On a Property of Continuous Operators in $$L^p (\mathbb{R}) $$ -Space

Mathematical Notes -     

Proper Holomorphic Legendrian Curves in $$SL_2(\mathbb {C})$$

In this paper, we prove that every open Riemann surface properly embeds in the special linear group \(SL_2(\mathbb {C})\) as a holomorphic Legendrian curve, where \(SL_2(\mathbb {C})\) is endowed with its standard contact structure. As a consequence, we derive the existence of proper, weakly complete, flat fronts with arbitrary complex structure in the real hyperbolic space \(\mathbb {H}^3\).     

Some Examples of Extremal Functions on the Dunkl-Type Fock Space $$F_k(\mathbb {C})$$

In this work, we recall some properties of the generalized Fock space \(F_k(\mathbb {C})\) and the Hilbert space \(H_k(\mathbb {C})\) (defined by the squared norm \(\Vert f\Vert ^2_{H_k(\mathbb {C})}:=\Vert f\Vert ^2_{F_k(\mathbb {C})}+\langle zf',f\rangle _{F_k(\mathbb {C})}\)). Next, we give the best approximation of the bounded operators \(L:F_k(\mathbb {C})\rightarrow H_k(\mathbb {C})\). As applications, we come up with some results regarding the approximate formulas for the difference operator, the Dunkl difference operator and the primitive operator.     

On differentiable compactifications of the hyperbolic plane and algebraic actions of $${\rm SL}_2(\mathbb{R})$$ on surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2;R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. The online version of the original article can be found under doi: .     

Renewal reward process for T-related fuzzy random variables on (\mathbb {R}^{p}, \mathbb {R}^{q})

In this paper, following our previous studies, we investigate the renewal rewards process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random rewards are characterized as weighted fuzzy numbers under \(t\) -norm-based fuzzy operations on \(\mathbb {R}^{p}\) and \(\mathbb {R}^{q}\,\,p,\,q \ge 1,\) respectively. Many versions of \(T\) -related fuzzy renewal rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on \(\mathbb {R}^{p}\) . An application example is provided to illustrate the utility of the results.     

Two-Wavelet Localization Operators on $$ L^p(\mathbb{R}^n) $$ for the Weyl-Heisenberg Group

We give results on the boundedness and compactness of localization operators with two admissible wavelets on for the Weyl-Heisenberg group.     

On correspondences between certain automorphic forms on Sp_{4n} (\mathbb{A}) and \widetilde{Sp}_{2n} (\mathbb{A})

In 2005, Ginzburg, Rallis and Soudry constructed, in terms of residues of certain Eisenstein series, and by use of the descent method, families of nontempered automorphic representations of $Sp_{4nm} (\mathbb{A})$ and $\widetilde{Sp}_{2n(2m - 1)} (\mathbb{A})$ , which generalized the classical work of Piatetski-Shapiro on Saito-Kurokawa liftings. In this paper, we introduce a new framework (Diagrams of Constructions) in order to establish explicit relations among the representations introduced in [GRS05]. In particular, we prove that these constructions yield bijections between a certain set of cuspidal automorphic forms on $\widetilde{Sp}_{2n} (\mathbb{A})$ and a certain set of square-integrable automorphic forms of $Sp_{4n} (\mathbb{A})$ . The proofs use new interpretations of composition of two consecutive descents with explicit identities, which we expect to be very useful to further investigation of the automorphic discrete spectrum of classical groups.     

On the Exponential Dichotomy on $$\mathbb{R}$$ of Linear Differential Equations in $$\mathbb{R}^n$$

We present new results on the exponential dichotomy on the entire axis of linear differential equations in .     

Scales of minimal complexity in {K(\mathbb{R})}

Using a Levy hierarchy and a fine structure theory for ${K(\mathbb{R})}$ , we obtain scales of minimal complexity in this inner model. Each such scale is obtained assuming the determinacy of only those sets of reals whose complexity is strictly below that of the scale constructed.     

Boundary functions in L2 H $$(\mathbb{B}^n )$$

We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball For a function u which is lower semi-continuous on we give necessary and sufficient conditions in order that there exists a holomorphic function f ∈ such that

Actions of SL $$(n,\mathbb{Z})$$ on Homology Spheres

Any continuous action of SL , where n > 2, on a r-dimensional mod 2 homology sphere factors through a finite group action if r < n −1. In particular, any continuous action of SL on the n-dimensional sphere factors through a finite group action.Mathematics Subject Classiffications (2000). Primary 57S25; Secondary 37C85, 57S17     

A covering lemma for {K(\mathbb {R})}

The Dodd–Jensen Covering Lemma states that “if there is no inner model with a measurable cardinal, then for any uncountable set of ordinals X, there is a ${Y\in K}$ such that ${X\subseteq Y}$ and |X| = |Y|”. Assuming ZF+AD alone, we establish the following analog: If there is no inner model with an ${\mathbb {R}}$ –complete measurable cardinal, then the real core model ${K(\mathbb {R})}$ is a “very good approximation” to the universe of sets V; that is, ${K(\mathbb {R})}$ and V have exactly the same sets of reals and for any set of ordinals X with ${|{X}|\ge\Theta}$ , there is a ${Y\in K(\mathbb {R})}$ such that ${X\subseteq Y}$ and |X| = |Y|. Here ${\mathbb {R}}$ is the set of reals and ${\Theta}$ is the supremum of the ordinals which are the surjective image of ${\mathbb {R}}$ .     

Dark $$f(\mathcal{R},\varphi,\chi)$$ universe with Noether symmetry

Theoretical and Mathematical Physics - Using the Noether symmetry approach, we investigate $$f( \mathcal{R} , \varphi ,\chi)$$ theories of gravity, where $$ \mathcal{R} $$ is the scalar curvature,...     

Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})

We prove real Paley-Wiener type theorems for the Dunkl transform ℱ _D on the space of tempered distributions. Let T∈S′(ℝ ^d ) and Δ _κ the Dunkl Laplacian operator. First, we establish that the support of ℱ _D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  _n→+∞ R ⁻²ⁿ Δ _κ ⁿ T=0 in S′(ℝ ^d ). Second, we prove that the support of ℱ _D (T) is included in ℝ ^d ∖B(0,M), M>0, if and only if for all R<M, we have lim  _n→+∞ R ²ⁿ  ℱ _D ⁻¹(‖y‖⁻²ⁿ ℱ _D (T))=0 in S′(ℝ ^d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.      

On critical nets in $${mathbb {R}}^k$$ R k

Geometriae Dedicata - Critical nets in $${mathbb {R}}^k$$ (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length...     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

Trajectory of an Observer Tracking Object Motion around Convex Obstacles in $${{\mathbb{R}}^{2}}$$ and $${{\mathbb{R}}^{3}}$$

Doklady Mathematics - An autonomous object moves at a constant speed along the shortest path, while bypassing an ordered collection of pairwise disjoint convex sets. The object is tracked by an...     

Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $

We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u₀ (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H² (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.