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In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

We have considered the process of inscribing triangles between parallel lines in the Euclidean plane using an iterative process. It is a familiar result that the limiting triangle is equilateral. In this Note, we demonstrate that the same result holds when the geometry is hyperbolic and the lines are asymptotically parallel. We further extend the process to the sphere, where it is found that the limiting triangle is isosceles.

On a recent paper Benchohra and Mostefai [2] presented some existence results for an integral equation of fractional order with multiple time delay in Banach spaces. In contrast to the classical case, when assumptions are expressed in terms of the strong topology, they considered another case, namely with the weak topology. It has some consequences for the proof. We present here some comments and corrections.

Many solid materials exhibit stress–induced phase transformations. Such phenomena can be modelled with the aid of the nonlinear elasticity theory with appropriate choices of the strain–energy function. It is known that if a two–phase deformation (with gradient F ) in a finite elastic body is a local energy minimizer, then given any point p of the surface of discontinuity, the piecewise–homogeneous deformation corresponding to the two values F ± ( p ) of F ( p ) is a global energy minimizer. Thus, instability of the latter state would imply instability of the former state. In this paper we investigate the stability properties of such piecewise–homogeneous deformations. More precisely, we are concerned with two joined half–spaces that correspond to two different phases of the same material. We first show how such a two–phase deformation can be constructed. Then the stability of the piecewise–homogeneous deformation is investigated with the aid of two test criteria. One is a kinetic stability criterion based on a quasi–static approach and on the growth/decay behaviour of the interface in the undeformed configuration when it is perturbed; the other, referred to as the energy criterion, is used to determine whether the deformation is a minimizer of the total energy with respect to perturbations of the interface in both the current and undeformed configurations. We clarify the differences between the two criteria, and provide a compact formula which can be used to establish the stability/instability of any two–phase piecewise–homogeneous deformations.

We study the problem of characterizing the effective (homogenized) properties of materials whose diffusive properties are modeled with random fields. Focusing on elliptic PDEs with stationary and ergodic random coefficient functions, we provide a gentle introduction to the mathematical theory of homogenization of random media. We also present numerical examples to elucidate the theoretical concepts and results.

Kolmogorov-Arnold-Moser (or KAM) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and KAM theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov's pioneering work.

Let S be a surface obtained from a plane polygon by identifying infinitely many pairs of segments along its boundary. A condition is given under which the complex structure in the interior of the polygon extends uniquely across the quotient of its boundary to make S into a closed Riemann surface. When this condition holds, a modulus of continuity is obtained for a uniformizing map on S., 36 pages, 11 figures

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.

In their 1993 paper, W. Goh and J. Wimp derive interesting asymptotics for the moments cn(?) ? cn = ?10tnd?(t), n = 0, 1, 2, ..., of some singular distributions ? (with support ? 0, 1]), which contain oscillatory terms. They suspect, that this is a general feature of singular distributions and that this behavior provides a striking contrast with what happens for absolutely continuous distributions. In the present note, however, we give an example of an absolutely continuous measure with asymptotics of moments containing oscillatory terms, and an example of a singular measure having very regular asymptotic behavior of its moments. Finally, we give a short proof of the fact that the drop-off rate of the moments is exactly the local measure dimension about 1 (if it exists).

This paper considers the existence of global weak solutions for the Navier-Stokes equations in the infinite cylinder R n × R + {{\mathbf {R}}^n} \times {{\mathbf {R}}_ + } with initial data in L r {L^r} , n ≥ 3 n \geq 3 , 1 > r > ∞ 1 > r > \infty . An imbedding theorem as well as related initial value problems are also studied, thus completing results in [2].

It is shown that an immersion of n dimensional compact oriented manifold without boundary into the n+1 dimensional Euclidean space, hyperbolic space or open half sphere is a totally umbilic immersion if one of the mean curvature function H_l does not vanish and the ratio H_k/H_l is constant, 1\leq k, l \leq n, k\ne l.

Let E be a real Banach space. A norm-one element e in E is said to be an isometric reflection vector if there exist a maximal subspace M of E and a linear isometry F : E → E fixing the elements of M and satisfying F (e) = −e. We prove that each of the conditions (i) and (ii) below implies that E is a Hilbert space. (i) There exists a nonrare subset of the unit sphere of E consisting only of isometric reflection vectors, (ii) There is an isometric reflection vector in E, the norm of E is convex transitive, and the identity component of the group of all surjective linear isometries on E relative to the strong operator topology is not reduced to the identity operator on E.

We disagree with Bucy’s comments (Bucy 1994) on our (Udwadia &; Kalaba 1992) paper. The reasons are as follows. 1. Just as the Gibbs-Appell equations (which use quasi-coordinates) are not the same as the Lagrange equations (which use Lagrange multipliers), the equations of motion obtained by us are not the same as the Gibbs-Appell equations. Some of the steps required to obtain the Gibbs-Appell equations are (Pars 1979; Whittaker 1937): (1) choice of quasi-coordinates; (2) elimination of certain quasicoordinates in terms of other preferred quasi-coordinates; (3) setting up of the Gibbs function; and (4) differentiation of the Gibbs function with respect to the preferred quasi-coordinates. None of these steps is used in obtaining our equations of motion. 2. Yet, the equations of motion obtained by us are equivalent to the Lagrange equations with multipliers (Kalaba et al. 1995) and to the Gibbs-Appell equations (Udwadia & Kalaba 1996). By equivalent we mean each set of equations implies the other. In fact all these different sets of equations are simply different, yet equivalent, ways of stating D’Alembert’s principle for bilinear constraints. Thus what applies to one set of equations applies to the others. Any deficiency in the equations derived in our paper (say regarding rank changes of the matrix A) is therefore present in the Gibbs-Appell equations and Lagrange’s equations as well, because of their equivalence.

In the paper [1], the authors obtain a KAM theorem for nearly integrable hamiltonian systems under the Russmann’s non-degeneracy condition, which is known to be sharpest one for small divisor conditions. However, the Remark 1.3 is wrong because we have ignored the null set − ∗, which may contain zeros of ω of high order such that (1.4) does not hold for all p ∈ . The Remark 1.3 might mislead the readers that the condition (1.5) of Theorem B is equivalent to the Russmann’s non-degeneracy condition. Actually, the Russmann’s non-degeneracy condition is equivalent to the condition (1.4) of Theorem A as proved in [1]. Under the Russmann’s non-degeneracy condition (1.4), as proved in the Remark 3.1 the condition (1.5) holds if we replace n − 1 by a sufficiently large number N depending on h, and then the conclusion of Theorem B remains valid if in the measure estimate n − 1 is replaced by N .

In this paper we point out an error in paper [2] and study the asymptotic behavior of the differential equation Lnx + f(t, x) = r(t). The results obtained are extensions of some of the results of [2].

A class of functions K(r) and the corresponding solutions are produced in the case of the differential equation d 2 u/dr 2 +1 du/r dr+K(r)e 2u(r) =1. The behavior of these solutions at an isolated singularity is also investigated

This paper deals with Schrodinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrodinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper).

Consider the first order differential equation (1) , where pi, and τi, for i = 1,…,n, are positive constants. To find necessary and sufficient conditions, in terms of the coefficients and the delays only, under which all solutions of (1) oscillate, is a problem of great importence. In a recent paper, Bowcock and Yu claimed that is a necessary and sufficient condition for all solutions of (1) to be oscillatory. In this paper a counterexample shows that the above result is not valid and the error in this paper is indicated.

Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

iF(y)=( where G c R* is a bounded domain, y: G -» R , y'(x) = (dyydx), and F: M -> R is continuous. Here M denotes the linear space of real N X k matrices. We suppose throughout that K > 2, N > 2. In [7] F is called T-conυex if there exists a convex function /, defined on R, r = (t) ~ 1, such that F(p) = f(τ(p)) for all/? e M, where τ(p) denotes the minors of p of all orders j , 1 y uniformly on G with supx χGG\yj(x) ~ yj(x)\ < C < oo for ally. (Equivalently, if G has sufficiently regular boundary then IF is lsc if and only if IF is sequentially weak* lower semicontinuous on the Sobolev space W(G; R).) A consequence of [7? Theorem 3.6] is that IF lsc implies F polyconvex; that this conclusion is false was pointed out implicitly by Morrey [4, p. 26]. Morrey's remark is based on an example due to Terpstra [8] of a quadratic form

We establish the relationship between certain invariant extension properties for linear functional of K. Fan when a semigroup S acts on a locally convex topological vector space and the existence of a left-invariant mean on the space of almost periodic and weakly almost periodic functions on S.

Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, Funder: Einstein Visiting Fellow Program, The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

© 2019, Pleiades Publishing, Ltd. It is well known that the classical and Sobolev wave fronts were extended to nonequivalent global versions by the use of the short-time Fourier transform. In this very short paper, we give complete characterizations of the former wave front sets in terms of the short-time Fourier transform.

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A subset A of a topological space is said to be semi-open if there exists an open set U such that U C A C Cl(U) where Cl(U) denotes the closure of U. The class of semi-open sets of a given topological space (X, J) is denoted S.O. (X, J). In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if J and V* are two topologies for a set X such that S.O.(X, 3) C S.O.(X, J*), then 'J C P. In a corollary to this theorem, Levine states that if S.O.(X, if) = S.O.(X,Jf*), then _T= f*. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subse,ts can arise from functions, and points out some of the implications of two topologies being related in this manner. In [6] Norman Levine defines a set A in a topological space X to be semi-open if there exists an open set U such that U C A C Cl (U), where Cl(U) denotes the closure of U. The class of semi-open sets for a given topological space (X, i) is denoted S.O. (X, sT). Levine states in Theorem 10 of [6] that if Jf and * are two topologies for a set X such that S.O. (X, i) C S.O. (X, J9 then iT C *. In a corollary to this theorem, Levine states that if S.O. (X, ) = S.O. (X, iT*), then Jf = -" The following example which is due to S. Gene Crossley and S. K. Hildebrand [1, Example 1.1] shows the above-mentioned theorem and its corollary are false. Example. Let X = la, b, cl, J; = t0, sal, la, bl, la, cl, XI, ;* = 10, sal, la, bl, XI. An exhaustion of all possibilities shows that S.O. (X, ) = S.O. (x, 5j*). Crossley and Hildebrand [3] defined two topologies iT and T* on a set X to be semi-correspondent if S.O. (X, J) = S.O. (X, 5f*). It is shown in [3] that semi-correspondence is an equivalence relation on the collection of Received by the editors March 3, 1974. AMS (MOS) subject classifications (1970). Primary 54B99; Secondary 54C10.

If BP denotes the space of smooth alternating p-forms on the C' manifold M, we are interested in finding the spaces of R-linear maps from iPP to be that commute with the diffeomorphisms of M. For a compact manifold M these spaces were found by R. S. Palais. In this note we find them for noncompact M. Let V, W be the spaces of sections of two tensor bundles over a connected Coo manifold M of dimension n. In a paper [3] appearing in Trans. Amer. Math. Soc. 92 (1959), R. S. Palais studied a number of the spaces 4 (V, W) of the R-linear maps V->W natural with respect to the action on V and W of the group G of diffeomorphisms on M. We sketch the definition of this action: If g-G we define R,:F(TM) -4F(TM) to be the differential of g, and Rg:r(T*M)-->F(T*M) to be the dual of the differential of g-1. For the other spaces of tensor fields Rg is defined as the obvious tensor product of these two R-linear isomorphisms. A map c: V->W is then called natural if cRg =R,c for all g G. Palais was particularly interested in the case when V and W are the space IP of p-forms and the space bq of q-forms respectively, 0 < p, q

It is the object of this paper to obtain theorems concerning the stability of the straight line solutions, and equilateral triangle solutions, respectively, in the problem of three bodies by means of the theorems and calculations of two preceding papers.t It is shown that the generalized theorems of Bohl can be applied to a neighborhood of the straight line solutions, with arbitrary masses, and to a neighborhood of the equilateral triangle solutions, if the masses are such that the characteristic exponents of the generating orbit are not all pure imaginaries. The mutual distances of the three masses are assumed constant on the generating orbit, in both cases.

The author regrets that there are certain errors in [1] and would like to give the following corrections.