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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.

Open access

# 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 $ψx$:= $∑g∈G,g≤xΛGg$, where $G$ is an arithmetical semigroup (a Beurling generalized system of integers) and $ΛG$ 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 $ζG$ , belonging to $G$, to the number of zeroes of $ζG$ in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of $ζG$, 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.

Open access

# 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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# Arithmetical Functions Commutable with Sums of Squares II

Mathematica Pannonica
Authors: Imre Kátai and Bui Minh Phong

We give all functions ƒ , E: ℕ → ℂ which satisfy the relation

for every a, b, c ∈ ℕ, where h ≥ 0 is an integers and K is a complex number. If n cannot be written as a2 + b2 + c2 + h for suitable a, b, c ∈ ℕ, then ƒ (n) is not determined. This is more complicated if we assume that ƒ and E are multiplicative functions.

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# Harvest Management Problem with a Fractional Logistic Equation

Mathematica Pannonica
Authors: Melani Barrios, Gabriela Reyero, and Mabel Tidball

In this article, we study a fractional control problem that models the maximization of the profit obtained by exploiting a certain resource whose dynamics are governed by the fractional logistic equation. Due to the singularity of this problem, we develop different resolution techniques, both for the classical case and for the fractional case. We perform several numerical simulations to make a comparison between both cases.

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# On the Negativity of the Walsh–Kaczmarz–Riesz Logarithmic Kernels

Mathematica Pannonica
Authors: György Gát and Gábor Lucskai

The main aim of this paper is to prove that the nonnegativity of the Riesz’s logarithmic kernels with respect to the Walsh– Kaczmarz system fails to hold.

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# From Binary Groups to Terminal Rings

Mathematica Pannonica
Author: Stuart D. Scott

Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P 0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order p n (p an odd prime and n ≥ 1 an integer ≤ 7).

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# On Zero Determinant Matrices that are Full

Mathematica Pannonica
Authors: Grigore Călugăreanu and Horia F. Pop

Column-row products have zero determinant over any commutative ring. In this paper we discuss the converse. For domains, we show that this yields a characterization of pre-Schreier rings, and for rings with zero divisors we show that reduced pre-Schreier rings have this property.

Finally, for the rings of integers modulo n, we determine the 2x2 matrices which are (or not) full and their numbers.

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# Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces

Mathematica Pannonica
Author: Silvestru Sever Dragomir

For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the followingmonotonic integral transform

where the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, then

$0≤124δM″(w,μ)(β)≤M(w,μ)A+B2−∫01M(w,μ)((1−t)A+tB)dt≤−124ΔM″(w,μ)(α)$

and

$0≤−112δM″(w,μ)(β)≤∫01M(w,μ)((1−t)A+tB)dt−M(w,μ)(A)+M(w,μ)(B)2≤112ΔM″(w,μ)(α),$

where$M″(w,μ)$ is the second derivative of$M(w,μ)$ as a real function.

Applications for power function and logarithm are also provided.

Open access