scholarly journals Period and toroidal knot mosaics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750031 ◽  
Author(s):  
Seungsang Oh ◽  
Kyungpyo Hong ◽  
Ho Lee ◽  
Hwa Jeong Lee ◽  
Mi Jeong Yeon
Keyword(s):  
Quantum System ◽  
Prime Number ◽  
Growth Rates ◽  
Exact Enumeration ◽  
Common Features ◽  
System A ◽  
Number Formula ◽  
And Mathematics ◽  
Definition Of ◽  
Positive Integers

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot [Formula: see text]-mosaic is an [Formula: see text] matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot [Formula: see text]-mosaics for any positive integers [Formula: see text] and [Formula: see text], toroidal knot [Formula: see text]-mosaics for co-prime integers [Formula: see text] and [Formula: see text], and furthermore toroidal knot [Formula: see text]-mosaics for a prime number [Formula: see text]. We also analyze the asymptotics of the growth rates of their cardinality.

scholarly journals Enumeration on graph mosaics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750032 ◽  
Author(s):  
Kyungpyo Hong ◽  
Seungsang Oh
Keyword(s):  
Quantum System ◽  
Research Program ◽  
Knot Theory ◽  
Quantum Physics ◽  
Recursion Formula ◽  
Jones Polynomial ◽  
Exact Enumeration ◽  
State Matrix ◽  
Embedded Graph ◽  
Definition Of

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using 16 graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. We extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.

Mosaic number of knots

2014 ◽  
Vol 23 (13) ◽  
pp. 1450069 ◽  
Author(s):  
Hwa Jeong Lee ◽  
Kyungpyo Hong ◽  
Ho Lee ◽  
Seungsang Oh
Keyword(s):  
Quantum System ◽  
Upper Bound ◽  
Crossing Number ◽  
System A ◽  
Definition Of ◽  
Hopf Link

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except [Formula: see text] link, then m(K) ≤ c(K) - 1.

scholarly journals Almost universal mixed sums of squares and polygonal numbers

2018 ◽  
Vol 14 (08) ◽  
pp. 2239-2256
Author(s):  
Hai-Liang Wu
Keyword(s):  
Prime Number ◽  
Quadratic Forms ◽  
Sums Of Squares ◽  
Ternary Quadratic Forms ◽  
Number Formula ◽  
Sum Formula ◽  
Polygonal Numbers ◽  
Positive Integers

For each integer [Formula: see text], let [Formula: see text] denote the generalized [Formula: see text]-gonal number [Formula: see text] with [Formula: see text]. Given positive integers [Formula: see text] and an odd prime number [Formula: see text] with [Formula: see text], we employ the theory of ternary quadratic forms to determine completely when the mixed sum [Formula: see text] represents all but finitely many positive integers.

scholarly journals Upper bound on the total number of knot n-mosaics

2014 ◽  
Vol 23 (13) ◽  
pp. 1450065 ◽  
Author(s):  
Kyungpyo Hong ◽  
Seungsang Oh ◽  
Ho Lee ◽  
Hwa Jeong Lee
Keyword(s):  
Quantum System ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Exact Number ◽  
System A ◽  
Definition Of

Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. Dn denotes the total number of all knot n-mosaics. Already known is that D1 = 1, D2 = 2 and D3 = 22. In this paper we establish the lower and upper bounds on Dn[Formula: see text] and find the exact number of D4 = 2594.

scholarly journals On Some Growth Properties of Entire Functions Using Their Maximum Moduli Focusingp,qth Relative Order

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Luis Manuel Sanchez Ruiz ◽  
Sanjib Kumar Datta ◽  
Tanmay Biswas ◽  
Golok Kumar Mondal
Keyword(s):  
Entire Function ◽  
Growth Rates ◽  
Lower Order ◽  
Entire Functions ◽  
Relative Order ◽  
Growth Properties ◽  
Definition Of ◽  
Positive Integers

We discuss some growth rates of composite entire functions on the basis of the definition of relativep,qth order (relativep,qth lower order) with respect to another entire function which improve some earlier results of Roy (2010) wherepandqare any two positive integers.

scholarly journals Zur exakten Behandlung von kollektiven Rotationen Teil A: Theorie / On the Exact Treatment of Collective Rotations Part A: Theory

1973 ◽  
Vol 28 (2) ◽  
pp. 206-215
Author(s):  
Hanns Ruder
Keyword(s):  
Coordinate System ◽  
Perturbation Expansion ◽  
Rotational Energy ◽  
The Body ◽  
System A ◽  
N Body Problem ◽  
Definition Of ◽  
Fixed System ◽  
Body Fixed Coordinate System ◽  
Groundstate Energy

Basic in the treatment of collective rotations is the definition of a body-fixed coordinate system. A kinematical method is derived to obtain the Hamiltonian of a n-body problem for a given definition of the body-fixed system. From this exact Hamiltonian, a consequent perturbation expansion in terms of the total angular momentum leads to two exact expressions: one for the collective rotational energy which has to be added to the groundstate energy in this order of perturbation and a second one for the effective inertia tensor in the groundstate. The discussion of these results leads to two criteria how to define the best body-fixed coordinate system, namely a differential equation and a variational principle. The equivalence of both is shown.

scholarly journals The restricted Burnside problem for Moufang loops

2021 ◽  
pp. 1-11
Author(s):  
ALEXANDER GRISHKOV ◽  
LIUDMILA SABININA ◽  
EFIM ZELMANOV
Keyword(s):  
Prime Number ◽  
Burnside Problem ◽  
Moufang Loops ◽  
Restricted Burnside Problem ◽  
Positive Integers

Abstract We prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite m-generated Moufang loops of exponent $p^n$ .

XV.—The Placoderm Fish Rhachiosteus pterygiatus Gross and its Relationships

1966 ◽  
Vol 66 (15) ◽  
pp. 377-392 ◽  
Author(s):  
Roger S. Miles
Keyword(s):  
Sensory System ◽  
System A ◽  
The Family ◽  
Definition Of

SynopsisThe holotype and only known specimen of Rhachiosteus pterygiatus Gross is partially redescribed and new restorations are given. Attention is drawn to important points in its osteology and the possible development of a cutaneous sensory system. A definition of the family Rhachiosteidsæ Stensiö is given. This family differs from all other described groups of euarthrodires in the lack of posterior lateral and posterior dorsolateral flank plates. Rhachiosteus is a pachyosteomorph brachythoracid, as defined in the text, and may be fairly closely related in some way to the (coccosteomorph) family Coccosteidsæ. There is no indication that it is closely related to any other known pachyosteomorph, or to other groups of arthrodires, such as the Rhenanida and Ptyctodontida, in which there are no posterior flank plates.

INDEX-DEPENDENT DIVISORS OF COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (07) ◽  
pp. 1841-1853 ◽  
Author(s):  
B. K. MORIYA ◽  
C. J. SMYTH
Keyword(s):  
Prime Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Survey Methods ◽  
Integer Sequences ◽  
Positive Integers

We evaluate [Formula: see text] for a certain family of integer sequences, which include the Fourier coefficients of some modular forms. In particular, we compute [Formula: see text] for all positive integers n for Ramanujan's τ-function. As a consequence, we obtain many congruences — for instance that τ(1000m) is always divisible by 64000. We also determine, for a given prime number p, the set of n for which τ(pn-1) is divisible by n. Further, we give a description of the set {n ∈ ℕ : n divides τ(n)}. We also survey methods for computing τ(n). Finally, we find the least n for which τ(n) is prime, complementing a result of D. H. Lehmer, who found the least n for which |τ(n)| is prime.

Approach Integrated of Science, Technology, Engineering, and Mathematics – STEM

2021 ◽  
Vol 4 (4) ◽  
pp. 99-136
Author(s):  
Ibrahiem Mohammed Abdullah ◽  
Keyword(s):  
Mathematics Education ◽  
Significant Role ◽  
Arab World ◽  
Research Paper ◽  
Educational Activities ◽  
Science Technology ◽  
Mathematics Curricula ◽  
Integrated Approaches ◽  
And Mathematics ◽  
Definition Of

The research paper aims to highlight the STEM approach as one of the modern integrated approaches in the field of mathematics education. STEM which means the integration of Science, Technology, Engineering, and Math has its significant role in the development of curricula in the Arab world generally and particularly in mathematics curricula. This paper addresses the definition of STEM, the justifications for its emergence and the causes for the attention it recently receives. Moreover, the paper sheds light on its objectives, content, related teaching strategies, educational activities, evaluation, characteristics, advantages and obstacles found in its application.

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