On the Exceptional Set of Goldbach’s Problem in Short Intervals

Denote by E[X,X+H] the set of even integers in [X,X+H] that are not a sum of two primes (i.e. that are not Goldbach numbers). Here we prove that there exists a (small) positive constant H 3 X ^[7/24]+7dH\ge X^{\,{7\over24}+7\delta} we have |E(X,H) | << H^1-d/600\vert E(X,H) \vert \ll H^{1-\delta/600} .     

Determination of a neighborhood of the imaginary axis which is disjoint from the spectrum of a real polynomial

The distance of the spectrum of f from the imaginary axis is estimated for a real polynomial with roots in the right (or as a corollary, in the left) half plane: f:min Resp(f) –1/tr(H₁H¹ > 0 where [a _n+i–2j]i, j=1,n] andH ₁:=[a _k]K:=n+i+2j+1 i, j=1,nTranslated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 161–166, August, 1977.In conclusion, the author thanks S. M. Lozinskii for assistance.     

Upper Bounds for Neumann–Schatten Norms

Let H and H _aux be Hilbert spaces, H a nonnegative self-adjoint operator in H,,s>0 and J a bounded linear transformation from the Hilbert space D(H ^s/2) (equipped with the graph scalar product of H ^s/2) to H _aux. It is shown that the operator J(H+)^–t belongs to the Neumann–Schatten class of order p=2+2(u–t)/(t–s/2) provided s/2<t<u,t–s/2<u–t and J(H+)^–u is Hilbert–Schmidt operator. An upper bound for the pth order Neumann–Schatten norm of J(H+)^–t is derived. If J is a closed operator from D(H ^1/2) to H _aux and D(J)D(H) then there exists a unique self-adjoint operator H ^J in H such that D(H ^J )D(J) and ( . Conditions which are sufficient in order that the operator (H ^J +)^–1–(H+)^–1 is compact and conditions which are sufficient in order that the wave operators W ^±(H ^J ,H) exist and are complete are derived. Instead of (Jf,Jg)_aux also certain other perturbation terms, not by necessity nonnegative, are considered. The special case when H equals the operator (–) ^r in L ²(R ^d ) for any strictly positive real number and H ^J equals (–) ^r + for some suitably chosen measure is discussed in detail. In particular, new results on existence and completeness of the wave operators W ^±(–+,–) are obtained.     

Sine-Laplace equation,sink - Laplace equation and harmonic maps

The relationship between harmonic maps from R² to S², H², S^T,1, S^1,1(–1) and the ± sinh — Laplace, ± sine — Laplace equation is found respectively. Existence theorems of some boundary value problems for the above harmonic maps are obtained. In the cases of H², S^1,1(+1), S^1,1(–1) the results are global.Research supported partially by the Institute for Applied Mathematics, Sonderforschungsbereich 72 of the University of Bonn     

A Recursive Construction for New Symmetric Designs

We introduce a recursive construction of regular Handamard matrices with row sum 2h for h=±3ⁿ. Whenever q=(2h – 1)² is a prime power, we construct, for every positive integer m, a symmetric designs with parameters (4h²(q^m+1 – 1)/(q – 1), (2h² – h)q^m, (h² – h)q^m).     

Behavior of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions at the Points s=1 and s=1/2

Let S_k(N)⁺ be the set of primitive cusp forms of even weight k for Γ₀(N) and let L(s, sym ²f) be the symmetric square L-function L(s, f) of a form f ∈ S_k(N)⁺. The moments of the variable L(1, sym ²f), f ∈ S₂(N)⁺, are computed for N = p, and the corresponding limiting distribution is determined in N-aspect. Let f ∈ S_k(1)⁺, g ∈ S_l(1)⁺, and ω_f = Γ(k - 1)/(4π)^k-1 〈f, f〉. Asymptotic formulas for and as k → ∞ are obtained. Bibliography: 17 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 149–167.     

On the sum of a prime and a k-th power of prime in short intervals

Let H_k\mathcal{H}_{k} denote the set {n∣2|n, n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when X^{\frac1120(1-\frac12k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n ? \allowbreak H_k ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when X^{\frac1120(1-\frac1k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.     

Ein Einschliessungsverfahren für Lösungen Fehlerbehafteter Linearer Gleichungen

In a partially ordered space, the method x_n+1 = L⁺x _n ⁺ – N⁺x _n ^- – L^–y⁺ + N^– y _n ^- + r, y_n+1 = N⁺y⁺ – L⁺y _n ^- – N^–x _n ⁺ + L^–x^– + t of successive approximation is developed in order to enclose the solutions of a set of linear fixed point equations monotonously. The method works if only the inequalities x₀ y₀, x₀ x₁, y₁ y₀ related to the starting elements are satisfied. In finite-dimensional spaces suitable starting vectors can be computed if a sufficiently good approximation for the fixed points is known.

     

Quotient algebras of Stanley-Reisner rings and local cohomology

We study certain subcomplexes Δ′ of an arbitrary simplicial complex Δ such that H_mⁱ(k[Δ])-H_mⁱ(k[Δ′]) for any 0i<dim(k[Δ′]). Here, H_mⁱ(k[Δ]) is the ith local cohomology module of the Stanley-Reisner ring k[Δ] of Δ over a field k. Our technique is an elegant approach to one of the most generalized versions of the rank selection theorems of J. Munkres (1984, Michigan Math. J.31, 113–128, Theorem 6.4) and R. Stanley (1979, Trans. Amer. Math. Soc.249, 139–157, Theorem 4.3).     

Controllability of the wave equation with moving point control

This paper deals with approximate and exact controllability of the wave equation in finite time with interior point control acting along a curve specified in advance in the system's spatial domain. The structure of the control input is dual to the structure of the observations which describe the measurements of velocity and gradient of the solution of the dual system, obtained from the moving point sensor. A relevant formalization of such a control problem is discussed, based on transposition. For any given timeinterval [0,T] the existence of the curves providing approximate controllability inH _D ^–[n/2]–1 ()×H _D ^–[n/2]–1 () (wheren stands for the space dimension) is established with controls fromL ²(0,T; R ⁿ +1). The same curves ensure exact controllability inL ²() × H^–1() if controls are allowed to be selected in [L (0,T; R ⁿ+1)]. Required curves can be constructed to be continuous on [0,T).This work was supported in part by NSF Grant ECS 89-13773 and NASA Grant NAG-1-1081.     

Limit of infinitely strong coupling in quantum chromodynamics and the model of dominance of anomalies of quark loops (the DAQL model)

Within the framework of quantum chromodynamics in the limit of infinitely strong coupling a field-theoretic justification is given for the phenomenological model of dominance of anomalies of quark loops (the DAQL model) which successfully descrbides interaction process at low meson energies of the meson J^p=0⁺, 0^–, 1^– and 1⁺.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 145, pp. 86–108, 1985.     

An identity on the maximum of a set of random variables

It is shown that if X₁, X₂, …, X_n are symmetric random variables and max(X₁, …, X_n)⁺ = max(0, X₁, …, X_n), then E[max(X₁,…,X_n)⁺]=[max(X₁,X₁,+X₂,+X₁,+X₃,…X₁,+X_n)⁺], and in the case of independent identically distributed symmetric random variables, E[max(X₁, X₂)⁺] = E[(X₁)⁺] + (1/2)E[(X₁ + X₂)⁺], so that for independent standard normal random variables, E[max(X₁, X₂)⁺] = (1/√2π)[1 + (1/√2)].     

On Spin Models, Modular Invariance, and Duality

A spin model is a triple (X, W ⁺, W ^–), where W ⁺ and W ^– are complex matrices with rows and columns indexed by X which satisfy certain equations (these equations allow the construction of a link invariant from(X, W ⁺, W ^–) ). We show that these equations imply the existence of a certain isomorphism between two algebras and associated with (X, W ⁺, W ^–) . When is the Bose-Mesner algebra of some association scheme, and is a duality of . These results had already been obtained in [15] when W ⁺, W ^– are symmetric, and in [5] in the general case, but the present proof is simpler and directly leads to a clear reformulation of the modular invariance property for self-dual association schemes. This reformulation establishes a correspondence between the modular invariance property and the existence of spin models at the algebraic level. Moreover, for Abelian group schemes, spin models at the algebraic level and actual spin models coincide. We solve explicitly the modular invariance equations in this case, obtaining generalizations of the spin models of Bannai and Bannai [3]. We show that these spin models can be identified with those constructed by Kac and Wakimoto [20] using even rational lattices. Finally we give some examples of spin models at the algebraic level which are not actual spin models.     

Oxidation of aldehydes by aqueous Ce (IV) in nitric acid

The kinetics of oxidation of acetaldehyde (at 20°, 31° and 40° C.), propionaldehyde (at 15°, 25° and 35° C.) and chloral hydrate (40°, 50° and 60° C.) by ceric nitrate in nitric acid medium [(H⁺) ≈ 0 · 5?1 · 5 M] were studied. The reactions were followed by determining rates of ceric disappearance for variations in [Ce⁴⁺], [RCHO], [H⁺], μ, temperature, etc. The stoichiometry, ΔCe⁴⁺ / Δ CH₃CHO ≈ 2 was established. The reactions were found to be 2nd Order—first order each with respect to [Ce⁴⁺] and [RCHO]. No complex formation between Ce⁴⁺ and aldehydes was observed. Neutral Ce (NO₃)₄ (H₂O)₂ and diol of the aldehyde were assumed the active species. The rate and thermodynamic data were calculated and discussed. The rates of oxidation were correlated with the structures of the aldehydes.     

Unique representation of primitive factors of 2 n −1,n ODD,in certain quadratic forms

Primitive factorsq of 2 ⁿ –1,n odd, are either of the form 8m–1 or 8m+1. In the first case there exists a unique representation ofq in the quadratic forma ²–2b ², (a, b)=1,a andb odd,b<[q/2], and in the latter a unique representation ofq in the quadratic formc ²+2 ^j d², (c, d)=1,c andd odd,j even and 6. Thus the uniqueness ofq=a ²–2b ² orc ²+2 ^j d² exhibits a proof of the primality ofq.A program that determinesa, b, c, andd for anyq not exceeding 16 decimal digits is described, and as an example the 13-digit prime 4432676798593 (a primitive factor of 2⁴⁹–1) is uniquely represented by 1374273²+2¹⁴·12461².     

On the Approximation of Positive Functions by Power Series, II

We consider positive functionsh=h(x) defined forxR⁺₀. Conditions for the existence of a power seriesN(x)=∑ c_nxⁿ,c_n0, with the propertyd₁h(x)/N(x)d₂, x0,for some constantsd₁, d₂R⁺, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erd s and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientsc_n,n ₀, for the above defined power seriesNand to find suitable constantsd₁andd₂. We also study the power seriesH(x)=∑ xⁿ/u_n, where we setu_n=sup{xⁿ/h(x), x0}, forn ₀, and the relation betweenhandHconcerning the above stated inequalities.     

On a Class of Symmetric Balanced Generalized Weighing Matrices

Let q be a prime power and m a positive integer. A construction method is given to multiply the parametrs of an -circulant BGW(v=1+q+q ₂+·+q ^m , q ^m , q ^m –q ^m–1) over the cyclic group C _n of order n with (q–1)/n being an even integer, by the parameters of a symmetric BGW(1+q ^m+1, q ^m+1, q ^m+1–q ^m ) with zero diagonal over a cyclic group C _vn to generate a symmetric BGW(1+q+·+q ^2m+1,q ^2m+1,q ^2m+1–q ^2m) with zero diagonal, over the cyclic group C _n . Applications include two new infinite classes of strongly regular graphs with parametersSRG(36(1+25+·+25^2m+1),15(25)^2m+1,6(25)^2m+1,6(25)^2m+1), and SRG(36(1+49+·+49^2m+1),21(49)^2m+1,12(49)^2m+1,12(49)^2m+1).     

Partitioned matrices and Seidel convergence

Summary Without using spectral resolution, an elementary proof of convergence of Seidel iteration. The proof is based on the lemma (generalizing a lemma of P. Stein): If (A+A ^*)–B ^*(A+A ^*)B>0, whereB=–(P+L) ^–1 R,A=P+L (Lower)+R (upper), then Seidel iteration ofAX=Y ₀ converges if and only ifA+A ^*>0. This lemma has as corollaries not only the well-known results of E. Reich and Stein, but also applications to a matrix that can be far from symmetric, e.g.M=[A _ij ] ₁ ² , whereA ₂₁=–A ₁₂ ^* ,A ₁₁,A ₂₂ are invertible;A ₁₁ +A ₁₁ ^* =A₂₂+A ₂₂ ^* ; and the proper values ofA ₁₂ ^–1 A ₁₁,A ₁₂ ^*–1 A ₂₂ are in the interior of the unit disk.Supported under NSF GP 32527.Supported under NSF GP 8758.     

Ordered sets with small width and large jump number

The following result is proved in this note: For any positive integers w and t, if an ordered set P has jump number at least (t+1) ^w–1, then either the width of P is more than w, or P has a tower, i.e., a linear sum of pairs of noncomparable elements, of height more than t.