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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.

Open access

# Another Look at Threshold Phenomena for Random Cones

Studia Scientiarum Mathematicarum Hungarica
Authors:
Daniel Hug
and
Rolf Schneider

In stochastic geometry there are several instances of threshold phenomena in high dimensions: the behavior of a limit of some expectation changes abruptly when some parameter passes through a critical value. This note continues the investigation of the expected face numbers of polyhedral random cones, when the dimension of the ambient space increases to infinity. In the focus are the critical values of the observed threshold phenomena, as well as threshold phenomena for differences instead of quotients.

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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

$0 ≤ 1 24 δ M ″ ( w , μ ) ( β ) ≤ M ( w , μ ) A + B 2 − ∫ 0 1 M ( w , μ ) ( ( 1 − t ) A + t B ) d t ≤ − 1 24 Δ M ″ ( w , μ ) ( α )$

and

$0 ≤ − 1 12 δ M ″ ( w , μ ) ( β ) ≤ ∫ 0 1 M ( w , μ ) ( ( 1 − t ) A + t B ) d t − M ( w , μ ) ( A ) + M ( w , μ ) ( B ) 2 ≤ 1 12 Δ M ″ ( w , μ ) ( α ) ,$

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.

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# Hankel Determinant of Second Order for Some Classes of Analytic Functions

Mathematica Pannonica
Authors:
and
Nikola Tuneski

Let ƒ be analytic in the unit disk B and normalized so that ƒ (z) = z + a2z2 + a3z3 +܁܁܁. In this paper, we give upper bounds of the Hankel determinant of second order for the classes of starlike functions of order α, Ozaki close-to-convex functions and two other classes of analytic functions. Some of the estimates are sharp.

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# Circles of Curvature at Points of Parabola in Isotropic Plane

Mathematica Pannonica
Authors:
,
Marija Šimić Horvath
, and
Ema Jurkin

The authors have studied the curvature of the focal conic in the isotropic plane and the form of the circle of curvature at its points has been obtained. Hereby, we discuss several properties of such circles of curvature at the points of a parabola in the isotropic plane.

Open access

# A Combinatorial Approach to the Stirling Numbers of the First Kind with Higher Level

Studia Scientiarum Mathematicarum Hungarica
Authors:
Takao Komatsu
,
José L. Ramírez
, and
Diego Villamizar

In this paper, we investigate a generalization of the classical Stirling numbers of the first kind by considering permutations over tuples with an extra condition on the minimal elements of the cycles. The main focus of this work is the analysis of combinatorial properties of these new objects. We give general combinatorial identities and some recurrence relations. We also show some connections with other sequences such as poly-Cauchy numbers with higher level and central factorial numbers. To obtain our results, we use pure combinatorial arguments and classical manipulations of formal power series.

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