Bilinear and quadratic variants on the Littlewood-Offord problem

If f(x ₁, …, x _n ) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a ₁, …,a _n , what is the maximum number of sums of the form ±a ₁ ± a ₂ ± … ± a _n which take on any single value? Here we consider the case where f is either a bilinear form or a quadratic form. For the bilinear case, we show that the only forms having concentration significantly larger than n ^?1 are those which are in a certain sense very close to being degenerate. For the quadratic case, we show that no form having many nonzero coefficients has concentration significantly larger than n ^?1/2. In both cases the results are nearly tight.     

Some results on the capitulation problem

Let K be an unramified abelian extension of a number field F with Galois group G. K corresponds to a subgroup H of the ideal class group of F. We study the subgroup J of ideal classes in H which become trivial in K. There is an epimorphism from the cohomology group H^?1(G, Cl_K) to J which is an isomorphism if G is cyclic; Cl_K is the ideal class group of K. Some results on the structure of J and Cl_K are obtained.     

Some results on the Oberwolfach problem

Aequationes mathematicae -     

Some results on the spectral reconstruction problem

This paper is concerned with the spectral version of the reconstruction conjecture: Whether a graph with n>2 vertices is determined (up to isomorphism) by the collection of its spectrum and the spectrum of its vertex-deleted graphs? Some positive results as well as a method for constructing counterexamples to the problem are provided.     

The Littlewood-Offord problem and invertibility of random matrices

We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n−1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables Xk and real numbers ak, determine the probability p that the sum k_∑akXk lies near some number v. For arbitrary coefficients ak of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p.     

Some results on the entire function sharing problem

Our main result is as follows: let f and a be two entire functions such that \(\max \{ \rho _2 (f),\rho _2 (a)\} < \tfrac{1} {2}\) . If f and f ^(k) a CM, and if ρ(a ^(k) ? a) < ρ(f ? a), then f ^(k) ? a = c(f ? a) for some nonzero constant c. This result is applied to improve a result of Gundersen and Yang.     

Some dichotomy results for the quenching problem

It is shown that any solution to the semilinear problem{ ut = uxx + δ(1-u)-p , (x, t) ∈ (-1 , 1) × (0 , T ), u( ±1 , t) = 0, t ∈ (0 , T ), u(x, 0) = u0(x) 1, x ∈ [ 1 , 1] either touches 1 in finite time or converges smoothly to a steady state as t →∞. Some extensions of this result to higher dimensions are also discussed.     

Some new results on subset sums

Let n be a large integer and A be a subset of [n]={1,…,n}. The set SA is the collection of the subset sums of A. In this note, we discuss new results (and proofs) on few well-known problems concerning SA. In particular, we improve an estimate of Alon and Erd?s concerning monochromatic representations.     

Some new results on transition probability

In this paper, we study the basic properties of stationary transition probability of Markov processes on a general measurable space （E, δ）, such as the continuity, maximum probability, zero point, positive probability set,standardization, and obtain a series of important results such as Continuity Theorem, Representation Theorem, Levy Theorem and so on. These results are very useful for us to study stationary tri-point transition probability on a general measurable space （E, δ）. Our main tools such as Egoroff＇s Theorem, Vitali-Hahn-Saks＇s Theorem and the theory of atomic set and well- posedness of measure are also very interesting and fashionable.     

Some new results on weak compactness

We consider the scattering problem in Lsu2$\text{Rsu3}$ for Schrödinger operators with momentum-dependent interactions, i.e., for the pair of Hamiltonians H₀ = ? Δ and H = H₀ + T, where T is a pseudodifferential operator. The existence of the wave operators is proved by estimating the integrals appearing in the CookHack convergence criterion, and their asymptotic completeness is established under more restrictive conditions on the symbol of T by using a local trace criterion.     

Some new results on simulation functions

In this paper, we consider, discuss, and update some recent results on simulation functions established by several authors. By using one lemma of Radenovi? et al. (Bull. Iran. Math. Soc., 2012, 38 (3), 625–645), we suggest much shorter and nicer proofs of some statements than the ones available in the literature.     

Some new results on sums of primes

We study sums of primes by means of a generalization of Euler’s prime number theorem and use the values of the Riemann zeta function for the approximation. We also give the truncation error of these approximations.     

Some new results on planar harmonic mappings

This survey paper contains some new results on the Landau theorem, Bloch theorem and Schwarz-Pick lemma for planar harmonic mappings.     

Some results on a heat conduction problem by Myshkis

An infinite homogeneous d-dimensional medium initially is at zero temperature. A heat impulse is applied at the origin, raising the temperature there to a value greater than a constant value u₀>0. The temperature at the origin then decays, and when it reaches u₀, another equal-sized heat impulse is applied at a normalized time τ₁=1. Subsequent equal-sized heat impulses are applied at the origin at the normalized times τ_n, n=2,3,…, when the temperature there has decayed to u₀. This sequence of normalized waiting times τ_n can be defined recursively by a difference equation and its asymptotic behavior was known recently. This heat conduction problem was first studied in [J. Difference Equations Appl. 3 (1997) 89–91].

A natural subsequent question is what happens if the problem is set in a finite region, like in a laboratory, with the temperature at the boundary being kept zero forever. In this paper we obtain the asymptotic behavior of the heating times for the one-dimensional case.     

Some new results on integration for multifunction

It has been proven in Di Piazza and Musia? (Set Valued Anal 13:167–179, 2005, Vector measures, integration and related topics, Birkhauser Verlag, Basel, vol 201, pp 171–182, 2010) that each Henstock–Kurzweil–Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable (Theorem 3.4). Moreover, in case of strongly measurable (multi)functions, a characterization of the Birkhoff integrability is given using a kind of Birkhoff strong property.     

Some new results on the class of AM-compact operators

We establish some new necessary and sufficient conditions under which each regular operator is AM-compact if and only if its adjoint is AM-compact. Also, we give some consequences.     

Some asymptotic results for the combination of evidence problem

The combination of evidence problem is treated here as the construction of a posterior possibility function (or probability function, as a special case) describing an unknown state parameter vector of interest. This function exhibits the appropriate components contributing to knowledge of the parameter, including conditions or inference rules, relating the parameter with observable characteristics or attributes, and errors or confidences of observed or reported data. Multivalued logic operators - in particular, disjunction, conjunction, and implication operators, where needed – are used to connect these components and structure the posterior function. Typically, these operators are well-defined for only a finite number of arguments. Yet, often in the problem at hand, a number of observable attributes represent probabilistic concepts in the form of probability density functions. This occur, for example, for attributes representing ordinary numerical measurements- as opposed to those attributes representing linguistic-based information, where non-probabilistic possibility functions are used. Thus the problem of discretization of probabilistic attributes arises, where p.d.f.'s are truncated and discretized to probability functions. As the discretization process becomes finer and finer, intuitively the posterior function should better and better represent the information available. Hence, the basic question that arises is: what is the limiting behavior of the resulting posterior functions when the level of discretization becomes infinitely fine, and, in effect, the entire p.d.f.'s are used?It is shown in this paper that under mild analytic conditions placed upon the relevant functions and operators involved, nontrivial limits in the above sense do exist and involve monotone transforms of statistical expectations of functions of random variable corresponding to the p.d.f.'s for the probabilistic attributes.