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Korselt rational bases of prime powers

Studia Scientiarum Mathematicarum Hungarica
Author: Nejib Ghanmi

Abstract

Let N be a positive integer, $A$ be a subset of ℚ and $α=α1α2∈A\{0, N}$. N is called an α-Korselt number (equivalently α is said an N-Korselt base) if α 2 pα 1 divides α 2 Nα 1 for every prime divisor p of N. By the Korselt set of N over $A$, we mean the set $A−KS(N)$ of all $α∈A\{0, N}$ such that 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 $A-KS(ql)$ and we establish some connections between the ql -Korselt bases in ℚ and others in ℤ. The case of $A=ℚ∩[−1, 1[$ is studied where we prove that $(ℚ∩[−1, 1[)-KS(ql)$ is empty if and only if l = 2.

Moreover, we show that each nonzero rational α is an N-Korselt base for infinitely many numbers N = ql where q is a prime number and l ∈ ℕ.

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Lemniscate convexity of generalized Bessel functions

Studia Scientiarum Mathematicarum Hungarica
Authors: Vibha Madaan, Ajay Kumar, and V. Ravichandran

Abstract

Sufficient conditions on associated parameters p, b and c are obtained so that the generalized and “normalized” Bessel function up(z) = up,b,c(z) satisfies the inequalities ∣(1 + (zup(z)/up(z)))2 − 1∣ &lt; 1 or ∣((zu p(z))′/up(z))2 − 1∣ &lt; 1. We also determine the condition on these parameters so that $−(4(p+(b+1)/2)/c)up'(x)≺1+z$. Relations between the parameters μ and p are obtained such that the normalized Lommel function of first kind hμ,p(z) satisfies the subordination $1+(zhμ,p''(z)/hμ,q'(z))≺1+z$. Moreover, the properties of Alexander transform of the function hμ,p(z) are discussed.

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On exponential decay of the variance of BLUE for the mean of a stationary sequence

Studia Scientiarum Mathematicarum Hungarica
Authors: Nikolay Babayan and Mamikon S. Ginovyan

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.

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On the multi-dimensional modal logic of substitutions

Studia Scientiarum Mathematicarum Hungarica
Authors: Tarek Sayed Ahmed and Mohammad Assem Mahmoud

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.

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Perfect Pell and Pell–Lucas numbers

Studia Scientiarum Mathematicarum Hungarica
Authors: Jhon J. Bravo and Florian Luca

Abstract

The Pell sequence $(Pn)n=0∞$ is given by the recurrence Pn = 2Pn −1 + Pn −2 with initial condition P 0 = 0, P 1 = 1 and its associated Pell-Lucas sequence $(Qn)n=0∞$ is given by the same recurrence relation but with initial condition 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.

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A remark on large gaps between primes in arithmetic progressions

Studia Scientiarum Mathematicarum Hungarica
Author: Deniz Ali Kaptan

Abstract

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

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Ruin probability for discrete risk processes

Studia Scientiarum Mathematicarum Hungarica
Author: Ivana Geček Tuden

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.

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Stronger forms of sensitivity on product dynamical system via Furstenberg families

Studia Scientiarum Mathematicarum Hungarica
Authors: Rahul Thakur and Ruchi Das

Abstract

In this paper, it has been investigated that how various stronger notions of sensitivity like 𝓕-sensitive, multi-𝓕-sensitive, (𝓕1, 𝓕2)-sensitive, etc., where 𝓕, 𝓕1, 𝓕2 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.

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A variant of the Fejér–Jackson inequality

Studia Scientiarum Mathematicarum Hungarica
Authors: Horst Alzer and Man Kam Kwong

Abstract

We prove: For all natural numbers n and real numbers x ∈ [0, π] we have $−548130−585≤∑k=1n(−1)k+1(sin((2k−1)x)2k−1+sin(2kx)2k)$.

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

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Cohomology of the vector fields lie algebras on ℝℙ1 acting on bilinear differential operators

Studia Scientiarum Mathematicarum Hungarica
Author: Abdaoui Meher

Abstract

Let Vect (ℝℙ1) be the Lie algebra of smooth vector fields on ℝℙ1. In this paper, we classify $aff(1)$-invariant linear differential operators from Vect (ℝℙ1) to $Dλ,μ;ν$ vanishing on $aff(1)$, where $Dλ,μ;ν :=Homdiff(ℱλ⊗ℱμ;ℱν)$ is the space of bilinear differential operators acting on weighted densities. This result allows us to compute the first differential $aff(1)$-relative cohomology of Vect (ℝℙ1) with coefficients in $Dλ,μ;ν$.

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