Cardinal inequalities for S(n)S(n)-spaces

Hajnal and Juh\'asz proved that if $X$ is a $T_1$-space, then $|X|\le 2^{s(X)\psi(X)}$, and if $X$ is a Hausdorff space, then $|X|\le 2^{c(X)\chi(X)}$ and $|X|\le 2^{2^{s(X)}}$. Schr\"oder sharpened the first two estimations by showing that if $X$ is a Hausdorff space, then $|X|\le 2^{Us(X)\psi_c(X)}$, and if $X$ is a Urysohn space, then $|X|\le 2^{Uc(X)\chi(X)}$. In this paper, for any positive integer $n$ and some topological spaces $X$, we define the cardinal functions $\chi_n(X)$, $\psi_n(X)$, $s_n(X)$, and $c_n(X)$, called respectively $S(n)$-character, $S(n)$-pseudocharacter, $S(n)$-spread, and $S(n)$-cellularity, and using these new cardinal functions we show that the above-mentioned inequalities could be extended to the class of $S(n)$-spaces. We recall that the $S(1)$-spaces are exactly the Hausdorff spaces and the $S(2)$-spaces are exactly the Urysohn spaces.Comment: 13 page

Cardinalities of weakly Lindel\"of spaces with regular GκG_\kappa-diagonals

For a Urysohn space $X$ we define the regular diagonal degree $\overline{\Delta}(X)$ of $X$ to be the minimal infinite cardinal $\kappa$ such that $X$ has a regular $G_\kappa$-diagonal i.e. there is a family $(U_\eta:\eta<\kappa)$ of open neighborhoods of $\Delta_X=\{(x,x)\in X^2:x\in X\}$ in $X^2$ such that $\Delta_X = \bigcap_{\eta<\kappa} \overline{U}_\eta$. In this paper we show that if $X$ is a Urysohn space then: (1) $|X|\leq 2^{c(X)\cdot\overline{\Delta}(X)}$; (2) $|X|\leq 2^{\overline{\Delta}(X)\cdot 2^{wL(X)}}$; (3) $|X|\le wL(X)^{\overline{\Delta}(X)\cdot\chi(X)}$; and (4) $|X|\le aL(X)^{\overline{\Delta}(X)}$; where $\chi(X)$, $c(X)$, $wL(X)$ and $aL(X)$ are respectively the character, the cellularity, the weak Lindel\"of number and the almost Lindel\"of number of $X$. The first inequality extends to the uncountable case Buzyakova's result that the cardinality of a ccc-space with a regular $G_\delta$-diagonal does not exceed $2^\omega$. It follows from (2) that every weakly Lindel\"of space with a regular $G_\delta$-diagonal has cardinality at most $2^{2^\omega}$. Inequality (3) implies that when $X$ is a space with a regular $G_\delta$-diagonal then $|X|\le wL(X)^{\chi(X)}$. This improves significantly Bell, Ginsburg and Woods inequality $|X|\le 2^{\chi(X)wL(X)}$ for the class of normal spaces with regular $G_\delta$-diagonals. In particular (3) shows that the cardinality of every first countable space with a regular $G_\delta$-diagonal does not exceed $wL(X)^\omega$. For the class of spaces with regular $G_\delta$-diagonals (4) improves Bella and Cammaroto inequality $|X|\le 2^{\chi(X)\cdot aL(X)}$, which is valid for all Urysohn spaces. Also, it follows from (4) that the cardinality of every space with a regular $G_\delta$-diagonal does not exceed $aL(X)^\omega$.Comment: 12 page

On the essential spectrum of the sum of self-adjoint operators and the closedness of the sum of operator ranges

We get a criterion for 0 to be in the essential spectrum of a sum of self-adjoint operators whose pairwise products are compact. Using this result, we obtain necessary and sufficient conditions for the sum of ranges of such operators to be closed.Comment: 9 page

Metric-based Hamiltonians, null boundaries, and isolated horizons

We extend the quasilocal (metric-based) Hamiltonian formulation of general relativity so that it may be used to study regions of spacetime with null boundaries. In particular we use this generalized Brown-York formalism to study the physics of isolated horizons. We show that the first law of isolated horizon mechanics follows directly from the first variation of the Hamiltonian. This variation is not restricted to the phase space of solutions to the equations of motion but is instead through the space of all (off-shell) spacetimes that contain isolated horizons. We find two-surface integrals evaluated on the horizons that are consistent with the Hamiltonian and which define the energy and angular momentum of these objects. These are closely related to the corresponding Komar integrals and for Kerr-Newman spacetime are equal to the corresponding ADM/Bondi quantities. Thus, the energy of an isolated horizon calculated by this method is in agreement with that recently calculated by Ashtekar and collaborators but not the same as the corresponding quasilocal energy defined by Brown and York. Isolated horizon mechanics and Brown-York thermodynamics are compared.Comment: 28 pages, LaTex, 2 figures, minor changes - typos fixed, a few sentences rephrased, one reference added. Content essentially unchange

On the closedness of the sum of ranges of operators AkA_k with almost compact products Ai∗AjA_i^* A_j

Let $\mathcal{H}_1,\ldots,\mathcal{H}_n,\mathcal{H}$ be complex Hilbert spaces and $A_k:\mathcal{H}_k\to\mathcal{H}$ be a bounded linear operator with the closed range $Ran(A_k)$, $k=1,\ldots,n$. It is known that if $A_i^*A_j$ is compact for any $i\neq j$, then $\sum_{k=1}^n Ran(A_k)$ is closed. We show that if all products $A_i^*A_j$, $i\neq j$ are "almost" compact, then the subspaces $Ran(A_1),\ldots,Ran(A_n)$ are essentially linearly independent and their sum is closed.Comment: 9 page

Generalizations of two cardinal inequalities of Hajnal and Juh\'asz

A non-empty subset $A$ of a topological space $X$ is called \emph{finitely non-Hausdorff} if for every non-empty finite subset $F$ of $A$ and every family $\{U_x:x\in F\}$ of open neighborhoods $U_x$ of $x\in F$, $\cap\{U_x:x\in F\}\ne\emptyset$ and \emph{the non-Hausdorff number $nh(X)$ of $X$} is defined as follows: $nh(X):=1+\sup\{|A|:A\subset X$ is finitely non-Hausdorff$\}$. Clearly, if $X$ is a Hausdorff space then $nh(X)=2$. We define the \emph{non-Urysohn number of $X$ with respect to the singletons}, $nu_s(X)$, as follows: $nu_s(X):=1+\sup\{\mathrm{cl}_\theta(\{x\}):x\in X\}$. In 1967 Hajnal and Juh\'asz proved that if $X$ is a Hausdorff space then: (1) $|X|\le 2^{c(X)\chi(X)}$; and (2) $|X|\le 2^{2^{s(X)}}$; where $c(X)$ is the cellularity, $\chi(X)$ is the character and $s(X)$ is the spread of $X$. In this paper we generalize (1) by showing that if $X$ is a topological space then $|X|\le nh(X)^{c(X)\chi(X)}$. Immediate corollary of this result is that (1) holds true for every space $X$ for which $nh(X)\le 2^\omega$ (and even for spaces with $nh(X)\le 2^{c(X)\chi(X)}$). This gives an affirmative answer to a question posed by M. Bonanzinga in 2013. A simple example of a $T_1$, first countable, ccc-space $X$ is given such that $|X|>2^\omega$ and $|X|=nh(X)^\omega=nh(X)$. This example shows that the upper bound in our inequality is exact and that $nh(X)$ cannot be omitted (in particular, $nh(X)$ cannot always be replaced by $2$ even for $T_1$-spaces). In this paper we also generalize (2) by showing that if $X$ is a $T_1$-space then $|X|\le 2^{nu_s(X)\cdot 2^{s(X)}}$. It follows from our result that (2) is true for every $T_1$-space for which $nu_s(X)\le 2^{s(X)}$. A simple example shows that the presence of the cardinal function $nu_s(X)$ in our inequality is essential.Comment: 10 page

Even Fourier multipliers and martingale transforms in infinite dimensions

In this paper we show sharp lower bounds for norms of even homogeneous Fourier multipliers in $\mathcal L(L^p(\mathbb R^d; X))$ for $1<p<\infty$ and for a UMD Banach space $X$ in terms of the range of the corresponding symbol. For example, if the range contains $a_1,\ldots,a_N \in \mathbb C$, then the norm of the multiplier exceeds $\|a_1R_1^2 + \cdots + a_NR_N^2\|_{\mathcal L(L^p(\mathbb R^N; X))}$, where $R_n$ is the corresponding Riesz transform. We also provide sharp upper bounds of norms of Ba\~{n}uelos-Bogdan type multipliers in terms of the range of the functions involved. The main tools that we exploit are $A$-weak differential subordination of martingales and UMD$_p^A$ constants, which are introduced here

Brownian representations of cylindrical continuous local martingales

In this paper we give necessary and sufficient conditions for a cylindrical continuous local martingale to be the stochastic integral with respect to a cylindrical Brownian motion. In particular we consider the class of cylindrical martingales with closed operator-generated covariations. We also prove that for every cylindrical continuous local martingale $M$ there exists a time change $\tau$ such that $M\circ \tau$ is Brownian representable.Comment: Minor revisio

Magneto-Plasmonic Nanoantennas: Basics and Applications (Review)

Plasmonic nanoantennas is a hot and rapidly expanding research field. Here we overview basic operating principles and applications of novel magneto-plasmonic nanoantennas, which are made of ferromagnetic metals and driven not only by light, but also by external magnetic fields. We demonstrate that magneto-plasmonic nanoantennas enhance the magneto-optical effects, which introduces additional degrees of freedom in the control of light at the nano-scale. This property is used in conceptually new devices such as magneto-plasmonic rulers, ultra-sensitive biosensors, one-way subwavelength waveguides and extraordinary optical transmission structures, as well as in novel biomedical imaging modalities. We also point out that in certain cases 'non-optical' ferromagnetic nanostructures may operate as magneto-plasmonic nanoantennas. This undesigned extra functionality capitalises on established optical characterisation techniques of magnetic nanomaterials and it may be useful for the integration of nanophotonics and nanomagnetism on a single chip.Comment: 15 pages, 12 figure

Figures of equilibrium of an inhomogeneous self-gravitating fluid

This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with density stratification and a steady-state velocity field. As in the classical setting, it is assumed that the figures or their layers uniformly rotate about an axis fixed in space