We study pseudospectral and spectral functions for Hamiltonian system J y ′ − B ( t ) = λ Δ $Jy^{\prime }-B(t)=\lambda \Delta (t)y$ differential equation l [ ] $l[y]=\lambda with matrix-valued coefficients defined on an interval I a , b $\mathcal {I}=[a,b)$ the regular endpoint a. It is not assumed that matrix weight ≥ 0 $\Delta (t)\ge 0$ invertible a.e. {I}$ . In this case function always exis...

A prime number p is called b-elite if only finitely many generalized Fermat numbers Fb,n = b 2 + 1 are quadratic residues modulo p. Let p be a prime. Write p − 1 = 2h with r ≥ 0 and h odd. Define the length of the b-Fermat period of p to be the minimal natural number L such that Fb,r+L ≡ Fb,r (mod p). Recently Müller and Reinhart derived three conjectures on b-elite primes, two of them being th...

Kirby and Lickorish [3] have previously obtained the second result using a different construction. This answers problem 13 in Gordon [2]. Results of Schubert [6] show that the knots constructed are prime. In the second half of this paper we will generalize his results to characterize prime "generalized" cable knots. If JSΓX is a knot in S x B and K2 is a knot in S , we can form the Kx cable of ...

We consider trinomials of the form zn+azm+b, where 0<m<n are relatively prime integers and a b nonzero complex numbers. Typically (but not exclusively), will be an integer with |a|≥2, while b=±1. Our main results cover irreducibility, Mahler measure house such trinomials.

1. Let f, g ∈ C[T ] be nonconstant and relatively prime. a) If f−g 6= 0, show deg(f−g) ≥ (1/2) deg f+1, or equivalently deg f ≤ 2(deg(f−g)−1). b) Find infinitely many examples where equality occurs in the conclusion of part a. Start with an example where deg f = 3 and deg g = 2. c) When the hypothesis of relative primality is dropped, is part a still true? d) Find a lower bound on deg(f − g) in...

The recent measurements of $R_{K^+}$, $R_{K_S^0}$, $R_{K^{*+}}$, $B_s\to\mu^+\mu^-$, a set CP-averaged angular observables for the $B^0\to K^{*0}\mu^+\mu^-$ decay, and its isospin partner $B^+\to K^{*+}\mu^+\mu^-$ by LHCb Collaboration, consistently hint at lepton universality violation in $b\to s\ell\ell$ transitions. In this work, we first perform global fits to data show that five one-dimens...

let  be a banach algebra and  a derivation. in this paper, it is proved, under certain conditions, that , where  is the jacobson radical of . moreover, we prove that if  is unital and  is a continuous derivation, then , where  denotes the set of all primitive ideals such that  is commutative,  denotes the set of all maximal (modular) ideals such that  is commutative, and  is the set of all non-...

Inspired by the recent observations of $T_{c\bar{s}0}^{0/++}$ in processes $B^0\to\bar{D}^0 D_s^+ \pi^-$ and $B^+\to D^- \pi^+$ LHCb Collaboration, we investigate decay properties $T_{c\bar{s}0}^{0}$ a $D^{*}K^{*}$ molecule scenario, widths $T_{c\bar{s}0}^{0}\to D^{0}K^{0}$, $D_{s}^{+}\pi^{-}$, $D_{s}^{*+}\rho^{-}$, $D_{s1}^{(\prime)+}\pi^{-}$, $D^{*0}(D\pi)^{0}$ are estimated. Our estimations ...

proved that there is an absolute constant V > 0 such that the linear equation a 1 p 1 + a 2 p 2 + a 3 p 3 = b has prime solutions p j 's if b (max j a j) V and a j > 0. Apart from the numerical value of V , the bound is sharp. In this manuscript, we obtain a numerical bound for V. We also obtain a numerical bound for the small prime solutions of the above equation if the a j 's are not all of t...