Definability of Boolean Algebras in \mathbb{H}\mathbb{F} -Superstructures

Within the frames of the -definability approach propounded by Yu. L. Ershov, we study into the definability of Boolean algebras and their Frechet ranks in hereditarily finite superstructures. Examples are constructed of a superatomic Boolean algebra whose Frechet rank is not -definable in the hereditarily finite superstructure over that algebra, and of an admissible set in which the atomless Boolean algebra is not autostable.     

Dynamical orders of decentralized H{infty}; controllers

The problem of decentralized control is addressed, i.e. theproblem of designing a controller where each control input isallowed to use only some of the measurements. It is shown that,for such problems, there does not always exist a sequence ofcontrollers of bounded order which obtains near-optimal control.Neither does there exist an infinite-dimensional optimal controller.Using the insight of the line of proof of these results, a heuristicdesign algorithm is proposed for designing near-optimal controllersof increasing orders.     

$$\pmb {{\mathcal {H}}_{p}}$$Hp-Theory of General Dirichlet Series

Journal of Fourier Analysis and Applications - Inspired by results of Bayart on ordinary Dirichlet series $$\sum a_n n^{-s}$$, the main purpose of this article is to start an $${\mathcal...     

Hypersurfaces with {H_{r+1}} = 0 in {\mathbb{H}^n \times \mathbb{R}}

We prove the existence of rotational hypersurfaces in \({\mathbb{H}^n \times \mathbb{R}}\) with \({H_{r+1} = 0}\) (r-minimal hupersurfaces) and we classify them. Then we prove some uniqueness theorems for r-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type theorem for two ended complete hypersurfaces.     

Ruled Weingarten Hypersurfaces in Hyperbolic Space {{\mathbb{H}}^{n+1}}

In this work we give a local classification of connected ruled Weingarten hypersurfaces M ⁿ , n ≥ 3 in the hyperbolic space ${{\mathbb{H}}^{n+1} \subset {\mathbb{L}}^{n+2}}$ .     

Two Inner Sequences Based Invariant Subspaces in {{H}^{2} (\mathbb{D}^{2})}

Let M be a shift invariant subspace in the vector-valued Hardy space ${H_{E}^{2}(\mathbb{D})}$ H E 2 ( D ) . The Beurling–Lax–Halmos theorem says that M can be completely characterized by ${\mathcal{B}(E)}$ B ( E ) -valued inner function ${\Theta}$ Θ . When ${E = H^{2}(\mathbb{D}),\,H_{E}^{2}(\mathbb{D})}$ E = H 2 ( D ) , H E 2 ( D ) is the Hardy space on the bidisk ${H^{2}(\mathbb{D}^2)}$ H 2 ( D 2 ) . Recently, Qin and Yang (Proc Am Math Soc, 2013) determines the operator valued inner function ${\Theta(z)}$ Θ ( z ) for two well-known invariant subspaces in ${H^{2}(\mathbb{D}^{2})}$ H 2 ( D 2 ) . This paper generalizes the ${\Theta(z)}$ Θ ( z ) by Qin and Yang (Proc Am Math Soc, 2013) and deal with the structure of ${M = {\Theta}(z)H^{2}(\mathbb{D}^{2})}$ M = Θ ( z ) H 2 ( D 2 ) when M is an invariant subspace in ${H^{2}(\mathbb{D}^{2})}$ H 2 ( D 2 ) . Unitary equivalence, spectrum of the compression operator and core operator are studied in this paper.     

On the admissible sets of type $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})$ over recursively saturated models

Some effective expression is obtained for the elements of an admissible set \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) as template sets. We prove the Σ-reducibility of \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) to \(\mathbb{H}\mathbb{F}(\mathfrak{M})\) for each recursively saturated model \(\mathfrak{M}\) of a regular theory, give a criterion for uniformization in \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) for each recursively saturated model \(\mathfrak{M}\), and establish uniformization in \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{N})\) and \(\mathbb{H}\mathbb{Y}\mathbb{P}(\Re ')\), where \(\mathfrak{N}\) and \(\Re '\) are recursively saturated models of arithmetic and real closed fields. We also prove the absence of uniformization in \(\mathbb{H}\mathbb{F}(\mathfrak{M})\) and \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) for each countably saturated model \(\mathfrak{M}\) of an uncountably categorical theory, and give an example of this type of theory with definable Skolem functions. Furthermore, some example is given of a model of a regular theory with Σ-definable Skolem functions, but lacking definable Skolem functions in every extension by finitely many constants.     

Multilevel interpolation of scattered data using H ${\mathcal{H}}$ -matrices

Numerical Algorithms - Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the...     

Unirationality of the Hurwitz space $$\varvec{\mathcal {H}_{9,8}}$$

In this paper we prove that the Hurwitz space \(\mathcal {H}_{9,8}\), which parameterizes 8-sheeted covers of \({\mathbb P }^1\) by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.     

二部图形式的Erd\H{O}s-S\'{o}s猜想

二部图形式的Erd\H{O}s-S\''{o}s猜想     

Approximation of solution operators of elliptic partial differential equations by {\mathcal{H}}- and {\mathcal{H}^2}-matrices

We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are non-local, the inverse matrices will in general be dense, therefore representing them by standard techniques will require prohibitively large amounts of storage. In the field of integral equations, a successful technique for handling dense matrices efficiently is to use a data-sparse representation like the popular multipole method. In this paper we prove that this approach can be generalized to cover inverse matrices corresponding to partial differential equations by switching to data-sparse ${\mathcal{H}}$ - and ${\mathcal{H}^2}$ -matrices. The key results are existence proofs for local low-rank approximations of the solution operator and its discrete counterpart, which give rise to error estimates for ${\mathcal{H}}$ - and ${\mathcal{H}^2}$ -matrix approximations of the entire matrices.     

Robust H{infty} control for Markovian jump nonlinear systems

This paper deals with the robust H control problem of uncertainnonlinear stochastic systems with Markovian jump parameters.We assume that the uncertainties are structured such that suitablebounding functions can be found. A robust controller is thendesigned that will guarantee disturbance attenuation and asymptoticstability for all admissible uncertainties and L₂-bounded disturbances.The solution to the problem is characterized in terms of a setof smooth-positive semidefinite functions satisfying certainHamilton-Jacobi-Isaac (HJI) inequalities with some appropriatescaling functions. Both the case of matched and unmatched uncertaintiesare considered.     

The reducible Specht modules for the Hecke algebra \mathcal{H}_{\mathbb{C},{-}1}(\mathfrak{S}_{n})

The reducible Specht modules for the Hecke algebra $\mathcal {H}_{\mathbb{F},q}(\mathfrak{S}_{n})$ have been classified except when q=?1. We prove one half of a conjecture which we believe classifies the reducible Specht modules when q=?1.     

On quasi $${\mathcal{H}}$$ -closed subspaces

In this paper some properties concerning the notion of quasi H{\mathcal{H}} -closedness in topological spaces are obtained. Locally quasi H{\mathcal{H}} -closed spaces are studied. Properties of these types of spaces are investigated, when the subsets involved are α-open, semi-open, clopen, or preopen.     

Finite-dimensional H{infty} control of flexible beam equation systems

In this paper, we consider H control problem with measurementfeedback for flexible beam equation systems. The aim is to constructa finite-dimensional H controller with a given level for theflexible beam equation system. For that purpose, we first formulatethe system as an infinite-dimensional system in l² and derivea finite-dimensional reduced-order system for the infinite-dimensionalsystem. Then, an H controller with level d less than is constructedfor the reduced-order model. The finite-dimensional controllertogether with a residual mode filter plays a role of a finite-dimensionalH controller with level for the original flexible beam equationsystem, if the order of the residual mode filter is chosen sufficientlylarge.     

Orbits in the Maximal Ideal Space of H{infty}

Let H be the Banach algebra of bounded analytic functions inthe open unit disc D. We can define the rotation in the maximalideal space M(H). For a point x in M(H)\D, an orbit O(x) isnot closed in M(H). It is proved that there exists a point xin M(H) such that x is not contained in the Shilov boundaryX and cl O(x), the closure of O(x), contains X, and there existsa point y in M(H)\(D X) such that cl O(y) X. The rotationpresents many problems concerning H. The purpose of this paperis to discuss these problems.     

Programmable criteria for strong $\mathcal {H}$-tensors

Strong \(\mathcal {H}\)-tensors play an important role in identifying positive semidefiniteness of even-order real symmetric tensors. We provide several simple practical criteria for identifying strong \(\mathcal {H}\)-tensors. These criteria only depend on the elements of the tensors; therefore, they are easy to be verified. Meanwhile, a sufficient and necessary condition of strong \(\mathcal {H}\)-tensors is obtained. We also propose an algorithm for identifying the strong \(\mathcal {H}\)-tensors based on these criterions. Some numerical results show the feasibility and effectiveness of the algorithm.