Corrigendum to the paper “Hermitian forms over division algebras over real function fields”

In the previous paper [T] we gave a classification of hermitian forms over the real function fieldk=R(t) and its completionsk _v with respect to valuationsv trivial onR. Unfortunately in the local case the arguments given for cases A and D, in general, were not correct. Therefore the resulting local and local-global classifications obtained were incorrect. I would like also to thank Dr. D. Hoffmann for pointing out these mistakes and the referee for useful comments. Here we would like to make necessary corrections to [T]. We keep the same notation used there, except that in the first paragraph,J is not the standard involution of a quaternion division algebraD (with basis {1,i,j,ij}). All hermitian forms will be hermitian forms with respect toJ, with values inD.     

A characterization of the exponential distribution

Summary A necessary and sufficient condition is given under which the minimum ofX _k /a _k ,k=1, 2, ...,n has the same distribution asX ₁, whereX's are i.i.d. positive random variables anda's are given positive constants. The Institute of Statistical Mathematics     

On the Cheney and Sharma Operator

In this paper, we consider the double-indexed Bernstein power series operatorP_{a, t}introduced by Cheney and Sharma and related to the classical Laguerre polynomials. We obtain sharp estimates of the first two moments which allow us to prove that, when acting on a continuous functionf, the convergence ofP_{a, t}ftofis uniform on the whole interval [0, 1]. Moreover, we show that the rates of convergence depend on the way in whicht/agoes to 0. We also show thatP_{a, t}preserves monotonicity and global smoothness. Finally, we consider the monotonicity properties ofP_{a, t}with respect to both parameters. To achieve the mentioned results, we use a probabilistic approach based on the representation of the operator in terms of a suitable multi-indexed stochastic process. The path properties and the martingale-type properties of this process are key points to give short proofs.     

Coincidence of Two Classes of Universal Laurent Series

There exist functions, called U.L.S. (Universal Laurent Series), holomorphic on finitely connected domains Ω in C, whose Laurent-type partial sums approximate everything we can hope for, on compact subsets outside Ω ∪ {a ₁,…,a}, for certain prescribed points a ₁,…,a _k. In this paper we prove that, under additional assumptions, for every U.L.S. there exists a subsequence of its Laurent-type partial sums, which converges to the function itself in the whole of Ω and which approximates everything we can hope for outside Ω ∪ {a ₁,…,a, _k}.     

Some bounds for coefficients of schlicht pseudo-conformal mappings

Summary The author considers a schlicht pseudo-conformal mapping of a domainB in the (z ₁ , z ₂ )-space onto the Reinhardt circular domainC in the (ξ ₁ , ξ ₂ )-space by a pair of functions (see (1)§ 1). The domainB is assumed to include the bicylinder ((2)§ 2) and to omit four planes z_k=a_k, z_k=b_k, k=1, 2. Upper bounds for a sequence of the coefficients _μv ^(k) of the developments (1) are given, see p. 304. The upper bounds depend only on a_k, b_k, k=1, 2, and on the radii r_k of the bicylinder ((2)§ 2). The bounds are obtained by using the method of the kernel function. The result can be considered as an analogue to the inequalities of Grunsky in the theory of functions of one complex variable. To Enrico Bompiani on his scientific Jubiles This paper was prepared under the sponsorship of the N. S. F.     

Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics

A generator of the complex algebra within the framework of general formulation obeys the quadratic equation of the type e² = a₁e − a₀. In this paper we construct the general complex algebras of the n-th order where the generators obey n-order polynomial equation of the type eⁿ = a_{n - 1}e^{n - 1} − a_{n - 2}e^{n - 2} + ... + (−)^{n + 1}a₀, with real coefficients a_k, k = 0, 1, ... n − 1. This algebra induces a generalized trigonometry ((n + 1)-gonometry), subyacent to the n-th order oscillator model and to the n-th order Hamilton equations.     

Erdos-Ko-Rado Theorems of Labeled Sets

For k = (k1, ··· , kn) ∈ Nn, 1 ≤ k1 ≤···≤ kn, let Lkr be the family of labeled r-sets on k given by Lkr := {{(a1, la1), ··· , (ar, lar)} : {a1, ··· , ar} ■[n],lai ∈ [kai],i = 1, ··· , r}. A family A of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.     

On the determination and construction of optimal row-column designs having unequal row and column sizes

In this paper we consider experimental situations requiring usage of a row-column design where v treatments are to be applied to experimental units arranged in b ₁ rows and b ₂ columns where row i has size k _1i , i=1,..., b ₁ and column j has size k _2j , j=1,..., b ₂. Conditions analogous to those given in Kunert (1983, Ann. Statis., 11, 247–257) and Cheng (1978, Ann. Statist., 6, 1262–1272) are given which can often be used to establish the optimality of a given row-column design from the optimality of an associated block design. In addition, sufficient conditions are derived which guarantee the existence of an optimal row-column design which can be constructed by appropriately arranging treatments within blocks of an optimal block design.Visiting from the Indian Statistical Institute.     

On the full automorphism group of a graph

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ _p wrZ _p , almost all Cayley graphs ofG have automorphism group isomorphic toG.     

Polytopes and the mean value property

LetP be any (not necessarily convex nor connected) solid polytope in then-dimensional Euclidean space ℝ_n, and letP(k) be thek-skeleton of P. LetH _P(k) be the set of all continuous functions satisfying the mean value property with respect toP (k). For anyk = 0,1,...,n, we show thatH _P(k) is a finite-dimensional linear space of polynomials. This settles an open problem posed by Friedman and Littman [37] in 1962. Moreover, we show that ifP admits ample symmetry, thenH _P(k) is a finite-dimensional linear space of harmonic polynomials. Some interesting examples are also given     

L 1-convergence of double cosine- and Walsh-Fourier series

We prove the convergence inL ¹([−gp, π)²)-norm of the double Fourier series of an integrable functionf(x, y) which is periodic and even with respect tox andy, with coefficientsa _jk satisfying certain conditions of Hardy-Karamata kind, and such thata _jk logj logk→0 asj, k→∞. These sufficient conditions become quite natural in particular cases. Then we extend these results to the convergence of double Walsh-Fourier series inL ¹ (0, 1)²)- norm. As a by-product, we obtain Tauberian conditions ensuring the convergence of a double numerical series provided it is Cesàro summable. This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # 234.     

0,1 distribution in the highest level sequences of primitive sequences over<Emphasis Type="Italic">Z</Emphasis><Subscript>2e</Subscript>

In this paper, we discuss the 0,1 distribution in the highest level sequence a_e-1 of primitive sequence over Z_2e generated by a primitive polynomial of degreen. First we get an estimate of the 0,1 distribution by using the estimates of exponential sums over Galois rings, which is tight fore relatively small ton. We also get an estimate which is suitable fore relatively large ton. Combining the two bounds, we obtain an estimate depending only onn, which shows that the largern is, the closer to 1/2 the proportion of 1 will be.     

Problem of the maximum of the product of the conformal radii of nonoverlapping domains in a circle

Let a_k, k=1,2,3, be distinct points of the circle U={z:¦z¦<1}, a_3+k=1/¯a_k, k=1,2,3. Let D₁,...,D₆ be a system of nonoverlapping simply connected domains D₁,...,D₆ on,a_k D_k, k=1,...,6. Let R(D_k,a_k) be the conformal radius of the domain D_k with respect to the point a_k. One formulates the following theorem. For any points a_k U, k=1,2,3, and any system of the indicated domains one has the sharp inequality One points out all the cases when equality prevails in (1). One indicates the main steps of the proof of this theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 99–113, 1983.     

Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere

In this paper we study eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere and obtain estimates for eigenvalues. In particular, the optimal result of Cheng and Yang (Math Ann 331:445–460, 2005) is included in our ones. In order to prove our results, we introduce 2(l + 1) functions a _i and b _i , for i = 0, 1, . . . , l and two operators μ and η. First of all, we study properties of functions a _i and b _i and the operators μ and η. By making use of these properties and introducing k free constants, we obtain estimates for eigenvalues.     

A higher-order large-scale regularity theory for random elliptic operators

We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. The large-scale regularity of a-harmonic functions is encoded by Liouville principles: The space of a-harmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale C^k,α-regularity theory, which in the present work is developed in the form of a corresponding C^k,α-“excess decay” estimate: For a given a-harmonic function u on a ball B_R, its energy distance on some ball B_r to the above space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r₀.

Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization a of the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kth-order correctors” and thereby the space of a-harmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle.     

The problem of product of conformal radii of nonoverlapping domains

Let a₁, a₂, a₃, b be distinct points in and let D be the family of all triples of nonoverlapping domains D₁, D₂, D₃ in \ {b} such that a_k∈ D_k, k=1,2,3. For this family we consider the problem on the maximum of the functional I=R₁R₂R₃, where R_k=R(D_k, a_k) is the conformal radius of D_k with respect to a_k. Geometrical properties of the extremal triple of domains are described. We prove that the maximum of I monotonically depends on the position of the point b and find the maximum in some special cases Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 114–128 Translated by N. Yu. Netsvetaev     

On cubic polynomials I. Hua's estimate of exponential sums

Hua andChen gave estimates of sums wheree(z)=e ^2iz and is a polynomial of the typef(x)/q wheref(x)=a _k x ^k +...+a ₁ x with integer coefficients having gcd (q, a _k ,...,a ₁)=1 But no good estimates hold for these sums whenq is small in comparison tok. We therefore consider here a related but different class of polynomials. Special emphasis is given to the cubic case.In subsequent papers of this series we shall deal with cubic exponential sums in many variables and withp-adic and rational zeros of systems of cubic forms.Partially supported by NSF contract NSF-MCS-8015356.     

On the structure of the sumsets

Let A be a set of nonnegative integers. For h≥2, denote by hA the set of all the integers representable by a sum of h elements from A. In this paper, we prove that, if k≥3, and A={a₀,a₁,…,a_k−1} is a finite set of integers such that 0=a₀<a₁<?<a_k−1 and (a₁,…,a_k−1)=1, then there exist integers c and d and sets C⊆[0,c−2] and D⊆[0,d−2] such that hA=C∪[c,ha_k−1−d]∪(ha_k−1−D) for all . The result is optimal. This improves Nathanson’s result: h≥max{1,(k−2)(a_k−1−1)a_k−1}.     

Numerical results on the Goldbach conjecture

The number of Goldbach partitions has been computed for all even numbers 350,000 and compared to well-known theoretical estimates. The random fluctuations are slowly decreasing and less than ± 5 per cent at the upper end of the interval. The number of partitions is given explicitly for 2 ⁿ ,n=3(1)22. Further, if for a givenN the smallest prime in all partitions of 2N isa=a(2N) we have also determineda ₁(2N ₁)<a ₂(2N ₂)<... withN ₁<N ₂<... such thatn<N _k impliesa(2n)<a _k (2N _k ) up to 2n=40,000,000.     

One class of singular complex-valued random variables of the Jessen-Wintner type

We study the structure of the distribution of a complex-valued random variable ξ = Σa _k ξ _k , where ξ _k are independent complex-valued random variables with discrete distribution and a _k are terms of an absolutely convergent series. We establish a criterion of discreteness and sufficient conditions for singularity of the distribution of ξ and investigate the fractal properties of the spectrum. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1653–1660, December, 1997.