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Implicit function formulas for differentiating the solutions of mathematical programming problems satisfying the conditions of the Kuhn--Tucker theorem are motivated and rigorously demonstrated. The special case of a convex objective function with linear constraints is also treated with emphasis on computational details. An example, an application to chemical equililibrium problems, is given. Implicit function formulas for differentiating the unique solution of a system of simultaneous inequalities are also derived.

The paper deals with questions of programming problems of the isoperimetric type in the geometry of convex surfaces using vector structures generated by the addition of Blaschke sets.

This paper suggests the solution of geometric programs with degrees of difficulty by means of an auxiliary problem which is the reduced equivalent of the primal problem. The auxiliary problem is derived from the primal program by direct algebraic transformation and has a highly simplified convex separable structure. Using a condensation technique introduced by Duffin, it is shown that generalized polynomial programs can be solved via a sequence of approximating problems with a similar convex separable structure.

Results showing the damage arising from water jet impact on convex, concave, and flat-inclined polymethylmethacrylate targets are presented and discussed. The direction and distribution of surface cracks and pits within the zone of impact are related to liquid flow during the impact and the essential features of the damage are analyzed in terms of stress wave phenomena.

An important development in the theory of (convex) polytopes was the determination by Barnette and McMullen of the minimum and maximum of v(P) (number of vertices of P) as P ranges over all simple d-polytopes with n facets. Their results are here extended to certain pairs consisting of a polytope and one of its facets. Corollaries of our main results are the determination of the minimum and maximum of v(P) as P ranges over all simple d-polyhedra with u unbounded and n - u bounded facets, and of the minimum and maximum of v(P not F)/v(F) as (P,F) ranges over all pairs consisting of a simple d-polytope P with n facets and a facet F intersecting all other facets of P. Such pairs, called Kirkman pairs of class (d,n), are related to several aspects of linear programming, including a recent algorithm of Mattheiss for finding all vertices of a polytope defined by a system of linear inequalities. (Author)

We solve the problem of the best quadrature formula of the form for the following classes of differentiable periodic functions: , (where is a convex modulus of continuity and is odd), and .

The design of prestressed cable trusses using a nonlinear computer analysis is investigated. An approximate method is used to select a trial design. Both concave and convex trusses with rigid and flexible supports are examined; symmetrical and unsymmetrical loads are considered. Other parameters investigated are cable rise and sag, area of top and bottom cable, initial prestress force, and number and pattern of web members. The results are compared to approximate solutions where possible. /ASCE/

In an expansion flow of a dusty gas around a convex corner, a dustfree region is formed downstream near the wall by the separation of the dust. The case of a small deflection angle is investigated by making use of a perturbation method. It is found that, when the gas traverses the separation line, the vorticity is increased suddenly to a large value, which depends on the angle between the separation line and the direction of the relative velocity of the gas to the dust. Interaction between the gas and the dust particles causes a decay of the centered expansion wave emanating from the corner. The development of the equilibrium centered expansion wave is also studied by means of the method of matched expansions.

This note shows that convexity cuts defined relative to polyhedral convex sets can utilize negative as well as positive edge extensions under appropriate circumstances, yielding stronger cuts than customarily available. We also show how to partially collapse the polyhedron to further improve these cuts.

SUMMARY The Flat Generating System considered has been used on automatic lathes for many years. Its function is to generate parallel flats, squares, hexagons and other regular patterns of flats on turned components as part of the turning operation. Figure 1 illustrates two typical components with hexagons produced by this System. It has been generally accepted that the system does not produce truly flat sides on the form being generated. However there is no apparent evidence than an analysis of the accuracy of the system has been carried out. This may be because under certain established empirical conditions the errors have not been of sufficient magnitude to warrant attention. A preliminary investigation does reveal that varying degrees of error occur with variations in the machining condition and that at a certain stage a change from a concave error to a convex error takes place. On this basis, there would appear to be merit in examining the system with a view to optimizing the machining conditions to a...

In the spaceLp (ℒ),p > 1, we consider the operatorA=aϕ +bSϕ +cPϕ +Tϕ, wherea(t), b(t), and c(t) are piecewise-continuous functions on the contour ℒ, T is a completely continuous operator, Pϕ=1/2πi∫ ϕ(τ) dτ/ℒ τ −t − 1, Sϕ=1/gpi∫ ϕ(τ) dτ/ℒ τ − i, ℒ is a closed convex Lyapunov contour having no rectilinear portions. We study the properties of the operator P and we show that the Noether property conditions and the index of the operator A do not depend on the term cP.

An expansion is derived for the solution to the transport equation in two dimensions subject to boundary conditions given for an arbitrary convex region. Questions of high-energy transport are considered along with the properties of the dose response function. The expansion of the solution of the transport equation is presented in terms of a parameter which measures the lateral dispersion of an unidirectional beam. This parameter is usually small and the expansion is expected to converge rapidly. The dominant term in the expansion is related to fluence-to-dose conversion factors in a semiinfinite slab for normal incidence. A convenient parameterization of the conversion factors is provided along with numerical examples.

We concern ourselves with problems of the best one-sided approximation of classes of continuous functions. We obtain estimates of the best one-sided approximation of one class of functions by another, and we find exact values of the upper bounds of the best one-sided approximations on the classes H Ω of 2π-periodic functions [given by an arbitrary convex modulus of continuityΩ(t)] by trigonometric polynomials of order not higher thann−1 in the L2π metric.

For a convex continuous function f:X→R, the smoothness of the set {x∈X|f(x)≤r} is characterized by a differentiability property of f. Questions of this kind arise from optimization problems and gauge functionals, e.g. in the theory of Orlicz spaces.

We show that in an arrangement ofn curves in the plane (or on the sphere) there are at leastn/2 points where precisely 2 curves cross (ordinary points). Furthermore there are at least (4/3)n triangular regions in the complex determined by the arrangement. Triangular regions and ordinary vertices are both connected with boundary vertices of certain distinguished subcomplexes. By analogy with rectilinear planar polygons we distinguish concave and convex vertices of these subcomplexes. Our lower bounds arise from lower bounds for convex vertices in the distinguished subcomplexes.

We study various characteristics and generalizations of approximative compact and approximative weak compact sets. We generalize a result of Asplund concerning sets whose intersection with each halfspace is an existence set. In particular, in smooth Efimov-Stechkin spaces, such a set, if it is a Chebyshev set, must be convex.

The thickness shear mode forms and frequency spectrum of AT-cut plano-convex and biconvex quartz crystal resonators are obtained by an approximate analytic method. The results show good agreement with the mode forms observed by x-ray topography and the measured frequencies.

Some definitions and mathematical notions useful in the analysis and synthesis of relaxed control as applied to asynchronous quenching and dynamical optimization are given. Existence and nonexistence of proper relaxed steady states are discussed. An example, a continuous stirred tank reactor, is analyzed with respect to convexity of the velocity set, stability etc. Finally, the role of convex velocity sets in dynamical optimization is discussed, and suggestions for the synthesis of chattering feed back controllers are given.

The ray method of Kelley and Rubinow [1] determines asymptotically certain of the large eigenvalues and corresponding eigenfunctions of the Laplace operator in convex domains, but the expression for each eigenfunction is singular at a certain caustic of the rays. We shall obtain uniform asymptotic expansions of these eigenfunctions in the two-dimensional case, and improved expressions for the eigenvalues, by using the uniform expansion of Kravtsov [2], [3] and Ludwig [4]. For caustics near the boundary the results agree with these of Buldyrev [5]. We also treat the case of caustics near the boundary by using the uniform expansion of Lewis, Bleistein and Ludwig [6] and the boundary layer method used by Matkowsky [7], both of which enable us to treat the impedance boundary condition and to obtain further terms more easily.

Let S be a 3-convex subset of the plane. If (cl S ~ £>) S int (cl S) or if (cl S ~ S) g bdry (cl S), then S is expressible as a union of four or fewer convex sets. Otherwise, S is a union of six or fewer convex sets. In each case, the bound is best possible. 1* Introduction. Let S be a subset of Rd. Then S is said to be 3-convex iff for every three distinct points in S, at least one of the segments determined by these points lies in S. Valentine [2] has proved that for S a closed, 3-convex subset of the plane, S is expressible as a union of three or fewer closed convex sets. We are interested in obtaining a similar decomposition without requiring the set S to be closed. The following definitions and results obtained by Valentine will be useful. For S S Rd> a point x in S is a point of local convexity of S iff there is some neighborhood U of x such that, if y, zeSCiU, then [y, A S S. If S fails to be locally convex at some point q in S, then q is called a point of local nonconvexity (lnc point) of S. Let S be a closed, connected, 3-convex subset of the plane, and let Q denote the closure of the set of isolated lnc points of S. Valentine has proved that for S not convex, then card Q ^ 1, Q lies in the convex kernel of S, and Q § bdry (conv Q). An edge of bdry (conv Q) is a closed segment (or ray) in bdry (conv Q) whose endpoints are in Q. We define a leaf of S in the following manner: In case card Q ^ 3, let L be the line determined by an edge of bdry (conv Q), Llf L2 the corresponding open half spaces. Then L supports conv Q, and we may assume conv Q § cl (LJ. We define W — cl (L2 f] S) to be a leaf of S. For 2 ^ card Q ^ 1, constructions used by Valentine may be employed to decompose S into two closed convex sets, and we define each of these convex sets to be a leaf of S. By Valentine's results, every point of S is either in conv Q or in some leaf W of S (or both), and every leaf W is convex. Moreover, Valentine obtains his decomposition of S by showing that for any collection {sj of disjoint edges of bdry (conv Q), with {Wt} the corresponding collection of leaves, conv Q U (U Wt) is closed and convex. Finally, we will use the following familiar definitions: For x, y in S, we say x see y via S iff the corresponding segment [x, y] lies in S. A subset T of S is visually independent via S iff for every

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.

Cyclic N-particl.e exchange terms for a hard sphere fluid are analysed m the limit that the thermal wavelength- is much shorter than the hard sphere diameter (a). This is a path integral analysis of cluster integrals in virial expansions in equilibrium statistical mechanics. In the results, the dominant dependence of the N-particle cycle is given by the classical action of a rotation of a planar N-sided regular polygon (side a) through an angle 2n/N. Extension of these results to general permutations of N particles and the relation of the results to theories of the Actransition in 4He are discussed.

A general theorem is proved which gives an estimate of the rate of convergence on convex sets in the multidimensional central limit theorem for identically distributed summands. The estimate depends on the distance of the boundary of the convex set from the origin (the larger the distance, the better the estimate). The estimate makes sense under minimal requirements on the moments. Furthermore, the dependence on the distribution of a summand in it is in terms of pseudo-moment type quantities which may be small even if the moments are large.

Let a control system be described by a continuous linear map ℒ* from the input spaceU* (some dual Banach space) into the output spaceX* (some finite-dimensional normed space). Within the class of control problems where the constraints and cost are expressed in terms of the norms on the input and output spaces, the following two have had extensive coverage: (i)minimum effort problem: find, from amongst all inputs which have corresponding outputs lying in some closed sphere inX* centered on some desired outputx d *, an output of minimum norm; and (ii)minimum deviation problem: find, from amongst all inputs lying in some closed sphere inU*, an input having corresponding output at a minimum distance fromx d *. However, thecomposite cost problem, where we seek to minimizeF(∥u*∥, ∥x d * −x*∥) over elements satisfyingx* = ℒ*u* (F a certain kind of convex functional), has not received the same attention. This paper presents results for the composite cost problem paralleling known results for the minimum effort and deviation problems. It is hoped that a gap in the literature is thereby filled. We show that (a) a solution exists, (b) the solution can be characterized in terms of some closed hyperplaneH inX, and (c)H can be computed as being an element on which some concave functional over closed hyperplanes inX achieves its maximum. The treatment allows of infinite-dimensional output spaces. We make extensive use of recently developed duality theory.

Points of concavity constitute an important feature of every digital picture. A 1-1 correspondence exists between them and any significant negative inflection of the digitized contour. Each point of concavity may be seen as a point marking the superposition region of two convex contours. A sufficient condition is given in order that a simple arc of the contour be concave and a parallel process is shown by means of which one point of concavity is detected in correspondence to every concave arc. An iterative array of the type of CLIP 3 has been used to implement the process and a description of the behavior of the machine is given. A detailed program is included.

A new family of clipping algorithms is described. These algorithms are able to clip polygons against irregular convex plane-faced volumes in three dimensions, removing the parts of the polygon which lie outside the volume. In two dimensions the algorithms permit clipping against irregular convex windows. Polygons to be clipped are represented as an ordered sequence of vertices without repetition of first and last, in marked contrast to representation as a collection of edges as was heretofore the common procedure. Output polygons have an identical format, with new vertices introduced in sequence to describe any newly-cut edge or edges. The algorithms easily handle the particularly difficult problem of detecting that a new vertex may be required at a corner of the clipping window. The algorithms described achieve considerable simplicity by clipping separately against each clipping plane or window boundary. Code capable of clipping the polygon against a single boundary is reentered to clip against subsequent boundaries. Each such reentrant stage of clipping need store only two vertex values and may begin its processing as soon as the first output vertex from the preceeding stage is ready. Because the same code is reentered for clipping against subsequent boundaries, clipping against very complex window shapes is practical. For perspective applications in three dimensions, a six-plane truncated pyramid is chosen as the clipping volume. The two additional planes parallel to the projection screen serve to limit the range of depth preserved through the projection. A perspective projection method which provides for arbitrary view angles and depth of field in spite of simple fixed clipping planes is described. This method is ideal for subsequent hidden-surface computations.

A completely graphical procedure for constructing plane strain slip-line fields for anisotropic materials, with arbitrary convex yield loci, is described. It can be viewed as a generalization of Prager's well-known procedure for isotropic materials. In illustration the method is applied to the problem of compression of a finite strip between perfectly rough parallel dies. In addition the corresponding analytical solution for an infinite strip is found for an arbitrary yield condition.

Since 1970, it has been known that open loop Nash equilibrium points exist for N-person differential games when there are integral bounds on the control functions. These results were obtained using various fixed-point theorems, and required that the duration of the game be “small”.In this paper it is assumed that the controls are constrained to take values in compact, convex subsets of $R^{q_i} $. A fixed-point theorem due to Tychonov is used, and as a result there are no restrictions on the duration of the game.

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 ...

Recently, results concerning existence of the first conjugate point of the equation $y^{(iv)} - py = 0,p(x) > 0$ for $x \in [ a,b ]$ when p is linear, concave and convex have been published. Leighton and Nehari have shown that a first conjugate point exists if and only if the lowest eigenvalue $\lambda _1 $ of $u^{(iv)} - \lambda pu = 0$, $u(a) = u'(a) = u(b) = u'(b)$, satisfies the inequality $\lambda _1 \leqq 1$. By using this result and known properties of $\lambda _1 $. as a functional of p, the known results for existence of conjugate points are strengthened and new results are obtained. This method may also be used to obtain existence theorems for conjugate points of other problems.

It is proved that ifX is a uniformly convex Banach space such thatX * is also uniformly convex andX is not a Hilbert space then there always exists a subsetA ofX×X which is maximal accretive but notm-accretive.

Recently it was pointed out that the generalized Lagrange-multiplier method provides a sharper bound on a nonconvex program than does the solution of the program formed by taking convex envelopes of all functions involved in the original problem. This note points out that the bounds are the same if the original problem has only linear constraints, or if the problem is separable with certain properties, and give an example in which the bounds differ. The better bound is currently being used in locating global solutions of nonconvex, separable, piecewise-linear problems.

Let f ( t , x , y ) : [ 0 , ∞ ) × R n × R n → R n f(t,x,y):[0,\infty ) \times {R^n} \times {R^n} \to {R^n} be continuous and 1-periodic in t, τ ( t ) : [ 0 , ∞ ) → [ 0 , h ] ( 0 > h ≦ 1 ) \tau (t):[0,\infty ) \to [0,h](0 > h \leqq 1) continuous and 1-periodic. A simple geometric condition (Theorem 1) is given for the existence of 1-periodic solutions x ( t ) x(t) of the nonlinear delay-differential system x ′ ( t ) = f ( t , x ( t ) , x ( t − τ ( t ) ) ) x’(t) = f(t,x(t),x(t - \tau (t))) , with x ( t ) x(t) in a given bounded convex open set G in R n {R^n} . The addition of a Lipschitz condition in x and monotonicity in y allows one to calculate x ( t ) x(t) by a monotone sequence of successive approximations (Theorem 2). Extensions to a more general functional differential equation x ′ ( t ) = g ( t , x ( t ) , x t ) x’(t) = g(t,x(t),{x_t}) are given.

Necessary and sufficient conditions for admissibility are given for test procedures based on a preliminary test of significance. Three types of problems are studied—testing the normal mean, fixed effects models of the analysis of variance and random effects models. Admissibility in this instance is equivalent to the intuitive and practical condition that acceptance regions of the procedures have convex sections in certain variables, while other variables are fixed. It is easy to check when the conditions hold. A discussion of optimality properties of these and other types of pooling procedures is given.

The exterior formulation of the Helmholtz equation for a scattered wave is reduced to a line integral via the Maggi transformation. This integration is performed over the line dividing a three‐dimensional scatterer into an illuminated region and a “shadow region.” The scattered wave was found for “shadow lines” with shapes of regular polygons and a circle. As the symmetry is broken, the amplitude of the scattered wave directly in back of the scatterer (on the symmetry axis) decreases.

This paper is devoted to an application of the methods of geometric function theory of a single complex variable and some results from the theory of conformal and quasiconformal mappings to a solution of the direct problem of the constructive theory of functions. Direct theorems are obtained for the theory of approximation of functions regular in regions bounded by quasiconformal curves admitting a geometric description. In particular, direct theorems are obtained for arbitrary bounded convex regions.Bibliography: 11 items.

A simple calculation of the Hall coefficient for a polyhedral Fermi surface having both convex and concave angles between its faces gave a result having the opposite sign from conventional low or high-field calculations. This is shown to be due to a breakdown in the low-field condition over parts of the energy surface. The possibility of observing the effect experimentally is considered.