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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.

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

Mathematica Pannonica
Authors: Milutin Obradović 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: Vladimir Volenec, 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.

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# 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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# Corrigendum: Ideals of Residuated Lattices

Studia Scientiarum Mathematicarum Hungarica
Authors: Liviu-Constantin Holdon and Arsham Borumand Saeid
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# A Corson Compact Space is Countable if the Complement of its Diagonal is Functionally Countable

Studia Scientiarum Mathematicarum Hungarica

A space X is called functionally countable if ƒ (X) is countable for any continuous function ƒ : X → Ø. Given an infinite cardinal k, we prove that a compact scattered space K with d(K) > k must have a convergent k+-sequence. This result implies that a Corson compact space K is countable if the space (K × K) \ ΔK is functionally countable; here ΔK = {(x, x): x ϵ K} is the diagonal of K. We also establish that, under Jensen’s Axiom ♦, there exists a compact hereditarily separable non-metrizable compact space X such that (X × X) \ ΔX is functionally countable and show in ZFC that there exists a non-separable σ-compact space X such that (X × X) \ ΔX is functionally countable.

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# Criterion for the Coincidence of Strong and Weak Orlicz Spaces

Studia Scientiarum Mathematicarum Hungarica
Authors: Maria Rosaria Formica and Eugeny Ostrovsky

We provide necessary and sufficient conditions for the coincidence, up to equivalence of the norms, between strong and weak Orlicz spaces. Roughly speaking, this coincidence holds true only for the so-called exponential spaces.

We also find the exact value of the embedding constant which appears in the corresponding norm inequality.

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# Heegaard Floer Homology, Degree-One Maps and Splicing Knot Complements

Studia Scientiarum Mathematicarum Hungarica
Authors: Narges Bagherifard and Eaman Eftekhary

Suppose that K and K' are knots inside the homology spheres Y and Y', respectively. Let X = Y (K, K') be the 3-manifold obtained by splicing the complements of K and K' and Z be the three-manifold obtained by 0 surgery on K. When Y' is an L-space, we use the splicing formula of [1] to show that the rank of $HY^$(X ) is bounded below by the rank of $HY^$(Y ) if τ(K 2) = 0 and is bounded below by rank($HY^$(Z)) − 2 rank($HY^$(Y)) + 1 if τ(K') ≠ 0.

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