Riesz transforms and multipliers for the Grushin operator

We show that Riesz transforms associated to the Grushin operator G = ?Δ ? |x|²～ _t ² are bounded on L ^p (? ⁿ⁺¹). We also establish an analogue of the Hörmander-Mihlin Multiplier Theorem and study Bochner-Riesz means associated to the Grushin operator. The main tools used are Littlewood-Paley theory and an operator-valued Fourier multiplier theorem due to L. Weis.     

Characterization and catalytic performances of three-dimensional mesoporous FeSBA-1 catalysts

Iron substituted cubic cage type mesoporous molecular sieves (FeSBA-1) were synthesized for the first time in a highly acidic media using cetyltriethylammonium bromide as a template. The amount of Fe incorporation in SBA-1 can easily be controlled by the simple adjustment of the molar hydrochloric acid-to-silicon ratio. All the materials were unambiguously characterized by AAS, XRD, N2 adsorption, UV-Vis DRS, XPS, and ESR spectroscopy. The results from AAS, XRD, and N2 adsorption reveal that the iron atom can be incorporated in the framework of SBA-1 matrix without altering the structural order and the textural parameters. The nature and the coordination of iron atoms were extensively studied by XPS spectroscopy, and the results revealed that most of the iron atoms in FeSBA-1 are in +3 coordination state. UV-Vis DRS and ESR studies confirmed that the majority of the Fe atoms in FeSBA-1 exist in a tetrahedral coordination environment (most probably occupying framework positions). tert-Butylation of phenol employing tert-butanol as the alkylation agent was carried out over FeSBA-1 catalysts with different iron content and the results are compared with one-dimensional mesoporous catalysts. The influence of various reaction parameters such as reaction temperature, reactant feed ratio, weight hourly space velocity, and time-on-stream affecting the activity and selectivity of FeSBA-1 were also studied. Under the optimized reaction conditions, the FeSBA-1(36) catalyst showed superior catalytic performance for the tert-butylation of phenol as compared to the uni-dimensional mesoporous catalysts.     

Composition of variable co-efficient singular integral operators

In this paper we give a formula for composition of two singular integral operators with variable co-efficients by explicitly calculating the lower order terms. Also we discuss the boundedness of the lower order terms inL ^p-spaces.     

Some uncertainty inequalities

We prove an uncertainty inequality for the Fourier transform on the Heisenberg group analogous to the classical uncertainty inequality for the Euclidean Fourier transform. Inequalities of similar form are obtained for the Hermite and Laguerre expansions.     

Studies on thermal stability and knot formation in the TG curves of Co(III) model complexes

TG studies of vitamin B₁₂ coenzyme model complexes, BrCo(diacetylmonoximeiminatodiacetylmonoximatoiminobenzene-1,6)L have been carried out with different heating rates and sample sizes. The trans ligand, L, is bromine, imidazole, methylimidazole, pyridine or triphenylphosphine. The decompositions of the complexes (except for the triphenylphosphine complex) are associated with the formation of a knot at the point of completion of the inflection of the TG curve at a heating rate of 20 deg/min; such a phenomenon is not observed for rates of 10 and 5 deg/min. The formation of a knot with increase in weight and decrease in temperature is attributed to the transition from high-energy six-coordinate Co(III) species to low-energy five-coordinate Co (II) species. The triphenylphosphine complex is found to deviate from these characteristics of decomposition.

---

Zusammenfassung TG-Untersuchungen von Vitamin-B₁₂-Coenzymkomplexen — BrCo (Diacetylmonoxim-iminato-diacetylmonoximato-iminobenzol-1,6)L wurden bei verschiedenen Aufheizgeschwindigkeiten und mit unterschiedlichen Probenmengen ausgeführt. Der trans-Ligand L ist mit Brom, Imidazol, Methylimidazol, Pyridin bzw. Triphenylphosphin modifiziert. Mit Ausnahme des Triphenylphosphinkomplexes geht die Zersetzung der Komplexe bei einer Aufheizgeschwindigkeit von 20 °/min mit der Ausbildung einer Schleife an der Stelle des TG Kurve einher, an der die Inflektion beendet ist. Solch ein Effekt ist nicht bei Aufheizgeschwindigkeiten von 10 und 5 °/min zu beobachten. Das eten einer Schleife mit zunehmenden Gewicht und abnehmender Temperatur ist der Umwandlung von 6-fach koordinierten Co(III)-Species hoher Energie in 5-fach koordinierte Co (II) -Species niedriger Energie zuzuschreiben. Der Triphenylphosphinkomplex weicht von dieser Zersetzungscharakteristik ab.

---

- ₁₂, r(- - -- -1,6)L, . - L , , , . , , 20°/. 10 5°/. () ((II) . , .

---

---

The author acknowledges the help of Mr. V. Alexander, who provided gift coenzyme model complex samples.     

Holomorphic Sobolev spaces associated to compact symmetric spaces

Using Gutzmer's formula, due to Lassalle, we characterise the images of Sobolev spaces under the Segal-Bargmann transform on compact Riemannian symmetric spaces. We also obtain necessary and sufficient conditions on a holomorphic function to be in the image of smooth functions and distributions under the Segal-Bargmann transform.     

Synthetic, spectroscopic and structural studies of two novel phosphanyl(organyl)borane compounds

The titanium (II) complex, Cp₂Ti(HBcat)₂, catalyzes the hydroboration of diphenylvinylphosphine by catecholborane and pinacolborane to afford exclusive anti-Markovnikov phosphanyl(organyl)boranes cleanly, rapidly and in good yields. Spectroscopic and X-ray studies of the phosphanyl(organyl)borane products show no evidence of boron-phosphorus interaction, indicating that the compounds preclude formation of Lewis pairs.     

On the equisummability of Hermite and Fourier expansions

We prove an equisummability result for the Fourier expansions and Hermite expansions as well as special Hermite expansions. We also prove the uniform boundedness of the Bochner-Riesz means associated to the Hermite expansions for polyradial functions.     

Generation of extended trains of compressed picosecond pulses with stable energy and time characteristics in a ND:YAG laser with passive negative feedback

A comparative analysis of the parameters of picosecond laser pulses generated under conditions of pure passive mode-locking and passive negative feedback was made. A twofold pulse compression (from 31 to 13.9 ps) was obtained. For the case of generation of extended flattened pulse trains under conditions of passive negative feedback, a considerable decrease in energy (from 24 to 5.7%) and time (from 20 to 3%) instabilities was achieved.