scholarly journals The degree sequences of a graph with restrictions

2021 ◽  
Vol 5 (2) ◽  
pp. 68
Author(s):  
Rikio Ichishima ◽  
Francesc A. Muntaner-Batle ◽  
Miquel Rius-Font ◽  
Yukio Takahashi
Keyword(s):  
Bipartite Graph ◽  
Sufficient Condition ◽  
Degree Sequences ◽  
Necessary And Sufficient Condition ◽  
Finite Sequences ◽  
Necessary And Sufficient

<p>Two finite sequences <em>s</em><sub>1 </sub>and <em>s</em><sub>2</sub> of nonnegative integers are called bigraphical if there exists a bipartite graph <em>G</em> with partite sets <em>V</em><sub>1</sub> and <em>V</em><sub>2</sub> such that <em>s</em><sub>1</sub> and <em>s</em><sub>2</sub> are the degrees in <em>G </em>of the vertices in <em>V</em><sub>1</sub> and <em>V</em><sub>2</sub>, respectively. In this paper, we introduce the concept of <em>1</em>-graphical sequences and present a necessary and sufficient condition for a sequence to be <em>1</em>-graphical in terms of bigraphical sequences.</p>

Interpretations of Kleene's metamathematical predicate Γ ∣ a in intuitionistic arithmetic

1965 ◽  
Vol 30 (2) ◽  
pp. 140-154 ◽  
Author(s):  
T. Thacher Robinson
Keyword(s):  
Predicate Calculus ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Recursive Functions ◽  
Finite Sequences ◽  
Constructive Proofs ◽  
Necessary And Sufficient ◽  
Intuitionistic Arithmetic

Let Pp, Pd, and N be the intuitionistic systems of prepositional calculus, predicate calculus, and elementary arithmetic, respectively, described in [3].Kleene [4] introduces a metamathematical predicate Γ ∣ A for each of the systems Pp, Pd, and N, where Γ ranges over finite sequences of wffs, and A ranges over wffs, of that system. In the case of N, if Γ is consistent, then ‘ Γ ∣ A’ is essentially the result of deleting all references to recursive functions from the metamathematical predicate ‘A is realizable-(Γ ⊦)’ described in [3], pp. 502–503.Through use of this predicate, Kleene [4] obtains elegant constructive proofs of the following results for N:Metatheorem 0.1. If B ∨ C is a closed theorem of N, then ⊦ B or ⊦ C.Metatheorem 0.2. If (∃a)D(a) is a closed theorem of N, then there is a numeral n such that ⊦D(n).Metatheorem 0.3. If A is a closed wff of N, then A ∣ A is a necessary and sufficient condition that, for all closed B, C, (∃a)D(a) inN:(0.3.1) ⊦ A ⊃ B ∨ C implies ⊦ A ⊃ B or ⊦A ⊃ Cand(0.3.2) ⊦A ⊃ (∃)D(a) implies there is a numeral n such that ⊦ A ⊃ D(n).

scholarly journals Some Properties of Regular Line Graphs

2008 ◽  
pp. 44-49
Keyword(s):  
Key Words ◽  
Bipartite Graph ◽  
Line Graph ◽  
Bipartite Graphs ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Regular Graphs ◽  
Necessary And Sufficient Condition ◽  
Regular Line ◽  
Necessary And Sufficient

In this paper, the concept of regular line graph has been introduced. The maximum number of vertices with different degrees in the regular line graphs has also been studied. Further, the necessary and sufficient condition for regular line graph to be bipartite graph have also been proved. Key words: Line Graphs, Regular graphs, Connected graphs, Bipartite Graphs.

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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