Small gaps between primes

Combining Goldston-Yildirim’s method on k-correlations of the truncated von Mangoldt function with Maier’s matrix method, we show that for all r ≧ 1 where p _n denotes the nth prime number and γ is Euler’s constant. This is the best known result for any r ≧ 11.      

Distribution of the maxima of random Takagi functions

This paper concerns the maximum value and the set of maximum points of a random version of Takagi’s continuous, nowhere differentiable function. Let F(x):=∑ _n=1^∞ ε _n ϕ(2 ⁿ⁻¹ x), x ∈ R, where ɛ ₁, ɛ ₂, ... are independent, identically distributed random variables taking values in {−1, 1}, and ϕ is the “tent map” defined by ϕ(x) = 2 dist (x, Z). Let p:= P (ɛ ₁ = 1), M:= max {F(x): x ∈ R}, and := {x ∈ [0, 1): F(x) = M}. An explicit expression for M is given in terms of the sequence {ɛ _n }, and it is shown that the probability distribution μ of M is purely atomic if p < , and is singular continuous if p ≧ . In the latter case, the Hausdorff dimension and the multifractal spectrum of μ are determined. It is shown further that the set is finite almost surely if p < , and is topologically equivalent to a Cantor set almost surely if p ≧ . The distribution of the cardinality of is determined in the first case, and the almost-sure Hausdorff dimension of is shown to be (2p − 1)/2p in the second case. The distribution of the leftmost point of is also given. Finally, some of the results are extended to the more general functions Σa ^{n − 1} ɛ _n ϕ(2 ^{n − 1} x), where 0 < a < 1.      

Polynomial solutions of quasi-homogeneous partial differential equations

By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a₁, …, a_n). Assume that either a₁, …, a_n are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝⁿ must be infinite     

Arithmetic Interpretability Types of Varieties and Some Additive Problems with Primes

We deal with varieties with one basic operation f(x₁,...,x_n) and one defining identity f(x₁,..., x_n) = f(x_π(1),...,x_π(n)), where π is a permutation whose cyclic set consists of distinct primes p₁,...,p_r, with the sum p₁+...+p_r = n. Their interpretability types, together with the greatest element 1 in a lattice ^int, are said to be arithmetic. It is proved that the arithmetic types constitute a distributive lattice _ar, which is dual to a lattice Sub _fΠ of finite subsets of the set Π of all primes. It is shown that for n ⩾ 2, the poset _ar( _n) of arithmetic types defined by permutations in _n, for n fixed, is a lattice iff n = 2, 3, 4, 6, 8, 9, 11. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 622–630, September–October, 2005.     

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0     

Diffeomorphism classification of smooth weighted complete intersections

Xn(d1, . . . , dr-1, dr; w) and Xn(e1, . . . , er-1, dr; w) are two complex odd-dimensional smooth weighted complete intersections defined in a smooth weighted hypersurface Xn+r-1(dr; w). We prove that they are diffeomorphic if and only if they have the same total degree d, the Pontrjagin classes and the Euler characteristic, under the following assumptions: the weights w = (ω0, . . . , ωn+r) are pairwise relatively prime and odd, νp(d/dr) ≥ 2n+1/ 2(p-1) + 1 for all primes p with p(p-1) ≤ n + 1, where νp(d/dr) satisfies d/dr =Ⅱp prime pνp (d/dr).     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

Vive la différence III

We show that, consistently, there is an ultrafilter on ω such that if N _n^ℓ = (P _n^ℓ ∪ Q _n^ℓ, P _n^ℓ, Q _n^ℓ, R _n^ℓ) (for ℓ = 1, 2, n < ω), P _n^ℓ ∪ Q _n^ℓ ⊆ ω, and are models of the canonical theory t ^ind of the strong independence property, then every isomorphism from onto is a product isomorphism. The first version of this work done in 93; First typed: May 1993. This research was partially supported by the United States-Israel Binational Science Foundation. Publication 509     

Two Inequalities for Convex Functions

Let a ₀ < a ₁ < ··· < a _n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a ₀ < a ₁ < ··· < a _n be positive integers with sums distinct. P. Erd?s conjectured that The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then In this paper, we obtain two generalizations of the above result; each is of special interest in itself. We prove: Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n , α' ₀, α' ₁, ... , α' _n , β' ₀ , β' ₁ , ... , β' _n be real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , α' ₀ ≤ α' ₁ ≤ ··· ≤ α' _n , Then Theorem 2 Let f be any given convex decreasing function on [A, B] with k ₀, k ₁, ... , k _n being nonnegative real numbers and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then      

A generalization of rummer’s congruences

Suppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersB _n(n = 1,2,3,...) assert that (1-p ^m-1)B _m/m isp-integral and

On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator

We describe the set of parameters γ for which there exists a decomposition of the operator γI _H in a sum of n self-adjoint operators with spectra from the sets M ₁, …, M _n, M _i = 0, 1, …, k _i, for n ≥ 4 and, in some cases, for n = 3. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 470–477, April, 2008.     

Domination inequality for martingale transforms of a Rademacher sequence

Letf _n = Σ _k=1 ⁿ v _k r _k ,n=1,…, be a martingale transform of a Rademacher sequence (r _n)and let (r _n ^′ ) be an independent copy of (r _n).The main result of this paper states that there exists an absolute constantK such that for allp, 1≤p<∞, the following inequality is true: In order to prove this result, we obtain some inequalities which may be of independent interest. In particular, we show that for every sequence of scalars (a _n)one has where is theK-interpolation norm between ℓ₁ and ℓ₂. We also derive a new exponential inequality for martingale transforms of a Rademacher sequence. This research was supported in part by an NSF grant and an FRPD grant at NCSU.     

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (x_n) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (x_n) be aT-martingale taking on values inH. If then x _n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (x_n) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X ^* of norm 1 such that If, in addition, the spaceX is strictly convex, x _n /n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x _n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093     

Geometry of finite-dimensional normed spaces,and continuous functions on the Euclidean sphere

Let ℝⁿ be the n-dimensional Euclidean space, and let { · } be a norm in Rⁿ. Two lines ℓ₁ and ℓ₂ in ℝⁿ are said to be { · }-orthogonal if their { · }-unit direction vectors e ₁ and e ₂ satisfy {e ₁ + e ₂} = {e ₁ − e ₂}. It is proved that for any two norms { · } and { · }′ in ℝⁿ there are n lines ℓ₁, ..., ℓ_n that are { · }-and { · }′-orthogonal simultaneously. Let be a continuous function on the unit sphere with center O. It is proved that there exists an (n − 1)-cube C centered at O, inscribed in , and such that all sums of values of f at the vertices of (n − 3)-faces of C are pairwise equal. If the function f is even, then there exists an n-cube with the same properties. Furthermore, there exists an orthonormal basis e ₁, ..., e _n such that for 1 ≤ i ≤ j ≤ n we have . Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 107–117.     

Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S _T |^γ<∞, γ>2 whereT is a stopping time and {S _n=∑ ₁ ⁿ ,X _j ℱ _n ,n⩾1} is a martingale and these ensure when (**)X _n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET ^γ/2<∞, sup _n≥1 E|X _n |^γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S _k,T |^γ<∞ and of the second moment equationES _k,T ² =σ ² EΣ _j=k ^T S _k−1,j−1 ² where and {X _n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X _n , n≥1} withEX=0,EX ²=1 that the a.s. limit set of {(n log logn)^−k/2 S _k,n ,n≥k} is [0,2 ^k/2/k!] or [−2 ^k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .     

On one extremal problem for numerical series

Let Γ be the set of all permutations of the natural series and let α = {α _j} _j∈ℕ, ν = {ν_j} _j∈ℕ, and η = {η_j} _j∈ℕ be nonnegative number sequences for which

Asymptotic behavior of the irrational factor

We study the irrational factor function I(n) introduced by Atanassov and defined by , where is the prime factorization of n. We show that the sequence {G(n)/n} _n≧1, where G(n) = Π _ν=1 ⁿ I(ν)^1/n , is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n). Research of the third author is supported in part by NSF grant number DMS-0456615.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

On the parity of the class number of the 7<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>th cyclotomic field

For an odd prime number p and an integer n ≥ 0, let h _n be the class number of the p ⁿ⁺¹st cyclotomic field Q(). It is known that when p = 3 or 5, h _n is odd for all n ≥ 0. We prove that the same holds also when p = 7. The author was partially supported by Grant-in-Aid for Scientific Research (C), No. 19540005, Japan Society for the Promotion of Science.     

Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables

We investigate the correlation between the constants K(ℝⁿ) and , where