We show that the inequality for all P, QGA n ={PG^n:P = (p I ,p 2 , ,p n ) where ΣΓ=i p, = 1 and p t > 0 …
We show that the inequality for all P, QGA n ={PG^n:P = (p I ,p 2 , ,p n ) where ΣΓ=i p, = 1 and p t > 0 for / = 1,2, , n} and some integer n ^ 3, implies that /(p) = Ap c where A is an arbitrary nonzero constant and either c^-1 or c^O.The converse holds as well, so that this result yields a characterization of the information gain.
Journal Article ON THE INEQUALITY │f″│2≤ K│f││f(4)│ Get access J. S. BRADLEY, J. S. BRADLEY University of Tennessee KnoxvilleTennessee 37916 Search for other works by this author on: Oxford Academic …
Journal Article ON THE INEQUALITY │f″│2≤ K│f││f(4)│ Get access J. S. BRADLEY, J. S. BRADLEY University of Tennessee KnoxvilleTennessee 37916 Search for other works by this author on: Oxford Academic Google Scholar W. N. EVERITT W. N. EVERITT University of Dundee DundeeScotland Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 25, Issue 1, 1974, Pages 241–252, https://doi.org/10.1093/qmath/25.1.241 Published: 01 January 1974 Article history Received: 16 May 1973 Published: 01 January 1974
Everyone knows that x − x = 0, but how many people realise the power of reversing this equality and writing 0 = x − x ? What I mean …
Everyone knows that x − x = 0, but how many people realise the power of reversing this equality and writing 0 = x − x ? What I mean is that we can always add and subtract the same quantity to and from any algebraic expression without altering its value. In general, we may choose infinitely many values of x , and the skill lies in choosing the most suitable x so that progress can be made in a specific problem. Beginners in mathematics are often mystified when a particular x is chosen and almost invariably respond with something like “I agree that what you are doing is correct, but why on earth did you pick that particular x ?” There is no answer to this except perhaps “because it works!”.
(1966). The Relation of f′+(a) to f′(a+) Mathematics Magazine: Vol. 39, No. 2, pp. 112-120.
(1966). The Relation of f′+(a) to f′(a+) Mathematics Magazine: Vol. 39, No. 2, pp. 112-120.
First we will briefly define the [ f , g ] and A λ summability methods. Let K = { w : | w | < 1}. T. H. Gronwall …
First we will briefly define the [ f , g ] and A λ summability methods. Let K = { w : | w | < 1}. T. H. Gronwall [3] introduced a general class of summability methods each of which involves a pair of functions f and g with the following properties. The function z = f ( w ) is analytic on \{1}, continuous and univalent on , with f (0) = 0, f (1) = 1, | f ( w )| < 1 if w ∊ K. The inverse function w = f -1 ( z ) is analytic on f ( K )\{1}, and at z = 1 1.1 where γ ≧ 1, a > 0, and the quantity in brackets is a power series in 1 — z with positive radius of convergence.
In this article, the operator inequalities and with linear positive operator K are analyzed in a real Banach space. The obtained results are applied to linear Fredholm and Volterra integral …
In this article, the operator inequalities and with linear positive operator K are analyzed in a real Banach space. The obtained results are applied to linear Fredholm and Volterra integral inequalities of the second kind with general continuous nonnegative kernel k(x, s). These results are all new. For a special case of a Volterra inequality, Gronwall’s lemma is deduced, however, in a manner that is different from the known one. The general results may also be applied to Fredholm-type and Volterra-type matrix inequalities.
(1961). A Note on the Limit of f(x)/f'(x) Mathematics Magazine: Vol. 34, No. 5, pp. 268-268.
(1961). A Note on the Limit of f(x)/f'(x) Mathematics Magazine: Vol. 34, No. 5, pp. 268-268.
We study the initial trace problem on Ω at t = o for the positive solutions of the following parabolic equation(PE).We prove that the initial trace is an outer regular …
We study the initial trace problem on Ω at t = o for the positive solutions of the following parabolic equation(PE).We prove that the initial trace is an outer regular Borel measure, not necessarily locally bounded. Conversely, we study the initial trace problem with a given Borel measure. Whenwe prove that there is a one to one correspondence between regular Borel measures and positive solutions of (PE). Whenwe give necessary and sufficient conditions for the existence of a maximal solution with a given initial trace, but uniqueness does not hold
We obtain exact conditions guaranteeing that any global weak solution of the differential inequality is trivial, where are integers and and g are some functions.
We obtain exact conditions guaranteeing that any global weak solution of the differential inequality is trivial, where are integers and and g are some functions.
<p style='text-indent:20px;'>Our purpose in this paper is to classify the non-topological solutions of equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta u +\frac{4e^u}{1+e^u} = 4\pi\sum\limits_{i = 1}^k n_i\delta_{p_i}-4\pi\sum^l\limits_{j = 1}m_j\delta_{q_j} \quad{\rm …
<p style='text-indent:20px;'>Our purpose in this paper is to classify the non-topological solutions of equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta u +\frac{4e^u}{1+e^u} = 4\pi\sum\limits_{i = 1}^k n_i\delta_{p_i}-4\pi\sum^l\limits_{j = 1}m_j\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2,\;\;\;\;\;\;(E) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \{\delta_{p_i}\}_{i = 1}^k $\end{document}</tex-math></inline-formula> (resp. <inline-formula><tex-math id="M2">\begin{document}$ \{\delta_{q_j}\}_{j = 1}^l $\end{document}</tex-math></inline-formula>) are Dirac masses concentrated at the points <inline-formula><tex-math id="M3">\begin{document}$ \{p_i\}_{i = 1}^k $\end{document}</tex-math></inline-formula>, (resp. <inline-formula><tex-math id="M4">\begin{document}$ \{q_j\}_{j = 1}^l $\end{document}</tex-math></inline-formula>), <inline-formula><tex-math id="M5">\begin{document}$ n_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ m_j $\end{document}</tex-math></inline-formula> are positive integers. Denote <inline-formula><tex-math id="M7">\begin{document}$ N = \sum^k_{i = 1}n_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ M = \sum^l_{j = 1}m_j $\end{document}</tex-math></inline-formula> satisfying that <inline-formula><tex-math id="M9">\begin{document}$ N-M&gt;1 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>Problem <inline-formula><tex-math id="M10">\begin{document}$ (E) $\end{document}</tex-math></inline-formula> arises from gauged sigma models and we first construct an extremal non-topological solution <inline-formula><tex-math id="M11">\begin{document}$ u $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M12">\begin{document}$ (E) $\end{document}</tex-math></inline-formula> with asymptotic behavior</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ u(x) = -2\ln |x|-2\ln\ln|x|+O(1)\quad{\rm as}\quad |x|\to+\infty $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and with total magnetic flux <inline-formula><tex-math id="M13">\begin{document}$ 4\pi (N-M-1) $\end{document}</tex-math></inline-formula>. And then we do the classification for non-topological solutions of <inline-formula><tex-math id="M14">\begin{document}$ (E) $\end{document}</tex-math></inline-formula> with finite magnetic flux. This solves a challenging long standing problem. We believe that our approach is novel and applies to other types of equations.</p>
If $Ω$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of …
If $Ω$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-Δu+f(u)=0$ in $Q_\infty^Ω:=Ω\times (0,\infty)$, $u=\infty$ on the parabolic boundary $\partial_{p}Q$. We prove that in most cases, the existence and uniqueness is reduced to the same property for the associated stationary equation in $Ω$.
Due to an oversight the authors make a statement (page 422) which amounts to saying that every generalized absolute retract is an absolute G β .This statement is not true, …
Due to an oversight the authors make a statement (page 422) which amounts to saying that every generalized absolute retract is an absolute G β .This statement is not true, see for instance, Dugundji, Pacific J. Math., 1951.
Abstract This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic …
Abstract This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem urn:x-wiley:0025584X:media:mana201600231:mana201600231-math-0001 where either with is a smooth bounded domain or . The function ϕ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics. The proofs are based on comparison principle, variational methods and topological arguments on the Orlicz–Sobolev spaces.
Sufficient conditions are given which ensure nonexistence of spherically symmetric entire solutions of Δ p u = f ( u ), p ≧ 2. Sufficient conditions for existence of spherically …
Sufficient conditions are given which ensure nonexistence of spherically symmetric entire solutions of Δ p u = f ( u ), p ≧ 2. Sufficient conditions for existence of spherically symmetric entire solutions of Δ p u = f ( r, u ) are also given.
Synopsis Semilinear elliptic equations of the form are considered, where Δ m is the m -th iterate of the two-dimensional Laplacian Δ, p ( t ) is continuous in [0, …
Synopsis Semilinear elliptic equations of the form are considered, where Δ m is the m -th iterate of the two-dimensional Laplacian Δ, p ( t ) is continuous in [0, ∞), and f ( u is continuous and positive either in (0, ∞) or in ℝ. Our main objective is to present conditions on p and f which imply the existence of radial entire solutions to (*), that is, those functions of class C 2 m (ℝ 2 ) which depend only on |x| and satisfy (*) pointwise in ℝ 2 . First, necessary and sufficient conditions are established for equation (*), with p ( t ) > 0 in [0, ∞), to possess infinitely many positive radial entire solutions which are asymptotic to positive constant multiples of | x | 2 m −2 log | x as | x | → ∞. Secondly, it is shown that, in the case p ( t < 0, in [ 0, ∞) and f ( u ) > 0 is nondecreasing in ℝ, equation (*) always has eventually negative radial entire solutions, all of which decrease at least as fast as negative constant multiples of | x | 2 m −2 log | x | as | x | → ∞. Our results seem to be new even when specialised to the prototypes where γ is a constant.
Let $b(x)$ be a positive function in a bounded smooth domain$\Omega\subset R^N$, and let $f(t)$ be a positive non decreasingfunction on $(0,\infty)$ such that $\lim_{t\to\infty}f(t)=\infty$.We investigate boundary blow-up solutions of …
Let $b(x)$ be a positive function in a bounded smooth domain$\Omega\subset R^N$, and let $f(t)$ be a positive non decreasingfunction on $(0,\infty)$ such that $\lim_{t\to\infty}f(t)=\infty$.We investigate boundary blow-up solutions of the equation $\Deltau=b(x)f(u)$. Under appropriate conditions on $b(x)$ as $x$approaches $\partial\Omega$ and on $f(t)$ as $t$ goes to infinity,we find a second order approximation of the solution $u(x)$ as $x$goes to $\partial\Omega$. We also investigate positive solutions of the equation $\Deltau+(\delta(x))^{2\ell}u^{-q}=0$ in $\Omega$ with $u=0$ on$\partial\Omega$, where $\ell\ge 0$, $q>3+2\ell$ and $\delta(x)$denotes the distance from $x$ to $\partial\Omega$. We find a secondorder approximation of the solution $u(x)$ as $x$ goes to$\partial\Omega$.
We obtain classification, solvability, and nonexistence theorems for positive stationary states of reaction-diffusion and Schrödinger systems involving a balance between repulsive and attractive terms. This class of systems contains PDE …
We obtain classification, solvability, and nonexistence theorems for positive stationary states of reaction-diffusion and Schrödinger systems involving a balance between repulsive and attractive terms. This class of systems contains PDE arising in biological models of Lotka--Volterra type, in physical models of Bose--Einstein condensates, and in models of chemical reactions. We show, with different proofs, that the results obtained in [A. Montaru, B. Sirakov, and P. Souplet, Arch. Ration. Mech. Anal., 213 (2014), pp. 129--169] for models with homogeneous diffusion are valid for general heterogeneous media, and even for controlled inhomogeneous diffusions.
In the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the …
In the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the leaves and transversally continuous is the curvature function of a unique laminated metric in the corresponding conformal class. We give an interpretation of this result as a continuity result about the solutions of some elliptic PDEs in the so called Cheeger–Gromov topology on the space of complete pointed riemannian manifolds.
The goal of this paper is to prove the boundary asymptotic behavior of solutions for weighted -Laplacian equations that take infinite value on a bounded domain.
The goal of this paper is to prove the boundary asymptotic behavior of solutions for weighted -Laplacian equations that take infinite value on a bounded domain.
We report on the solution of two long-standing conjectures on the boundary behavior of maximal solutions of semilinear elliptic equations, focusing on the proof of the boundary regularity of the …
We report on the solution of two long-standing conjectures on the boundary behavior of maximal solutions of semilinear elliptic equations, focusing on the proof of the boundary regularity of the hyperbolic radius in higher dimensions. The main tool is the reduction of the problem to a degenerate equation of Fuchsian type, for which new Schauder-type estimates are proved. We also sketch an algorithm suitable for large classes of applications.
A sufficient and necessary condition for existence of solution for the boundary blow-up problem in one dimensional case is obtained. This problem can be seen as the Keller-Osserman conjecture, which …
A sufficient and necessary condition for existence of solution for the boundary blow-up problem in one dimensional case is obtained. This problem can be seen as the Keller-Osserman conjecture, which comes from the study on elliptic equations.
Jowhere C k (x), the kth.fractional integral of cos x, is commonly known as Young's function [6, p. 564].We shall say that the infinite series Σ a n is summable …
Jowhere C k (x), the kth.fractional integral of cos x, is commonly known as Young's function [6, p. 564].We shall say that the infinite series Σ a n is summable (r, k) if 0 oo (i) ΣαWfc(^) converges for 0<t<A 0 and oo (ii) lim Yja n γ k {nt) -S , where S is finite.ί-»0 0We see that (γ, l)=(i?, 1) and (γ, 2) = (R, 2), where {R, 1) and (R } 2) are the well known Riemann summability methods.Hence the (γ, &)-summability methods constitute, in a sense, an extension of (R, 1) and (R, 2) summability methods to {R, k) methods where k may be non-integral.But this extension is not linked with the ideas which lie at the root of the Riemann summability methods, that is, taking generalised symmetric derivatives of repeatedly integrated Fourier series, so that the equivalence of (γ, k) and (R,k) for k=l,2 may be considered to be somewhat accidental, and the extension artificial.However, the {γ, k) methods are also connected with certain aspects of the summability problems of oo Fourier series.For, let Σ A n {x) be the Fourier series of a periodic and 0 Lebesgue integrable function f(x) and let
drawn my attention to a paper by F. Hausdorff, Die Aquivalenz der Holderschen und Cesarbschen Grenzwerte negativer Ordnung
drawn my attention to a paper by F. Hausdorff, Die Aquivalenz der Holderschen und Cesarbschen Grenzwerte negativer Ordnung
Due to an oversight the authors make a statement (page 422) which amounts to saying that every generalized absolute retract is an absolute G β .This statement is not true, …
Due to an oversight the authors make a statement (page 422) which amounts to saying that every generalized absolute retract is an absolute G β .This statement is not true, see for instance, Dugundji, Pacific J. Math., 1951.
In the paper of the title [1], a number of problems are posed. Ne gative solutions of two of them (Problems 2 and 3) are derived in a straightforward way …
In the paper of the title [1], a number of problems are posed. Ne gative solutions of two of them (Problems 2 and 3) are derived in a straightforward way from a paper of L. Gillman and the present author [2]. Motivation will not be supplied since it is given amply in [1], but enough definitions are given to keep the presentation reasonably self contained. 1. A Hausdorff space X is said to satisfy (Qm), where m is an in finite cardinal, if, whenever U and V are disjoint open subsets of X such that each is a union of the closures of less than m open subsets of X, then U and V have disjoint closures. In particular, a normal (Hausdorff) space X satisfies (Q*,) if and only if disjoint open F.-SUbsets of X have disjoint closures. (For, an open set that is the union of less than ~, closed sets is a fortiori an F.. Conversely if U is the union of countably many closed subsets Fo> then since X is normal, for each n there is an open set Un containing Fn whose closure is contained in U. Thus U is the union of the closures of the open sets Un.) In Prob lem 3 of [1], it is asked if every compact (HaUSdorff) space satisfying (Qm) for some m>~o is necessarily totally disconnected, and it is re marked that this is the case if the first axiom of countability is also as sumed. If X is a completely regular space, let C(X) denote the ring of all continuous real-valued functions on X, and let Z(f)= {x EX: f(x)=O}, let P(f)={xEX:f(x»O}, and let N(f)=P(-f). As usual, let (IX denote the Stone-Cech compactification of X. If every finitely generated ideal of C(X) is a principal ideal, then X is called an F-space. The fol lowing are equivalent. ( i) X is an F-space. ( ii) If f E C(X), then P(f) and NU) are completely separated [2, Theorem 2.3]. (iii) If f E C(X), then every bounded g E C(X-ZU» has an ex tension g E C(X) [2, Theorem 2.6]. A good supply of compact F-spaces is provided by the fact that if X is locally compact and ,,-compact, then {lX-X is an F-space [2, Theo rem 2.7].
1 Namely, a conditionally cr-complete vector lattice.In this paper we use the terminology and notation of [5].2 In this paper, Amemiya also proved the following lemma: Let R be a …
1 Namely, a conditionally cr-complete vector lattice.In this paper we use the terminology and notation of [5].2 In this paper, Amemiya also proved the following lemma: Let R be a monotone complete normed semi-ordered linear space.Then there exists a number f>0 such that x implies * For the definition of this sequence space, see [6].