ON THE PAIRWISE COMPATIBILITY PROPERTY OF SOME SUPERCLASSES OF THRESHOLD GRAPHS

2013 ◽  
Vol 05 (02) ◽  
pp. 1360002 ◽  
Author(s):  
TIZIANA CALAMONERI ◽  
ROSSELLA PETRESCHI ◽  
BLERINA SINAIMERI
Keyword(s):  
The Other ◽  
Real Numbers ◽  
Weighted Tree ◽  
Positive Edge ◽  
Threshold Graphs ◽  
Compatibility Graph ◽  
Unique Path ◽  
Compatibility Property

A graph G is called a pairwise compatibility graph (PCG) if there exists a positive edge weighted tree T and two non-negative real numbers d min and d max such that each leaf lu of T corresponds to a node u ∈ V and there is an edge (u, v) ∈ E if and only if d min ≤ dT (lu, lv) ≤ d max , where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T. In this paper we study the relations between the pairwise compatibility property and superclasses of threshold graphs, i.e., graphs where the neighborhoods of any couple of nodes either coincide or are included one into the other. Namely, we prove that some of these superclasses belong to the PCG class. Moreover, we tackle the problem of characterizing the class of graphs that are PCGs of a star, deducing that also these graphs are a generalization of threshold graphs.

DISCOVERING PAIRWISE COMPATIBILITY GRAPHS

2010 ◽  
Vol 02 (04) ◽  
pp. 607-623 ◽  
Author(s):  
MUHAMMAD NUR YANHAONA ◽  
MD. SHAMSUZZOHA BAYZID ◽  
MD. SAIDUR RAHMAN
Keyword(s):  
Bipartite Graphs ◽  
Real Numbers ◽  
Weighted Tree ◽  
Compatibility Graph

Let T be an edge weighted tree, let dT(u, v) be the sum of the weights of the edges on the path from u to v in T, and let d min and d max be two non-negative real numbers such that d min ≤ d max . Then a pairwise compatibility graph of T for d min and d max is a graph G = (V, E), where each vertex u' ∈ V corresponds to a leaf u of T and there is an edge (u', v') ∈ E if and only if d min ≤ dT(u, v) ≤ d max . A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers d min and d max such that G is a pairwise compatibility graph of T for d min and d max . Kearney et al. conjectured that every graph is a PCG [3]. In this paper, we refute the conjecture by showing that not all graphs are PCG s . Moreover, we recognize several classes of graphs as pairwise compatibility graphs. We identify two restricted classes of bipartite graphs as PCG. We also show that the well known tree power graphs and some of their extensions are PCGs.

scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

THE CLASSICAL HARMONIC OSCILLATOR ON GALOIS AND P-ADIC FIELDS

1989 ◽  
Vol 04 (09) ◽  
pp. 2211-2233 ◽  
Author(s):  
YANNICK MEURICE
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Difference Equation ◽  
Harmonic Oscillator ◽  
Continuum Limit ◽  
The Other ◽  
Quantum Field ◽  
Real Numbers ◽  
Technical Tool ◽  
Classical Motion

Starting from a difference equation corresponding to the harmonic oscillator, we discuss various properties of the classical motion (cycles, conserved quantity, boundedness, continuum limit) when the dynamical variables take their values on Galois or p-adic fields. We show that these properties can be applied as a technical tool to calculate the motion on the real numbers. On the other hand, we also give an example where the motions over Galois and p-adic fields have a direct physical interpretation. Some perspectives for quantum field theory and strings are briefly discussed.

Set-theoretical basis for real numbers

1950 ◽  
Vol 15 (4) ◽  
pp. 241-247
Author(s):  
Hao Wang
Keyword(s):  
Upper Bound ◽  
Theoretical Basis ◽  
The Other ◽  
Real Numbers ◽  
Full Generality ◽  
Ordered Field ◽  
Other Hand

In [1] we have considered a certain system L and shown that although its axioms are considerably weaker than those of [2], it suffices for purposes of the topics covered in [2]. The purpose of the present paper is to consider the system L more carefully and to show that with suitably chosen definitions for numbers, the ordinary theory of real numbers is also obtainable in it. For this purpose, we shall indicate that we can prove in L a certain set of twenty axioms used by Tarski which are sufficient for the arithmetic of real numbers and are to the effect that real numbers form a complete ordered field. Indeed, we cannot prove in L all Tarski's twenty axioms in their full generality. One of them, stating in effect that every bounded class of real numbers possesses a least upper bound, can only be proved as a metatheorem which states that every bounded nameable class of real numbers possesses a least upper bound. However, all the other nineteen axioms can be proved in L without any modification.This result may be of some interest because the axioms of L are considerably weaker than those commonly employed for the same purpose. In L variables need to take as values only classes each of whose members has no more than two members. In other words, only classes each with no more than two members are to be elements. On the other hand, it is usual to assume for the purpose of natural arithmetic that all finite classes are elements, and, for the purpose of real arithmetic, that all enumerable classes are elements.

Some James numbers of Stiefel manifolds

1982 ◽  
Vol 92 (1) ◽  
pp. 139-161 ◽  
Author(s):  
Hideaki Ōshima
Keyword(s):  
The Other ◽  
Stiefel Manifold ◽  
Complex Numbers ◽  
Real Numbers ◽  
The Real ◽  
Stiefel Manifolds ◽  
Other Hand ◽  
Usual Norm

The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.

Quadratic forms ‘à la’ local theory

1967 ◽  
Vol 63 (3) ◽  
pp. 579-586 ◽  
Author(s):  
A. Fröhlich
Keyword(s):  
Quadratic Forms ◽  
The Other ◽  
Local Theory ◽  
Real Numbers ◽  
The Real ◽  
Other Hand ◽  
Non Local ◽  
Characteristic 2 ◽  
The One ◽  
Adic Case

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

The Introduction of Negative Numbers

1940 ◽  
Vol 33 (6) ◽  
pp. 262-269
Author(s):  
Walter J. Bruns
Keyword(s):  
Young People ◽  
Elementary Mathematics ◽  
The Other ◽  
Real Numbers ◽  
Negative Numbers ◽  
Scientific Methods ◽  
Know How ◽  
Mathematical Situation ◽  
The Moment ◽  
Do So

It Is one of the most difficult tasks in teaching elementary mathematics in high schools to be both, understandable and correct. Most textbook authors know how to write simply and so to speak popularly, but, on the other hand, there is no doubt that many of them fall short of perfect scientific correctness. It would be ridiculous to try putting in school classes such modern scientific methods as, for example, the formal introduction of real fractions as couples of integral numbers, negative numbers as couples of positive numbers, imaginary numbers as couples of real numbers. However, we are often forced as teachers to meet a mathematical situation where we have to pay attention to two points: first, no conclusion must be gained surreptitously or by tricks, and secondly, when some of the steps within a statement are too difficult to be understood by young people the gaps must not be concealed but, on the contrary, pointed out to the students with the explanation that it is possible to accomplish the proof although we may not do so for the moment.

scholarly journals On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1670
Author(s):  
Ali Enayat ◽  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Set Theory ◽  
Equivalence Relation ◽  
Lower Class ◽  
Equivalence Classes ◽  
The Other ◽  
Real Numbers ◽  
Generic Extensions ◽  
Models Of Set Theory

Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface Π21 equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes, neither one of which is ordinal definable, and therefore the classes are OD-indiscernible. A similar but somewhat weaker result is obtained for Silver extensions. The other main result is that for any n, starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface Πn1, does not imply the existence of such a pair with the associated relation in Σn1 or in a lower class.

scholarly journals ON GENERALIZED FIBONACCI DIFFERENCE SPACE DERIVED FROM THE ABSOLUTELY p− SUMMABLE SEQUENCE SPACES

2019 ◽  
pp. 903
Author(s):  
Gülsen Kılınç
Keyword(s):  
Sequence Space ◽  
Bounded Sequence ◽  
Sequence Spaces ◽  
The Other ◽  
Matrix Transformations ◽  
Real Numbers ◽  
Positive Real ◽  
Topological Features ◽  
Summable Sequence ◽  
Alpha Beta

In this study, it is specified \emph{the sequence space} $l\left( F\left( r,s\right),p\right) $, (where $p=\left( p_{k}\right) $ is any bounded sequence of positive real numbers) and researched some algebraic and topological features of this space. Further, $\alpha -,$ $\beta -,$ $\gamma -$ duals and its Schauder Basis are given. The classes of \emph{matrix transformations} from the space $l\left( F\left( r,s\right) ,p\right) $ to the spaces $l_{\infty },c,$ and $% c_{0}$ are qualified. Additionally, acquiring qualifications of some other \emph{matrix transformations} from the space $l\left( F\left( r,s\right) ,p\right) $ to the \emph{Euler, Riesz, difference}, etc., \emph{sequence spaces} is the other result of the paper.

scholarly journals Otherness and Affectivity - in Dialogue with Being and Time

Phainomenon ◽  
2021 ◽  
Vol 31 (1) ◽  
pp. 127-151
Author(s):  
Maria Adelaide Pacheco
Keyword(s):  
Mystical Experience ◽  
The Other ◽  
Being And Time ◽  
The World ◽  
Unique Path ◽  
The Future ◽  
Working Out

Abstract In Sein und Zeit, the Dasein, thrown in the world by Geworfenheit and relaunched by Entwurf (projection) into the future, experiences itself as a “Self”. This exercise of existence cannot escape the critique of solipsism. However, paragraph 29 — about the existentiale of Befindlichkeit — opens an access way to the Other, which later will be ceaselessly explored by Heidegger, after having found the Stimmungen of the Greek beginning in Holderlin’s poetry and the Grund Stimmungen of “the night of the gods” and of the forgetfulness of being of our time. Not until the fifties and the mystical experience of the fourfold (das Geviert) will Heidegger find the possibility of transmuting the experience of the Other from a πόλεμος into a being-in-harmony (stimmen). This unique path of affectivity, asserted in paragraph 29 of Sein und Zeit, will enable Heidegger to reach an essential way of working out the question of Otherness, that only right now we may be starting to understand.

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