This paper deals with the oscillation problems on nonlinear differential equation $(a(t)|x^{\prime }|^{p(t)-2}x^{\prime })^{\prime }+b(t)|x|^{\lambda -2}x=0$ involving $p(t)$-Laplacian. Sufficient conditions are given under which all proper solutions oscillatory. In addition, we give a-priori estimates for nonoscillatory and propose an open problem.

We have analyzed the kinematics of 9750 OB2 stars with proper motions and parallaxes selected by Xu et al. from Gaia EDR3 catalogue. The relative parallax errors for these do not exceed 10 $${\%}$$ . Based on entire sample stars, we found velocities $$(U,V)_{\odot}=(7.17,7.37)\pm(0.16,0.24)$$ km s $${}^{-1}$$ components angular velocity Galactic rotation: $$\Omega_{0}=29.700\pm 0.076$$ kpc , $$...

Fix α > 0, and sample N integers uniformly at random from { 1, 2, . . . , ⌊ eαN ⌋} . Given η > 0, the probability that the maximum of the pairwise GCDs lies between N2−η and N2+η converges to 1 as N → ∞. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order N2 / log(N). The proof generalizes to any arithmetical...

A 3-equitable prime cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, ..., |V |} such that if an edge uv is assigned the label 1 if gcd(f(u), f(v)) = 1 and gcd(f(u) + f(v), f(u)−f(v)) = 1, the label 2 if gcd(f(u), f(v)) = 1 and gcd(f(u) + f(v), f(u)− f(v)) = 2 and 0 otherwise, then the number of edges labeled with i and the number of edges labeled with j differ b...

In this paper we construct a cover {as(mod ns)}s=1 of Z with odd moduli such that there are distinct primes p1, . . . , pk dividing 2 n1 − 1, . . . , 2k − 1 respectively. Using this cover we show that for any positive integer m relatively prime to 15015 there exists an infinite arithmetic progression of positive odd integers the mth powers of whose terms are never of the form 2±p with a, n ∈ {0...

Let G be an abelian group and let R be a commutative G-graded super-ring (briefly, graded super-ring) with unity 1 6= 0. We say that a ∈ h(R), where h(R) is the set of homogeneous elements in R, is weakly prime to a graded superideal I of R if 0 6= r a ∈ I , where r ∈ h(R), then r ∈ I . If ν(I ) is the set of homogeneous elements in R that are not weakly prime to I , then we define I to be weak...

In this paper we construct a cover {as(mod ns)}ks=1 of Z with odd moduli such that there are distinct primes p1, . . . , pk dividing 2 n1 − 1, . . . , 2k − 1 respectively. Using this cover we show that for any positive integer m relatively prime to 15015 there exists an infinite arithmetic progression of positive odd integers the mth powers of whose terms are never of the form 2 + p with p a pr...

Abstract In this paper, we are concerned with the oscillation of solutions to a class fourth-order delay differential equations p -Laplacian like operators $( r ( t ) \vert x^{\prime \prime } ^{p_{1}-2}x^{\prime ^{\prime }+q x \tau ^{p_{2}-2}x =0$ ( r t )</m...

We consider the intuitionistic fuzzification of the concept of prime ideals in commutative BCK-algebras, and investigate some of their properties. We show that if P is a prime ideal of commutative BCK-algebra X iff ∼ P = 〈XP , − XP 〉 is an intuitionistic fuzzy prime ideal of X. We also prove that An IFS A = 〈μA, λA〉 of commutative BCK-algebra X is an intuitionistic fuzzy prime ideal of X if and...

Numbers that are not prime are defined as composite. From this definition, we can see that a prime number can never be even, since it would have the divisor of 2. Of course, the only exception to that rule is the prime number 2, which has no other divisors but itself. Therefore, we can extend the definition to say that every prime number greater than 2 must be odd. From the above definition, we...