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## Abstract

We provide a new proof of Hua's result that every sufficiently large integer *N* ≡ 5 (mod 24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of *L*-functions. Here, we present a proof based on the transference principle first introduced in[5]. Using a sieve theoretic approach similar to ([10]), we do not require any results related to the distributions of zeros of *L*- functions. The main technical difficulty of our approach lies in proving the pseudo-randomness of the majorant of the characteristic function of the *W*-tricked primes which requires a precise evaluation of the occurring Gaussian sums and Jacobi symbols.

## Abstract

For each even classical pretzel knot *P*(2*k*
_{1} + 1, 2*k*
_{2} + 1, 2*k*
_{3}), we determine the character variety of irreducible SL (2, ℂ)-representations, and clarify the steps of computing its A-polynomial.

## Abstract

We present a technique to construct Cohen–Macaulay graphs from a given graph; if this graph fulfills certain conditions. As a consequence, we characterize Cohen–Macaulay paths.

## Abstract

We prove that, for any cofinally Polish space *X*, every locally finite family of non-empty open subsets of *X* is countable. It is also established that Lindelöf domain representable spaces are cofinally Polish and domain representability coincides with subcompactness in the class of *σ*-compact spaces. It turns out that, for a topological group *G* whose space has the Lindelöf Σ-property, the space *G* is domain representable if and only if it is Čech-complete. Our results solve several published open questions.

## 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.