scholarly journals The Connection between the PQ Penny Flip Game and the Dihedral Groups

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1115
Author(s):  
Theodore Andronikos ◽  
Alla Sirokofskich
Keyword(s):  
Infinite Number ◽  
Dihedral Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Original Game ◽  
Dihedral Groups ◽  
Winning Strategies ◽  
Significant Change ◽  
Necessary And Sufficient

This paper is inspired by the PQ penny flip game. It employs group-theoretic concepts to study the original game and its possible extensions. In this paper, it is shown that the PQ penny flip game can be associated, in a precise way, with the dihedral group D8 and that within D8 there exist precisely two classes of equivalent winning strategies for Q. This is achieved by proving that there are exactly two different sequences of states that can guarantee Q’s win with probability 1.0. It is demonstrated that the game can be played in every dihedral group D8n, where n≥1, without any significant change. A formal examination of what happens when Q can draw their moves from the entire U(2), leads to the conclusion that, again, there are exactly two classes of winning strategies for Q, each class containing an infinite number of equivalent strategies, but all of them sending the coin through the same sequence of states as before. Finally, when general extensions of the game, with the quantum player having U(2) at their disposal, are considered, a necessary and sufficient condition for Q to surely win against Picard is established: Q must make both the first and the last move in the game.

scholarly journals Isomorphisms in switching classes of graphs

1978 ◽  
Vol 26 (4) ◽  
pp. 475-486 ◽  
Author(s):  
David Harries ◽  
Hans Liebeck
Keyword(s):  
Permutation Group ◽  
Dihedral Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Switching Class

AbstractWe investigate conditions on a permutation group G sufficient to ensure that G fixes a graph in any switching class of graphs that it stabilizes. Our main result gives a necessary and sufficient condition for a dihedral group G to have this property.

scholarly journals Orthogonal bases of Brauer symmetry classes of tensors for groups having cyclic support on non-linear Brauer characters

2016 ◽  
Vol 31 ◽  
pp. 263-285 ◽  
Author(s):  
Mahdi Hormozi ◽  
Kijti Rodtes
Keyword(s):  
Cyclic Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dihedral Groups ◽  
Symmetry Classes ◽  
Dimension Formula ◽  
Non Linear ◽  
Brauer Characters ◽  
Vanishing Elements ◽  
Necessary And Sufficient

This paper provides some properties of Brauer symmetry classes of tensors. A dimension formula is derived for the orbital subspaces in the Brauer symmetry classes of tensors corresponding to the irreducible Brauer characters of the groups whose non-linear Brauer characters have support being a cyclic group. Using the derived formula, necessary and sufficient condition are investigated for the existence of an o-basis of dicyclic groups, semi-dihedral groups, and also those things are reinvestigated on dihedral groups. Some criteria for the non-vanishing elements in the Brauer symmetry classes of tensors associated to those groups are also included.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

scholarly journals Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

Physics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 352-366
Author(s):  
Thomas Berry ◽  
Matt Visser
Keyword(s):  
Point Of View ◽  
Sufficient Condition ◽  
Use Of Self ◽  
Necessary And Sufficient Condition ◽  
Wigner Rotation ◽  
Inertial Frames ◽  
Explicit Formulae ◽  
Specific Implementation ◽  
Necessary And Sufficient ◽  
Lorentz Boosts

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

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