scholarly journals Minimal degree sequence for 2-bridge knots

2006 ◽  
Vol 190 ◽  
pp. 191-210 ◽  
Author(s):  
Prabhakar Madeti ◽  
Rama Mishra
Keyword(s):  
Degree Sequence ◽  
Minimal Degree

MINIMAL DEGREE SEQUENCE FOR TORUS KNOTS OF TYPE (p, q)

2009 ◽  
Vol 18 (04) ◽  
pp. 485-491 ◽  
Author(s):  
PRABHAKAR MADETI ◽  
RAMA MISHRA
Keyword(s):  
Positive Integer ◽  
Degree Sequence ◽  
Minimal Degree ◽  
Torus Knots ◽  
Torus Knot ◽  
Positive Integers ◽  
Coprime Positive Integers

In this paper we prove the following result: for coprime positive integers p and q with p < q, if r is the least positive integer such that 2p-1 and q + r are coprime, then the minimal degree sequence for a torus knot of type (p, q) is the triple (2p - 1, q + r, d) or (q + r, 2p - 1, d) where q + r + 1 ≤ d ≤ 2q - 1.

MINIMAL DEGREE SEQUENCE FOR TORUS KNOTS OF TYPE (p,2p-1)

2006 ◽  
Vol 15 (09) ◽  
pp. 1141-1151 ◽  
Author(s):  
PRABHAKAR MADETI ◽  
RAMA MISHRA
Keyword(s):  
Algebraic Geometry ◽  
Degree Sequence ◽  
Minimal Degree ◽  
Torus Knots

In this paper, we explore the issue of minimizing the degree sequence for torus knots. We find the minimal degree sequence for torus knots of type (p, 2p-1) for any integer p ≥2. We use some results from algebraic geometry to prove our main result.

scholarly journals Non-minimal Degree-Sequence-Forcing Triples

2014 ◽  
Vol 31 (5) ◽  
pp. 1189-1209 ◽  
Author(s):  
Michael D. Barrus ◽  
Stephen G. Hartke ◽  
Mohit Kumbhat
Keyword(s):  
Degree Sequence ◽  
Minimal Degree

The configuration model

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Sequence ◽  
Graph Model ◽  
Network Models ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Configuration Model ◽  
Bipartite Networks ◽  
Small Component ◽  
Random Graph Model ◽  
Average Size

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a number of basic properties are derived, including the probability of an edge, the expected number of multiedges, the excess degree distribution, the friendship paradox, and the clustering coefficient. This is followed by derivations of some more advanced properties including the condition for the existence of a giant component, the size of the giant component, the average size of a small component, and the expected diameter. Generating function methods for network models are also introduced and used to perform some more advanced calculations, such as the calculation of the distribution of the number of second neighbors of a node and the complete distribution of sizes of small components. The chapter ends with a brief discussion of extensions of the configuration model to directed networks, bipartite networks, networks with degree correlations, networks with high clustering, and networks with community structure, among other possibilities.

What sort of representation is conscious?

2002 ◽  
Vol 25 (3) ◽  
pp. 336-337 ◽  
Author(s):  
Zoltan Dienes ◽  
Josef Perner
Keyword(s):  
Mental Representations ◽  
Minimal Degree ◽  
Central Claim

We consider Perruchet & Vinter's (P&V's) central claim that all mental representations are conscious. P&V require some way of fixing their meaning of representation to avoid the claim becoming either obviously false or unfalsifiable. We use the framework of Dienes and Perner (1999) to provide a well-specified possible version of the claim, in which all representations of a minimal degree of explicitness are postulated to be conscious.

scholarly journals FINDING THE LIMIT OF INCOMPLETENESS I

2020 ◽  
Vol 26 (3-4) ◽  
pp. 268-286
Author(s):  
YONG CHENG
Keyword(s):  
Model Theory ◽  
Recursion Theory ◽  
Minimal Degree ◽  
Incompleteness Theorem ◽  
Turing Degree ◽  
Turing Reducibility ◽  
Inseparable Pair

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

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