Weak convergence of nonparametric estimators of the multidimensional and multidimensional-multivariate renewal functions on Skorohod topology spaces

Statistical Inference for Stochastic Processes - This paper deals with the weak convergence of nonparametric estimators of the multidimensional and multidimensional-multivariate renewal functions...     

Positive Lyapunov exponents for a class of deterministic potentials

LetV() be a smooth, non-constant function on the torus and letT be a hyperbolic toral automorphism. Consider a discrete one dimensional Schrödinger operatorH, whose potential at sitej is given bygV _j =gV(T ^j ). We prove that wheng0 is small andg ^1/2 |E|2–g ^1/2 , the Lyapunov exponent for the cocycle generated byH-E is proportional tog ². The proof relies on a formula of Pastur and Figotin and on symbolic dynamics.     

Electric field effects on roto-vibrational transitions of13CD3OH1

We used a Stark-Optoacoustic cell and hybrid waveguide resonators to perform an Infrared and Far Infrared Stark Spectroscopy study on some transitions of¹³CD₃OH. Different behaviours of the transitions in the presence of a d.c. electric field were observed. The Stark splittings of six FIR laser lines ranging from 34 to 136 MHz/kVcm^–1 were determined. The analysis of the behaviour of the IR and FIR transitions in the presence of the external electric fields gives important and exclusive information on the levels involved in the transitions.Work Supported by FAPESP, CNPq, FAEP - Brazil, and CNR - Italy.     

Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties

We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV ₁ ,...,V _r in a toric varietyP and the system of differential operators annihilating the special generalized hypergeometric series ⁰ constructed from the fan . Using this generalized hypergeometric series, we propose conjectural mirrorsV ofV and the canonicalq-coordinates on the moduli spaces of Calabi-Yau manifolds.In the second part of the paper we consider some examples of Calabi-Yau 3-folds having Picard number >1 in products of projective spaces. For conjectural mirrors, using the recurrent relation among coefficients of the restriction of the hypergeometric function ⁰ on a special line in the moduli space, we determine the Picard-Fuchs equation satisfied by periods of this special one-parameter subfamily. This allows to obtain some sequences of integers which can be conjecturally interpreted in terms of Gromov-Witten invariants. Using standard techniques from enumerative geometry, first terms of these sequence of integers are checked to coincide with numbers of rational curves on Calabi-Yau 3-folds.     

Design considerations and computer modeling related to the development of molecular scaffolds and peptide mimetics for combinatorial chemistry

Summary A critical issue in drug discovery utilizing combinatorial chemistry as part of the discovery process is the choice of scaffolds to be used for a proper presentation, in a three-dimensional space, of the critical elements of structure necessary for molecular recognition (binding) and information transfer (agonist/ antagonist). In the case of polypeptide ligands, considerations related to the properties of various backbone structures (-helix, -sheets, etc.; , space) and those related to three-dimensional presentation of side-chain moieties (topography; (chi) space) must be addressed, although they often present quite different elements in the molecular recognition puzzle. We have addressed aspects of this problem by examining the three-dimensional structures of chemically different scaffolds at various distances from the scaffold to evaluate their putative diversity. We find that chemically diverse scaffolds can readily become topographically similar. We suggest a topographical approach involving design in chi space to deal with these problems.     

Constructing nonresidues in finite fields and the extended Riemann hypothesis

We present a new deterministic algorithm for the problem of constructing th power nonresidues in finite fields , where is prime and is a prime divisor of . We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed and , our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomial-time bound holds even if is exponentially large. More generally, assuming the ERH, in time we can construct a set of elements that generates the multiplicative group . An extended abstract of this paper appeared in Proc. 23rd Ann. ACM Symp. on Theory of Computing, 1991.

     

On a problem of H. N. Gupta

It is shown that the axiom For any points x, y, z such that y is between x and z, there is a right triangle having x and z as endpoints of the hypotenuse and y as foot of the altitude to the hypotenuse, when added to three-dimensional Euclidean geometry over arbitrary ordered fields, is weaker than the axiom Every line which passes through the interior of a sphere intersects that sphere.     

Splitting the Pasch axiom

On the basis of the theory ^– of Pasch-free 2-dimensional geometry, Pasch's axiom is shown to be equivalent to the conjunction of the following two axioms: In any right triangle the hypotenuse is greater than the leg and If AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is greater than the segment BD. This represents an attempt to split the Pasch axiom with respect to ^–. Only the question whether the second of the above two axioms is really weaker than Pasch's axiom, remains open.     

Dithiacrown Ether Substituted Porphyrazines: Synthesis, Single-Crystal Structure, and Control of Aggregation in Solution by Complexation of Transition-Metal Ions

The synthesis of novel magnesium, copper, and metal-free porphyrazines, peripherally substituted with dithia-7-crown-2 (MPz(7)), dithia-15-crown-5 (MPz(15)), and dithia-18-crown-6 (MPz(18)) macrocycles is reported. These compounds are prepared starting from dicyanoethylene containing crown ethers 3, 2(1), and 2(2), respectively, which contain sulfur as well as oxygen heteroatoms. The "crowned" porphyrazines bind silver(I) and mercury(II) perchlorates. UV/vis spectroscopy and electron paramagnetic resonance measurements reveal that addition of the transition-metal ions leads to dimerization of the porphyrazine complexes. In the case of the dithia-18-crown-6-substituted porphyrazines, the dimers break up to form monomeric 6:1 guest-host complexes when more than 2 equiv of the metal ion is added. The single-crystal structures of the crown ether 2(2) and the porphyrazine MgPz(18) are presented. Compound C(14)H(20)N(2)O(4)S(2) (2(2)) crystallizes in the monoclinic space group P2(1)/c with a = 10.9310(13) ?, b = 19.383(3) ?, c = 8.6976(14) ?, beta = 108.898(11) degrees, V = 1743.5(5) ?(3), and Z = 4. The structure refinement converged to R = 0.0366 and R(w) = 0.0504. Compound C(56)H(82)MgN(8)O(17)S(8) (MgPz(18)) crystallizes in the triclinic space group P&onemacr; with a = 9.584(3) ?, b = 17.672(2) ?, c = 19.620(4) ?, alpha = 84.904(14) degrees, beta = 85.21(2) degrees, gamma = 89.29(2) degrees, V = 3298.4(13) ?(3), and Z = 2. The structure refinement converged to R1 = 0.0839 and wR2 = 0.2196. The electrical properties of H(2)Pz(18) have been studied by complex impedance spectroscopy. The bulk electrical conductivity of this compound is approximately 1 order of magnitude higher than that of the corresponding 18-crown-6 phthalocyanine.     

Common supports as fixed points

A family S of sets in R ^d is sundered if for each way of choosing a point from rd+1 members of S, the chosen points form the vertex-set of an (r–1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R ^d , and for each partition (S , S ), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S from the members of S . This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.