. Let V be a proper irreducible subvariety of dimension d in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. Then, for V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup Γ of finite rank, are non-Zariski-dense in V. If Γ has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function.

scholarly journals New lower bounds for the Shannon capacity of odd cycles

2016 ◽  
Vol 84 (1-2) ◽  
pp. 13-22 ◽  
Author(s):  
K. Ashik Mathew ◽  
Patric R. J. Östergård
Keyword(s):  
Lower Bounds ◽  
Shannon Capacity ◽  
Odd Cycles

scholarly journals Cubic Graphs with Small Independence Ratio

10.37236/7272 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Xujun Liu
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Independence Number ◽  
Regular Graphs ◽  
Cubic Graphs

Let $i(r,g)$ denote the infimum of the ratio $\frac{\alpha(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $\alpha(G)$ is the independence number of $G$, and  let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \geqslant 0.4375,$ and Csóka improved it to $i(3,\infty) \geqslant 0.44533$ in 2016. Bollobás proved the upper bound  $i(3,\infty) < \frac{6}{13}$  in 1981, and McKay improved it to $i(3,\infty) < 0.45537$in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \leqslant 0.454.$

scholarly journals On the Shannon Capacity of Triangular Graphs

10.37236/3214 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Ashik Mathew Kizhakkepallathu ◽  
Patric RJ Östergård ◽  
Alexandru Popa
Keyword(s):  
Independence Number ◽  
Shannon Capacity ◽  
Dimensional Torus ◽  
Induced Subgraph ◽  
N Cycle ◽  
Kneser Graph ◽  
Triangular Graph ◽  
Current Context

The Shannon capacity of a graph $G$ is $c(G)=\sup_{d\geq 1}(\alpha(G^d))^{\frac{1}{d}},$ where $\alpha(G)$ is the independence number of $G$. The Shannon capacity of the Kneser graph $\mathrm{KG}_{n,r}$ was determined by Lovász in 1979, but little is known about the Shannon capacity of the complement of that graph when $r$ does not divide $n$. The complement of the Kneser graph, $\overline{\mathrm{KG}}_{n,2}$, is also called the triangular graph $T_n$. The graph $T_n$ has the $n$-cycle $C_n$ as an induced subgraph, whereby $c(T_n) \geq c(C_n)$, and these two families of graphs are closely related in the current context as both can be considered via geometric packings of the discrete $d$-dimensional torus of width $n$ using two types of $d$-dimensional cubes of width $2$. Bounds on $c(T_n)$ obtained in this work include $c(T_7) \geq \sqrt[3]{35} \approx 3.271$, $c(T_{13}) \geq \sqrt[3]{248} \approx 6.283$, $c(T_{15}) \geq \sqrt[4]{2802} \approx 7.276$, and $c(T_{21}) \geq \sqrt[4]{11441} \approx 10.342$.

scholarly journals On the Independence Numbers of the Cubes of Odd Cycles

10.37236/2598 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Tom Bohman ◽  
Ron Holzman ◽  
Venkatesh Natarajan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Independence Number ◽  
Odd Cycle ◽  
Odd Cycles ◽  
Independence Numbers

We give an upper bound on the independence number of the cube of the odd cycle $C_{8n+5}$. The best known lower bound is conjectured to be the truth; we prove the conjecture in the case $8n+5$ prime and, within $2$, for general $n$.

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