scholarly journals A non-levi branching rule in terms of Littelmann paths

2018 ◽  
Vol 117 (5) ◽  
pp. 1077-1100 ◽  
Author(s):  
Bea Schumann ◽  
Jacinta Torres
Keyword(s):  
Branching Rule ◽  
Littelmann Paths

scholarly journals On a conjecture of Naito-Sagaki: Littelmann paths and Littlewood-Richardson Sundaram tableaux

2021 ◽  
pp. 305-319
Author(s):  
Jacinta Torres
Keyword(s):  
Fixed Point ◽  
Irreducible Representation ◽  
Recent Work ◽  
Lie Algebra ◽  
Fixed Point Set ◽  
New Approach ◽  
Branching Rule ◽  
Branching Rules ◽  
Point Set ◽  
Littelmann Paths

In recent work with Schumann we have proven a conjecture of Naito-Sagaki giving a branching rule for the decomposition of the restriction of an irreducible representation of the special linear Lie algebra to the symplectic Lie algebra, therein embedded as the fixed-point set of the involution obtained by the folding of the corresponding Dyinkin diagram. It provides a new approach to branching rules for non-Levi subalgebras in terms of Littelmann paths. In this paper we motivate this result, provide examples, and give an overview of the combinatorics involved in its proof.

scholarly journals Dynamic Shortest Paths Methods for the Time-Dependent TSP

Algorithms ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 21
Author(s):  
Christoph Hansknecht ◽  
Imke Joormann ◽  
Sebastian Stiller
Keyword(s):  
Column Generation ◽  
Traveling Salesman Problem ◽  
Shortest Path ◽  
Valid Inequalities ◽  
Shortest Paths ◽  
Traveling Salesman ◽  
Time Dependent ◽  
Full Generality ◽  
Branching Rule ◽  
The Traveling Salesman Problem

The time-dependent traveling salesman problem (TDTSP) asks for a shortest Hamiltonian tour in a directed graph where (asymmetric) arc-costs depend on the time the arc is entered. With traffic data abundantly available, methods to optimize routes with respect to time-dependent travel times are widely desired. This holds in particular for the traveling salesman problem, which is a corner stone of logistic planning. In this paper, we devise column-generation-based IP methods to solve the TDTSP in full generality, both for arc- and path-based formulations. The algorithmic key is a time-dependent shortest path problem, which arises from the pricing problem of the column generation and is of independent interest—namely, to find paths in a time-expanded graph that are acyclic in the underlying (non-expanded) graph. As this problem is computationally too costly, we price over the set of paths that contain no cycles of length k. In addition, we devise—tailored for the TDTSP—several families of valid inequalities, primal heuristics, a propagation method, and a branching rule. Combining these with the time-dependent shortest path pricing we provide—to our knowledge—the first elaborate method to solve the TDTSP in general and with fully general time-dependence. We also provide for results on complexity and approximability of the TDTSP. In computational experiments on randomly generated instances, we are able to solve the large majority of small instances (20 nodes) to optimality, while closing about two thirds of the remaining gap of the large instances (40 nodes) after one hour of computation.

scholarly journals Tree Trimming: Four Non-Branching Rules for Priest’s Introduction to Non-Classical Logic

2017 ◽  
Vol 12 (2) ◽  
Author(s):  
Marilynn Johnson
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Variable Domain ◽  
Free Logic ◽  
Intuitionist Logic ◽  
Branching Rule ◽  
Branching Rules ◽  
Graham Priest

In An Introduction to Non-Classical Logic: From If to Is Graham Priest (2008) presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest’s rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest’s rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest’s branching universal and particular instantiation rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest grapples with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest.

scholarly journals Littelmann paths and Brownian paths

2005 ◽  
Vol 130 (1) ◽  
pp. 127-167 ◽  
Author(s):  
Philippe Biane ◽  
Philippe Bougerol ◽  
Neil O'connell
Keyword(s):  
Littelmann Paths

Method of choosing the element base and design solutions to ensure the required reliability of promising radio equipment

2020 ◽  
Vol 49 (5) ◽  
pp. 71-75
Author(s):  
Y. S. Kucherov ◽  
R. V. Dopira ◽  
D. V. Yagolnikov ◽  
I. E. Yanochkin
Keyword(s):  
Linear Programming ◽  
Product Development ◽  
Branch And Bound ◽  
Practical Problem ◽  
Main Idea ◽  
Branch And Bound Method ◽  
Minimal Cost ◽  
Branching Rule ◽  
Radio Equipment ◽  
Element Base

The article proposes a method for solving the problem of choosing the element base and constructive solutions to ensure the required reliability of promising radio equipment at minimal cost. The problem belongs to the class of Boolean linear programming and is solved using the branch and bound method. The main idea of the branch and bound method is to determine the branching rule for assigning options and further evaluating the objective function on these subsets, which allows us to exclude from consideration subsets that do not contain optimal points. The task of increasing reliability can be solved by choosing more reliable elements and using the method of structural reservation of elements at the stage of product development. The results of using the proposed method to solve the practical problem of choosing the elements are presented.

THE PARAMETER-DEPENDENT FERMION STATES FOR MOLECULES WITH SYMPLECTIC SYMMETRY

2006 ◽  
Vol 05 (04) ◽  
pp. 779-799
Author(s):  
C. C. SUN ◽  
B. F. LI ◽  
Z. S. LI ◽  
H. X. ZHANG ◽  
X. R. HUANG
Keyword(s):  
Quantum Chemistry ◽  
Explicit Form ◽  
Symplectic Group ◽  
Similarity Transformation ◽  
Optimization Procedure ◽  
Complete Classification ◽  
Branching Rule ◽  
Parameter Dependent

Under a certain kind of similarity transformation, a parameter-dependent (abbreviated as PD) symplectic group chain Sp(2M) ⊃ Sp(2M - 2) ⊃ ⋯ ⊃ Sp(2) that is characterized by a set of pairing parameters is introduced to build up the PD antisymmetrized fermion states for molecules with symplectic symmetry, and these states will be useful in carrying out the optimization procedure in quantum chemistry. In order to make a complete classification of the states, a generalized branching rule associated with the symplectic group chain is proposed. Further, we are led to the result that the explicit form of the PD antisymmetrized fermion states is obtained in terms of M one-particle operators and M geminal operators.

scholarly journals Outcome space range reduction method for global optimization of sum of affine ratios problem

2016 ◽  
Vol 14 (1) ◽  
pp. 736-746 ◽  
Author(s):  
Hongwei Jiao ◽  
Sanyang Liu ◽  
Jingben Yin ◽  
Yingfeng Zhao
Keyword(s):  
Global Optimization ◽  
Reduction Method ◽  
Optimal Solution ◽  
Space Region ◽  
Equivalent Problem ◽  
Solution Quality ◽  
Range Reduction ◽  
Branching Rule ◽  
Outcome Space ◽  
Space Range

Abstract Many algorithms for globally solving sum of affine ratios problem (SAR) are based on equivalent problem and branch-and-bound framework. Since the exhaustiveness of branching rule leads to a significant increase in the computational burden for solving the equivalent problem. In this study, a new range reduction method for outcome space of the denominator is presented for globally solving the sum of affine ratios problem (SAR). The proposed range reduction method offers a possibility to delete a large part of the outcome space region of the denominators in which the global optimal solution of the equivalent problem does not exist, and which can be seen as an accelerating device for global optimization of the (SAR). Several numerical examples are presented to demonstrate the advantages of the proposed algorithm using new range reduction method in terms of both computational efficiency and solution quality.

scholarly journals A study of Kostant-Kumar modules via Littelmann paths

2021 ◽  
Vol 381 ◽  
pp. 107614
Author(s):  
Mrigendra Singh Kushwaha ◽  
K.N. Raghavan ◽  
Sankaran Viswanath
Keyword(s):  
Littelmann Paths
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