scholarly journals Improved Thermal Area Law and Quasilinear Time Algorithm for Quantum Gibbs States

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Tomotaka Kuwahara ◽  
Álvaro M. Alhambra ◽  
Anurag Anshu
Keyword(s):  
Time Algorithm ◽  
Gibbs States ◽  
Area Law

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

Ultrashort pulses propagation through different approaches of the Split-Step Fourier method

2018 ◽  
Vol 1 (3) ◽  
pp. 2
Author(s):  
José Stênio De Negreiros Júnior ◽  
Daniel Do Nascimento e Sá Cavalcante ◽  
Jermana Lopes de Moraes ◽  
Lucas Rodrigues Marcelino ◽  
Francisco Tadeu De Carvalho Belchior Magalhães ◽  
...  
Keyword(s):  
Energy Conservation ◽  
Optical Pulse ◽  
Time Window ◽  
Fourier Method ◽  
Ultrashort Pulses ◽  
Nonlinear Effects ◽  
Single Mode ◽  
Time Algorithm ◽  
Optical Pulses ◽  
Running Time

Simulating the propagation of optical pulses in a single mode optical fiber is of fundamental importance for studying the several effects that may occur within such medium when it is under some linear and nonlinear effects. In this work, we simulate it by implementing the nonlinear Schrödinger equation using the Split-Step Fourier method in some of its approaches. Then, we compare their running time, algorithm complexity and accuracy regarding energy conservation of the optical pulse. We note that the method is simple to implement and presents good results of energy conservation, besides low temporal cost. We observe a greater precision for the symmetrized approach, although its running time can be up to 126% higher than the other approaches, depending on the parameters set. We conclude that the time window must be adjusted for each length of propagation in the fiber, so that the error regarding energy conservation during propagation can be reduced.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

scholarly journals PHASE TRANSITIONS AND QUANTUM STABILIZATION IN QUANTUM ANHARMONIC CRYSTALS

2008 ◽  
Vol 20 (05) ◽  
pp. 529-595 ◽  
Author(s):  
ALINA KARGOL ◽  
YURI KONDRATIEV ◽  
YURI KOZITSKY
Keyword(s):  
Phase Transitions ◽  
Gibbs Measures ◽  
Quantum Effects ◽  
Gibbs States ◽  
Translation Invariance ◽  
Unified Theory ◽  
The Relationship ◽  
Anharmonic Crystals

A unified theory of phase transitions and quantum effects in quantum anharmonic crystals is presented. In its framework, the relationship between these two phenomena is analyzed. The theory is based on the representation of the model Gibbs states in terms of path measures (Euclidean Gibbs measures). It covers the case of crystals without translation invariance, as well as the case of asymmetric anharmonic potentials. The results obtained are compared with those known in the literature.

scholarly journals Target space entanglement in quantum mechanics of fermions and matrices

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Sotaro Sugishita
Keyword(s):  
Mutual Information ◽  
Ground State ◽  
Entanglement Entropy ◽  
Algebraic Approach ◽  
Upper Bounds ◽  
Wave Functions ◽  
Target Space ◽  
Single Interval ◽  
Area Law ◽  
Formula Expression

Abstract We consider entanglement of first-quantized identical particles by adopting an algebraic approach. In particular, we investigate fermions whose wave functions are given by the Slater determinants, as for singlet sectors of one-matrix models. We show that the upper bounds of the general Rényi entropies are N log 2 for N particles or an N × N matrix. We compute the target space entanglement entropy and the mutual information in a free one-matrix model. We confirm the area law: the single-interval entropy for the ground state scales as $$ \frac{1}{3} $$ 1 3 log N in the large N model. We obtain an analytical $$ \mathcal{O}\left({N}^0\right) $$ O N 0 expression of the mutual information for two intervals in the large N expansion.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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