Linear Combinations of Composition Operators on the Fock-Sobolev Spaces

We study linear combinations of composition operators acting on the Fock-Sobolev spaces of several variables. We show that such an operator is bounded only when all the composition operators in the combination are bounded individually. In other words, composition operators on the Fock-Sobolev spaces do not possess the same cancelation properties as composition operators on other well-known function spaces over the unit disk. We also show the analogues for compactness and the membership in the Schatten classes. In particular, compactness and the membership in some/all of the Schatten classes turn out to be the same.     

Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions

We consider the conormal bundle of a Schubert variety \(S_I\) in the cotangent bundle \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) of the Grassmannian \({{\mathrm{\mathrm {Gr}}}}\) of \(k\) -planes in \({{\mathrm{\mathbb {C}}}}^n\) . This conormal bundle has a fundamental class \({\kappa _I}\) in the equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . Here \({{\mathrm{\mathbb T}}}=({{\mathrm{\mathbb {C}}}}^*)^n\times {{\mathrm{\mathbb {C}}}}^*\) . The torus \(({{\mathrm{\mathbb {C}}}}^*)^n\) acts on \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) in the standard way and the last factor \({{\mathrm{\mathbb {C}}}}^*\) acts by multiplication on fibers of the bundle. We express this fundamental class as a sum \(Y_I\) of the Yangian \(Y(\mathfrak {gl}_2)\) weight functions \((W_J)_J\) . We describe a relation of \(Y_I\) with the double Schur polynomial \([S_I]\) . A modified version of the \(\kappa _I\) classes, named \(\kappa '_I\) , satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) . This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on \({{\mathrm{\mathrm {Gr}}}}\) . The classes \((\kappa '_I)_I\) form a basis in the suitably localized equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . This basis depends on the choice of the coordinate flag in \({{\mathrm{\mathbb {C}}}}^n\) . We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.     

Application of doubly reflected BSDEs to an impulse control problem

We establish existence results for adapted solutions of infinite horizon backward stochastic differential equations with two reflected barriers. We also apply these results to get the existence of an optimal impulse control strategy for the infinite horizon impulse control problem. The properties of the Snell envelope reduce our problem to the existence of a pair of continuous processes.     

Potential-controlled filtering in quantum star graphs

We study the scattering in a quantum star graph with a Fülöp–Tsutsui coupling in its vertex and with external potentials on the lines. We find certain special couplings for which the probability of the transmission between two given lines of the graph is strongly influenced by the potential applied on another line. On the basis of this phenomenon we design a tunable quantum band-pass spectral filter. The transmission from the input to the output line is governed by a potential added on the controlling line. The strength of the potential directly determines the passband position, which allows to control the filter in a macroscopic manner. Generalization of this concept to quantum devices with multiple controlling lines proves possible. It enables the construction of spectral filters with more controllable parameters or with more operation modes. In particular, we design a band-pass filter with independently adjustable multiple passbands. We also address the problem of the physical realization of Fülöp–Tsutsui couplings and demonstrate that the couplings needed for the construction of the proposed quantum devices can be approximated by simple graphs carrying only δ$\delta$ potentials.     

Weak Hopf lemma for the invariant Laplacian and related elliptic operators

We obtain a weak version of the Hopf lemma for the invariant Laplacian on the unit ball of the complex n$n$-space. We also show that our result is sharp in some sense. Motivated by this result, we also consider a class of degenerate elliptic operators with the degeneracy depending on the distance to the boundary of the domain. We study the dependence of the validity of Hopf lemma on the degree of degeneracy of the operator. We show that Hopf lemma holds if the degeneracy is small and fails in general if the degeneracy is large. What is more interesting is the critical case for which we show that certain weak version of Hopf lemma holds.     

Neuronal activation increases the density of eukaryotic translation initiation factor 4E mRNA clusters in dendrites of cultured hippocampal neurons

Activity-dependent dendritic translation in CNS neurons is important for the synapse-specific provision of proteins that may be necessary for strengthening of synaptic connections. A major rate-limiting factor during protein synthesis is the availability of eukaryotic translation initiation factor 4E (eIF4E), an mRNA 5''-cap-binding protein. In this study we show by fluorescence in situ hybridization (FISH) that the mRNA for eIF4E is present in the dendrites of cultured rat hippocampal neurons. Under basal culture conditions, 58.7 ± 11.6% of the eIF4E mRNA clusters localize with or immediately adjacent to PSD-95 clusters. Neuronal activation with KCl (60 mM, 10 min) very significantly increases the number of eIF4E mRNA clusters in dendrites by 50.1 and 74.5% at 2 and 6 h after treatment, respectively. In addition, the proportion of eIF4E mRNA clusters that localize with PSD-95 increases to 74.4 ± 7.7% and 77.8 ± 7.6% of the eIF4E clusters at 2 and 6 h after KCl treatment, respectively. Our results demonstrate the presence of eIF4E mRNA in dendrites and an activity-dependent increase of these clusters at synaptic sites. This provides a potential mechanism by which protein translation at synapses may be enhanced in response to synaptic stimulation.     

B cell activation factor (BAFF) is a novel adipokine that links obesity and inflammation

B cell activation factor (BAFF) is a novel member of the TNF ligand superfamily, mainly produced by myeloid cells. BAFF has been shown to participate in B-cell survival and B- and T-cell maturation. BAFF expression in adipocytes has been recently demonstrated. In the current study, we verified that BAFF expression is increased during adipocyte differentiation. BAFF expression was augmented by TNF-α treatment and was decreased by rosiglitazone treatment. BAFF secretion in lean and in ob/ob mice sera were compared and smaller amount of BAFF was secreted in ob/ob mice. mRNA and protein expression were different between epididymal and visceral adipose tissue. BAFF expression was also increased in ob/ob mouse adipose tissue. We sought to identify known BAFF receptors (BAFF-R, BCMA, and TACI) in adipocytes, and determined that all three were present and upregulated during adipocyte differentiation. However, the expression of TACI was distinct from that of BAFF-R and BCMA under TNF-α and BAFF ligand treatment. BAFF-R and BCMA expression levels were upregulated under pro-inflammatory conditions, but TACI was reduced. Conversely, BAFF-R and BCMA expression levels were downregulated by rosiglitazone treatment, but TACI was increased. Taken together, our results suggest that BAFF may be a new adipokine, representing a link between obesity and inflammation.     

Micro-electrodeposition techniques for the preparation of small actinide counting sources for ultra-high resolution alpha spectrometry by microcalorimetry

Special considerations and techniques are desired for the preparation of small actinide counting sources. Counting sources have been prepared on metal disk substrates (planchets) with an active area of only 0.079 mm². This represents a 93.75% reduction in deposition area from standard electrodeposition methods. The actinide distribution upon the smaller planchet must remain thin and uniform to allow alpha particle emissions to escape the counting source with a minimal amount of self-attenuation. This work describes the development of micro-electrodeposition methods and optimization of the technique with respect to deposition time and current density for various planchet sizes.     

A generalization of Lucas polynomial sequence

In this paper, we obtain a generalized Lucas polynomial sequence from the lattice paths for the Delannoy numbers by allowing weights on the steps (1,0),(0,1) and (1,1). These weighted lattice paths lead us to a combinatorial interpretation for such a Lucas polynomial sequence. The concept of Riordan arrays is extensively used throughout this paper.