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## Abstract

Let *N* be a positive integer, *N* is called an *α-Korselt number* (equivalently *α* is said an *N-Korselt base*) if *α*
_{2}
*p* − *α*
_{1} divides *α*
_{2}
*N* − *α*
_{1} for every prime divisor *p* of *N.* By the *Korselt set* of *N* over *N* is an *α*-Korselt number.

In this paper we determine explicitly for a given prime number *q* and an integer *l* ∈ ℕ \{0, 1}, the set *q ^{l}* -Korselt bases in ℚ and others in ℤ. The case of

*l*= 2.

Moreover, we show that each nonzero rational *α* is an *N*-Korselt base for infinitely many numbers *N* = *q ^{l}* where

*q*is a prime number and

*l*∈ ℕ.

## Abstract

Sufficient conditions on associated parameters *p, b* and *c* are obtained so that the generalized and “normalized” Bessel function *u _{p}*(

*z*) =

*u*(

_{p,b,c}*z*) satisfies the inequalities ∣(1 + (

*zu*″

*(*

_{p}*z*)/

*u*′

*(*

_{p}*z*)))

^{2}− 1∣ < 1 or ∣((

*zu*

_{p}(

*z*))′/

*u*(

_{p}*z*))

^{2}− 1∣ < 1. We also determine the condition on these parameters so that $$-(4(p+(b+1)/2)/c){u}_{p}^{\text{'}}(x)\prec \sqrt{1+z}$$. Relations between the parameters

*μ*and

*p*are obtained such that the normalized Lo

__mmel__function of first kind

*h*(

_{μ,p}*z*) satisfies the subordination $$1+(z{h}_{\mu ,p}^{\text{'}\text{'}}(z)/{h}_{\mu ,q}^{\text{'}}(z))\prec \sqrt{1+z}$$. Moreover, the properties of Alexander transform of the function

*h*(

_{μ,p}*z*) are discussed.

## Abstract

In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of the variance of the best linear unbiased estimator (BLUE) for the unknown mean of a stationary sequence possessing a spectral density. In particular, we show that a necessary condition for variance of BLUE to decrease to zero exponentially is that the spectral density vanishes on a set of positive Lebesgue measure in any vicinity of zero.

## Abstract

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an *N P* complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

## Abstract

The Pell sequence *P _{n}* = 2

*P*

_{n}_{−1}+

*P*

_{n}_{−2}with initial condition

*P*

_{0}= 0,

*P*

_{1}= 1 and its associated Pell-Lucas sequence

*Q*

_{0}

*=*2,

*Q*

_{1}= 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

## Abstract

We record an implication between a recent result due to Li, Pratt and Shakan and large gaps between arithmetic progressions.

## Abstract

We study the discrete time risk process modelled by the skip-free random walk and derive results connected to the ruin probability and crossing a fixed level for this type of process. We use the method relying on the classical ballot theorems to derive the results for crossing a fixed level and compare them to the results known for the continuous time version of the risk process. We generalize this model by adding a perturbation and, still relying on the skip-free structure of that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk process. We further derive the famous Pollaczek-Khinchine type formula for this generalized process, using the decomposition of the supremum of the dual process at some special instants of time.

## Abstract

In this paper, it has been investigated that how various stronger notions of sensitivity like 𝓕-sensitive, multi-𝓕-sensitive, (𝓕_{1}, 𝓕_{2})-sensitive, etc., where 𝓕, 𝓕_{1}, 𝓕_{2} are Furstenberg families, are carried over to countably infinite product of dynamical systems having these properties and vice versa. Similar results are also proved for induced hyperspaces.

## Abstract

We prove: For all natural numbers n and real numbers *x* ∈ [0, π] we have

The sign of equality holds if and only if *n* = 2 and *x* = 4π/5.

## Abstract

Let Vect (ℝℙ^{1}) be the Lie algebra of smooth vector fields on ℝℙ^{1}. In this paper, we classify ^{1}) to ^{1}) with coefficients in