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# Lemniscate and exponential starlikeness of regular Coulomb wave functions

Studia Scientiarum Mathematicarum Hungarica
Author: İbrahim Aktaş

## Abstract

In this study, a normalized form of regular Coulomb wave function is considered. By using the differential subordinations method due to Miller and Mocanu, we determine some conditions on the parameters such that the normalized regular Coulomb wave function is lemniscate starlike and exponential starlike in the open unit disk, respectively. In additon, by using the relationship between the regular Coulomb wave function and the Bessel function of the first kind we give some conditions for which the classical Bessel function of the first kind is lemniscate and exponential starlike in the unit disk 𝔻.

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# Ninety years of k-tridiagonal matrices

Studia Scientiarum Mathematicarum Hungarica
Authors: Carlos M. da Fonseca, Victor Kowalenko, and László Losonczi

## Abstract

This survey revisits Jenő Egerváry and Otto Szász’s article of 1928 on trigonometric polynomials and simple structured matrices focussing mainly on the latter topic. In particular, we concentrate on the spectral theory for the first type of the matrices introduced in the article, which are today referred to as k-tridiagonal matrices, and then discuss the explosion of interest in them over the last two decades, most of which could have benefitted from the seminal article, had it not been overlooked.

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# On power integral bases for certain pure number fields defined by x 24 – m

Studia Scientiarum Mathematicarum Hungarica

## Abstract

Let K = ℚ(α) be a number field generated by a complex root α of a monic irreducible polynomial f(x) = x 24m, with m ≠ 1 is a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≢∓1 (mod 9), then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡ 1 (mod 9), then the number field K is not monogenic.

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# Quadrilateral cell graphs of the normalizer with signature (2,4,∞)

Studia Scientiarum Mathematicarum Hungarica
Authors: Nazli Yazici Gözütok and Bahadir Özgür Güler

## Abstract

In this study, we investigate suborbital graphs G u,n of the normalizer ΓB (N) of Γ0 (N) in PSL(2, ℝ) for N = 2α3β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of ΓB (N) on $ℚ∧$ using the group $ГΒ+$ (N) and then obtain some properties of the graphs arising from this action.

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# The unit balls of $ℒ ( n l ∞ m )$ and $ℒ s ( n l ∞ m )$

Studia Scientiarum Mathematicarum Hungarica
Author: Sung Guen Kim

## Abstract

For n,m≥ 2 this paper is devoted to the description of the sets of extreme and exposed points of the closed unit balls of $ℒ(ln∞m)$ and $ℒs(ln∞m)$, where $ℒ(ln∞m)$ is the space of n-linear forms on $ℝm$ with the supremum norm, and $ℒs(ln∞m)$ is the subspace of $ℒ(ln∞m)$ consisting of symmetric n-linear forms. First we classify the extreme points of the unit balls of $ℒ(ln∞m)$ and $ℒs(ln∞m)$, respectively. We show that ext $Bℒ(ln∞m)$ ⊂ ext $Bℒ(ln∞m+1)$, which answers the question in []. We show that every extreme point of the unit balls of $ℒ(ln∞m)$ and $ℒs(ln∞m)$ is exposed, correspondingly. We also show that
$expBℒS(ln∞2)=expBℒ(ln∞2)∩ℒs(ln∞2)$

and $expBℒs(l2∞m+1)≠expBℒ(l2∞m+1)∩ℒs(l2∞m+1),$

which answers the questions in [].

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# Descartes’ rule of signs, Rolle’s theorem and sequences of compatible pairs

Studia Scientiarum Mathematicarum Hungarica
Authors: Hassen Cheriha, Yousra Gati, and Vladimir Petrov Kostov

## Abstract

Consider the sequence s of the signs of the coefficients of a real univariate polynomial P of degree d. Descartes’ rule of signs gives compatibility conditions between s and the pair (r + ,r ), where r + is the number of positive roots and r the number of negative roots of P. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (s; r + ,r ) which begins at degree d = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for $(s;r0+,r0−;r1+,r1−;…;rd−1+,rd−1−)$, where$ri+$ (resp.$ri−$) is the number of positive (resp. negative) roots of the i-th derivative of P. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each i, and the trivial conditions given by Rolle’s theorem.

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# A note on the ternary purely exponential diophantine equation A x + B y = C z with A + B = C 2

Studia Scientiarum Mathematicarum Hungarica
Authors: Elif kizildere, Maohua le, and Gökhan Soydan

## Abstract

Let l,m,r be fixed positive integers such that 2$∤$| l, 3$∤$ lm, l > r and 3 | r. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{rlm 2 − 1,(lr)lm 2 + 1} > 30, then the equation (rlm 2 − 1)x + ((lr)lm 2 + 1)y = (lm)z has only the positive integer solution (x,y,z) = (1,1,2).

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# Numerical solution of linear differential equations by Walsh polynomials approach

Studia Scientiarum Mathematicarum Hungarica
Authors: György Gát and Rodolfo Toledo

## Abstract

In 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.

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# On sharpening of inequalities for a class of polynomials satisfying $p ( z ) ≡ z n p ( 1 / z )$

Studia Scientiarum Mathematicarum Hungarica
Authors: Ritu Dhankhar, Narendra Kumar Govil, and Prasanna Kumar

## Abstract

Let $p(z)=∑j=0najzj$ be a polynomial of degree n. Further, let$M(p, R)=max|z|=R≥1|p(z)| ,$ and $‖p‖=M(p, 1)$. Then according to the well-known Bernstein inequalities, we have $‖p′‖≤n‖p‖$ and $M(p,R)≤Rn‖p‖$. It is an open problem to obtain inequalities analogous to these inequalities for the class of polynomials satisfying p(z) ≡ z n p(1/z). In this paper we obtain some inequalites in this direction for polynomials that belong to this class and have all their coefficients in any sector of opening γ, where 0 $≤_$ γ < π. Our results generalize and sharpen several of the known results in this direction, including those of Govil and Vetterlein [3], and Rahman and Tariq [12]. We also present two examples to show that in some cases the bounds obtained by our results can be considerably sharper than the known bounds.

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# On star Lindelöf spaces

Studia Scientiarum Mathematicarum Hungarica
Authors: Wei-Feng Xuan and Yan-Kui Song

## Abstract

In this paper, we prove that if X is a space with a regular G δ-diagonal and X 2 is star Lindelöf then the cardinality of X is at most 2c. We also prove that if X is a star Lindelöf space with a symmetric g-function such that $∩${g 2(n, x): nω} = {x} for each xX then the cardinality of X is at most 2c. Moreover, we prove that if X is a star Lindelöf Hausdorff space satisfying (X) = κ then e(X) $≦$ 22κ; and if X is Hausdorff and we(X) = (X) = κsubset of a space then e(X) $≦$ 2κ. Finally, we prove that under V = L if X is a first countable DCCC normal space then X has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in Spaces with property (DC(ω 1)), Comment. Math. Univ. Carolin., 58(1) (2017), 131-135.

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