One-parameter Group Research Articles

Self-similarity in fate and transport of contaminants in groundwater

Similarity within non-linear geophysical processes under different space and time has been investigated in literature to understand and model the sophisticated nature of these processes. This study deals with the self-similarity of the governing processes of fate and transport of contaminants in groundwater, which include advection, dispersion, sorption, and degradation, either by chemical reaction or microbiological interaction. As such, self-similarity conditions of three-dimensional advective-dispersive-reactive transport (ADR) equation with various initial and boundary conditions are obtained by employing one-parameter Lie group of point scaling transformations. The strength of the proposed scaling approach lies in its ability to handle initial and boundary conditions of the process together with the governing equation. Then, utilizing reported conditions in Borden Field experiment domain, numerical simulations are performed to demonstrate examples of self-similarity when first-order degradation rate constant is zero and nonzero under three sorption conditions. The results of the numerical examples showed the effectiveness of the derived self-similarity conditions and the strength of the proposed approach in handling initial and boundary conditions of the governing transport process. It is possible to obtain both smaller and larger size self-similar (scaled-up or scaled-down) domains, which result in faster and slower transport processes, respectively. The obtained self-similarity criteria can offer further understanding of ADR transport phenomena and can provide spatial, temporal, and economical flexibility in constructing scaled models of field conditions governed by three-dimensional ADR transport process.

Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey-type pseudo-differential operators

We deduce one-parameter group properties for pseudo-differential operators [Formula: see text], where [Formula: see text] belongs to the class [Formula: see text] of certain Gevrey symbols. We use this to show that there are pseudo-differential operators [Formula: see text] and [Formula: see text] which are inverses to each other, where [Formula: see text] and [Formula: see text]. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorphisms between them. For each weight functions [Formula: see text] moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) [Formula: see text] is an isomorphism from [Formula: see text] to [Formula: see text] for every [Formula: see text].

Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

Abstract We study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient M ≃ G / K {M\simeq G/K} . Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

Integral generators of Archimedean n-copulas

ABSTRACT We introduce two algebraic structures induced by an Archimedean n-copula: a one-parameter group of transformations and a one-parameter local group of transformations. We then study the natural functional characteristics (integral generators) of these structures. We discuss differential equations joining the integral generators and the additive generators of copulas. We show that the algebraic structures can be recovered by using a certain parametric family of differential equations. Finally, we give some applications of integral generators, and we indicate potential applications of the algebraic and analytical results obtained.

KMS conditions, standard real subspaces and reflection positivity on the circle group

In the present paper we continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions \psi on the additive group (R,+) satisfying a suitably defined KMS condition. These functions take values in the space Bil(V) of bilinear forms on a real vector space V. As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic continuation of \psi to the strip { z \in C\: 0 \leq Im z \leq b} with a coupling condition \psi (ib + t) = \oline{\psi (t)} on the boundary. Our first main result consists of a characterization of these functions in terms of modular objects (\Delta, J) (J an antilinear involution and \Delta > 0 selfadjoint with J\Delta J = \Delta^{-1}) and an integral representation. Our second main result is the existence of a Bil(V)-valued positive definite function f on the group R_\tau = R \rtimes {\id_\R,\tau} with \tau(t) = -t satisfying f(t,\tau) = \psi(it) for t \in R. We thus obtain a 2b-periodic unitary one-parameter group on the GNS space H_f for which the one-parameter group on the GNS space H_\psi is obtained by Osterwalder--Schrader quantization. Finally, we show that the building blocks of these representations arise from bundle-valued Sobolev spaces corresponding to the kernels 1/(\lambda^2 - (d^2)/(dt^2}) on the circle R/bZ of length b.

Structure and thermodynamics of charged nonrotating black holes in higher dimensions

We analyze the structural and thermodynamic properties of $D$-dimensional ($D \geq 4$), asymptotically flat or Anti-de-Sitter, electrically charged black hole solutions, resulting from the minimal coupling of general nonlinear electrodynamics to General Relativity. This analysis deals with static spherically symmetric (elementary) configurations with spherical horizons. Our methods are based on the study of the behaviour (in vacuum and on the boundary of their domain of definition) of the Lagrangian density functions characterizing the nonlinear electrodynamic models in flat spacetime. These functions are constrained by some admissibility conditions endorsing the physical consistency of the corresponding theories, which are classified in several families, some of them supporting elementary solutions in flat space which are non topological solitons. This classification induces a similar one for the elementary black hole solutions of the associated gravitating nonlinear electrodynamics, whose geometrical structures are thoroughly explored. A consistent thermodynamic analysis can be developed for the subclass of families whose associated black hole solutions behave asymptotically as the Schwarzschild metric (in absence of a cosmological term). In these cases we obtain the behaviour of the main thermodynamic functions, as well as important finite relations among them. In particular, we find the general equation determining the set of extreme black holes for every model, and a general Smarr formula, valid for the set of elementary black hole solutions of such models. We also consider the one-parameter group of scale transformations, which are symmetries of the field equations of any nonlinear electrodynamics in flat spacetime.

Unbounded derivations and *-automorphisms groups of Banach quasi *-algebras

This paper is devoted to the study of unbounded derivations on Banach quasi *-algebras with a particular emphasis to the case when they are infinitesimal generators of one-parameter automorphisms groups. Both of them, derivations and automorphisms are considered in a weak sense, i.e., with the use of a certain families of bounded sesquilinear forms. Conditions for a weak *-derivation to be the generator of a *-automorphisms group are given.

Linear groups of degree 3 over a field that contain a one-parametric group of elements with minimal polynomial

The article classifies groups of 3 × 3 matrices over a field k of characteristic not 2 containing a subgroup which is a conjugate of the groupwith a ranging over k. Nonidentity elements of this group are modeled after short root elements of the orthogonal group of a quadratic form with Witt index 1 and have the minimal polynomial

Canonical sphere bundles of the Grassmann manifold

For a given Hilbert space $$\mathcal H$$ , consider the space of self-adjoint projections $$\mathcal P(\mathcal H)$$ . In this paper we study the differentiable structure of a canonical sphere bundle over $$\mathcal P(\mathcal H)$$ given by $$\begin{aligned} \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \Vert f\Vert =1\, \}. \end{aligned}$$ We establish the smooth action on $$\mathcal R$$ of the group of unitary operators of $$\mathcal H$$ , and it thereby turns out that the connected components of $$\mathcal R$$ are homogeneous spaces. Then we study the metric structure of $$\mathcal R$$ by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into $$\mathcal R$$ by the natural action of the unitary group. Then we study the restricted bundle $$\mathcal R_2^+$$ given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow $$\mathcal R_2^+$$ with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi–Civita connection of this metric and establish a Hopf–Rinow theorem for $$\mathcal R_2^+$$ , again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.

An integrable coupled Alice–Bob modified Korteweg de-Vries system: Lax pairs, Bäcklund transformations, residual symmetries and exact solutions

ABSTRACTA coupled Alice–Bob modified Korteweg de-Vries (mKdV) system is established from the mKdV equation in this paper, which is nonlocal and suitable to model two-place entangled events. The Lax integrability of the coupled Alice–Bob mKdV system is proved by demonstrating three types of Lax pairs. By means of the truncated Painlevé expansion, auto-Bäcklund transformation of the coupled Alice–Bob mKdV system and Bäcklund transformation between the coupled Alice–Bob mKdV system and the Schwarzian mKdV equation are demonstrated. Nonlocal residual symmetries of the coupled Alice–Bob mKdV system are researched. To obtain localized Lie point symmetries of residual symmetries, the coupled Alice–Bob mKdV system is extended to a system consisting six equations. Calculation on the prolonged system shows that it is invariant under the scaling transformations, space-time translations, phase translations and Galilean translations. One-parameter group transformation and one-parameter subgroup invariant solutions are obtained. The consistent Riccati expansion (CRE) solvability of the coupled Alice–Bob mKdV system is proved and some interaction structures between soliton–cnoidal waves are obtained by CRE. Moreover, Jacobi periodic wave solutions, solitary wave solutions and singular solutions are obtained by elliptic function expansion and exponential function expansion.

Absence of Nontrivial Symmetries to the Heat Equation in Goursat Groups of Dimension at Least 4

Using the extension method, we study the one-parameter symmetry groups of the heat equation ∂tp = Δp, where $$\Delta=X_1^2+X_2^2$$ is the sub-Laplacian constructed by a Goursat distribution span({X1, X2}) in ℝn, where the vector fields X1 and X2 satisfy the commutation relations [X1, Xj] = Xj+1 (where Xn+1 = 0) and [Xj, Xk] = 0 for j ≥ 1 and k ≥ 1. We show that there are no such groups for n ≥ 4 (with exception of the linear transformations of solutions which are admitted by every linear equation).

On 3-Transitive Transformation Groups of the Lobachevskii Plane

In this paper, we consider three transformation groups of the Lobachevskii plane that are generated by the group of all motions and one-parameter transformation groups, which preserve an elliptic, a hyperbolic or a parabolic bundle of straight lines of this plane, respectively. It is proved that each of these groups acts 3-transitively on the Lobachevskii plane. The transformation groups and their generalizations can be applied an research of quasi-conformal mappings of the Lobachevskii space, in the special theory of relativity and in the fractal geometry.

Invariant surfaces in Euclidean space with a log-linear density

A λ-translating soliton with density vector v→ is a surface in Euclidean space whose mean curvature H satisfies 2H=2λ+〈N,v→〉, where N is the Gauss map. We classify all λ-translating solitons that are invariant by a one-parameter group of translations and a one-parameter group of rotations.

On domains of unbounded derivations of generalized B $^{*}$ -algebras

We study properties under which the domain of a closed derivation δ:D(δ)→A of a generalized B∗-algebra A remains invariant under analytic functional calculus. For a complete, generalized B∗-algebra with jointly continuous multiplication, two sufficient conditions are assumed: that the unit of A belongs to the domain of the derivation, along with a condition related to the coincidence σA(x)=σD(δ)(x) of the (Allan) spectra for every element x∈D(δ). Certain results are derived concerning the spectra for a general element of the domain, in the realm of a domain which is advertibly complete or enjoys the Q-property. For a closed ∗-derivation δ of a complete GB∗-algebra with jointly continuous multiplication such that 1∈D(δ) and x a normal element of the domain, f(x)∈D(δ) for every analytic function on a neighborhood of the spectrum of x. We also give an example of a closed derivation of a GB∗-algebra which does not contain the identity element. A condition for a closed derivation of a GB∗-algebra A to be the generator of a one-parameter group of automorphisms of A is provided along with a generalization of the Lumer–Phillips theorem for complete locally convex spaces.

Symmetries of the KMS Simplex

A continuous groupoid homomorphism $c$ on a locally compact second countable Hausdorff \'etale groupoid $\mathcal{G}$ gives rise to a $C^{*}$-dynamical system in which every $\beta$-KMS state can be associated to a $e^{-\beta c}$-quasi-invariant measure $\mu$ on $\mathcal{G}^{(0)}$. Letting $\Delta_{\mu}$ denote the set of KMS states associated to such a $\mu$, we will prove that $\Delta_{\mu}$ is a simplex for a large class of groupoids, and we will show that there is an abelian group that acts transitively and freely on the extremal points of $\Delta_{\mu}$. This group can be described using the support of $\mu$, so our theory of symmetries can be used to obtain a description of all KMS states by describing the $e^{-\beta c}$-quasi-invariant measures. To illustrate this we will describe the KMS states for the Cuntz-Krieger algebras of all finite higher rank graphs without sources and a large class of continuous one-parameter groups.

Geodesic orbit Riemannian structures on [formula omitted

A geodesic orbit manifold is a complete Riemannian manifold all of whose geodesics are orbits of one-parameter groups of isometries. We give both a geometric and an algebraic characterization of geodesic orbit manifolds that are diffeomorphic to Rn. Along the way, we establish various structural properties of more general geodesic orbit manifolds.

The one dimensional case of the singular minimal surfaces with density

Given $$\alpha ,\lambda \in \mathbb {R}$$ , a singular minimal surface with density vector $$\mathbf {v}$$ is a surface $$\Sigma $$ in Euclidean space whose mean curvature H satisfies $$2H(p)= \alpha \langle N(p),\mathbf {v}\rangle /\langle p,\mathbf {v}\rangle +2\lambda $$ , $$p\in \Sigma $$ , being N the Gauss map of $$\Sigma $$ . In this paper we classify the class of these surfaces that are invariant by a one-parameter group of translations.

Four-dimensional pseudo-Riemannian g.o. spaces and manifolds

Abstract A g.o. manifold is a homogeneous pseudo-Riemannian manifold whose geodesics are all homogeneous, that is, they are orbits of a one-parameter group of isometries. A g.o. space is a realization of a homogeneous pseudo-Riemannian manifold ( M , g ) as a coset space M = G ∕ H , such that all the geodesics are homogeneous. We prove that apart from the already classified non-reductive examples (Calvaruso et al., 2015), any four-dimensional pseudo-Riemannian g.o. manifold is naturally reductive. To obtain this result, we shall also provide a complete description up to isometries of four-dimensional pseudo-Riemannian g.o. spaces, and show explicit realizations of the four-dimensional pseudo-Riemannian naturally reductive spaces classified in Batat et al. (2015).

Two diverse models of embedding class one

Embedding theorems have continued to be a topic of interest in the general theory of relativity since these help connect the classical theory to higher-dimensional manifolds. This paper deals with spacetimes of embedding class one, i.e., spacetimes that can be embedded in a five-dimensional flat spacetime. These ideas are applied to two diverse models, a complete solution for a charged wormhole admitting a one-parameter group of conformal motions and a new model to explain the flat rotation curves in spiral galaxies without the need for dark matter.

Cylinders as left invariant CMC surfaces in [formula omitted] and E(κ,τ)-spaces diffeomorphic to [formula omitted

In the present paper we give a geometric proof for the existence of cylinders with constant mean curvature H>H(X) in certain simply connected homogeneous three-manifolds X diffeomorphic to R3, which always admit a Lie group structure. Here, H(X) denotes the critical value for which constant mean curvature spheres in X exist. Our cylinders are generated by a simple closed curve under a one-parameter group of isometries, induced by left translations along certain geodesics. In the spaces Sol3 and PSL˜2(R) we establish existence of new properly embedded constant mean curvature annuli. We include computed examples of cylinders in Sol3 generated by non-embedded simple closed curves.