On conditional expectations in $$L^p(\mu ;L^q(\nu ;X))$$ L p ( μ ; L q ( ν ; X ) )

Positivity - Let $$(A,{\mathscr {A}},\mu )$$ and $$(B,{\mathscr {B}},\nu )$$ be probability spaces, let $${\mathscr {F}}$$ be a sub- $$\sigma $$ -algebra of the product $$\sigma $$ -algebra...     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

Constructing saturating sets in projective spaces using subgeometries

Designs, Codes and Cryptography - A $$\varrho $$ -saturating set of $$\text {PG}(N,q)$$ is a point set $${\mathcal {S}}$$ such that any point of $$\text {PG}(N,q)$$ lies in a subspace of dimension...     

Partial Decidable Presentations in Hyperarithmetic

Siberian Mathematical Journal - We study the problem of the existence of decidable and positive $$\Pi_1^1$$ - and $$\Sigma_1^1$$ -numberings of the families of $$\Pi_1^1$$ - and $$\Sigma_1^1$$...     

Certain graphs with exactly one irreducible T-module with endpoint 1, which is thin

Journal of Algebraic Combinatorics - Let $$\Gamma $$ denote a finite, simple and connected graph. Fix a vertex x of $$\Gamma $$ and let $$T=T(x)$$ denote the Terwilliger algebra of $$\Gamma $$ with...     

Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II

Positivity - Let $${{\mathcal {M}}}$$ be a von Neumann algebra of operators on a Hilbert space $${\mathcal {H}}$$ and $$\tau $$ be a faithful normal semifinite trace on $$\mathcal {M}$$ . Let...     

Characterizing linear mappings through zero products or zero Jordan products

Periodica Mathematica Hungarica - Let $${\mathcal {A}}$$ be a $$*$$ -algebra and $${{\mathcal {M}}}$$ be a $$*$$ - $${\mathcal {A}}$$ -bimodule. We study the local properties of $$*$$ -derivations...     

Sparse graphs and an augmentation problem

Mathematical Programming - For two integers $$k>0$$ and $$\ell $$ , a graph $$G=(V,E)$$ is called $$(k,\ell )$$ -tight if $$|E|=k|V|-\ell $$ and $$i_G(X)\le k|X|-\ell $$ for each...     

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Monatshefte für Mathematik - Let $$\Omega $$ be a $$C^2$$ -smooth bounded pseudoconvex domain in $$\mathbb {C}^n$$ for $$n\ge 2$$ and let $$\varphi $$ be a holomorphic function on $$\Omega $$...     

Optimal selection for good polynomials of degree up to five

Designs, Codes and Cryptography - An $$(r,\ell )$$ -good polynomial is a polynomial of degree $$r+1$$ that is constant on $$\ell $$ subsets of $$\mathbb F_q$$ , each of size $$r+1$$ . For any...     

Remarks on Rawnsley’s $$\varvec{\varepsilon }$$ε -function on the Fock–Bargmann–Hartogs domains

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kähler metric associated with the Kähler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.     

One-Sided Invertibility Criteria for Binomial Functional Operators with Shift and Slowly Oscillating Data

Mediterranean Journal of Mathematics - Let $${\alpha}$$ be an orientation-preserving homeomorphism of $${[0,\infty]}$$ onto itself with only two fixed points at 0 and $${\infty}$$ , whose...     

On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis

Mathematical Notes - Let $$f(x)$$ be a function belonging to the Lebesgue class $$L^p({\mathbb R}_+)$$ on the semiaxis $${\mathbb R}_+=[0,+\infty)$$ , $$1\le p\le 2$$ , and let $$\widehat{f}$$ be...     

Exponent of a finite group of odd order with an involutory automorphism

Let $$w = w(x_1, \ldots , x_n)$$ be a non-trivial word of n-variables. The word map on a group G that corresponds to w is the map $$\widetilde{w}: G^n\rightarrow G$$ where $$\widetilde{w}((g_1, \ldots , g_n)) := w(g_1, \ldots , g_n)$$ for every sequence $$(g_1, \ldots , g_n)$$ . Let $$\mathcal G$$ be a simple and simply connected group which is defined and split over an infinite field K and let $$G =\mathcal G(K)$$ . For the case when $$w = w_1w_2 w_3 w_4$$ and $$w_1, w_2, w_3, w_4$$ are non-trivial words with independent variables, it has been proved by Hui et al. (Israel J Math 210:81–100, 2015) that $$G{\setminus } Z(G) \subset {{\text { Im}}}\,\widetilde{w}$$ where Z(G) is the center of the group G and $${{\text { Im}}}\, {\widetilde{w}}$$ is the image of the word map $$\widetilde{w}$$ . For the case when $$G = {{\text {SL}}}_n(K)$$ and $$n \ge 3$$ , in the same paper of Hui et al. (2015) it was shown that the inclusion $$G{\setminus } Z(G)\subset {{\text { Im}}}\,\widetilde{w}$$ holds for a product $$w = w_1w_2 w_3$$ of any three non-trivial words $$ w_1, w_2, w_3$$ with independent variables. Here we extent the latter result for every simple and simply connected group which is defined and split over an infinite field K except the groups of types $$B_2, G_2$$ .     

Weak law of large numbers for iterates of random-valued functions

Aequationes mathematicae - Given a probability space $$ (\Omega , {\mathcal {A}}, P) $$ , a complete and separable metric space X with the $$ \sigma $$ -algebra $$ {\mathcal {B}} $$ of all its...     

Characterization of distributive semigroup operations on the positive real numbers

Given any cancellative continuous semigroup operation $$\star $$ on the positive real numbers $$\mathbf {R}_+$$ with the ordinary topology, we completely characterize the set $$\mathcal {D}_\star (\mathbf {R}_+)$$ of all cancellative continuous semigroup operations on $$\mathbf {R}_+$$ which are distributed by $$\star $$ in terms of homeomorphism. As a consequence, we show that an arbitrary semigroup operation in $$\mathcal {D}_\star (\mathbf {R}_+)$$ is homeomorphically isomorphic to the ordinary addition $$+$$ on $$\mathbf {R}_+$$.     

Generation in singularity categories of hypersurfaces of countable representation type

The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category $$\textsf {D} _{\textsf {sg} }(R)$$ of a hypersurface R of countable representation type. For a thick subcategory $${\mathcal {T}}$$ of $$\textsf {D} _{\textsf {sg} }(R)$$ and a full subcategory $$\mathcal {X}$$ of $${\mathcal {T}}$$, we calculate the Rouquier dimension of $${\mathcal {T}}$$ with respect to $$\mathcal {X}$$. Furthermore, we prove that the level in $$\textsf {D} _{\textsf {sg} }(R)$$ of the residue field of R with respect to each nonzero object is at most one.     

The universal minimal one parameter system with quasi-discrete spectrum

The universal minimal one parameter system will be characterized as the space $$\Gamma ^{\infty }$$, in which $$\Gamma$$ is the Bohr compactification of the additive group $${\mathbb {R}}$$ of real numbers. In this way, we need to show that $$\Gamma ^\infty$$ is isomorphic to the spectrum of $$W({\mathbb {R}})$$, the norm closure of the invariant algebra generated by the maps $$\exp q(t)$$, where q(t) is a real polynomial on $${\mathbb {R}}$$.     

The Liouville-type theorem for problems with nonstandard growth derived by Caccioppoli-type estimate

Let u be a nonnegative solution to the PDI $$-\,\mathrm{div} \mathcal {A}(x, u, \nabla u)\geqslant \mathcal {B}(x,u, \nabla u)$$ in $$\Omega $$, where $$\mathcal {A}$$ and $$\mathcal {B}$$ are differential operators with p(x)-type growth. As a consequence of the Caccioppoli-type inequality for the solution u, we obtain the Liouville-type theorem under some integral condition. We simplify the assumptions on functions $$ \mathcal {A}$$ and $$ \mathcal {B}$$, and we do not restrict the range of p(x) by the dimension n, therefore we can cover quite general family of problems.