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# On the Mean Value of Arithmetic Error Terms

Mathematica Pannonica
Author:
János Pintz

In the 1980’s the author proved lower bounds for the mean value of the modulus of the error term of the prime number theorem and other important number theoretic functions whose oscillation is in connection with the zeros of the Riemann zeta function. In the present work a general theorem is shown in a simple way which gives a lower bound for the mentioned mean value as a function of a hypothetical pole of the Mellin transform of the function. The conditions are amply satisfied for the Riemann zeta function. In such a way the results recover the earlier ones (even in a slightly sharper form). The obtained estimates are often optimal apart from a constant factor, at least under reasonable conditions as the Riemann Hypothesis. This is the case, in particular, for the error term of the prime number theorem.

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# Some Inequalities of Ostrowski Type for Double Integral Mean of Absolutely Continuous Functions

Mathematica Pannonica
Author:
Silvestru Sever Dragomir

In this paper we establish some Ostrowski type inequalities for double integral mean of absolutely continuous functions. An application for special means is given as well.

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# Trimmed Least Square Estimators for Stable Ar(1) Processes

Mathematica Pannonica
Authors:
Alina Bazarova
,
István Berkes
, and
Lajos Horváth

We prove the weak consistency of the trimmed least square estimator of the covariance parameter of an AR(1) process with stable errors.

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# Interval Chains and Completeness in Ultrapowers of Ordered Sets

Mathematica Pannonica
Authors:
Zoltán Boros
and
Péter Tóth

The ultrapower T* of an arbitrary ordered set T is introduced as an infinitesimal extension of T. It is obtained as the set of equivalence classes of the sequences in T, where the corresponding relation is generated by a free ultrafilter on the set of natural numbers. It is established that T* always satisfies Cantor’s property, while one can give the necessary and sufficient conditions for T so that T* would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of T* is determined by the cardinality of T.

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# Seiberg–Witten Floer Homotopy Contact Invariant

Studia Scientiarum Mathematicarum Hungarica
Authors:
Nobuo Iida
and
Masaki Taniguchi

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer–Mrowka–Ozsváth–Szabó. Moreover, we prove a gluing formula relating our invariant with the first author’s Bauer–Furuta type invariant, which refines Kronheimer–Mrowka’s invariant for 4-manifolds with contact boundary. As an application, we give a constraint for a certain class of symplectic fillings using equivariant KO-cohomology.

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# Upsilon Invariants from Cyclic Branched Covers

Studia Scientiarum Mathematicarum Hungarica
Authors:
Antonio Alfieri
,
Daniele Celoria
, and
András Stipsicz

We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m-fold cyclic branched covers with m a prime power, this extension provides new knot concordance invariants $Y m C ( K )$ of knots in S3. We give computations of some of these invariants for alternating knots and reprove independence results in the smooth concordance group.

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# Inequalities for the First and Second Derivatives of Algebraic Polynomials on an Ellipse

Mathematica Pannonica
Author:
Tatiana M. Nikiforova

We prove a theorem on the preservation of inequalities between functions of a special form after differentiation on an ellipse. In particular, we obtain generalizations of the Duffin–Schaeffer inequality and the Vidensky inequality for the first and second derivatives of algebraic polynomials to an ellipse.

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# A Riemann–von Mangoldt-Type Formula for the Distribution of Beurling Primes

Mathematica Pannonica
Author:
Szilárd Gy. Révész

In this paper we work out a Riemann–von Mangoldt type formula for the summatory function $ψ x$ :=  $∑ g ∈ G , g ≤ x Λ G g$ , where $G$ is an arithmetical semigroup (a Beurling generalized system of integers) and $Λ G$ is the corresponding von Mangoldt function attaining with a prime element $p ∈ G$ and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function $ζ G$ , belonging to $G$ , to the number of zeroes of $ζ G$ in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of $ζ G$ , under the sole additional assumption that Knopfmacher’s Axiom A is satisfied. We also construct a technically useful broken line contour to which the technic of integral transformation can be well applied. The whole work serves as a first step towards a further study of the distribution of zeros of the Beurling zeta function, providing appropriate zero density and zero clustering estimates, to be presented in the continuation of this paper.

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# A Theory of Congruences and Birkhoff’s Theorem for Matroids

Mathematica Pannonica
Author:
Stefan Veldsman

A congruence is defined for a matroid. This leads to suitable versions of the algebraic isomorphism theorems for matroids. As an application of the congruence theory for matroids, a version of Birkhoff’s Theorem for matroids is given which shows that every nontrivial matroid is a subdirect product of subdirectly irreducible matroids.

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# The Metrics G = agS + bgH + cgV on the Tangent Bundle of a Weyl Manifold

Mathematica Pannonica
Author:
Murat Altunbaş

Let (M, [g]) be a Weyl manifold and TM be its tangent bundle equipped with Riemannian g−natural metrics which are linear combinations of Sasaki, horizontal and vertical lifts of the base metric with constant coefficients. The aim of this paper is to construct a Weyl structure on TM and to show that TM cannot be Einstein-Weyl even if (M, g) is fiat.

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