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## Abstract

Let **𝔄** be a unital Banach algebra and **ℜ** its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra **𝔄** satisfies the descending chain condition on closed left ideals then **𝔄**/**ℜ** is finite dimensional. We also prove that a *C*
^{*}-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

## Abstract

We give two new simple characterizations of the Cauchy distribution by using the Möbius and Mellin transforms. They also yield characterizations of the circular Cauchy distribution and the mixture Cauchy model.

## Abstract

In this paper we present different variants of the well-known Hermite–Hadamard inequality, in a generalized context. We consider general fractional integral operators for *h*-convex and *r*-convex functions.

## 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.

## 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.

## Abstract

Let *K* = ℚ(*α*) be a number field generated by a complex root *α* of a monic irreducible polynomial *f*(*x*) = *x*
^{24} – *m*, 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.

## Abstract

In this study, we investigate suborbital graphs *G*
_{
u,n
} of the normalizer Γ_{
B
} (*N*) of Γ_{0} (*N*) in *PSL*(2, ℝ) for *N* = 2^{
α
}3^{
β
} 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 Γ_{
B
} (*N*) on *N*) and then obtain some properties of the graphs arising from this action.

## Abstract

*n,m*≥ 2 this paper is devoted to the description of the sets of extreme and exposed points of the closed unit balls of

*n*-linear forms on

*n*-linear forms. First we classify the extreme points of the unit balls of

and

which answers the questions in [31].

## 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 *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.

## 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*,*(*l* − *r*)*lm*
^{2} + 1} *>* 30, then the equation (*rlm*
^{2} − 1)^{
x
} + ((*l* − *r*)*lm*
^{2} + 1)^{
y
} = (*lm*)^{
z
} has only the positive integer solution (*x,y,z*) = (1*,*1*,*2).