Search

The statistical initial value problem for the envelope equation \[ \frac{{\partial w}}{{\partial t}} - \gamma \frac{{\partial ^2 w}}{{\partial x^2 }} = \chi w - \beta w^2 w^ * ,\quad \gamma ,\beta \,{\text{complex}} \], is discussed. It is shown that there exists an instability (stability) criterion \[ \beta _r \gamma _r + \beta _i \gamma _i \gtreqless 0,\quad \beta _r > 0 \], which determines whether the system underlying the above equation achieves or does not achieve a monochromatic state.

The resultant intensity distribution due to random interference of two m-vectors is derived under the conditions : (1) Intensities of two m-vectors follow the correlated m-distribution with equal mean square and an arbitrary natural number of m : (2) The phase difference is uniformly distributed over the interval (0, 2π). An expression for the resultant intensity distribution is obtained in a positive finite series which consists of the generalized q-distributions ( or the m-distributions, in limiting cases of the correlation).

It is shown that there is a constantK so that, for every separable metric spaceX, there is a mapT:X →co satisfyingd(x, y)≦‖Tx−Ty‖≦Kd(x, y) for everyx, y ∈ X.

The manifolds in an additive basis for the cobordism ring of manifolds with involution are Z2-equivariantly immersed and embedded into representation spaces having the smallest possible number of nontrivial factors up to cobordism. Let a be the Z2-action on RP' which is given in homogeneous coordinates by o4[xo, xl, * , x]=[-xo, x1, **, xj]. Define r n+1 by r[M, T] xS [m, z] [ > [m,z]], M(m, z) (Tm, -z) where qZ2 iS the bordism group of unrestricted involutions. F can be defined in the same way for manifolds which are not compact. {l}u {Fi(RPnl, or) x (RPrt2, a) x*** x (RPnk, c) x (Mm, l)ii>O, k> 1, ni >n2?> * nk> 1, [M]2 0O}, where by ro we mean the identity, and by [MJ2 we mean the class of the closed c' manifold M in the unoriented cobordism ring (Thom [8]), is an additive set of generators for qZ2 (Alexander [1], Stong [7]). Let (-1)re31s denote Rrs furnished with the Z2-action (xl, ,xr+s) (-xi, * *, _X,r, Xr+l, * * *, Xr+s). For each element (N, S) in the additive basis for ?jZ2, we wish to find an integer rN (resp. r') such that there is a manifold with involution in the bordism class of (N, S) which can be Z2equivariantly embedded (immersed) in a representation space (-I)r,v (D Is ( ( _ I)r2v' (D ls) for some s and such that no manifold with involution in the bordism class of (N, S) can be Z2-equivariantly embedded (immersed) in a representation space (_1)r,-1 D lt ((1 )r 1 D lt) for any t. This is a generalization to the case of Z2-manifolds of the work Received by the editors June 28, 1973. AAMS (MOS) sub]ect classifications (1970). Primary 57D40; Secondary 57D85, 57E25. ? American Mathematical Society 1974

We consider finding the best possible constants $C = C(n,\alpha ,\beta ,p,q)$ such that $\| f \|_p \leqq C(b - a)^{n + 1/p -1/q} \| {f^{(n)} } \|_q $ under several differen sets of boundary conditions. Specifically in one case, f is required to have a zeros at $a $ and $\beta $ zeros at b where $n \leqq \alpha + \beta \leqq 2n$, $\alpha \leqq n$ and $\beta \leqq n$. In the other case, f has $\alpha $ zeros at $a $, $\beta $ zeros at b and $\int _a^b f(x)x^r (x - a)^\alpha (x - \beta )^\beta dx = 0$, $r = 0, \cdots ,n - 1 - \alpha - \beta $, where $0 \leqq \alpha $, $0 \leqq \beta $ and $\alpha + \beta < n$. The various problems are related and the numbers C are calculated exactly in some cases. Finally, we show how some of these can be applied to disconjugacy problems in differential equations.

Sufficient conditions on the coefficients p 2 n , p 2 n − 1 , ⋯ , p 0 {p_{2n}},{p_{2n - 1}}, \cdots ,{p_0} are given which guarantee that no nontrivial solution of p 2 n y ( 2 n ) + p 2 n − 1 y ( 2 n − 1 ) + ⋯ + p 0 y = 0 {p_{2n}}{y^{(2n)}} + {p_{2n - 1}}{y^{(2n - 1)}} + \cdots + {p_0}y = 0 has two distinct zeros each of order at least n n . These conditions are in the form of n n inequalities which are satisfied by linear combinations of the coefficients and their derivatives.

Let X be a Hilbert space with inner product (x, y) for x, y E X. For fixed y the distance from x to y is p&x) = 11 x y 11 = (x y, x y)lls and for a nonempty set KC X the distance from x to K is P&) = inf(p,(x) 1 y E K}. It is known [2] (and it follows from Theorem 1 below by taking K = {y}) that pV is FrCchet differentiable at all x # y. We consider here some circumstances under which pK is FrCchet differentiable. Since

Let\((x_\alpha )_{\alpha \epsilon A}\), where A is a directed set containing cofinal chains — a generalized sequence in a complete chain. It is established that every such sequence contains a monotonic cofinal sub-sequence. For a monotonically increasing (decreasing) bounded sequence\((x_\alpha )_{\alpha \epsilon A}\), by definition, we put\((i) - \mathop {\lim }\limits_{\alpha \in A} x_\alpha = \mathop {\sup }\limits_{\alpha \in A} (x_\alpha ) \cdot ((i) - \mathop {\lim }\limits_{\alpha \in A} x_\alpha = \mathop {\inf }\limits_{\alpha \in A} (x_\alpha ))\). For an arbitrary sequence\((x_\alpha )_{\alpha \epsilon A}\) the (i)-limit is defined as the common (i)-limit of its monotonic cofinal sub-sequences. The properties of (i)-convergence and some of its applications to generalized sequences of mappings are discussed.

It is known that the restricted circular Three Body Problem with the perturbing function being twice averaged (by the longitudes of the point with zero-mass and the one with the least finite mass) could be integrated. Due to the complexity of one of the known integrals the full investigation of this problem so far does not exist. Only investigations of limiting cases are known. This paper shows a new limiting class of solution of this problem and gives its qualitative investigation. The space orbits of the ‘asteroids’ everywhere close to the circular orbit of the body with the least finite mass are considered. Certain qualitative singularities of the behavior of solutions, connected with the critical inclination (Krasinsky, 1972) are noted. The paper also shows an analogous class of solutions of the twice-averaged problem, dealing with the perturbing of the satellite's orbit by the ‘mascon’.

The simplest initial value problem for delay differential equations takes the form \[ {\text{(1)}}\qquad \dot y(t) = - \lambda y(t - \tau ),\quad t > 0;\qquad y(t) = \psi (t),\quad - \tau \leqq t \leqq 0.\] If the step size h is chosen so that $\tau = mh$ where m is an integer, then the linear multistep method $\{ \rho ,\sigma \} $ may be applied to (1) to yield an approximate solution $y^h $ defined at the gridpoints $t_j = jh$.It is known that $y(t) \to 0$ as $t \to \infty $ for all initial data $\psi $ iff $\lambda \in (0,{\pi / 2})$. In analogy with the concept of A-stability, the method $\{ \rho ,\sigma \} $ is called $DA_0 $-stable if $y^h (t_j ) \to 0$ as $j \to \infty $ for all initial data $\psi $ and all positive integers m. The properties of $DA_0 $-stable methods are studied.

The closed convex hull and extreme points are obtained for the functions which are convex, starlike, and close-to-convex and in addition are real on(−1,1)( - 1,1). We also obtain this result for the functions which are convex in the direction of the imaginary axis and real on(−1,1)( - 1,1). Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. We also obtain such a representation for the functionsf(z)f(z)analytic in the unit disk, normalized and satisfyingRe⁡f′(z)>α\operatorname {Re} f’(z) > \alphaforα>1\alpha > 1. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to some function satisfyingRe⁡f′(z)>α\operatorname {Re} f’(z) > \alpha.

An alternate proof is presented of Kershaw’s result that the $L_\infty $-norm of the error in natural spline interpolation to a function $f \in C^4 [a,b]$ is $O(h^4 )$ in a closed subinterval which is asymptotic to $[a,b]$ as $h \to 0$. The case $f \in C^m [a,b],m = 2$ or 3, is also considered. These univariate results are then used to investigate natural bicubic spline interpolation over a rectangle $\mathcal{R}$. For $f \in C^m [\mathcal{R}]$, the $L_\infty $-norm of the error is shown to be $O(h^{m - 2} )$ throughout $\mathcal{R}$ and $O(h^m )$ in a closed subrectangle which is asymptotic to $\mathcal{R}$ as $h \to 0$. Arbitrary sequences of partitions are considered throughout.

RADIAL AVERAGING OF DOMAINS, ESTIMATES FOR DIRICHLET INTEGRALS AND APPLICATIONS by Moshe Marcus Let & = {D,,...,D } be a family of domains in the plane, containing the origin. We define a radial averaging transformation R. on S by which we obtain a starlike domain D . When & is such that the domains D,,...^ are obtained from a fixed domain D by rotation or reflexion, ft. becomes j\ a radial symmetrization. One of the results we present is an inequality relating the conformal radius of D to the conformal radii of D, ,...,D at the origin. This result includes, as particular cases, the radial symmetrization results of Szego [11] (for starlike domains), Marcus [7] (for general domains) and Aharonov and Kirwan [1]. The inequality for the conformal radii is obtained via an inequality for conformal capacities, which seems to be of independent interest. A number of applications in the theory of functions are discussed. Here we introduce a definition of a class of functions {f}, analytic in the unit disk | §| 0 the set CL (f^) is open (relative to M) and that for 0 0.) We now keep j fixed. Let P1 = (x- y-Jefi . Kt(f.) and x x x a ^ D j f . ( P ) = A , . Denote by Kc(P,) the open d i s k of r a d i u s 6, c e n t e r e d at I>1. If 0 X ("X A ) 4r6 (x -x ) Since this holds for every j, we obtain (under the same assumptions) (1-9) **(x ,A ) ^l*(x ^ ) + [S (x.-x ) ] 1 / . From (1.9) it follows that: (1.10) |f*(P)-f*(P' ) | (i = 1,2), in na, b,(f*) such that: (1.11) |PX-P2 | = 6 k6. Suppose f*(P1) A2 and A + k5 £*(x2,A2). On the other hand inequality (1.9) holds for these values of X T > X 2 > ^ T * ^ 9 ' H e n c e we obtain: 2 > l + t 6 -( i2 ) 2 ] 1 / 2 -* I1"2 > ' which is a contradiction to (1.11). Definition 1.3. Let feB(M) and denote: = {(x5y) |0 < f(x,y) < 1} PI M; YA(f) = {(x,y) |f(x,y) = A} 0 M, 0 < A < 1. A Suppose that f€C(Q(f)). Let PQ = (xQ,y0) be an interior point of Cl(f) and f (PQ) = AQ. We shall say that PQ is a regular point of f, if Sf/5y j4 0 at all the points of the set Y-x PI {x = xQ} and if this set is contained in the interior of fi(f). Otherwise we shall say that PQ is a critical point and AQ a critical value of f on x = x . Lemma 1.2. Let feB(M) n C(n(f)). Suppose that AQ, (0 < 7\Q < 1) , is not a critical value of f on x = x , (0 < xQ < 1) . Then l(x,~k;±) eC in a neighborhood of (xQ,A ) . Proof. Since feB(M), y-v intersects the line x = xn at a 0 finite number of points {p. , . . . ,P.} . Let P. = (x ,y.) and suppose that y, < y2

The operator Lƒ(t, λ) = e−iλ(t, λ) − 2e−iλ ∝t2π ∝Tƒ(s, x) e(s, t) dvs(x) dm(s) acting on H=∝02π⊕L2(vt), where m and vt, 0 ⩽ t ⩽ 2π are measures on [0, 2π] with m smooth and e(s, t) = exp[−∝ts∝Tdvλ(θ) dm(λ)], satisfies rank(I − LL∗) = rank(I − L∗L) = 1. It is, therefore, unitarily equivalent to a scalar Sz.-Nagy-Foias canonical model. The purpose of this paper is to determine the model explicitly and to give a formula for the unitary equivalence.

A new proof is given of the unpublished results of Morlet on the relation between the homeomorphism group and the diffeomorphism group of a smooth manifold. In particular, the result Diff ( D n , ∂ ) ≃ Ω n + 1 ( Top n / O n ) {\operatorname {Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text {Top}_n}/{O_n}) is obtained. The main technique is fibrewise smoothing.

The solution $x(t)$ of the Volterra integral equation of the second kind $x(t) = f_1 (t) + \sqrt t f_2 (t) + \int _0^t g(t,s,x(s))(t - s)^{ - {1 / 2}} ds$ is examined. It is shown that $x(t) = u(t) + \sqrt t v(t)$, where $u(t)$ and $v(t)$ are smooth under appropriate smoothness conditions on $f_1 (t)$, $f_2 (t)$ and $g(t,s,x)$ and satisfy a system of Volterra integral equations of the second kind.

LetS i have the Wishart distributionW p(∑i,ni) fori=1,2. An asymptotic expansion of the distribution of\( - 2\log \lambda = - 2\log \left[ {\prod\limits_{\alpha = 1}^2 {\left| {{{S_\alpha } \mathord{\left/ {\vphantom {{S_\alpha } {n_\alpha }}} \right. \kern-\nulldelimiterspace} {n_\alpha }}} \right|^{{{n_\alpha } \mathord{\left/ {\vphantom {{n_\alpha } 2}} \right. \kern-\nulldelimiterspace} 2}} } \cdot \left| {{{\left( {S_1 + S_2 } \right)} \mathord{\left/ {\vphantom {{\left( {S_1 + S_2 } \right)} n}} \right. \kern-\nulldelimiterspace} n}} \right|^{ - {n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} } \right]\) for large n=n1+n2 is derived, when∑ 1∑ 2 −1 =I+n−1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function2 F 1, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log|S 2(S 1+S 2)−1| under fixed∑ 1∑ 2 −1 , which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for1 F 1 were investigated by Sugiura [7].

For M M a PL n n -manifold, n ≧ 5 n \geqq 5 , let I ( M ) I(M) be the subset of torsions σ ∈ Wh ⁡ ( π 1 M ) \sigma \in \operatorname {Wh} ({\pi _1}M) such that the h h -cobordism W W constructed from M M with torsion σ \sigma has its other boundary component PL homeomorphic to M M . We present three techniques dealing with the determination of I ( M ) I(M) and apply them when π 1 M = Z q {\pi _1}M = {Z_q} . We prove: (1) If n n is even, π 1 M ≃ Z q , q {\pi _1}M \simeq {Z_q},q odd, then I ( M ) = Wh ⁡ ( π 1 M ) I(M) = \operatorname {Wh} ({\pi _1}M) . (2) If n n is odd, then there exists M M with π 1 M ≃ Z q {\pi _1}M \simeq {Z_q} such that I ( M ) = Wh ⁡ ( π 1 M ) I(M) = \operatorname {Wh} ({\pi _1}M) .

This paper deals with the behaviour of power series with coefficients in a non-archimedean valued field Ω for arguments in a non-archimedean Banachalgebra E/Ω. Beyond a trivial augmentation, the domain of convergence is extended in relation to Ω by certain elements b ∈ E for which\(\mathop {\lim }\limits_{n \to \infty } \left( {\frac{{||b^n ||}}{{||b||^n }}} \right) = 0\). These elements are called pseudo-nilpotent and characterized by\(\mathop {\lim }\limits_{n \to \infty } (||b^n ||^{\frac{1}{n}} )< ||b||\). The examples of exponential functions, logarithms and powers show the changes in relation to the methods and results in Ω caused by the extension of the domain of convergence and the absence of norm multiplicativity and invertibility in E. Finally the algebraic-topological structure of a part of the domain of convergence of the exponential function is presented.

The concept of a heat polynomial defined earlier by Rosenbloom and Widder is extended and the polynomials are characterized using integrals of Poisson type. One particular class of polynomials is examined in detail and leads to expansions of solutions of the problem \[u_{xx} = u_t ,\qquad u(x,0) = f(x)\] which are different from those obtained by Rosenbloom and Widder. In the last section some applications of polynomials to heat equation problems are given. In most results, much use is made of the close relationship between these polynomials and Hermite polynomials.

Let $\phi$ be a function from a normed linear space X into a finite-dimensional Euclidean space Y and let A be a continuous linear mapping from XontoY. We assume that $\phi$ is continuous in a neighborhood of some point $\hat x \in X$ and that $\phi$ admits A as its differential at the point $\hat x$. In this paper we prove that under those conditions there exist a neighborhood U of $\hat x $ and a mapping $\zeta $ from U into X such that $\phi (x + \zeta (x)) = \phi (\hat x) + A(x - \hat x)$ for all $x \in U$ and such that $\lim _{\eta \to 0 + } \sup _{|x - \hat x| \leqq \eta } |\zeta (x)|/\eta = 0$. Besides the above “correction function theorem” this paper contains related implicit and inverse function theorems. The correction function theorem is applied to the proof of the multiplier rule for mathematical programming problems with equality and inequality constraints without assuming the continuous differentiability of the data in a neighborhood of the optimal solution.

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup i→∞{sup y∈K ‖t ix−Tiy∼−‖x−y‖}≦0. IfT N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tix−Tiy‖≦k i‖x−y∼,x,y∈K, wherek i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ0 (X), be less than one.

Let X,(to) be a process in R a with stationary independent increments and let X(E, 09) denote the image under Xt(to) of a time set E. It is shown that dim X(E, og) 1 in R l is found to be a(1-1/~).

We consider a game problem [1,2] similar in formulation to the problems in [3–5] and being a direct continuation of the results in [6], Two material points of unit mass (the first and second players) move in a three-dimensional space under the action of controls F 1 , F 2 alone. The control u = F 1 is bounded in total momentum, while the control − / gv = F 2 is bounded in absolute value. The game termination set M is an arbitrary fixed point in the space of relative positions and velocities of the players, while the payoff is the time taken to lead a relative trajectory to this point. The first player minimizes this time and the second maximizes it. The solution is in many respects analogous to the solution in [6] wherein the minimax time up to “hard” (with respect to the coordinates) and “soft” (with respect to the coordinates and velocities) contact of the points was determined. In the conclusion we consider the problem of soft contact of two controlled points in a linear position central gravity field. In the course of solving the problem in the title we form a vector-valued function q ( w , p ) depending upon the game's position w and on a parameter p , and we divide the whole space W of possible positions into the regions W ° and W 0 . In region W ° there exists a function p 2 (w) >0 , defined as the smallest root of the equation q (w, p)= 0. The first and second players' optimal controls are formed and the minimax time is computed in region W °. In the region W 0 , where the equation q (w, p) = 0 either is entirely free of roots or admits of only nonnegative roots, the second player's control is formed, permitting him to evade falling onto set M under any action by the first player.

This paper presents error estimates for finite difference approximate solutions of initial-boundary problems for parabolic partial differential equations. Methods having matrices of essentially positive type are considered using many techniques from Bramble, Hubbard and Thomee [2] where elliptic partial differential equations were examined.It will be shown that if the approximation has “accuracy” $\nu $, the right-hand side of the differential equation f has “smoothness” $\lambda $ and the initial and boundary data $\varphi $ and $\Phi $ have “smoothness” $\mu $, then letting$| \cdot |_A^{(r)} $ denote a Holder-type norm of order r in the set A we have for $\mu \ne \nu $ that the maximum error is bounded by \[ C\{ h^{\min (\lambda ,v)} | f |_\mathcal{L}^{(\lambda )} + h^{\min (\mu ,\nu )} | {\varphi ,\Phi } |_{\mathcal{R} \times \mathcal{L}}^{(\mu )} \} \] and for $\mu = \nu $ and $f \equiv 0$, this error is dominated by \[ Ch^\nu \left( {\ln \frac{1}{h}} \right)| {(\varphi ,\Phi )} |_{\mathcal{R} \times ...

We prove a Marx conjecture for the closed convex hull of the family of functions which are starlike of order α \alpha and k-fold symmetric. We obtain precise results for the functions which are starlike, starlike of order 1 2 \tfrac {1}{2} , and starlike with 2-fold symmetry in their power series expansions.

Consider an operator $Q:X_1 \to X_2 $ where $X_1 $ and $X_2 $ are normed linear spaces. Let $\delta _1 , \cdots ,\delta _m $ be functionals on $X_1 $ and let $H_2 $ be a finite-dimensional affine subspace of $X_2 $. In this paper we consider the problem \[ Q(x) \in H_2 \quad {\text{subject to}}\quad \delta _i (x) = \beta _i, \quad i = 1, \cdots ,m.\]A generalized form of Newton’s method which approximates a solution of this problem by a sequence of the form \[ x_{n + 1} = x_n - \Gamma _n p_n (x_n ),\qquad n = 0,1, \cdots , \] is presented. It is shown that many important constrained problems including the ordinary differential equation boundary value problem and the Euler–Poisson equation from the calculus of variations fall very naturally within this theory.

It is shown that the nonnegative, real-valued function of two nonnegative real variables \[ ({\text{i}})\qquad J(u,v) = 1 - \int_0^u {\exp [ { - (v + \omega )} ]} I_0 (2\sqrt {v\omega } )d\omega\] may be used as an approximant for generalized functions. In particular, the following asymptotic limits are valid in a generalized sense: \[ ({\text{iia}})\qquad \mathop {\lim }\limits_{t \to 0} J\left( {\frac{{a^2 }}{{4st}},\frac{{x^2 }}{{4st}}} \right) = u( | x | - | a |) , \]\[ ({\text{iib}})\qquad \mathop {\lim }\limits_{t \to 0} t^{ - 1} \,\exp \,\left( { - \frac{{a^2 + x^2 }} {{4st}}} \right)\,I_1 \left( {\frac{{| {ax} |}}{{2st}}} \right) = \frac{{2s}}{{| a |}}\delta ( {| x | - | a |} ), \] where sis a positive real number, a is a nonzero constant and $u(z)$ and $\delta (z)$ are the unit-step and Dirac delta functions, respectively.The limits are proved by showing that, under all $L_2 $-inner products of the respective limits in (ii) with an arbitrary testing function, the resultant functionals are equal t...

Gravity data cannot usually be inverted to yield unique structures from incomplete data; however, there is a smallest density compatible with the data or, if the density is known, a deepest depth of burial. A general theory is derived which gives the greatest lower bound on density or the least upper bound on depth. These bounds are discovered by consideration of a class of “ideal” bodies which achieve the extreme values of depth or density. The theory is illustrated with several examples which are solved by analytic methods. New maximum depth rules derived by this theory are, unlike some earlier rules of this type, optimal for the data they treat.

Sufficient conditions on the functions $a_i ,i = 1, \cdots ,N$, and f are determined which ensure, via a monotonicity argument, the existence of weak periodic solutions of the Dirichlet problem for \[ u_{tt} - \frac{d} {{dx_i }}a_i (x,t,u_{xi} ) - {\bf \Delta } _N \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{u} _t = f,\qquad ,(x,t) \in \Omega \times E^1 , \] where ${\bf \Omega }$ is a bounded domain in $E^N $. In addition to possessing some smoothness, each $a_i $ is required to satisfy \[ \begin{gathered} \left| {a_i (x,t,\eta )} \right| \leqq k_2 \left\{ | \eta |^{p - 1} + 1\right\} \quad ({\text{boundedness}}) \hfill \\ a_i (x,t,\eta )\gtrless k_0 | \eta |^{p - 2} \eta ,\quad \eta \gtrless 0,\quad ({\text{coercivity}}) \hfill \\ (\partial/\partial t)a_i (x,t,\eta )\gtrless 0,\quad \eta \lessgtr 0, \hfill \\ (a_i (x,t,\eta ) - a_i (x,t,\xi ))(\eta - \xi )\geqq 0\quad ({\text{monotonicity}}) \hfill \\ \end{gathered} \] for some $p\geqq 2$, some constants $k_0 $ and $k_2 $ and all real $...

For r ≧ 1 r \geqq 1 and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes p ≦ N p \leqq N such that p + 2 r p + 2r is the least prime greater than p is asymptotic to \[ ∫ 2 N ( ∑ k = 1 r A r , k ( log ⁡ x ) k + 1 ) d x , \int _2^N {\left ( {\sum \limits _{k = 1}^r {\frac {{{A_{r,k}}}}{{{{(\log x)}^{k + 1}}}}} } \right )} \;dx, \] where the A r , k {A_{r,k}} are certain constants. We describe a method for computing these constants. Related constants are given to 10D for r = 1 ( 1 ) 40 r = 1(1)40 , and some empirical evidence supporting the conjecture is mentioned.

Let $\mathcal{L}^i (i = 1,2, \cdots ,n)$ denote n linear functionals on $C^n [a,b]$, $f \in C^n [a,b]$, $p \in P_{n - 1} $ (the space of polynomials of degree $ < n$), where $\mathcal{L}^i p = \mathcal{L}^i f$. Then one has the formula $| {f(x) - p(x)} | \leqq U(x)\| {f^{(n)} } \|_\infty $, where U is the upper envelope of all functions $R \in C^n [a,b]$ such that $\left\| {R^{(n)} } \right\|_\infty \leqq 1 $ and $\mathcal{L}^i R = 0$. $U(x)$ can be determined for each x to be the total variation of a certain measure calculated directly from the $\mathcal{L}^i $. One can, however, bypass this procedure and still obtain easily calculable formulas for $V(x)$, an upper bound for $U(x)$. In addition, situations in which $f(x) - p(x) = L(x)f^{(n)} (\xi _x )$, $\xi _x \in [a,b]$, will be investigated, where L is the unique element of $C^n [a,b]$ such that $L^{(n)} \equiv 1$ and $\mathcal{L}^{(i)} L = 0$.

SUMMARY This paper presents necessary and sufficient conditions under which a random variable X may be generated as the ratio ZI V where Z and V are independent and Z has a standard normal distribution. This representation is useful in Monte Carlo calculations. It is established that when 7 V2 is exponential, X is double exponential; and that when WV has the asymptotic distribution of the Kolmogorov distance statistic, X is logistic.

The analytical method of calculation of a three-dimensional boundary layer in a compressible fluid stream is considered. The method is based on the use of successive approximations and is similar to that used in the case of incompressible fluid [1], Results of this approximate analysis are compared with those obtained by the method of finite differences.

The eigenvalues of a structure are its natural frequencies (vibration problems) or its critical load factors (buckling problems). It is often desirable that no eigenvalues should lie in a prescribed ‘prohibited range’, i.e. within a specified range of values of F, where F is the frequency (vibration problems) or load factor (buckling problems). Three alternative procedures, which are all based on a recent algorithm [1, 2], are presented. They enable the presence of eigenvalues within the prohibited range to be detected, and permit rapid determination of the effectiveness of structural modifications aimed at making this range free of eigenvalues. These procedures may involve considerably less computation than finding a single eigenvalue of the unmodified structure. The procedures are exact within their context, which is the stiffness matrix approach. They allow the use of sub-structures. Furthermore, they apply when the mathematical model used to represent the structure possesses a finite number of degrees-of-freedom (e.g. finite element representations) and when the model has an infinite number of degrees-of-freedom (e.g. models assembled from beams treated ‘exactly’ [7–9]).

This paper extends the notion of convex programming with set-inclusive constraints as set forth by Soyster [Opns. Res. 21, 1154–1157 (1973)] by replacing the objective vector c with a convex set C and formulating a dual problem. The primal problem to be considered is \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sup\Bigl\{\inf_{c\in C} c \cdot x \mid x_{1}K_{1}+x_{2}K_{2}+ \ldots x_{n}K_{n}\subseteq K(b), x_{i}\geq 0\Bigr\}$$ \end{document} where the sets {Kj} are convex activity sets, K(b) is a polyhedral resource set, C is a convex set of objective vectors, and the binary operation + refers to addition of sets. Any feasible solution to the dual problem provides an upper bound to (I) and, at optimality conditions, the value of (I) is equal ...

The joint density function of the sample mean and sample variance is recursively derived for samples from a population with density function f where f (x) > 0 cilinost syervi,vheie, everywhere continuous and has certain integral properties. For populations where f does not have these integral properties, this joint density is an approximation. This joint density-function is used to derive the density function of the t-statistic for samples from f. The family of generalized normal density functions is used for an example. The approximation for the t-density is given for that family. For some specific members of the family, the true a probabilities for the approximations are tabled and compared to the results of a simulation study.

Let P r {P_r} be r-dimensional real projective space with r odd, and let π 0 Diff + : P r {\pi _0}{\text {Diff}^ + }:{P_r} be the group of orientation preserving diffeomorphisms P r → P r {P_r} \to {P_r} factored by the normal subgroup of those concordant (= pseudoisotopic) to the identity. The main theorem of this paper is that for r ≡ 11 mod 16 r \equiv 11 \bmod 16 the group π 0 Diff + : P r {\pi _0}{\text {Diff}^ + }:{P_r} is isomorphic to the homotopy group π r + 1 + k ( P ∞ / P k − 1 ) {\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}}) , where k = d 2 L − r − 1 k = d{2^L} - r - 1 with L ≥ φ ( ( r + 1 ) / 2 ) L \geq \varphi ((r + 1)/2) and d 2 L ≥ r + 1 d{2^L} \geq r + 1 . The function φ \varphi is denned by φ ( l ) = { i | 0 > i ≤ l , i ≡ 0 , 1 , 2 , 4 mod ( 8 ) } \varphi (l) = \{ i|0 > i \leq l,i \equiv 0,1,2,4 \bmod (8)\} . The theorem is proved by introducing a cobordism version of the mapping torus construction; this mapping torus construction is a homomorphism t : π 0 Diff + : P r → Ω r + 1 ( v ) t:{\pi _0}{\text {Diff}^ + }:{P_r} \to {\Omega _{r + 1}}(v) for r ≡ 11 mod 16 r \equiv 11 \bmod 16 and Ω r + 1 ( v ) {\Omega _{r + 1}}(v) a suitable Lashof cobordism group. It is shown that t is an isomorphism onto the torsion subgroup Ω r + 1 ( v ) {\Omega _{r + 1}}(v) , and that this subgroup is isomorphic to π r + 1 + k ( P ∞ / P k − 1 ) {\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}}) as above. Then one reads off from Mahowald’s tables of π n + m ( P ∞ / P m − 1 ) {\pi _{n + m}}({P_\infty }/{P_{m - 1}}) that π 0 Diff + : P 11 = Z 2 {\pi _0}{\text {Diff}^ + }:{P_{11}} = {Z_2} and π 0 Diff + : P 27 = 6 Z 2 {\pi _0}{\text {Diff}^ + }:{P_{27}} = 6{Z_2} .

It has been known for some time that the trapezoidal rule Tnf = 12f(0) + f(1) + … + f(n − 1) + 12f(n) is the best quadrature formula in the sense of Sard for the space W1,p, all functions such that f′ ϵ Lp. In other words, the norm of the error functional Ef = ∝0n f(x) dx − ∑k = 0n λkf(k) in W1,p is uniquely minimized by the trapezoidal sum. This paper deals with quadrature formulas of the form ∑k = 0n ∑lϵJ cklf(l)(k) where J is some subset of {0, 1,…, m − 1}. For certain index sets J we identify the best quadrature formula for the space Wm,p, all functions such that f(m) ϵ Lp. As a result, we show that the Euler-Maclaurin quadrature formula Tnf + ∑o

This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces. 1. Preliminaries. In order to make this paper self-contained we include the definition of the two types of derivative used and the definition of an 8-category. We will employ the definitions of the derivative in topological linear spaces investigated in detail by Averbukh and Smolyanov [2], [31. See also [14], [15] and [191. Let TLS denote the class of all Hausdorff topological linear spaces over the real field R. Let 2 1(E, F) = 2(E, F) denote the set of all continuous linear maps from E into F, where E, F e TLS. We define by induction p(E, F) = 2(E, p 1(E, F)). Each ?1p(E, F) is given the topology of uniform convergence on bounded subsets of E. If X is a topological space, W(X) will denote the class of all open subsets of X. Let f: U -' V, where U e ((E), V E C(F), E e TLS and F c TLS. Then we say f is Fre6cbet differentiable at x e U, if there exists u e S(E, F) such that for each bounded subset B of E and for each o-neighbourhood W in F, there exists 8 > 0 such that f(x + th) f(x) u * th E tW, whenever b E B and ItL

In this paper we introduce a method to obtain all generating functions of Boas and Buck type for any given orthogonal polynomial set. We also characterize the orthogonal polynomials that satisfy \[xP'_n (x) = nP_n (x) + \sum\limits_0^{n - 1} {( - 1)^{n - k} \rho _k P_k (x)\quad {\text{with }}\rho _k \ne 0,\quad k = 0,1, \cdots .} \]

Let F be the collection of finite dimensional tori, F n the tori of dimension n or less, and S n the n -sphere. A complete shape classification for F -like Hausdorff continua, F n . like Hausdorff continua, and S n -like Hausdorff continua is obtained. If M is a F -like continuum, then there exists a unique compact connected abelian group having the same shape as M . If M is an S n -like Hausdorff continuum, then there exists a unique compact connected abelian group G M of dimension 1 or less such that the ( n −1)-fold suspension of G M has the same shape as M . As a consequence of these results, every F n -like Hausdorff continuum and every S n -like Hausdorff continuum has the shape of a metric continuum.