Martingale property of empirical processes

It is shown that for a large collection of independent martingales, the martingale property is preserved on the empirical processes. Under the assumptions of independence and identical finite-dimensional distributions, it is proved that a large collection of stochastic processes are martingales essentially if and only if the empirical processes are also martingales. These two results have implications on the testability of the martingale property in scientific modeling. Extensions to submartingales and supermartingales are given.

     

BERNOULLI CONVOLUTIONS ASSOCIATED WITH CERTAIN NON-PISOT NUMBERS

1　IntroductionForany 0 <λ <1 ,letνλ　denotethedistributionof ∞n=0 εnλn wherethecoefficientsεnareeither0or1 ,chosenindependentlywithprobability12 foreach .Itistheinfiniteconvo lutionproductofthedistributions 12 (δ0 +δλn) ,givingrisetotheterm“infiniteBernoulliconvolution”orsimply“Bernoulliconvolution” .TheBernoulliconvolutioncanbeexpressedasaself similarmeasureνλsatisfyingtheequationνλ =12 νλ φ- 10 + 12 νλ φ- 11,( 1 .1 )whereφ0 (x) =λxand φ1(x) =λx + 1 .Thisme…     

On Ramsey numbers of uniform hypergraphs with given maximum degree

For every ?>0 and every positive integers Δ and r, there exists C=C(?,Δ,r) such that the Ramsey number, R(H,H) of any r-uniform hypergraph H with maximum degree at most Δ is at most C|V(H)|^1+?.     

Hidden structure in the randomness of the prime number sequence?

We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy–Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal.     

Finding numerical solution of linear partial differential equations of first order with real coefficients

Applying Bittner's operational calculus we present a method to give approximate solutions of linear partial differential equations of first order

The effectivity of solvents as electron pair donors

The effectivity of solvents as electron pair donors, their donicity as expressed by their donor numbers DN, is reexamined. The linear dependence of the enthalpy and the Gibbs free energy for the reaction of donor solvents with antimony pentachloride is affirmed. Extension of the DN scale by other measured quantities, via their linear correlations for sets of solvents where both kinds of data are known is applied to Drago's E-C scale, Kamlet's scale, and Koppel's B scale. This extension, added to previous extensions employing Selbin's D_{I, II} and Popov's ²³Na, produces a DN scale for 170 solvents. The relation of DN measured for isolated solvent molecules in 1,2-dichloroethane to DN values measured for bulk solvents is examined by means of the scaled particle theory and solute-solvent interaction terms. A scale of normalized donor numbers DN ^N is presented. These values are practically the same as the scale, and for those solvents for which DN values are presented here and values are lacking, they can serve in their place for those preferring to use the scale for donor solvents.     

Doubly excited3Se,3De and3Ge states of two-electron atomic systems

Summary Time-dependent perturbation theory has been applied to calculate the doubly excited triplet statesNsns:³S^e,Npnp:³D^e andNdnd:³G^e (N=2, 3, 4,n=N+1, ... ,5) for He, Li⁺, Be²⁺ and B³⁺. A time-dependent harmonic perturbation causes simulataneous excitation of both the electrons with a change of spin state. The doubly excited energy levels have been identified as the poles of an appropriately constructed linearized variational functional with respect to the driving frequency. In addition to the transition energies, effective quantum numbers of these doubly excited states have been calculated and analytic representations of their wave functions are obtained. These are utilized to estimate the Coulomb repulsion term for these states which checks the consistency of the wave functions. These wave functions may also be used for calculating other physical properties of the systems.     

Clustering of nucleosides in the presence of alkali metals: Biologically relevant quartets of guanosine,deoxyguanosine and uridine observed by ESI-MS/MS

Electrospray ionization (ESI) mass spectra of nucleosides, recorded in the presence of alkali metals, display alkali metal ion-bound quartets and other clusters that may have implications for understanding non-covalent interactions in DNA and RNA. The tetramers of guanosine and deoxyguanosine and also their metaclusters (clusters of clusters), cationized by alkali metals, were observed as unusually abundant magic number clusters. The observation of these species in the gas phase parallels previous condensed-phase studies, which show that guanine derivatives can form quartets and metaclusters of quartets in solution in the presence of metal cations. This parallel behavior and also internal evidence suggest that bonding in the guanosine tetramers involves the bases rather than the sugar units. The nucleobases thymine and uracil are known to form magic number pentameric adducts with K+, Cs+ and NH4+ in the gas phase. In sharp contrast, we now show that the nucleosides uridine and deoxythymidine do not form the pentameric clusters characteristic of the corresponding bases. More subtle effects of the sugars are evident in the fact that adenosine and cytidine form numerous higher order clusters with alkali metals, whereas deoxyadenosine and deoxycytidine show no clustering. It is suggested that hydrogen bonding between the bases in the tetramers of dG and rG are the dominant interactions in the clusters, hence changing the ribose group to deoxyribose (and vice versa) generally has little effect. However, the additional hydroxyl group of RNA nucleosides enhances the non-selective formation of higher-order aggregates for adenosine and cytidine and results in the lack of highly stable magic number clusters. Some clusters are the result of aggregation in the course of ionization (ESI) whereas others appear to be intrinsic to the solution being examined.     

Range-Renewal Processes: SLLNs and Power Laws*

Given n samples(viewed as an n-tuple) of a γ-regular discrete distribution π,in this article the authors concern with the weighted and unweighted graphs induced by the n samples. They first prove a series of SLLN results(of Dvoretzky-Erd¨os’ type). Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1 + γ. They also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distribu...     

The peak and descent statistics over ballot permutations

A ballot permutation is a permutation π such that in any prefix of π the descent number is not more than the ascent number. By using a reversal-concatenation map, we (i) give a formula for the joint distribution (pk, des) of the peak and descent statistics over ballot permutations, (ii) connect this distribution and the joint distribution (pk, des) over ordinary permutations in terms of generating functions, and (iii) confirm Spiro's conjecture which finds the equidistribution of the descent statistic for ballot permutations and an analogue of the descent statistic for odd order permutations.