Heisenberg Space Research Articles

Nonlinear optical dynamics in Heisenberg space: Directional curves and recursive inquiry

This paper delves into the exploration of directional recursion operators within the realm of regular space curves modeled by Heisenberg systems. The central objective is to introduce a myriad of recursive flows, encompassing ferromagnetic and antiferromagnetic solutions, alongside a family of general normalization operators in the normal and binormal directions. The study employs the extended compatible and inextensible flow model of curves to examine the evolution models, providing a comprehensive understanding of their dynamics. A significant aspect of the investigation involves elucidating the evolution model in terms of anholonomy shapes and their density. The directional recursive operator, a focus of this study, demonstrates distinct results compared to traditional approaches. The reliability and applicability of the obtained results extend to the examination of various linear and nonlinear continuous dynamical systems.

Delaunay Surfaces of Prescribed Mean Curvature in $$\mathrm {Nil}_3$$ and $$\widetilde{SL_2}(\mathbb {R})$$

We obtain a classification result for rotational surfaces in the Heisenberg space and the universal cover of the special linear group, whose mean curvature is given as a prescribed \(C^1\) function depending on their angle function. We show that these surfaces behave like the Delaunay surfaces of constant mean curvature, under some assumptions on the prescribed function. In contrast with the constant mean curvature case, we exhibit the existence of rotational, embedded tori, providing counterexamples of the Alexandrov problem for this class of immersed surfaces.

New quasi uniformly accelerated motion with hidden quasi momentum

We firstly characterize uniformly quasi accelerated motion (UQAM) with Fermi Walker derivation. We obtain new energy of quasi tangent magnetic biharmonic particle and Lorentz field. Thus, we give new relationship between Fermi Walker parallelism and uniformly quasi accelerated motion in Heisenberg group. Moreover, we give applied geometric characterization for a uniformly quasi accelerated motion of velocity magnetic biharmonic particles in Heisenberg space.

Quasi uniformly accelered motion of quasi normal magnetic biharmonic particles in Heisenberg space with cold plasma

In this paper, we firstly review the quasi uniformly accelerated motion (QUAM) with Fermi derivative by cold plasma in Heisenberg space. We define new quasi uniformly accelerated potential electric energy of quasi normal magnetic biharmonic particles and some Lorentz fields. Moreover, we design the new relationship between the quasi uniformly accelerated motion and the Fermi parallel transportation in cold plasma. Also, we collect new physical geometric designs for a quasi uniformly accelerated motion of normal magnetic biharmonic particles in Heisenberg space. Finally, we construct quasi uniformly accelerated potential electric energy with respect to its electric field and some quasi curvatures.

Optical spherical electroosmotic phase and optical energy for spherical [formula omitted]-magnetic fibers

In this paper, we introduce a different concept of optical spherical α-magnetic fibers with some optical conditions in spherical Heisenberg space. Besides, we give some relationships between optical spherical optimistic density and spherical optical energy for Lorentz forces. By using these optical spherical optimistic density, we give some methods to obtain optical spherical electroosmotic ϕ(α),ϕ(t),ϕ(s)-phase. Finally, we give some related optical electroosmotic microfluidics in spherical Heisenberg space.

Riemannian approximation scheme in sub-Riemannian Heisenberg space H1 and rotation surfaces of constant Gaussian curvature

Riemannian approximation scheme in sub-Riemannian Heisenberg space H1 and rotation surfaces of constant Gaussian curvature

Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model

We construct and analyze the phase diagram of a self-interacting matrix field in two dimensions coupled to the curvature of the non-commutative truncated Heisenberg space. In the infinite size limit, the model reduces to the renormalizable Grosse-Wulkenhaar's. The curvature term proves crucial for the diagram's structure: when turned off, the triple point collapses into the origin as matrices grow larger; when turned on, the triple point recedes from the origin proportionally to the coupling strength and the matrix size. The coupling attenuation that turns the Grosse-Wulkenhaar model into a renormalizable version of the $\phi^4_\star$-model cannot stop the triple point recession. As a result, the stripe phase escapes to infinity, removing the problems with UV/IR mixing.

New approach for optical electroostimistic phase with optical quasi potential energy

In this paper, we firstly review the optical circular motion with optical optimistic density in Heisenberg space. We define new quasi optical potential electromagnetic energy of quasi normal magnetic biharmonic fields. Moreover, we design the new characterization of uniformly optical electroosmotic phase and optical parallel transportation in cold plasma. Also, we collect new physical geometric designs for a optical optimistic density of normal magnetic biharmonic fields in Heisenberg space. Finally, we construct optical potential electromagnetic energy with respect to its electric field and sonic crystals.

Optical energy of spherical velocity with optical magnetic density in Heisenberg sphere space [formula omitted

In this article, we define a new concept of spherical optical magnetic density with biharmonic particles in spherical Heisenberg space. Moreover, we construct the optical energy of spherical tangent magnetic biharmonic particles and new Lorentz fields. Furthermore, we show that optical energy and optical magnetic flux density can be stated by Frenet fields in Heisenberg space.

Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

PurposeIn the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r(x, y) = γ1(x)*γ2(y), where γ1 and γ2 are two planar curves lying in planes, which are not orthogonal. In this article, we classify translation surfaces in H3, which satisfy some algebraic equations in terms of the coordinate functions and the Laplacian operator with respect to the first fundamental form of the surface.Design/methodology/approachIn this paper, we classify some type of space-like translation surfaces of H3 endowed with flat metric g3 under the conditionΔri = λiri. We will develop the system which describes surfaces of type finite in H3. For solve the system thus obtained, we will use the calculation variational. Finally, we will try to give performances geometric surfaces that meet the condition imposed.FindingsClassification of six types of translation surfaces of finite type in the three-dimensional Lorentz Heisenberg group H3.Originality/valueThe subject of this paper lies at the border of geometry differential and spectral analysis on manifolds. Historically, the first research on the study of sub-finite type varieties began around the 1970 by B.Y.Chen. The idea was to find a better estimate of the mean total curvature of a compact subvariety of a Euclidean space. In fact, the notion of finite type subvariety is a natural extension of the notion of a minimal subvariety or surface, a notion directly linked to the calculation of variations. The goal of this work is the classification of surfaces in H3, in other words the surfaces which satisfy the condition/Delta (ri) = /Lambda (ri), such that the Laplacian is associated with the first, fundamental form.

The Equivalence of Operator Norm between the Hardy-Littlewood Maximal Function and Truncated Maximal Function on the Heisenberg Group

In this article, we define a kind of truncated maximal function on the Heisenberg space by M γ c f x = sup 0 < r < γ 1 / m B x , r ∫ B x , r f y d y . The equivalence of operator norm between the Hardy-Littlewood maximal function and the truncated maximal function on the Heisenberg group is obtained. More specifically, when 1 < p < ∞ , the L p norm and central Morrey norm of truncated maximal function are equal to those of the Hardy-Littlewood maximal function. When p = 1 , we get the equivalence of weak norm L 1 ⟶ L 1 , ∞ and M ̇ 1 , λ ⟶ W ̇ M 1 , λ . Those results are generalization of previous work on Euclid spaces.

Approximate solutions for optical magnetic and electric phase with fractional optical Heisenberg ferromagnetic spin by RPSM

In this paper, we obtain a new representation of optical spherical electric and magnetic flow density of magnetic spherical Heisenberg SHt particle flows by spherical frame on spherical Heisenberg space SH32 with the residual power series method (RPSM). Then, we have approximately new optical conditions of magnetic spherical Heisenberg SHt electric and magnetic phase with optical spherical electric and magnetic flow density. Moreover, we obtain new fractional equation by spherical magnetic Lorentz flux with the residual power series method. Finally, magnetic spherical Heisenberg SHt electric and magnetic phase of spherical surface is demonstrated in uniform optical magnetic surface applying analytical and fractional numerical consequences in spherical Heisenberg space SH32. Characteristics of presented fractional equations are shown by using some precise values of optical arguments on obtained solutions.

New approach to optical spherical [formula omitted]t- flux by Heisenberg optical Landau Lifshitz condition

In this article, we study spherical optical Lorentz flux of spherical St − optical flows with a spherical frame on the spherical optical Heisenberg space SH2. We get optical new conditions for spherical St − optical Lorentz Heisenberg flux with spherical fields. Thus, we define spherical Lorentz Heisenberg flux for optical magnetic fields. We provide new construction for optical spherical curvatures of spherical St − flows with spherical Landau Lifshitz condition. Additionally, magnetic optical flux surface is constructed in certain uniform optical surfaces with numerical and analytical results by spherical Landau Lifshitz condition.

Optical spherical electromotive density with some fractional applications with Laplace transform in spherical Heisenberg space [formula omitted

In this paper, we construct a new theory of spherical electromotive force of spherical Ss −magnetic flows with the spherical frame in spherical Heisenberg space SH2. Also, we obtain some optical conditions of spherical Ss −magnetic Lorentz flux by using optical spherical fields in spherical Heisenberg space SH2. Moreover, we give some applications of spherical Heisenberg electromotive forces fEϕαH,fEϕtH,fEϕsH for spherical vector fields. Additionally, we compute representation of spherical curvatures of spherical Ss −magnetic flows by considering Heisenberg spherical ferromagnetic spin in spherical Heisenberg space SH2. Finally, spherical Heisenberg electromotive forces fEϕαH,fEϕtH,fEϕsH are given in a static and uniform magnetic surface with the q -Homotopy analysis transform method in spherical Heisenberg space SH2.

New version of optical spherical electric and magnetic flow phase with some fractional solutions in [formula omitted

In this paper, we introduce a new characterization of spherical electric and magnetic flow density of magnetic spherical Heisenberg SHα particle by spherical frame in spherical Heisenberg space SH32. Thus, we get some new optical qualifications of magnetic spherical Heisenberg SHα electric and magnetic phase by using spherical magnetic fields. Also, we demonstrate spherical electric and magnetic flow density for spherical vector fields. Then, their fractional solutions are examined by using magnetic and optical geometric quantities with the conformable fractional derivative modeling. Finally, magnetic spherical HeisenbergSHα electric and magnetic phase of spherical surface are demonstrated in uniform spherical magnetic surface by using analytical and fractional numerical developments.

Optical fractional spherical magnetic flux flows with Heisenberg spherical Landau Lifshitz model

In this paper, we first offer the approach of spherical magnetic Lorentz flux of spherical Sα-magnetic particle flows by the spherical frame in spherical Heisenberg space Sℍ2. Eventually, we obtain some optical conditions of spherical Sα-magnetic Lorentz flux by using directional spherical fields. Moreover, we determine spherical magnetic Lorentz flux for spherical vector fields. Also, we give new construction for spherical curvatures of spherical Sα-magnetic particle flows by considering Heisenberg spherical Landau Lifshitz model. Finally, magnetic flux surface is demonstrated in a static and uniform magnetic surface by using the analytical and numerical results with Heisenberg spherical Landau Lifshitz model.

Minimal translation surfaces in Lorentz Heisenberg space (ℋ3, g 2)

The aim of this work is the classification of some types of minimal translation surfaces in the 3–dimensional Lorentzian Heisenberg space ℋ3 endowed with the left invariant metric

Detecting scaling in phase transitions on the truncated Heisenberg algebra

We construct and analyze a phase diagram of a self-interacting matrix field coupled to curvature of the non-commutative truncated Heisenberg space. The model reduces to the renormalizable Grosse-Wulkenhaar model in an infinite matrix size limit and exhibits a purely non-commutative non-uniformly ordered phase. Particular attention is given to scaling of model’s parameters. We additionally provide the infinite matrix size limit for the disordered to ordered phase transition line.

New approach to uniformly quasi circular motion of quasi velocity biharmonic magnetic particles in the Heisenberg space

In this paper, we define concept of the uniformly quasi circular motion (UQCM) with biharmonicity condition in the Heisenberg space. That is, we aim to define a new class of UQCM in the three‐dimensional Heisenberg space. We further improve an alternative method to find uniformly quasi circular potential electric energy of biharmonic velocity magnetic particles in the Heisenberg space. We also give the relationships between physical and geometrical characterizations of uniformly quasi circular potential electric energy. Finally, we illustrate important figures for uniformly quasi circular potential electric energy with respect to its electric field in the radial direction.

New Uniform Motion and Fermi–Walker Derivative of Normal Magnetic Biharmonic Particles in Heisenberg Space

In the present paper, we firstly discuss the normal biharmonic magnetic particles in the Heisenberg space. We express new uniform motions and its properties in the Heisenberg space. Moreover, we obtain a new uniform motion of Fermi–Walker derivative of normal magnetic biharmonic particles in the Heisenberg space. Finally, we investigate uniformly accelerated motion (UAM), the unchanged direction motion (UDM), and the uniformly circular motion (UCM) of the moving normal magnetic biharmonic particles in Heisenberg space.