Browse

You are looking at 11 - 20 of 11,008 items

Lemniscate convexity of generalized Bessel functions

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∣ < 1 or ∣((zu p(z))′/up(z))2 − 1∣ < 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.

Full access

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.

Full access
On the multi-dimensional modal logic of substitutions

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.

Full access
Perfect Pell and Pell–Lucas numbers

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.

Full access

Abstract

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

Full access
Ruin probability for discrete risk processes

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.

Full access

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.

Full access
A variant of the Fejér–Jackson inequality

Abstract

We prove: For all natural numbers n and real numbers x ∈ [0, π] we have 548130585k=1n(1)k+1(sin((2k1)x)2k1+sin(2kx)2k).

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

Full access

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(λμ;ν) 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.

Full access
Counting surface branched covers

Abstract

To a branched cover f between orientable surfaces one can associate a certain branch datum D(f), that encodes the combinatorics of the cover. This D(f) satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum D there exists a branched cover f such that D(f)=D. One can actually refine this problem and ask how many these f's exist, but one must of course decide what restrictions one puts on such f’s, and choose an equivalence relation up to which one regards them. As it turns out, quite a few natural choices for this relation are possible. In this short note we carefully analyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different from each other we employ Grothendieck's dessins d'enfant.

Full access