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The Dirac-[Formula: see text] distribution may be realized through sequences of convolutions, the latter being also regarded as approximation to the identity. This paper proposes the real form of pre-orthogonal adaptive Fourier decomposition (POAFD) method to realize fast approximation to the identity. It belongs to sparse representation of signals having potential applications in signal and image analysis.

Let $(Z,\omega)$ be a connected Kahler manifold with an holomorphic action of the complex reductive Lie group $U^{\mathbb C}$, where $U$ is a compact connected Lie group acting in a hamiltonian fashion. Let $G$ be a closed compatible Lie group of $U^{\mathbb C}$ and let $M$ be a $G$-invariant connected submanifold of $Z$. Let $x\in M$. If $G$ is a real form of $U^{\mathbb C}$, we investigate conditions such that $G\cdot x$ compact implies $U^{\mathbb C} \cdot x$ is compact as well. The vice-versa is also investigated. We also characterize $G$-invariant real submanifolds such that the norm square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of $(Z,\omega)$ generalizing a result proved in \cite{pg}, see also \cite{bg,bs}., Comment: 9 opages

We study the Hitchin map for Gℝ-Higgs bundles on a smooth curve, where Gℝ is a quasi-split real form of a complex reductive algebraic group G. By looking at the moduli stack of regular Gℝ-Higgs bundles, we prove that it induces a banded gerbe structure on a slightly larger stack, whose band is given by sheaves of tori. This characterization yields a cocyclic description of the fibres of the corresponding Hitchin map by means of cameral data. According to this, fibres of the Hitchin map are categories of principal torus bundles on the cameral cover. The corresponding points inside the stack of G-Higgs bundles are contained in the substack of points fixed by an involution induced by the Cartan involution of Gℝ. We determine this substack of fixed points and prove that stable points are in correspondence with stable Gℝ-Higgs bundles.

We obtain a necessary and sufficient condition for the existence of equivariant real structures on complex symmetric spaces for semisimple groups and discuss how to determine the number of equivalence classes for such structures., 14 pages, final version

An absolute parallelism for $2$-nondegenerate CR manifolds $M$ of hypersurface type was recently constructed independently by Isaev-Zaitsev, Medori-Spiro, and Pocchiola in the minimal possible dimension ($\dim M=5$), and for $\dim M=7$ in certain cases by the first author. We develop a bigraded analog of Tanaka's prolongation procedure to construct a canonical absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol. Under regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of this bigraded prolongation. We show that there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded prolongation of the symbol. In the case of $1$-dimensional Levi kernel we classify all regular symbols and calculate their bigraded prolongations. In this case the regular symbols can be subdivided into nilpotent, strongly non-nilpotent and weakly non-nilpotent. The bigraded prolongation of strongly non-nilpotent symbols is isomorphic to $\mathfrak{so}\left(m,\mathbb C\right)$ where $m=\tfrac{1}{2}(\dim M+5)$. Any real form of this algebra, except $\mathfrak{so}\left(m\right)$ and $\mathfrak{so}\left(m-1,1\right)$, corresponds to the real part of the bigraded prolongation of exactly one strongly non-nilpotent symbol. However, for a fixed $\dim M\geq 7$ the dimension of the bigraded prolongations achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is equal to $\tfrac{1}{4}(\dim M-1)^2+7$., 44 pages; Accepted to Journal f\"ur die reine und angewandte Mathematik (Crelle's journal); the functorial property of the canonical frame is emphasized, some references were added/updated

In this paper, we focus on the robust partial quadratic eigenvalue assignment problem for vibrating structures by active feedback control. This problem is reformulated as an minimization problem, where the cost function can measure both the sensitivity of the closed-loop eigenvalues and the feedback norms. This can be seen as a modified version of the minimization problem proposed by Bai et al. (2016) [5] , where there exist additional equality constraints involving determinants of linear functions of the parameter matrix. By using the receptance measurements, the system matrices and a few undesired open-loop eigenvalues and associated eigenvectors, we propose a modified gradient-based optimization method for solving the minimization problem, where the explicit gradient formula of the cost function is derived. To implement our method in real operation, the real form of our method is also presented. Finally, some numerical examples are given to illustrate the validity of the proposed method.

We find two different families of $$\mathbf{Sp}(4,\mathbb{R})$$ Sp ( 4 , R ) symmetric $$G_2$$ G 2 structures in seven dimensions. These are $$G_2$$ G 2 structures with $$G_2$$ G 2 being the split real form of the simple exceptional complex Lie group $$G_2$$ G 2 . The first family has $$\tau _2\equiv 0$$ τ 2 ≡ 0 , while the second family has $$\tau _1\equiv \tau _2\equiv 0$$ τ 1 ≡ τ 2 ≡ 0 , where $$\tau _1$$ τ 1 , $$\tau _2$$ τ 2 are the celebrated $$G_2$$ G 2 -invariant parts of the intrinsic torsion of the $$G_2$$ G 2 structure. The families are different in the sense that the first one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_l$$ Sp ( 4 , R ) / SL ( 2 , R ) l , and the second one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_s$$ Sp ( 4 , R ) / SL ( 2 , R ) s . Here $$\mathbf{SL}(2,\mathbb{R})_l$$ SL ( 2 , R ) l is an $$\mathbf{SL}(2,\mathbb{R})$$ SL ( 2 , R ) corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$ sl ( 2 , R ) related to the long roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$ sp ( 4 , R ) , and $$\mathbf{SL}(2,\mathbb{R})_s$$ SL ( 2 , R ) s is an $$\mathbf{SL}(2,\mathbb{R})$$ SL ( 2 , R ) corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$ sl ( 2 , R ) related to the short roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$ sp ( 4 , R ) .

Let L be a contragredient Lie superalgebra. A symmetric pair of L is a pair $$({\mathfrak {g}},{\mathfrak {g}}^\sigma )$$ , where $${\mathfrak {g}}$$ is a real form of L, and $$\sigma $$ is a $${\mathfrak {g}}$$ -involution with invariant subalgebra $${\mathfrak {g}}^\sigma $$ . We show that a symmetric pair carries invariant symplectic forms if and only if $${\mathfrak {g}}^\sigma $$ has a 1-dimensional center. Furthermore, the symplectic form is pseudo-Kahler if and only if the center of $${\mathfrak {g}}^\sigma $$ is compact. As an application, we classify the symplectic symmetric pairs, as well as the pseudo-Kahler symmetric pairs.

It has been known for some time that there exist $5$ essentially different real forms of the complex affine Kac-Moody algebra of type $A_2^{(2)}$ and that one can associate $4$ of these real forms with certain classes of "integrable surfaces", such as minimal Lagrangian surfaces in $\mathbb {CP}^2$ and $\mathbb {CH}^2$, as well as definite and indefinite affine spheres in $\mathbb R^3$. In this paper we consider the class of timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space $\mathbb{CH}^2_1$. We show that this class of surfaces corresponds to the fifth real form. Moreover, for each timelike Lagrangian surface in $\mathbb {CH}^2_1$ we define natural Gauss maps into certain homogeneous spaces and prove a Ruh-Vilms type theorem, characterizing timelike minimal Lagrangian surfaces among all timelike Lagrangian surfaces in terms of the harmonicity of these Gauss maps., Comment: Typological errors have been fixed

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{\"u}ller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic $G$-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of connected components of moduli spaces of such objects. We further exhibit an alternative explanation of fundamental results on counting components in the absence of a parabolic structure., Comment: 48 pages, 3 tables