Kato square root problem with unbounded leading coefficients

We prove the Kato conjecture for elliptic operators, $L=-\nabla\cdot\left((\mathbf A+\mathbf D)\nabla\ \right)$, with $\mathbf A$ a complex measurable bounded coercive matrix and $\mathbf D$ a measurable real-valued skew-symmetric matrix in $\mathbb{R}^n$ with entries in $BMO(\mathbb{R}^n)$;\, i.e., the domain of $\sqrt{L}\,$ is the Sobolev space $\dot H^1(\mathbb{R}^n)$ in any dimension, with the estimate $\|\sqrt{L}\, f\|_2\lesssim \| \nabla f\|_2$

Smectic ordering in liquid crystal - aerosil dispersions II. Scaling analysis

Liquid crystals offer many unique opportunities to study various phase transitions with continuous symmetry in the presence of quenched random disorder (QRD). The QRD arises from the presence of porous solids in the form of a random gel network. Experimental and theoretical work support the view that for fixed (static) inclusions, quasi-long-range smectic order is destroyed for arbitrarily small volume fractions of the solid. However, the presence of porous solids indicates that finite-size effects could play some role in limiting long-range order. In an earlier work, the nematic - smectic-A transition region of octylcyanobiphenyl (8CB) and silica aerosils was investigated calorimetrically. A detailed x-ray study of this system is presented in the preceding Paper I, which indicates that pseudo-critical scaling behavior is observed. In the present paper, the role of finite-size scaling and two-scale universality aspects of the 8CB+aerosil system are presented and the dependence of the QRD strength on the aerosil density is discussed.Comment: 14 pages, 10 figures, 1 table. Companion paper to "Smectic ordering in liquid crystal - aerosil dispersions I. X-ray scattering" by R.L. Leheny, S. Park, R.J. Birgeneau, J.-L. Gallani, C.W. Garland, and G.S. Iannacchion

Ultrasonic studies of the magnetic phase transition in MnSi

Measurements of the sound velocities in a single crystal of MnSi were performed in the temperature range 4-150 K. Elastic constants, controlling propagation of longitudinal waves reveal significant softening at a temperature of about 29.6 K and small discontinuities at $\sim$28.8 K, which corresponds to the magnetic phase transition in MnSi. In contrast the shear elastic moduli do not show any softening at all, reacting only to the small volume deformation caused by the magneto-volume effect. The current ultrasonic study exposes an important fact that the magnetic phase transition in MnSi, occurring at 28.8 K, is just a minor feature of the global transformation marked by the rounded maxima or minima of heat capacity, thermal expansion coefficient, sound velocities and absorption, and the temperature derivative of resistivity.Comment: 4 pages, 4 figure

General massive one-loop off-shell three-point functions

In this work we compute the most general massive one-loop off-shell three-point vertex in D-dimensions, where the masses, external momenta, and exponents of propagators are arbitrary. This follows our previous paper in which we have calculated several new hypergeometric series representations for massless and massive (with equal masses) scalar one-loop three-point functions, in the negative dimensional approach.Comment: 16 pages, 2 figures, 4 table

Smectic ordering in liquid crystal - aerosil dispersions I. X-ray scattering

Comprehensive x-ray scattering studies have characterized the smectic ordering of octylcyanobiphenyl (8CB) confined in the hydrogen-bonded silica gels formed by aerosil dispersions. For all densities of aerosil and all measurement temperatures, the correlations remain short range, demonstrating that the disorder imposed by the gels destroys the nematic (N) to smectic-A (SmA) transition. The smectic correlation function contains two distinct contributions. The first has a form identical to that describing the critical thermal fluctuations in pure 8CB near the N-SmA transition, and this term displays a temperature dependence at high temperatures similar to that of the pure liquid crystal. The second term, which is negligible at high temperatures but dominates at low temperatures, has a shape given by the thermal term squared and describes the static fluctuations due to random fields induced by confinement in the gel. The correlation lengths appearing in the thermal and disorder terms are the same and show strong variation with gel density at low temperatures. The temperature dependence of the amplitude of the static fluctuations further suggests that nematic susceptibility become suppressed with increasing quenched disorder. The results overall are well described by a mapping of the liquid crystal-aerosil system into a three dimensional XY model in a random field with disorder strength varying linearly with the aerosil density.Comment: 14 pages, 13 figure

Hydrogen-bonded Silica Gels Dispersed in a Smectic Liquid Crystal: A Random Field XY System

The effect on the nematic to smectic-A transition in octylcyanobiphenyl (8CB) due to dispersions of hydrogen-bonded silica (aerosil) particles is characterized with high-resolution x-ray scattering. The particles form weak gels in 8CB creating a quenched disorder that replaces the transition with the growth of short range smectic correlations. The correlations include thermal critical fluctuations that dominate at high temperatures and a second contribution that quantitatively matches the static fluctuations of a random field system and becomes important at low temperatures.Comment: 10 pages, 4 postscript figures as separate file

Cerebellum Transcriptome of Mice Bred for High Voluntary Activity Offers Insights into Locomotor Control and Reward-Dependent Behaviors.

The role of the cerebellum in motivation and addictive behaviors is less understood than that in control and coordination of movements. High running can be a self-rewarding behavior exhibiting addictive properties. Changes in the cerebellum transcriptional networks of mice from a line selectively bred for High voluntary running (H) were profiled relative to an unselected Control (C) line. The environmental modulation of these changes was assessed both in activity environments corresponding to 7 days of Free (F) access to running wheel and to Blocked (B) access on day 7. Overall, 457 genes exhibited a significant (FDR-adjusted P-value < 0.05) genotype-by-environment interaction effect, indicating that activity genotype differences in gene expression depend on environmental access to running. Among these genes, network analysis highlighted 6 genes (Nrgn, Drd2, Rxrg, Gda, Adora2a, and Rab40b) connected by their products that displayed opposite expression patterns in the activity genotype contrast within the B and F environments. The comparison of network expression topologies suggests that selection for high voluntary running is linked to a predominant dysregulation of hub genes in the F environment that enables running whereas a dysregulation of ancillary genes is favored in the B environment that blocks running. Genes associated with locomotor regulation, signaling pathways, reward-processing, goal-focused, and reward-dependent behaviors exhibited significant genotype-by-environment interaction (e.g. Pak6, Adora2a, Drd2, and Arhgap8). Neuropeptide genes including Adcyap1, Cck, Sst, Vgf, Npy, Nts, Penk, and Tac2 and related receptor genes also exhibited significant genotype-by-environment interaction. The majority of the 183 differentially expressed genes between activity genotypes (e.g. Drd1) were under-expressed in C relative to H genotypes and were also under-expressed in B relative to F environments. Our findings indicate that the high voluntary running mouse line studied is a helpful model for understanding the molecular mechanisms in the cerebellum that influence locomotor control and reward-dependent behaviors

Invariants of Lie algebras extended over commutative algebras without unit

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.Comment: v3: added a footnote on p.10 about a wrong derivation of the correct statemen

Subtraction terms for one-loop amplitudes with one unresolved parton

Fully differential next-to-next-to-leading order calculations require a method to cancel infrared singularities. In a previous publication, I discussed the general setup for the subtraction method at NNLO. In this paper I give all subtraction terms for electron-positron annihilation associated with one-loop amplitudes with one unresolved parton. These subtraction terms are integrated within dimensional regularization over the unresolved one-particle phase space. The results can be used with all variants of dimensional regularization (conventional dimensional regularization, the 't Hooft-Veltman scheme and the four-dimensional scheme).Comment: 27 page