scholarly journals On Capacity-Filling and Substitutable Choice Rules

2021 ◽  
Author(s):  
Battal Doğan ◽  
Serhat Doğan ◽  
Kemal Yıldız
Keyword(s):  
Choice Rule ◽  
Critical Set ◽  
Critical Sets ◽  
Choice Set

Each capacity-filling and substitutable choice rule is known to have a maximizer-collecting representation: There exists a list of priority orderings such that from each choice set that includes more alternatives than the capacity, the choice is the union of the priority orderings’ maximizers. We introduce the notion of a critical set and constructively prove that the number of critical sets for a choice rule determines its smallest-size maximizer-collecting representation. We show that responsive choice rules require the maximal number of priority orderings in their smallest-size maximizer-collecting representations among all capacity-filling and substitutable choice rules. We also analyze maximizer-collecting choice rules in which the number of priority orderings equals the capacity. We show that if the capacity is greater than three and the number of alternatives exceeds the capacity by at least two, then no capacity-filling and substitutable choice rule has a maximizer-collecting representation of the size equal to the capacity.

Two remarks about Sudoku squares

2006 ◽  
Vol 90 (519) ◽  
pp. 425-430 ◽  
Author(s):  
A. D. Keedwell
Keyword(s):  
Current Problem ◽  
Latin Square ◽  
Latin Squares ◽  
Critical Set ◽  
Critical Sets ◽  
The Given ◽  
A Current ◽  
Defining Sets

Smallest defining setsA standard Sudoku square is a 9 × 9 latin square in which each of the nine 3 × 3 subsquares into which it can be separated contains each of the integers 1 to 9 exactly once.A current problem is to complete such a square when only some of the cells have been filled. These cells are often called ‘givens’. (Such problems are currently (2005) published daily in British newspapers.) In more mathematical terms, the given filled cells constitute a defining set or uniquely completable set for the square if they lead to a unique completion of the square. If, after deletion of any one of these givens, the square can no longer be completed uniquely, the givens form a critical set. The investigation of critical sets for ‘ordinary’ latin squares is a topic of current mathematical interest. (See [1] for more details.)

Gradient-like flows on high-dimensional manifolds

1999 ◽  
Vol 19 (2) ◽  
pp. 339-362 ◽  
Author(s):  
R. N. CRUZ ◽  
K. A. DE REZENDE
Keyword(s):  
Level Sets ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
High Dimensional ◽  
Topological Information ◽  
Critical Set ◽  
Critical Sets ◽  
Lyapunov Graph

The main purpose of this paper is to study the implications that the homology index of critical sets of smooth flows on closed manifolds M have on both the homology of level sets of M and the homology of M itself. The bookkeeping of the data containing the critical set information of the flow and topological information of M is done through the use of Lyapunov graphs. Our main result characterizes the necessary conditions that a Lyapunov graph must possess in order to be associated to a Morse–Smale flow. With additional restrictions on an abstract Lyapunov graph L we determine sufficient conditions for L to be associated to a flow on M.

scholarly journals Critical sets in Latin squares given that they are symmetric

2007 ◽  
pp. 38-45
Author(s):  
Ali Mojdeh ◽  
Jafari Rad
Keyword(s):  
Latin Square ◽  
Latin Squares ◽  
Main Diagonal ◽  
Critical Set ◽  
Critical Sets

A uniquely completable (UC) set U is a subset of a Latin square L such that L is the only superset of U which is a Latin square. A critical set C of L is a subset of L such that C is uniquely completable and no subset of C has this property. We show that there is a symmetric Latin square with fixed main diagonal entries for each even number, and obtain a uniquely completable partial symmetric Latin square of order 2n for each n and prove that, it is critical set for n = 3, 4, 5 and 6, and make a problem.

scholarly journals Convexity of momentum map, Morse index, and quantum entanglement

2014 ◽  
Vol 26 (03) ◽  
pp. 1450004 ◽  
Author(s):  
Adam Sawicki ◽  
Michał Oszmaniec ◽  
Marek Kuś
Keyword(s):  
Morse Index ◽  
Hausdorff Space ◽  
Quantum Systems ◽  
Total Variance ◽  
Geometric Invariant ◽  
Classical Communication ◽  
Critical Set ◽  
Momentum Map ◽  
Critical Sets

We analyze from the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry, we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many "asymptotically" equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well-defined measure of entanglement) of the state Var [v] attains maximum. We provide an algorithm for finding critical sets of Var [v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using some developments in the momentum map geometry known as the Kirwan–Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.

Finding Critical Sets.

1985 ◽  
Author(s):  
Donald W. Loveland
Keyword(s):  
Critical Sets

scholarly journals A Discussion of a Cryptographical Scheme Based in F-Critical Sets of a Latin Square

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 285
Author(s):  
Laura M. Johnson ◽  
Stephanie Perkins
Keyword(s):  
Necessary Conditions ◽  
Latin Square ◽  
Latin Squares ◽  
Critical Sets ◽  
Long Cycles ◽  
The Given

This communication provides a discussion of a scheme originally proposed by Falcón in a paper entitled “Latin squares associated to principal autotopisms of long cycles. Applications in cryptography”. Falcón outlines the protocol for a cryptographical scheme that uses the F-critical sets associated with a particular Latin square to generate access levels for participants of the scheme. Accompanying the scheme is an example, which applies the protocol to a particular Latin square of order six. Exploration of the example itself, revealed some interesting observations about both the structure of the Latin square itself and the autotopisms associated with the Latin square. These observations give rise to necessary conditions for the generation of the F-critical sets associated with certain autotopisms of the given Latin square. The communication culminates with a table which outlines the various access levels for the given Latin square in accordance with the scheme detailed by Falcón.

scholarly journals Choice Set Definition Issues in a Kuhn-Tucker Model of Recreation Demand

1999 ◽  
Vol 14 (4) ◽  
pp. 343-355 ◽  
Author(s):  
DANIEL J. PHANEUF ◽  
JOSEPH A. HERRIGES
Keyword(s):  
Recreation Demand ◽  
Choice Set
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