Koksma–Hlawka type inequalities of fractional order

The Koksma–Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded variation. Hence here we relax the smoothness assumptions required in the Koksma–Hlawka inequality. We introduce Banach spaces of functions whose fractional derivative of order is in . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma–Hlawka inequality to functions whose partial fractional derivatives are in . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation but satisfy weaker smoothness conditions.      

On the Coefficients of Some Subclasses of Univalent Functions

In this paper we consider the class L_c*(,,) consisting of analytic and univlent functions with negative coefficients and with fixed second coefficient. The object of the present paper is to show coefficient estimates, convex linear combinations, some distortion theoems and radii of starlikeness and convexity for f(z) in the class L_c*(,,). The results are generalized to families with finitely many fixed coefficients.AMS Subject Classification (1991) 30C45     

Doubly β-Derived Translation Planes

Existence and uniqueness of a doubly -derived translation plane of order 49 are proved. Furthermore, we give a complete classification of those translation planes of order 49 which can be obtained from the desarguesian plane of order 49 by a mixed double derivation, namely by applying a -derivation on and a classical derivation (also called Ostrom's derivation or -derivation) on .     

K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

On the linear isoperimetric inequality

This paper contains a sharp version of the well-known linear isoperimetric inequality for minimal surfacesX area(X)1/2oscillation(X)length(X).Supported by Sonderforschungsbereich 72 der Deutschen Forschungsgemeinschaft at Bonn University.     

Invariant Subspaces for Positive Operators Acting on a Banach Space with Markushevich Basis

We introduce weak quasinilpotence for operators. Then, by substituting Markushevich basis and weak quasinilpotence at a nonzero vector for Schauder basis and quasinilpotence at a nonzero vector, respectively, we answer a question on the invariant subspaces of positive operators in [3].     

An improved bound for the de Bruijn–Newman constant

The Riemann hypothesis is equivalent to the conjecture that the de Bruijn–Newman constant satisfies 0. However, so far all the bounds that have been proved for go in the other direction, and provide support for the conjecture of Newman that 0. This paper shows how to improve previous lower bounds and prove that –2.710^–9<. This can be done using a pair of zeros of the Riemann zeta function near zero number 10²⁰ that are unusually close together. The new bound provides yet more evidence that the Riemann hypothesis, if true, is just barely true.     

Zeros of the densities of infinitely divisible measures on R n

Let be an infinitely divisible probability measure onR ⁿ without Gaussian component and let be its Lévy measure. Suppose that is absolutely continuous with respect to the Lebesgue measure . We investigate the structure of the set ⁿ of admissible translates of . This yields a unified presentation of previously known results. We also show that if(S)>0 then is equivalent to , under the assumption that supp =R ⁿ , whereS is the closure of the semigroup generated by the support of .The research of this author is supported by KBN Grant.The research of this author is supported by AFSOR Grant No. 90-0168, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.     

Tamely Ramified Towers and Discriminant Bounds for Number Fields

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R _2m be the minimal root discriminant for totally complex number fields of degree 2m, and put ₀ = lim inf _m R _2m . One knows that ₀ 4e 22.3, and, assuming the Generalized Riemann Hypothesis, ₀ 8e 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that ₀ < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate ₀ < 83.9.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

Differentiation of fractional order onp-groups,approximation properties

G- p- . [5] - (G) L _r(G) (1r<), . . , - . , , , . . , X. , . (. [1], [2] [4]).     

Einschliessungsaussagen für Differentialoperatoren vierter Ordnung und ein Verfahren zur Berechnung von Schrankenfunktionen

For a linear fourth order ordinary differential operator M we study Range Domain Implications (RDI). Let C_o [O,1] be positive; we show under what conditions there exists a C_O[O,1] such that the following RDI holds: Mu(x) (x) (0x1) u(x) (0x1). In particular we provide a numerical procedure to calculate .RDI are used to obtain error estimations and to solve related nonlinear problems.The basic idea to prove RDI is to split M into a product of second order differential operators which are easier to handle. For the general case that there exists no global splitting the concept of a local splitting is introduced.

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The author would like to thank the European Research Office of the United States Army for their kind interest.     

Equivalent Sets of Solutions of the Klein–Gordon Equation with a Constant Electric Field

We argue extensively in favor of our earlier choice of the in and out states (among the solutions of a wave equation with one-dimensional potential). In this connection, we study the nonstationary and stationary families of complete sets of solutions of the Klein–Gordon equation with a constant electric field. A nonstationary set _Pv consists of the solutions with the quantum number p _v=p ⁰ v–p₃. It can be obtained from the nonstationary set _P3 with the quantum number p ₃ by a boost along the x ₃ axis (in the direction of the electric field) with the velocity –v. By changing the gauge, we can bring the solutions in all sets to the same potential without changing quantum numbers. Then the transformations of solutions in one set (with the quantum number p _v) to the solutions in another set (with the quantum number p _v) have group properties. The stationary solutions and sets have the same properties as the nonstationary ones and are obtainable from stationary solutions with the quantum number p ⁰ by the same boost. It turns out that each set can be obtained from any other by gauge manipulations. All sets are therefore equivalent, and the classification (i.e., assigning the frequency sign and the in and out indices) in any set is determined by the classification in the set _P3, where it is obvious.     

On strong summability of polynomial expansions

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Dedicated to Professor L. Leindler on his 50^th birthday     

The state density of elliptic operators with random potentials

Summary The existence of the state density N() is established for certain linear elliptic coercive selfadjoint operators subject to perturbation by a unbounded random potential and its asymptotic properties discussed as (and for certain Gaussian perturbations as — ). The estimates depend on recent worlc by Métivier on the approximation numbers of the classical Sobolev spaces.The research of the author was supported by F.I.N.E.P. at U.F.R.G.S. under contract B/31/79/183/00/00 and by the CNPq.     

Generalized Cranston–McConnell Inequalities for Discontinuous Superharmonic Functions

Let be an open set in R² with Green function G(x,y) for the Laplace equation. We give a generalization of the Cranston-McConnell inequality concerning the integrability of positive harmonic functions on .     

Eine Bemerkung zu den höheren Pythagoraszahlen reeller Körper

Becker has shown in [1] that for the 4-th Pythagoras number of the field (X) the inequality P₄ ((X)) 36 holds. In this paper we will show P₄ ((X)) 24 and P₄ (K) 3 for all real pythagorean fields K.     

Intégalités isopérimétriques pour les graphes

In this article we prove the equivalence between the strong isoperimetric inequality #A #A, for any subset A of countable graph cG, and the inequality for any function with finite variation on cG and null at infinity, with optimal constant. More generally, we prove the equivalence between the isoperimetric inequality # A cP ^-1(1/# A) # A and the inequality || ||_cM | |_var, where cM is a Young function and cP its conjugate, and we also obtain an isoperimetric inequality in as an application.     

Une approche transformationnelle a l'algebre universelle

In this paper we propose an axiomatization of the notion of system of terms of a theory by means of which we obtain a representation of equational classes (or varieties) of algebras. We define analgebraic transformational system (S.T.A.) as a quadruple (T,v,S,+) satisfying the axioms, where T is a set containing the variables v(n), n, and having operators S(): TT, . In addition there are operations Q⁺ on T commuting with the operators. A notion of morphism between S.T.A. 's is defined to obtain the category which is shown to be equivalent to the dual of the category of equational classes. In the last section we establish the equivalence between and Lawvere's category of algebraic theories in which every definable constant is present.

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Extrait de la Thèse de doctorat de l'auteur, Université de Montréal, 1971.     

Tate–Vogel Completions of Half-Exact Functors

We provide a general method to construct the Tate–Vogel homology theory for a general half-exact functor with one variable, aiming at a good generalization of Cohen–Macaulay approximations of modules over commutative Gorenstein rings. For a half exact functor F, using the left and right satellites (S _n and S ⁿ ), we define F (X)=lim S _n S ⁿ F(X) and F (X)=lim S _n S ⁿ F(X), and call F and F the Tate–Vogel completions of F. We provide several properties of F and F , and their relations with the G-dimension and the projective dimension of the functor F. A comparison theorem of Tate–Vogel completions with ordinary Tate–Vogel homologies is proved. If F is a half exact functor over the category of R-modules, where R is a commutative Noetherian local ring inspired by Martsinkovsky's works, we can define the invariants (F) and (F) of F. If F=Ext _R ⁱ (M, ), then they coincide with Martsinkovsky's -invariants and Auslander's delta invariants. Our advantage is that we can consider these invariants for any half exact functors. We also compute these invariants for the local cohomology functors.