On the almost sure central limit theorem and domains of attraction

We give necessary and sufficient criteria for a sequence (X _n) of i.i.d. r.v.'s to satisfy the a.s. central limit theorem, i.e.,

Construction of a solution of a mixed problem for the generalized heat conduction equation by the fourier method

By the Fourier method a solution of the equation

Estimates for Multiplicative Functions on the Set of Shifted Primes

Let fi, i = 1, ... k, be complex-valued multiplicative functions satisfying the conditions

An Elementary Proof of Pólya-Vinogradov's Inequality, II

Let be a primitive character mod k, k > 2. In [1], the following elementary estimate

An Ω-theorem for an error term related to the sum-of-divisors function

Let denote the sum-of-divisors function, and set . Gronwall and Wigert proved (independently) in 1913 and 1914, respectively, thatE ₁ (x)= (x log logx). In this paper we obtain the more preciseE ₁ (x)=_–(x log logx). The method consists in averaging over suitable arithmetic progressions, and was suggested by the work ofP. Erdös andH. N. Shapiro [Canad. J. Math. 3–4, 375–385 (1951)] on the error term corresponding to Euler's functions, .     

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

Algebraic Dilogarithm Identities

The Rogers L-function satisfies the functional equation .From this we derive several other such equations, including Euler's identity L(x)+L(1-x)=L(1) and various identities arising from summation and transformation formulas for basic hypergeometric series. We also obtain some new equations of the form where is algebraic and the c _k are integers.     

Use of complex analysis for deriving lower bounds for trigonometric polynomials

It is shown that for any distinct natural numbersk ₁,...,k _n and arbitrary real numbersa ₁,...,a _n the following inequality holds:

Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines

In what follows, $C$ is the space of -periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; is the mth modulus of continuity of a function f with step h and calculated with respect to P; , ( ), ,

On the number of faces of centrally-symmetric simplicial polytopes

I. Bárány and L. Lovász [Acta Math. Acad. Sci. Hung.40, 323–329 (1982)] showed that ad-dimensional centrally-symmetric simplicial polytopeP has at least 2 ^d facets, and conjectured a lower bound for the numberf _i ofi-dimensional faces ofP in terms ofd and the numberf ₀ =2n of vertices. Define integers A. Björner conjectured (unpublished) that (which generalizes the result of Bárány-Lovász sincef _d–1 = h _i ), and more strongly that , which is easily seen to imply the conjecture of Bárány-Lovász. In this paper the conjectures of Björner are proved.Partially supported by NSF grant MCS-8104855. The research was performed when the author was a Sherman Fairchild Distinguished Scholar at Caltech.     

Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis

Using the following notation: C is the space of continuous bounded functions f equipped with the norm , V is the set of functions f such that , the set E consists of fCV and possesses the following property:

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Let = (₁,...,_d) be a vector with positive components and let D be the corresponding mixed derivative (of order _j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that

Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then

The Binet-Pexider functional equation for rectangular matrices

IfK is a field of characteristic 0 then the following is shown. Iff, g, h: M _n (K) K are non-constant solutions of the Binet—Pexider functional equation

On Singular Functional Differential Inequalities

Classical theorems on differential inequalities [1, 2, 3] are generalized for initial value problems of the kind and where is a singular Volterra operator, is continuous and positive on ]a, b], is a norm in R ⁿ, and [u]₊ and [u]_– are respectively the positive and the negative part of the vector u R ⁿ.     

The Law of the Iterated Logarithm for Functionals of Harris Recurrent Markov Chains: Self Normalization

Let {X _n } _n0 be a Harris recurrent Markov chain with state space E, transition probability P(x, A) and invariant measure , and let f be a real measurable function on E. We prove that with probability one,

The Limit of Lebesgue Constants for Summation Methods of the Fourier―Legendre Series Defined by a Multiplier Function

Let C[-1,1] be the space of continuous functions f:[-1,1] with the uniform norm, let _Pk be the Legendre polynomials such that _Pk (1)=1, and let _J0 be the Bessel function of zero index. We consider sequences of linear operators (summation methods) _Un:C [-1,1] C[-1,1] defined by a multiplier function as follows:

The theorem of the arithmetic and geometric means and its application to a problem in gas dynamics

Zusammenfassung Durch das Theorem der arithmetischen und geometrischen Mitte kann man den maximalen Wert des Produktes von Variablen, die einer linearen Beschränkung unterworfen sind, feststellen. Die vorliegende Arbeit untersucht einige einfache Versionen des Theorems. Die allgemeinste Version ergibt den maximalen Wert des Produktes vonn Variablen, wenn jede Variable zu einer höheren Potenz erhoben wird und die Variablen einer Gruppe vonr linearen Beschränkungen unterworfen sind (r<n). Das Theorem wird dann auf das Problem des maximalen totalen Druckwiedergewinns über einem Stosswellensystem angewandt und schliesslich wird daraus ein ziemlich allgemeines Theorem über das adiabatische Fliessen eines Gases hergeleitet.

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List of Symbols a _ij given positive constants - b , - f _i , - g _i , - h , - k , - m the arithmetic mean ofn variables, see Equation (2) - M the Mach number - n the number of variables - P _T the total pressure - q the geometric mean ofn variables, see Equation (1) - r the number of constraining conditions on the function to be maximised - R the gas constant - S the entropy - x _i for the Oswatitsch analysis, otherwisex _i represents any variable - y _i - ratio of specific heats - _j Lagrange multiplier - _i given positive constant - W _i shock wave angle of (i–1)th shock     

Analog of the Akhiezer–Krein–Favard Sums for Periodic Splines of Minimal Defect

Let be the space of 2-periodic functions whose (r – 1)th-order derivative is absolutely continuous on any segment and rth-order derivative belongs to L _p, S _2n,m is the space of 2-periodic splines of order m of minimal defect over the uniform partition . In this paper, we construct linear operators such that