On mappings preserving pentagons

Summary We introduce a new characterization of linear isometries. More precisely, we prove that if a one-to-one mapping f:<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿ(2<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>≦n<<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>∞) maps every regular pentagon of side length a> 0 onto a pentagon with side length b> 0, then there exists a linear isometry I :<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿup to translation such that f(x) = (b/a) I(x).     

On topologization of transfinite convergence of sequences of real functions

Summary Let κ be an infinite regular cardinal. We are concerned with the question when the κ-convergence is topologyzable. In particular, we show that if<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>cis regular then<span style='font-size:10.0pt; font-family:"Lucida Sans Unicode"'>c-convergence is topologyzable by the <<span style='font-size:10.0pt;font-family: "Lucida Sans Unicode"'>c-box topology on R^R. Thus under CH the transfinite convergence is topologyzable. On the other hand, under MA+?CH the transfinite convergence is not topologyzable.     

On the curvature of a generalization of contact metric manifolds

Summary We consider a genaralization of contact metric manifolds given by assignment of 1-formsη¹, . . . ,η^sand a compatible metric gon a manifold. With some integrability conditions they are called almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva"; mso-bidi-font-family:"Monotype Corsiva"'>S-manifolds. We give a sufficient condition regarding the curvature of an almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva";mso-bidi-font-family: "Monotype Corsiva"'>S-manifold to be locally isometric to a product of a Euclidean space and a sphere.     

On centralizers of standard operator algebras and semisimple H*-algebras

Summary Let Abe a semisimple H*-algebra and let T: A→Abe an additive mapping such that T(x ^{n +1})<span lang=EN-US style='font-size:10.0pt;mso-ansi-language:EN-US'>=T(x)x ⁿ+x ⁿ T(x) holds for all x∈Aand some integer n≥1. In this case Tis a left and a right centralizer.     

Strongly θ-b-continuous functions

Summary We introduce a new class of functions called strongly θ-b-continuous function which is a generalization of both strongly θ-precontinuous functions<span lang=EN-US style='font-size:10.0pt; mso-ansi-language:EN-US'>[16]and strongly θ-semicontinuous functions [7]. Some characterizations and several properties concerning strongly θ-b-continuous functions are obtained.     

A Turán type inequality for rational functions with prescribed poles

Summary By employing a novel idea and simple techniques, we substantially generalize the Turán type inequality for rational functions with real zeros and prescribed poles established by Min [5] to include L^pspaces for 1≤p≤∞<span style='font-size:10.0pt'>while loosing the restriction ρ > 2 at the same time.     

Oscillation of second order quasilinear neutral delay differential equations

In this paper, by employing Riccati transformation technique, some new sufficient conditions for the oscillation criteria are given for the second order quasilinear neutral delay differential equations with delayed argument in the form $$\bigl(r(t)\bigl|z'(t)\bigr|^{\alpha-1}z'(t)\bigr)'+q(t)f\bigl(x\bigl(\sigma(t)\bigr)\bigr)=0,\quad t\geq t_0,$$ where z(t)=x(t)?p(t)x(??(t)), 0??p(t)??p<1, lim _t???? p(t)=p ₁<1, q(t)>0, ??>0. Two examples are considered to illustrate the main results.     

Totally geodesic submanifolds in Lie groups

Summary We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G_F(n,N) in the Lie group G_F(N) defined naturally makes G_F(n,N) a totally geodesic submanifold; (2) The imbedding S⁷→SO(8) defined by octonians makes S⁷a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G_F(N) in the sphere S^{cN^2-1}(√N) of gl(N,F)is minimal. Therefore the natural imbedding G_F(N)<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>→gl(N,F)is formed by the eigenfunctions of the Laplacian on G_F(N).     

The regular p-gonal prism tilings and their optimal hyperball packings in the hyperbolic 3-space

Summary We investigate the regular p-gonal prism tilings (mosaics) in the hyperbolic 3-space that were classified by I. Vermes in<span lang=EN-US style='font-size:10.0pt; mso-ansi-language:EN-US'>[12]and [13]. The optimal hyperball packings of these tilings are generated by the ``inscribed hyperspheres' whose metric data can be calculated by our method -- based on the projective interpretation of the hyperbolic geometry -- by the volume formulas of J. Bolyai and R. Kellerhals, respectively. We summarize in some tables the data and the densities of the optimal hyperball packings to each prism tiling in the hyperbolic space H³.     

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.     

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q₁(t)| y(t-s₁)|^a sgn y(t-s₁) +q₂(t)| y(t-s₂)|^b sgn y(t-s₂)=0,    t 3 t₀,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, q₂ ? C([t₀, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

Existence and iteration of positive solutions for a singular two-point boundary value problem with a <Emphasis Type="Italic">p</Emphasis>-Laplacian operator

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation (ϕ _p (u′))′+q(t)f(u) = 0, 0 < t < 1, where ϕ _p (s):= |s| ^p−2 s, p > 1, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0; 1.     

Oscillation theorems for second-order nonlinear differential equations with damped term

Several oscillation criteria are given for the second-order damped nonlinear differential equation (a(t)[y′(t)]^σ′i +p(t)[y′(t)]^σ +q(t)f(y(t)) = 0, where σ > 0 is any quotient of odd integers, a ϵ C(R, (0, ∞)), p(t) and q(t) are allowed to change sign on [t_o, ∞), and f ϵ Cl (R, R) such that xf (x) > 0 for x≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.     

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

Oscillation theorems for second order nonlinear differential equations

Some oscillation criteria are established for certain second order nonlinear differential equations of the form (a(t)ψ(x(t)) x^. (t))^. + p(t) x^. (t) + q(t)f(x(t)) = 0. These criteria improve upon some of the known results by Kura, Kamenev and Philos.     

A note on a result by M. Bognár concerning the Hahn--Mazurkiewicz Theorem

Summary In this note we use Theorem 2.4 in<span lang=EN-US style='font-size:10.0pt;mso-ansi-language:EN-US'>[1]to give a very short proof of a recent result due to M. Bognár (Theorem A in [3]). We also prove that Bognár's result is in fact equivalent to the classical Hahn-Mazurkiewicz Theorem. Finally we give generalizations in the non-compact setting.     

Oscillation criteria for nonlinear second-order differential equations with damping

Some new oscillation criteria are given for general nonlinear second-order ordinary differential equations with damping of the form x″+ p ( t ) x′+ q ( t ) f ( x ) = 0, where f is monotone or nonmonotone. Our results generalize and extend some earlier results of Deng. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 694–700, May, 2008.     

Nonlinear oscillation of second‐order neutral dynamic equations with distributed delay

In this paper, we consider the nonlinear oscillation of the following second‐order neutral delay dynamic equations with distributed delay on a time scale , where Z(t) = x(t) + p(t)x(τ(t)),α,β > 0 are constants. By using some new techniques, we obtain oscillation criteria for the equation when β > α,β = α, and β < α, respectively. Those results established here complete and develop the oscillation criteria in the literature. Also, our main results are illustrated with some examples. Copyright © 2015 John Wiley & Sons, Ltd.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.