A version of Sharkovskii's theorem for differential equations

We present a version of the Sharkovskii cycle coexistence theorem for differential equations. Our earlier applicable version is extended here to hold with the exception of at most two orbits. This result, which (because of counter-examples) cannot be improved, is then applied to ordinary differential equations and inclusions. In particular, if a time-periodic differential equation has -periodic solutions with , for all , then infinitely many subharmonics coexist.

     

A refinement of the Gauss-Lucas theorem

The classical Gauss-Lucas Theorem states that all the critical points (zeros of the derivative) of a nonconstant polynomial lie in the convex hull of the zeros of . It is proved that, actually, a subdomain of contains the critical points of .

     

A quantitative refinement of Rado's theorem

The fundamental result of the paper is the following. Theorem: Let be a k-quasiconformal Jordan curve and let be another Jordan curve (not necessarily quasiconformal). Assume that f maps conformallyext ontoext , f()=, f()>0. We assume that there exists a homeomorphism between and such that Then there exist numbers =(k)>0 and A=A(k), such that f(())– A, .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 103–112, 1987.     

A refinement of Kuratowski's theorem

The conjecture of Kelmans that any 3-connected non-planar graph with at least six vertices contains a cycle with three pairwise crossing chords is proved. Using this, a refinement of Kuratowski's theorem which also includes the result of Tutte that a graph is planar if and only if every cycle has a bipartite overlap graph is obtained.     

On the refinement of Cauchy's theorem and Pellet's theorem

Using the relationship of a polynomial and its associated polynomial, we derived a necessary and sufficient condition for determining all roots of a given polynomial on the circumference of a circle defined by its associated polynomial. By employing the technology of analytic inequality and the theory of distribution of zeros of meromorphic function, we refine two classical results of Cauchy and Pellet about bounds of modules of polynomial zeros. Sufficient conditions are obtained for the polynomial whose Cauchy's bound and Pellet's bounds are strict bounds. The characteristics is given for the polynomial whose Cauchy's bound or Pellet's bounds can be achieved by the modules of zeros of the polynomial.     

A further refinement of a three critical points theorem

In this paper, we complete the refinement process, made by Ricceri (2009) [4], of a result established by Ricceri (2000) [1], which is one of the most applied abstract multiplicity theorems in the past decade. A sample of application of our new result is as follows.Let (n≥3) be a bounded domain with smooth boundary and let .Then, for each ?>0 small enough, there exists λ_?>0 such that, for every compact interval , there exists ρ>0 with the following property: for every λ∈[a,b] and every continuous function satisfying for some , there exists δ>0 such that, for each ν∈[0,δ], the problem has at least three weak solutions whose norms in are less than ρ.     

Terminating q-Whipple summation theorem with two integer parameters

The Ramanujan Journal - By means of linearization method, we investigate the terminating q-Whipple summation theorem for $$_3F_2(1)$$ extended with two integer parameters. Three analytical formulae...     

A refinement of the complex convexity theorem via symplectic techniques

We apply techniques from symplectic geometry to extend and give a new proof of the complex convexity theorem of Gindikin-Krötz.     

A conditional functional limit theorem for decomposable branching processes with two types of particles

Consider a critical decomposable branching process with two types of particles in which particles of the first type give birth, at the end of their life, to descendants of the first type, as well as to descendants of the second type, while particles of the second type produce only descendants of the same type at the time of their death. We prove a functional limit theorem describing the distribution for the total number of particles of the second type appearing in the process in time Nt, 0 ≤ t < ∞, given that the number of particles of the first type appearing in the process during its evolution is N.     

A refinement of von Neumann—Heinz theorem and related inequalities

The paper gives a quantitative refinement of von Neumann’s theorem with some relevant inequalities permitting to estimate from below the distance between the origin of the coordinate system and the so-called normalized numerical range of an operator acting in the Hilbert space.     

A refinement of the existence and uniqueness theorem for the Vlasov-Poisson system

In a recent paper Horst shows that if a classical solution to the Vlasov-Poisson system ceases to exist then at this point in time not only does velocity support become unbounded but support in position space becomes unbounded also (assuming compactly supported initial data). In the present paper we formulate this result another way and give a different proof. It is shown that an a priori bound on the support of solutions in position space leads to an a priori bound on the support in velocity space and hence existence and uniqueness of solutions. Thus a necessary and sufficient condition for solvability is that the system admit an a priori bound on the support in position space alone. This gives a refinement of Wollman (J. Math. Anal. Appl., 90 1982, p. 141, Theorem 2.1).     

On a refinement of a theorem of Landau on Koebe domains

We state and prove a refinement of a classical theorem due to Landau on the Koebe domains for certain families of holomorphic functions introduced by A. W. Goodman. Our geometric approach in this article enables us to derive several statements of interest, which would not be produced via the methods in Goodman's paper, as immediate corollaries of the proof of the main theorem.     

A refinement of two effective inequalities of A. Baker

A refinement is made of A. Baker's effective estimates for ¦-p/q¦, where is algebraic of degree 3 and p/q is a rational fraction, and for the boundary of the integral solutions of the Diophantine equation f(x, y) = m, wheref is an irreducible form with integral coefficients of degree 3, and m is a rational integer.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 767–769, December, 1969.