Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diff _c (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S ¹, the geodesic distance on Diff _c (S ¹) vanishes if and only if ${s\leq\frac12}$ . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff _c (M) vanishes for ${M=\mathbb{R}\times N, s < \frac12}$ and for ${M=S^1\times N, s\leq\frac12}$ , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff _c (M) is positive for ${{\rm dim}(M)=1, s > \frac12}$ and dim(M) ≥ 2, s ≥ 1. For ${M=\mathbb{R}^n}$ , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin–Lax–Majda equation for ${s=\frac12}$ , and the Camassa–Holm equation for s = 1.     

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

Uniqueness of H-surfaces in {\mathbb{H}}^2 \times \mathbb{R},{{\vert H\vert \leq 1/2}} , with boundary one or two parallel horizontal circles

We prove that a H-surface M in ${\mathbb{H}}^2 \times {\mathbb{R}} ,\vert H\vert \leq 1/2$ , inherits the symmetries of its boundary $\partial M,$ when $\partial M$ is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.     

On a problem concerning the Banach algebra generated by the maps n\mapsto\lambda^{\binom{n}{k}}

In Jabbari and Namioka (Milan J. Math. 78:503?C522, 2010), the authors characterized the spectrum M(W) of the Weyl algebra W, i.e. the norm closure of the algebra generated by the family of functions $\{n\mapsto x^{n^{k}}; x\in\mathbb{T}, k\in\mathbb{N}\}$ , ( $\mathbb{T}$ the unit circle), with a closed subgroup of $E(\mathbb{T})^{\mathbb{N}}$ where $E(\mathbb{T})$ denotes the family of the endomorphisms of the multiplicative group $\mathbb{T}$ . But the size of M(W) in $E(\mathbb{T})^{\mathbb{N}}$ as well as the induced group operation were left as a problem. In this paper, we will give a solution to this problem.     

The Ricci tensor and structure Jacobi operator of real hypersurfaces in a complex projective space

Let M be a real hypersurface with almost contact metric structure ${(\phi, \xi, \eta, g)}$ in a complex projective space ${P_{n}\mathbb{C}}$ . A Real hypersurface M is said to be a Hopf hypersurface if ξ is principal. In this paper we investigate real hypersurfaces of ${P_{n}\mathbb{C}}$ whose Ricci tensors S satisfy ${\nabla_{\phi\nabla_{\xi}\xi}S = 0}$ . Under some further conditions we characterize Hopf hypersurfaces of ${P_{n}\mathbb{C}}$ .     

On the stability of minimal cones in warped products

In a seminal paper published in 1968, J. Simons proved that, for n ≤ 5, the Euclidean (minimal) cone CM, built on a closed, oriented, minimal and non totally geodesic hypersurface M ⁿ of \(\mathbb{S}^{n + 1} \) is unstable. In this paper, we extend Simons’ analysis to warped (minimal) cones built over a closed, oriented, minimal hypersurface of a leaf of suitable warped product spaces. Then, we apply our general results to the particular case of the warped product model of the Euclidean sphere, and establish the unstability of CM, whenever 2 ≤ n ≤ 14 and M ⁿ is a closed, oriented, minimal and non totally geodesic hypersurface of \(\mathbb{S}^{n + 1} \) .     

The moment problem for continuous positive semidefinite linear functionals

Let τ be a locally convex topology on the countable dimensional polynomial ${\mathbb{R}}$ -algebra ${\mathbb{R} [\underline{X}] := \mathbb{R} [X_1, \ldots, X_{n}]}$ . Let K be a closed subset of ${\mathbb{R} ^{n}}$ , and let ${M := M_{\{g_1, \ldots, g_s\}}}$ be a finitely generated quadratic module in ${\mathbb{R} [\underline{X}]}$ . We investigate the following question: When is the cone Psd(K) (of polynomials nonnegative on K) included in the closure of M? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of ${M = \sum \mathbb{R} [\underline{X}]^{2}}$ with respect to weighted norm-p topologies. We show that this closure coincides with the cone Psd(K) where K is a certain convex compact polyhedron.     

Permanent Versus Determinant over a Finite Field

Let $ \mathbb{F} $ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and for the whole matrix space M _n ( $ \mathbb{F} $ ). It is known that for n = 2, there are bijective linear maps Φ on $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ?), where Φ is an arbitrary bijective map on matrices and ? : $ \mathbb{F} $ → $ \mathbb{F} $ is an arbitrary map such that per A = ?(det Φ(A)) for all matrices A from the spaces $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ), respectively. Moreover, for the space M _n ( $ \mathbb{F} $ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $ \mathbb{F} $ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.     

Bessel multiwavelet sequences and dual multiframelets in Sobolev spaces

The dual 2I _d -framelets in $ (H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d})) $ , s?>?0, were introduced by Han and Shen (Constr Approx 29(3):369–406, 2009). In this paper, we systematically study the Bessel property of multiwavelet sequences in Sobolev spaces. The conditions for Bessel multiwavelet sequence in $ H^{-s}(\mathbb{R}^{d}) $ take great difference from those for Bessel wavelet sequence in this space. Precisely, the Bessel property of multiwavelet sequence in $ H^{-s}(\mathbb{R}^{d}) $ is not only related to multiwavelets themselves but also to the corresponding refinable function vector. We construct a class of Bessel M-refinable function vectors with M being an isotropic dilation matrix, which have high Sobolev smoothness, and of which the mask symbols have high sum rules. Based on the constructed Bessel refinable function vector, an explicit algorithm is given for dual M-multiframelets in $ (H^{s}(\mathbb{R}^{d}),H^{-s}(\mathbb{R}^{d})) $ with the multiframelets in $ H^{-s}(\mathbb{R}^{d}) $ having high vanishing moments. On the other hand, based on the dual multiframelets, an algorithm for dual M-multiframelets with symmetry is given. In Section 6, we give an example to illustrate the constructing procedures of dual multiframelets.     

On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities

We derive the inequality $$\int_\mathbb{R}M(|f'(x)|h(f(x))) dx\leq C(M,h)\int_\mathbb{R}M\left({\sqrt{|f''(x)\tau_h(f(x))|}\cdot h(f(x))}\right)dx$$ with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space ${W^{2,1}(\mathbb{R})}$ and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and ${\tau_h(\cdot)}$ is its given transform independent of M. When M(λ) =  λ ^p and ${h \equiv 1}$ we retrieve the well-known inequality ${\int_\mathbb{R}|f'(x)|^{p}dx \leq (\sqrt{p - 1})^{p}\int_\mathbb{R}(\sqrt{|f''(x) f(x)|})^{p}dx}$ . We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz’ya (1985).     

Entropy numbers of Besov classes of generalized smoothness on the sphere

We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ(Sd 1)of generalized smoothness on the sphere inL q(Sd 1)for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p(Sd 1)inL q(Sd 1)whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p(Sd 1)inL q(Sd 1)are concerned.     

The second main theorem of hypersurfaces in the projective space

We deal with a holomorphic map from the complex plane ${\mathbb{C}}$ to the n-dimensional complex projective space ${\mathbb{P}^{n}(\mathbb{C})}$ and prove the Nevanlinna Second Main Theorem for some families of non-linear hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ . This Second Main Theorem implies the defect relation. If the degree of the hypersurfaces are sufficiently large, the defect of the map is smaller than one. This means that holomorphic maps which omit the irreducible hypersurface of large degree is algebraically degenerate. To prove the Second Main Theorem, we use a meromorphic partial projective connection which is totally geodesic with respect to these hypersurfaces. A meromorphic partial projective connection is a family of locally defined meromorphic connections such which work as an entirely defined meromorphic connection under the Wronskian operator. By resolving the singularity and pulling back a meromorphic partial projective connection, we also prove the Second Main Theorem for singular hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ , and prove the Second Main Theorem for smooth hypersurfaces in ${\mathbb{P}^{2}(\mathbb{C})}$ which are not normal crossing.     

Bi-Lipschitz extension from boundaries of certain hyperbolic spaces

Tukia and Väisälä showed that every quasi-conformal map of ${\mathbb{R}^n}$ extends to a quasi-conformal self-map of ${\mathbb{R}^{n+1}}$ . The restriction of the extended map to the upper half-space ${\mathbb{R}^n \times \mathbb{R}_+}$ is, in fact, bi-Lipschitz with respect to the hyperbolic metric. More generally, every simply connected homogeneous negatively curved manifold decomposes as ${M = N \rtimes \mathbb{R}_+}$ where N is a nilpotent group with a metric on which ${\mathbb{R}_+}$ acts by dilations. We show that under some assumptions on N, every quasi-symmetry of N extends to a bi-Lipschitz map of M. The result applies to a wide class of manifolds M including non-compact rank one symmetric spaces and certain manifolds that do not admit co-compact group actions. Although M must be Gromov hyperbolic, its curvature need not be strictly negative.     

Extent partitions and context extensions

We prove that the extent partitions of a formal context $\mathbb{K}: = (G,M,I)$ can be constructed from the box extents of it, which form a complete atomistic lattice. $\mathbb{K}$ is called a one-object extension of the subcontext (H, M, J) if it is obtained by adding a new element with attributes in M to the set H. We investigate the interplay between the box extents of (H, M, J) and those of its one-object extension $\mathbb{K}$ , and describe those extent partitions of (H, M, J) which can be extended to $\mathbb{K}$ .     

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms ${M or_P (\mathbb{G}, M(n, \xi))}$ and the moduli space of stable bundles over ${X \times \mathbb{G}}$ , where ${\mathbb{G}}$ is the Grassmannian ${\mathbb{G}(n - r, \mathbb{C}^n)}$ . Moreover, we give sufficient conditions for ${M or_{2ns}(\mathbb{P}^1, M(n, \xi))}$ to be non-empty, when s ≥ 1.     

Noncommutative Multivariable Operator Theory

In this paper, we study noncommutative domains ${\mathbb{D}_f^\varphi(\mathcal{H}) \subset B(\mathcal{H})^n}$ generated by positive regular free holomorphic functions f and certain classes of n-tuples ${\varphi = (\varphi_1, \ldots, \varphi_n)}$ of formal power series in noncommutative indeterminates Z ₁, . . . , Z _n . Noncommutative Poisson transforms are employed to show that each abstract domain ${\mathbb{D}_f^\varphi}$ has a universal model consisting of multiplication operators (M _Z1, . . . , M _{Z n} ) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under M _Z1, . . . , M _{Z n} and show that all pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ are compressions of ${M_{Z_1} \otimes I, \ldots, M_{Z_n} \otimes I}$ to their coinvariant subspaces. We show that the eigenvectors of ${M_{Z_1}^*, \ldots, M_{Z_n}^*}$ are precisely the noncommutative Poisson kernels ${\Gamma_\lambda}$ associated with the elements ${\lambda}$ of the scalar domain ${\mathbb{D}_{f,<}^\varphi(\mathbb{C}) \subset \mathbb{C}^n}$ . These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra ${H^\infty(\mathbb{D}_f^\varphi)}$ . We introduce the characteristic function of an n-tuple ${T=(T_1, \ldots , T_n) \in \mathbb{D}_f^\varphi(\mathcal{H})}$ , present a model for pure n-tuples of operators in the noncommutative domain ${\mathbb{D}_f^\varphi(\mathcal{H})}$ in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ .     

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

Biharmonic properly immersed submanifolds in Euclidean spaces

We consider a complete biharmonic immersed submanifold M in a Euclidean space ${\mathbb{E}^N}$ . Assume that the immersion is proper, that is, the preimage of every compact set in ${\mathbb{E}^N}$ is also compact in M. Then, we prove that M is minimal. It is considered as an affirmative answer to the global version of Chen’s conjecture for biharmonic submanifolds.