Compactness Properties of Weighted Summation Operators on Trees

We investigate compactness properties of weighted summation operators $V_{\alpha,\sigma}$ as mapping from $\ell_1(T)$ into $\ell_q(T)$ for some $q\in (1,\infty)$. Those operators are defined by $$(V_{\alpha,\sigma} x)(t) :=\alpha(t)\sum_{s\succeq t}\sigma(s) x(s)\,,\quad t\in T\;,$$ where $T$ is a tree with induced partial order $t \preceq s$ (or $s \succeq t$) for $t,s\in T$. Here $\alpha$ and $\sigma$ are given weights on $T$. We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two--sided estimates for $e_n(V_{\alpha,\sigma})$, the (dyadic) entropy numbers of $V_{\alpha,\sigma}$. The results are applied for concrete trees as e.g. moderate increasing, biased or binary trees and for weights with $\alpha(t)\sigma(t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees

A new generalization of the Takagi function

We consider a one-parameter family of functions $\{F(t,x)\}_{t}$ on $[0,1]$ and partial derivatives $\partial_{t}^{k} F(t, x)$ with respect to the parameter $t$. Each function of the class is defined by a certain pair of two square matrices of order two. The class includes the Lebesgue singular functions and other singular functions. Our approach to the Takagi function is similar to Hata and Yamaguti. The class of partial derivatives $\partial_{t}^{k} F(t, x)$ includes the original Takagi function and some generalizations. We consider real-analytic properties of $\partial_{t}^{k} F(t, x)$ as a function of $x$, specifically, differentiability, the Hausdorff dimension of the graph, the asymptotic around dyadic rationals, variation, a question of local monotonicity and a modulus of continuity. Our results are extensions of some results for the original Takagi function and some generalizations.Comment: 22 pages, 2 figures. The structure of paper has been changed significantl

Evidence for structural and electronic instabilities at intermediate temperatures in κ\kappa-(BEDT-TTF)2_{2}X for X=Cu[N(CN)2_{2}]Cl, Cu[N(CN)2_{2}]Br and Cu(NCS)2_{2}: Implications for the phase diagram of these quasi-2D organic superconductors

We present high-resolution measurements of the coefficient of thermal expansion $\alpha (T)=\partial \ln l(T)/\partial T$ of the quasi-twodimensional (quasi-2D) salts $\kappa$-(BEDT-TTF)$_2$X with X = Cu(NCS)$_2$, Cu[N(CN)$_2$]Br and Cu[N(CN)$_2$]Cl. At intermediate temperatures (B), distinct anomalies reminiscent of second-order phase transitions have been found at $T^\ast = 38$ K and 45 K for the superconducting X = Cu(NCS)$_2$ and Cu[N(CN)$_2$]Br salts, respectively. Most interestingly, we find that the signs of the uniaxial pressure coefficients of $T^\ast$ are strictly anticorrelated with those of $T_c$. We propose that $T^\ast$ marks the transition to a spin-density-wave (SDW) state forming on minor, quasi-1D parts of the Fermi surface. Our results are compatible with two competing order parameters that form on disjunct portions of the Fermi surface. At elevated temperatures (C), all compounds show $\alpha (T)$ anomalies that can be identified with a kinetic, glass-like transition where, below a characteristic temperature $T_g$, disorder in the orientational degrees of freedom of the terminal ethylene groups becomes frozen in. We argue that the degree of disorder increases on going from the X = Cu(NCS)$_2$ to Cu[N(CN)$_2$]Br and the Cu[N(CN)$_2$]Cl salt. Our results provide a natural explanation for the unusual time- and cooling-rate dependencies of the ground-state properties in the hydrogenated and deuterated Cu[N(CN)$_2$]Br salts reported in the literature.Comment: 22 pages, 7 figure

Commensurate-Incommensurate transition in the melting process of the orbital ordering in Pr0.5Ca0.5MnO3: neutron diffraction study

The melting process of the orbital order in Pr0.5Ca0.5MnO3 single crystal has been studied in detail as a function of temperature by neutron diffraction. It is demonstrated that a commensurate-incommensurate (C-IC) transition of the orbital ordering takes place in a bulk sample, being consistent with the electron diffraction studies. The lattice structure and the transport properties go through drastic changes in the IC orbital ordering phase below the charge/orbital ordering temperature Tco/oo, indicating that the anomalies are intimately related to the partial disordering of the orbital order, unlike the consensus that it is related to the charge disordering process. For the same T range, partial disorder of the orbital ordering turns on the ferromagnetic spin fluctuations which were observed in a previous neutron scattering study.Comment: 5 pages, 2 figures, REVTeX, to be published in Phys. Rev.

Partial orders on transformation semigroups

In 1986, Kowol and Mitsch studied properties of the so-called 'natural partial order' less than or equal to on T(X), the total transformation semigroup defined on a set X. In particular, they determined when two total transformations are related under this order, and they described the minimal and maximal elements of (T(X), less than or equal to). In this paper, we extend that work to the semigroup P(X) of all partial transformations of X, compare less than or equal to with another 'natural' partial order on P(X), characterise the meet and join of these two orders, and determine the minimal and maximal elements of P(X) with respect to each order.Centro de Matemática da Universidade do Minho.Fundação para a Ciência e a Tecnologia (FCT)