scholarly journals Rotation-Equivalence Classes of Binary Vectors

2016 ◽  
Vol 67 (1) ◽  
pp. 93-98
Author(s):  
Otokar Grošek ◽  
Viliam Hromada
Keyword(s):  
Algebraic Approach ◽  
Equivalence Classes ◽  
Linear Feedback ◽  
Sufficient Condition ◽  
Hamming Weight ◽  
Necessary And Sufficient Condition ◽  
Binary Vectors ◽  
Linear Feedback Shift Registers ◽  
Feedback Shift Registers ◽  
Necessary And Sufficient

Abstract In this paper we study equivalence classes of binary vectors with regards to their rotation by using an algebraic approach based on the theory of linear feedback shift registers. We state the necessary and sufficient condition for existence of an equivalence class with given cardinality and provide two formulas. The first represents the sharp distribution of cardinalities for given length and Hamming weight of binary vectors and the second enables us to determine the number of different classes with the same cardinality.

scholarly journals Bracing Zonohedra With Special Faces

2015 ◽  
Vol 3 (1-2) ◽  
pp. 88-95 ◽  
Author(s):  
Gyula Nagy
Keyword(s):  
Convex Polyhedron ◽  
Three Dimensional ◽  
Equivalence Classes ◽  
Frame Structure ◽  
Sufficient Condition ◽  
Preliminary Design ◽  
Necessary And Sufficient Condition ◽  
Special Assumption ◽  
Centrally Symmetric ◽  
Necessary And Sufficient

Abstract The analysis of simpler preliminary design gives useful input for more complicated three-dimensional building frame structure. A zonohedron, as a preliminary structure of design, is a convex polyhedron for which each face possesses central symmetry. We considered zonohedron as a special framework with the special assumption that the polygonal faces can be deformed in such a way that faces remain planar and centrally symmetric, moreover the length of all edges remains unchanged. Introducing some diagonal braces we got a new mechanism. This paper deals with the flexibility of this kind of mechanisms, and investigates the rigidity of the braced framework. The flexibility of the framework can be characterized by some vectors, which represent equivalence classes of the edges. A necessary and sufficient condition for the rigidity of the braced rhombic face zonohedra is posed. A real mechanical construction, based on two simple elements, provides a CAD prototype of these new mechanisms.

On Engel Fuzzy Subpolygroups

2017 ◽  
Vol 13 (02) ◽  
pp. 195-206 ◽  
Author(s):  
R. A. Borzooei ◽  
E. Mohammadzadeh ◽  
Violeta Fotea
Keyword(s):  
Equivalence Classes ◽  
Sufficient Condition ◽  
Fundamental Relation ◽  
Necessary And Sufficient Condition ◽  
Engel Group ◽  
Fuzzy Subgroup ◽  
Necessary And Sufficient

In this paper, by considering the notions of polygroup and Engel group, we introduce the concept of Engel fuzzy subpolygroups. In this regard, by a normal Engel fuzzy subpolygroup [Formula: see text] of [Formula: see text] and [Formula: see text], the fundamental relation on a given polygroup [Formula: see text], we construct an Engel fuzzy subgroup [Formula: see text]. We obtain a necessary and sufficient condition between Engel fuzzy subpolygroups and the Engel group [Formula: see text]/[Formula: see text], the group of equivalence classes derived from a fuzzy subpolygroup of [Formula: see text]. Finally, by using these results, we get Zorn’s lemma, in the Engel fuzzy subpolygroups.

ON HAMILTONIAN GROUP OF MULTISYMPLECTIC MANIFOLDS

2011 ◽  
Vol 08 (05) ◽  
pp. 929-935 ◽  
Author(s):  
M. SHAFIEE
Keyword(s):  
Lie Algebra ◽  
Equivalence Classes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hamiltonian Paths ◽  
Hamiltonian Group ◽  
Identity Component ◽  
Necessary And Sufficient ◽  
Hamiltonian Forms

In this paper the Hamiltonian group Ham (M, Ω) is defined for a compact k-plectic manifold (M, Ω) and it is shown that its Lie algebra is the space of equivalence classes of Hamiltonian forms, modulo closed forms. Also if ψ be a multisymplectomorphism in the identity component Msymp 0(M, Ω) of the group of multisymplectomorphisms Msymp (M, Ω), we obtain a necessary and sufficient condition under which ψ belongs to Ham (M, Ω). As two consequences, we show that Hamiltonian paths are generated by Hamiltonian forms and if Hk (M, ℝ) = 0, then Ham (M, Ω) is equal to Msymp 0(M, Ω).

scholarly journals On Super-Strong Wilf Equivalence Classes of Permutations

10.37236/6808 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Demetris Hadjiloucas ◽  
Ioannis Michos ◽  
Christina Savvidou
Keyword(s):  
Equivalence Class ◽  
Equivalence Classes ◽  
Binary Trees ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Wilf Equivalence

Super-strong Wilf equivalence is a type of Wilf equivalence on words that was originally introduced as strong Wilf equivalence by Kitaev et al. [Electron. J. Combin. 16(2)] in $2009$. We provide a necessary and sufficient condition for two permutations in $n$ letters to be super-strongly Wilf equivalent, using distances between letters within a permutation. Furthermore, we give a characterization of such equivalence classes via two-colored binary trees. This allows us to prove, in the case of super-strong Wilf equivalence, the conjecture stated in the same article by Kitaev et al. that the cardinality of each Wilf equivalence class is a power of $2$.

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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