Symmetry

Research

11 pages, 455 KiB

Open AccessFeature PaperArticle

Nonlinear Management of Topological Solitons in a Spin-Orbit-Coupled System

by Hidetsugu Sakaguchi and Boris Malomed

Symmetry 2019, 11(3), 388; https://doi.org/10.3390/sym11030388 - 17 Mar 2019

Cited by 8 | Viewed by 2677

We consider possibilities to control dynamics of solitons of two types, maintained by the combination of cubic attraction and spin-orbit coupling (SOC) in a two-component system, namely, semi-dipoles (SDs) and mixed modes (MMs), by making the relative strength of the cross-attraction,

γ

, [...] Read more.

We consider possibilities to control dynamics of solitons of two types, maintained by the combination of cubic attraction and spin-orbit coupling (SOC) in a two-component system, namely, semi-dipoles (SDs) and mixed modes (MMs), by making the relative strength of the cross-attraction,

γ

, a function of time periodically oscillating around the critical value,

γ = 1

, which is an SD/MM stability boundary in the static system. The structure of SDs is represented by the combination of a fundamental soliton in one component and localized dipole mode in the other, while MMs combine fundamental and dipole terms in each component. Systematic numerical analysis reveals a finite bistability region for the SDs and MMs around

γ = 1

, which does not exist in the absence of the periodic temporal modulation (“management”), as well as emergence of specific instability troughs and stability tongues for the solitons of both types, which may be explained as manifestations of resonances between the time-periodic modulation and intrinsic modes of the solitons. The system can be implemented in Bose-Einstein condensates (BECs), and emulated in nonlinear optical waveguides. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

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15 pages, 311 KiB

Open AccessArticle

The Non-Relativistic Limit of the DKP Equation in Non-Commutative Phase-Space

by Ilyas Haouam

Symmetry 2019, 11(2), 223; https://doi.org/10.3390/sym11020223 - 14 Feb 2019

Cited by 17 | Viewed by 6708

Abstract

The non-relativistic limit of the relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only [...] Read more.

The non-relativistic limit of the relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only the non-commutativity in phases with a non-interacting fields case leads to the non-commutative Schrödinger equation; thereafter, considering the non-commutativity in phase and space with an external electromagnetic field thus leads to extract a phase-space non-commutative Schrödinger–Pauli equation; there, we examined the effect of the non-commutativity in phase-space on the non-relativistic limit of the DKP equation. However, with both Bopp–Shift linear transformation through the Heisenberg-like commutation relations, and the Moyal–Weyl product, we introduced the non-commutativity in phase and space. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

14 pages, 706 KiB

Open AccessArticle

The Existence of Symmetric Positive Solutions of Fourth-Order Elastic Beam Equations

by Münevver Tuz

Symmetry 2019, 11(1), 121; https://doi.org/10.3390/sym11010121 - 20 Jan 2019

Cited by 6 | Viewed by 3337

Abstract

In this study, we consider the eigenvalue problems of fourth-order elastic beam equations. By using Avery and Peterson’s fixed point theory, we prove the existence of symmetric positive solutions for four-point boundary value problem (BVP). After this, we show that there is at [...] Read more.

In this study, we consider the eigenvalue problems of fourth-order elastic beam equations. By using Avery and Peterson’s fixed point theory, we prove the existence of symmetric positive solutions for four-point boundary value problem (BVP). After this, we show that there is at least one positive solution by applying the fixed point theorem of Guo-Krasnosel’skii. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

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12 pages, 307 KiB

Open AccessArticle

by Xiangzhi Zhang and Yufeng Zhang

Symmetry 2019, 11(1), 112; https://doi.org/10.3390/sym11010112 - 18 Jan 2019

Cited by 8 | Viewed by 2790

Abstract

In the paper, we discuss some similarity solutions of the time-fractional Burgers system (TFBS). Firstly, with the help of the Lie-point symmetry and the corresponding invariant variables, we transform the TFBS to a fractional ordinary differential system (FODS) under the case where the [...] Read more.

In the paper, we discuss some similarity solutions of the time-fractional Burgers system (TFBS). Firstly, with the help of the Lie-point symmetry and the corresponding invariant variables, we transform the TFBS to a fractional ordinary differential system (FODS) under the case where the time-fractional derivative is the Riemann–Liouville type. The FODS can be approximated by some integer-order ordinary differential equations; here, we present three such integer-order ordinary differential equations (called IODE-1, IODE-2, and IODE-3, respectively). For IODE-1, we obtain its similarity solutions and numerical solutions, which approximate the similarity solutions and the numerical solutions of the TFBS. Secondly, we apply the numerical analysis method to obtain the numerical solutions of IODE-2 and IODE-3. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

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9 pages, 244 KiB

Open AccessArticle

On the Statistical Convergence of Order α in Paranormed Space

by Sinan ERCAN

Symmetry 2018, 10(10), 483; https://doi.org/10.3390/sym10100483 - 11 Oct 2018

Cited by 4 | Viewed by 2042

Abstract

The aim of the present work is to introduce notions of statistical convergence, strongly p-Cesàro summability and the statistically Cauchy sequence of order

α

in paranormed spaces. Some certain topological properties of these new concepts are examined. Furthermore, we introduce the some [...] Read more.

The aim of the present work is to introduce notions of statistical convergence, strongly p-Cesàro summability and the statistically Cauchy sequence of order

α

in paranormed spaces. Some certain topological properties of these new concepts are examined. Furthermore, we introduce the some inclusion relations among them. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

12 pages, 287 KiB

Open AccessArticle

Ground State Representations of Some Non-Rational Conformal Nets

by Yoh Tanimoto

Symmetry 2018, 10(9), 415; https://doi.org/10.3390/sym10090415 - 19 Sep 2018

Cited by 2 | Viewed by 2545

Abstract

We construct families of ground state representations of the

U (1)

-current net and of the Virasoro nets

{Vir}_{c}

with central charge

c \geq 1

. We show that these representations are not covariant with respect to the original dilations, [...] Read more.

We construct families of ground state representations of the

U (1)

-current net and of the Virasoro nets

{Vir}_{c}

with central charge

c \geq 1

. We show that these representations are not covariant with respect to the original dilations, and those on the

U (1)

-current net are not solitonic. Furthermore, by going to the dual net with respect to the ground state representations of

{Vir}_{c}

, one obtains possibly new family of Möbius covariant nets on

S^{1}

. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

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10 pages, 257 KiB

Open AccessArticle

On Fitting Ideals of Kahler Differential Module

by Nurbige Turan and Necati Olgun

Symmetry 2018, 10(9), 413; https://doi.org/10.3390/sym10090413 - 19 Sep 2018

Viewed by 2672

Abstract

Let

k

be an algebraically closed field of characteristic zero, and

R / I

and

S / J

be algebras over

k

.

Ω_{1} (R / I)

and

Ω_{1} (S / J)

denote universal module of first [...] Read more.

Let

k

be an algebraically closed field of characteristic zero, and

R / I

and

S / J

be algebras over

k

.

Ω_{1} (R / I)

and

Ω_{1} (S / J)

denote universal module of first order derivation over

k .

The main result of this paper asserts that the first nonzero Fitting ideal

Ω_{1} (R / I \otimes_{k} S / J)

is an invertible ideal, if the first nonzero Fitting ideals

Ω_{1} (R / I)

and

Ω_{1} (S / J)

are invertible ideals. Then using this result, we conclude that the projective dimension of

Ω_{1} (R / I \otimes_{k} S / J)

is less than or equal to one. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

9 pages, 740 KiB

Open AccessArticle

On Connection between Second-Degree Exterior and Symmetric Derivations of Kähler Modules

by Hamiyet Merkepçi

Symmetry 2018, 10(9), 365; https://doi.org/10.3390/sym10090365 - 27 Aug 2018

Viewed by 2298

Abstract

Mathematical physics looks for ways to apply mathematical ideas to problems in physics. In differential forms, the tensor form is first defined, and the definitions of exterior and symmetric differential forms are made accordingly. For instance, M is an R-module, [...] Read more.

Mathematical physics looks for ways to apply mathematical ideas to problems in physics. In differential forms, the tensor form is first defined, and the definitions of exterior and symmetric differential forms are made accordingly. For instance, M is an R-module,

M \otimes_{R} M

the tensor product of M with itself and H a submodule of

M \otimes_{R} M

generated by

x \otimes y - y \otimes x

, where

x, y

in M. Then,

\lor^{2} (M) = M \otimes_{R} M / H

is called the second symmetric power of M. A role of the exterior differential forms in field theory is related to the conservation laws for physical fields, etc. In this study, I present a new approach to emphasize the properties of second exterior and symmetric derivations on Kahler modules, and I find a connection between them. I constitute exact sequences of

\lor^{2} (Ω_{1} (S))

and

Λ^{2} (Ω_{1} (S))

, and I describe and prove a new isomorphism in the following: Let S be an affine algebra presented by

R / I

, where

R = k [x_{1}, \dots x_{s}]

is a polynomial algebra and

I = (f_{1}, \dots f_{m})

an ideal of R. Then, I have

J_{1} Ω_{1} (S) ≃ Ω_{1} (S) \oplus \lor^{2} (Ω_{1} (S)) \oplus Λ^{2} (Ω_{1} (S)

. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

17 pages, 358 KiB

Open AccessArticle

A Different Study on the Spaces of Generalized Fibonacci Difference bs and cs Spaces Sequence

by Fevzi Yaşar and Kuddusi Kayaduman

Symmetry 2018, 10(7), 274; https://doi.org/10.3390/sym10070274 - 11 Jul 2018

Cited by 3 | Viewed by 3052

Abstract

The main topic in this article is to define and examine new sequence spaces

b s (\hat{F} (s, r))

and

c s (\hat{F} (s, r)))

, where [...] Read more.

The main topic in this article is to define and examine new sequence spaces

b s (\hat{F} (s, r))

and

c s (\hat{F} (s, r)))

, where

\hat{F} (s, r)

is generalized difference Fibonacci matrix in which

s, r \in R \ \{0\}

. Some algebric properties including some inclusion relations, linearly isomorphism and norms defined over them are given. In addition, it is shown that they are Banach spaces. Finally, the

α

-,

β

- and

γ

-duals of the spaces

b s (\hat{F} (s, r))

and

c s (\hat{F} (s, r))

are appointed and some matrix transformations of them are given. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

39 pages, 477 KiB

Open AccessFeature PaperArticle

Reflection Negative Kernels and Fractional Brownian Motion

by Palle E. T. Jorgensen, Karl-Hermann Neeb and Gestur Ólafsson

Symmetry 2018, 10(6), 191; https://doi.org/10.3390/sym10060191 - 1 Jun 2018

Cited by 1 | Viewed by 2764

Abstract

In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space

E

and show in particular that [...] Read more.

In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space

E

and show in particular that fractional Brownian motion for Hurst index

0 < H \leq 1 / 2

is reflection positive and leads via reflection positivity to an infinite dimensional Hilbert space if

0 < H < 1 / 2

. We also study projective invariance of fractional Brownian motion and relate this to the complementary series representations of

{GL}_{2} (R)

. We relate this to a measure preserving action on a Gaussian

L^{2}

-Hilbert space

L^{2} (E)

. Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

9 pages, 269 KiB

Open AccessArticle

On the Second-Degree Exterior Derivation of Kahler Modules on X ⊗ Y

by Ali Karakuş, Necati Olgun and Mehmet Şahin

Symmetry 2018, 10(5), 166; https://doi.org/10.3390/sym10050166 - 16 May 2018

Cited by 2 | Viewed by 2673

Abstract

This article presents a new approach to stress the properties of Kahler modules. In this paper, we construct the Kahler modules of second-degree exterior derivations and we constitute an exact sequence of

X \otimes Y

-modules. Particularly, we examine Kahler modules on [...] Read more.

This article presents a new approach to stress the properties of Kahler modules. In this paper, we construct the Kahler modules of second-degree exterior derivations and we constitute an exact sequence of

X \otimes Y

-modules. Particularly, we examine Kahler modules on

X \otimes Y

, and search for the homological size of

Λ^{2} (Ω_{1} (X \otimes Y)) .

Full article

(This article belongs to the Special Issue Mathematical Physics and Symmetry)

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