scholarly journals Alexander- and Markov-type theorems for virtual trivalent braids

2019 ◽  
Vol 28 (01) ◽  
pp. 1950003 ◽  
Author(s):  
Carmen Caprau ◽  
Abigayle Dirdak ◽  
Rita Post ◽  
Erica Sawyer
Keyword(s):  
Type Theorem ◽  
Algebraic Approach ◽  
The Other ◽  
Markov Type ◽  
Trivalent Graphs

We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on [Formula: see text]-moves.

scholarly journals Lupaş-type inequality and applications to Markov-type inequalities in weighted Sobolev spaces

2020 ◽  
pp. 1950022
Author(s):  
Francisco Marcellán ◽  
José M. Rodríguez
Keyword(s):  
Sobolev Spaces ◽  
Jacobi Polynomials ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
The Other ◽  
Main Role ◽  
Markov Type ◽  
Sobolev Orthogonal Polynomials ◽  
Approximation Problems ◽  
Connection Formulas

Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. In particular, analytic properties of such polynomials have been extensively studied, mainly focused on their asymptotic behavior and the location of their zeros. On the other hand, the behavior of the Fourier–Sobolev projector allows to deal with very interesting approximation problems. The aim of this paper is twofold. First, we improve a well-known inequality by Lupaş by using connection formulas for Jacobi polynomials with different parameters. In the next step, we deduce Markov-type inequalities in weighted Sobolev spaces associated with generalized Laguerre and generalized Hermite weights.

scholarly journals Algebraic approach for the one-dimensional Dirac–Dunkl oscillator

2020 ◽  
Vol 35 (31) ◽  
pp. 2050255
Author(s):  
D. Ojeda-Guillén ◽  
R. D. Mota ◽  
M. Salazar-Ramírez ◽  
V. D. Granados
Keyword(s):  
Energy Spectrum ◽  
Irreducible Representation ◽  
Differential Equations ◽  
Representation Theory ◽  
Algebraic Approach ◽  
The Other ◽  
One Dimensional ◽  
The One

We extend the (1 + 1)-dimensional Dirac–Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac–Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate su(1, 1) algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the su(1, 1) irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.

scholarly journals ON A MAP FROM PURE BRAIDS TO KNOTS

2003 ◽  
Vol 12 (03) ◽  
pp. 417-425 ◽  
Author(s):  
JACOB MOSTOVOY ◽  
THEODORE STANFORD
Keyword(s):  
Type Theorem ◽  
Vassiliev Invariants ◽  
Markov Type

We introduce an oriented version of the plat closure and prove a Markov-type theorem for it. Some implications for Vassiliev invariants are discussed.

scholarly journals Rational approximation and arithmetic progressions

2015 ◽  
Vol 11 (02) ◽  
pp. 451-486
Author(s):  
Faustin Adiceam
Keyword(s):  
Rational Approximation ◽  
Uniform Approximation ◽  
Type Theorem ◽  
Point Of View ◽  
Arithmetic Progressions ◽  
The Other ◽  
Complete Theory ◽  
Other Hand ◽  
The One

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed.

Desargues' theorem in n-space

1960 ◽  
Vol 1 (3) ◽  
pp. 311-318 ◽  
Author(s):  
Sahib Ram Mandan
Keyword(s):  
Projective Space ◽  
Algebraic Approach ◽  
The Other ◽  
Desargues Theorem

Two sets of r + 2 points, Pi, P'i, each spanning a projective space of r + 1 dimensions, [r + 1], which has no solid ([3]) common with that spanned by the other, are said to be projective from an [r — 1], if here is an [r — 1] which meets the r + 2 joins Pi ′i. It is to be proved that the two sets are projective, if and only if the r + 2 intersections Ai of their corresponding [r]s lie in a line a. Ai are said to be the arguesian points and a the arguesian line of the sets. When r= 1, the proposition becomes the well- known Desargues' two-triangle theorem (3) in a plane. Therefore in analogy with the same we name it as the Desargues' theorem in [2r]. Following Baker (1, pp. 8—39), we may prove this theorem in the same synthetic style by making use of the axioms and the corresponding proposition of incidence in [2r + 1] or with the aid of the Desargues' theorem in a plane and the axioms of [2r] only. But the use of symbols makes its proof more concise; the algebraic approach adopted here is due to the referee (Arts. 2, 3, 5, 6, 7). Pairs of sets of r + p points each projective from an [r— 1] are also introduced to serve as a basis for a much more thorough investigation.

scholarly journals Algebraic Approach to Casimir Force Between Two $$\delta $$-like Potentials

2021 ◽  
Author(s):  
Kamil Ziemian
Keyword(s):  
Asymptotic Expansion ◽  
Casimir Force ◽  
Casimir Effect ◽  
Scaling Limit ◽  
Algebraic Approach ◽  
Casimir Energy ◽  
Universal Function ◽  
The Other ◽  
Three Space ◽  
Physical Quantities

AbstractWe analyse the Casimir effect of two nonsingular centers of interaction in three space dimensions, using the framework developed by Herdegen. Our model is mathematically well-defined and all physical quantities are finite. We also consider a scaling limit, in which the problem tends to that with two Dirac $$\delta $$ δ ’s. In this limit the global Casimir energy diverges, but we obtain its asymptotic expansion, which turns out to be model dependent. On the other hand, outside singular supports of $$\delta $$ δ ’s the limit of energy density is a finite universal function (independent of the details of the nonsingular model before scaling). These facts confirm the conclusions obtained earlier for other systems within the approach adopted here: the form of the global Casimir force is usually dominated by the modification of the quantum state in the vicinity of macroscopic bodies.

Implementing The Standards: Multiple Approaches to Geometry: Teaching Similarity

1990 ◽  
Vol 83 (4) ◽  
pp. 274-280
Author(s):  
Sharon L. Senk ◽  
Daniel B. Hirschhorn
Keyword(s):  
Mathematics Education ◽  
Secondary School ◽  
Algebraic Approach ◽  
Physical World ◽  
School Mathematics ◽  
The Other ◽  
Synthetic Approach ◽  
Evaluation Standards

Geometry as a subject uniquely furnishes a language for describing our physical world. It also gives a way visually to represent concepts and relations in other branches of mathematics. Although debate might always ensue on whether geometry should be a full-year secondary school course, the importance of geometry throughout a student's mathematics education seems to have broad acceptance. Consequently, it is not surprising that in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) we find an explicit standard on geometry for all levels K–4, 5–8, and 9–12. In fact, for grades 9–12, two standards on geometry are included—one focusing on a synthetic approach, the other on an algebraic approach.

A CAYLEY THEOREM FOR THE MULTIPLICATIVE SEMIGROUP OF A FIELD

2012 ◽  
Vol 11 (02) ◽  
pp. 1250042 ◽  
Author(s):  
E. NAZARI ◽  
Yu. M. MOVSISYAN
Keyword(s):  
Type Theorem ◽  
The Other ◽  
Binary Representation ◽  
Multiplicative Semigroup ◽  
First Order ◽  
Order Language ◽  
Multiplicative Semigroups

Since there exist two commutative elementarily equivalent semigroups of which one is the multiplicative semigroup of a field and the other is not a multiplicative semigroup of any field, it is impossible to characterize multiplicative semigroups of fields by formulas of the first order language (logic). In this work we characterize the multiplicative semigroup of a field by its binary representation (Cayley type theorem).

scholarly journals On the Group Structure in Homotopy Groups

1954 ◽  
Vol 7 ◽  
pp. 133-144
Author(s):  
Masatake Kuranishi
Keyword(s):  
Homotopy Group ◽  
Group Structure ◽  
Algebraic Approach ◽  
Point Of View ◽  
The Other ◽  
Homotopy Groups ◽  
Other Hand ◽  
Special Case

Usually the group structure in a homotopy group is defined directly and explicitly. But the algebraic approach to the topology, now common, seems to raise the following question : is that the only group sturcture which is natural from the algebraic topological point of view? On the other hand, several algebraists have begun to feel a necessity to construct a “homotopy or cohomotopy theory of groups,” and it may be allowed to say that one of the first steps to the problem is the axiomatization of homotopy groups. Our first question is of course a special case of the latter problem.

scholarly journals Voltage Graphs, Group Presentations and Cages

10.37236/1843 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
The Other ◽  
Trivalent Graphs ◽  
Group Presentations ◽  
Coset Enumeration

We construct smallest known trivalent graphs for girths 16 and 18. One construction uses voltage graphs, and the other coset enumeration techniques for group presentations.

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