LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART II): In terms of a given Hamiltonian function the 1-form w = dH + ?_j|dπ^j is defined, where {?_j:j = 1,…, n} denotes an invariant basis of the planes of the distribution D_n. The latter is said to be canonical if w = 0 (which is analogous to the definition of Hamiltonian vector fields in symplectic geometry). This condition is equivalent to two sets of canonical equations that are expressed explicitly in term of the derivatives of H with respect to its positional arguments. The distribution D_n is said to be pseudo-Lagrangian if dπ^j(?_j,V_h) = 0; if D_n, is both canonical and pseudo-Lagrangian it is integrable and such that H = const. on each leaf of the resulting foliation. The Cartan form associated with this construction [9] is defined a II = π² ? ? πⁿ. If π is closed, the distribution D_N is integrable, and the exterior system {π^j} admits the representation ψ^j = dS^j in terms of a set of 0-forms S^j on M. If, in addition, the distribution D_N is canonical, these functions satisfy a single first order Hamilton-Jacobi equation, and conversely. Finally, a complete figure is constructed on the basis of the assumptions that (i) the Cartan form be closed, and (ii) that the distribution D_n, be both canonical and integrable. The last of these requirements implies the existence of N functions ψ^A that depend on x^h and N parameters w^B, whose derivatives are given by ?ψ^A (x^h, w^B)/?x^j = B^A _j (x^h, ψ^B (x^h,w^B)). The complete figure then consists of two complementary foliations: the leaves of the first are described by the functions ψ^A and satisfy the standard Euler-Lagrange equations, while the second, that is, the transversal foliation, is represented by the aforementioned solution of the Hamilton-Jacobi equation. The entire configuration then gives rise in a natural manner to a generalized Hilbert independent integral and consequently also to a generalized Weierstrass excess function.     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART I): A field-theoretic treatment of variational problems in n independent variables {x^j} and N dependent variables {ψ^A)} is presented that differs substantially from the standard field theories, such as those of Carathéodory [4] and Weyl [10], inasmuch as it is not stipulated ab initio that the Lagrangian be everywhere positive. This is accomplished by the systematic use of a canonical formalism. Since the latter must necessarily be prescribed by appropriate Legendre transformations, the construction of such transformations is the central theme of Part I.—The underlying manifold is M = M_n x M_N, where M_n, M_N are manifolds with local coordinates {x^j}, {ψ^A}, respectively. The basic ingredient of the theory consists of a pair of complementary distributions D_n, D_N on M that are defined respectively by the characteristic subspaces in the tangent spaces of M of two sets of smooth 1-forms {π^A:A = 1,…, N}, {π^j = 1,…, n} on M. For a given local coordinate system on M the planes of 4, have unique (adapted) basis elements B_j = (?/?_x ^j) + B^A _j (?/?ψ^A), whose coefficients B^A _j will assume the role of derivatives such as ?ψ^A/?x^j in the final analysis of Part II. The first step toward a Legendre transformation is a stipulation that prescribes B^A _j as a function of the components {π^j _h,π^j _A} of {π^j}—these components being ultimately the canonical Variables—in such a manner that B^A _j is unaffected by the action of any unimodular transformation applied to the exterior system {π^j}. A function H of the canonical variables is said to be an acceptable Hamiltonian if it satisfies a similar invariance requirement, together with a determinantal condition that involves its Hessian with respect to π^j _A. The second part of the Legendre transformation consists of the identification in terms of H and the canonical variables of a function L that depends solely on B^A _j and the coordinates on M. This identification imposes a condition on the Hessian of L with respect to B^A _j. Conversely, any function L that satisfies these requirements is an acceptable Lagrangian, whose Hamiltonian is uniquely determined by the general construction.     

A note on uniqueness problem for meromorphic mappings with 2N + 3 hyperplanes

In this paper,a uniqueness theorem for meromorphic mappings partially sharing 2N+3 hyperplanes is proved.For a meromorphic mapping f and a hyperplane H,set E(H,f) = {z|ν(f,H)(z) 0}.Let f and g be two linearly non-degenerate meromorphic mappings and {Hj}j2=N1+ 3be 2N + 3 hyperplanes in general position such that dim f-1(Hi) ∩ f-1(Hj) n-2 for i = j.Assume that E(Hj,f) E(Hj,g) for each j with 1 j 2N +3 and f = g on j2=N1+ 3f-1(Hj).If liminfr→+∞ 2j=N1+ 3N(1f,Hj)(r) j2=N1+ 3N(1g,Hj)(r) NN+1,then f ≡ g.     

Tame Sets in the Complement of Algebraic Variety

Let A?? ^N be an algebraic variety with dim?A≤N?2. Given discrete sequences {a _j },{b _j }?? ^N \ A with slow growth ( $\sum_{j}{1\over|a_{j}|^{2}}<\infty,\sum_{j}{1\over |b_{j}|^{2}}<\inftyLet A⊂ℂ ^N be an algebraic variety with dim A≤N−2. Given discrete sequences {a _j },{b _j }⊂ℂ ^N \ A with slow growth ( ?_j[1/(|a_j|²)] < ￥,?_j[1/(|b_j|²)] < ￥\sum_{j}{1\over|a_{j}|^{2}}<\infty,\sum_{j}{1\over |b_{j}|^{2}}<\infty ) we construct a holomorphic automorphism F with F(z)=z for all z∈A and F(a _j )=b _j for all j∈ℕ. Additional approximation of a given automorphism on a compact polynomially convex set, fixing A, is also possible. Given unbounded analytic variety A there is a tame set E such that F(E)≠{(j,0 ^N−1):j∈ℕ} for all automorphisms F with F| _A =id. As an application we obtain an embedding of a Stein manifold into the complement of an algebraic variety in ℂ ^N with interpolation on a given discrete set.     

On a multiplicative functional transformation arising in nonlinear filtering theory

Summary This paper concerns the nonlinear filtering problem of calculating estimates E[f(x_t)¦y _s, st] where {x _t} is a Markov process with infinitesimal generator A and {y _t} is an observation process given by dy _t=h(x_t)dt +dw_twhere {w _t} is a Brownian motion. If h(x_t) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup T _s,t ^y obtained by a certain y-dependent multiplicative functional transformation of the signal process {x _t}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is A _t ^y f=e^y(t)h(A– 1/2h²)(e^–y(t)h f).Work partially supported by the U.S. Department of Energy (Contract ET-76-C-01-2295) at the Massachusetts Institute of Technology     

Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids

Let Ω = {t₀, t₁, …, t_N} and Ω_N = {x₀, x₁, …, x_N–1}, where x_j = (t_j + t_{j + 1})/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums S_{n, N}( f, x) with respect to the system of polynomials {p?_k,N(x)} _k=0 ^N–1 , forming an orthonormal system on nonuniform point systems Ω_N consisting of finite number N of points from the segment [–1, 1] with weight Δt_j = t_{j + 1}–t_j. We find the growth order for the Lebesgue function L_n,N (x) of the considered partial discrete Fourier sums S_n,N ( f, x) as n = O(δ _N ^?2/7 ), δ_N = max_{0≤ j≤N?1} Δt_j More exactly, we have a two-sided pointwise estimate for the Lebesgue function L_{n, N}(x), depending on n and the position of the point x from [–1, 1].     

On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

Uniform approximation on compact subsets of the real line

Let {? _i } _i=∩ ⁿ be continuous real functions on the compact set M?R. We consider the problem of best uniform approximation of the function? by polynomials \(\sum\nolimits_{i = 1}^n {c_i \varphi _i }\) on M. Let V(?₀, A) be a set of polynomials of best approximation on A ? M. We show that \(V(\varphi _0 ,M) = \mathop \cap \limits_{A_{n + 1} } V(\varphi _0 ,A_{n + 1} )\) , where A_n+1 represents all the possible sets of n+ 1 points {x₁, ..., x_n+1} in M, containing the characteristic set of the given problem of best approximation and for which the the rank of ∥?_i ∥ (i=1, ...,n; j=1,..., n+1) is equal to n. This theorem is applied to a problem of uniform approximation where {? _i } _i=1 ⁿ is a weakly Chebyshev system.     

The “Full Müntz Theorem” inL p[0, 1] for 0<p<∞

Denote by span {f ₁,f ₂, …} the collection of all finite linear combinations of the functionsf ₁,f ₂, … over ?. The principal result of the paper is the following. Theorem (Full Müntz Theorem in L_p(A) for p ∈ (0, ∞) and for compact sets A ? [0, 1] with positive lower density at 0). Let A ? [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ _j ) _j=1 ^∞ is a sequence of distinct real numbers greater than ?(1/p). Then span {x ^λ1,x ^λ2,…} is dense in L_p(A) if and only if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ . Moreover, if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ , then every function from the L_p(A) closure of {x ^λ1,x ^λ2,…} can be represented as an analytic function on {z ∈ ? \ (?∞,0] : |z| < r_A} restricted to A ∩ (0, r_A) where $r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x ^λ1,x ^λ2,…} are also considered.     

Some results on non-linear recurrence

We study the behavior of measure-preserving systems with continuous time along sequences of the form {n ^α}n∈#x2115;} where α is a positive real number¹. Let {S ^t } _t∈? be an ergodic continuous measure preserving flow on a probability Lebesgue space (X, β, μ). Among other results we show that:

1. For all but countably many α (in particular, for all α∈???) one can find anL ^∞-functionf for which the averagesA _N (f)(1/N)=Σ _n=1 ^N f(S ^nα x) fail to converge almost everywhere (the convergence in norm holds for any α!).
2. For any non-integer and pairwise distinct numbers α₁, α₂,..., α _k ∈(0, 1) and anyL ^∞-functionsf ₁,f ₂, ...,f _k , one has $$\mathop {lim}\limits_{N \to \infty } \left\| {\frac{1}{N}\sum\limits_{n - 1}^N {\prod\limits_{i - 1}^k {f_i (S^{n^{\alpha _i } } x) - \prod\limits_{i - 1}^k {\int_X {f_i d\mu } } } } } \right\|_{L^2 } = 0$$

We also show that Furstenberg’s correspondence principle fails for ?-actions by demonstrating that for all but a countably many α>0 there exists a setE?? having densityd(E)=1/2 such that, for alln∈?, $$d(E \cap (E - n^\alpha )) = 0$$ .     

A Tauberian theorem related to Weyl's criterion

Let {x_n} be a sequence of real numbers and let a(n) be a sequence of positive real numbers, with A(N) = Σ_n=1^Na(n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t₀ for which A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞. In case f(s) = Σ_n=1^∞a(n)e^?sx_n, s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f(σ)^?1f(σ + it₀) → 0 as σ → 0 + if and only if A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

Automatic Smoothing Spline Projection Pursuit

Abstract

A highly flexible nonparametric regression model for predicting a response y given covariates {x_k}^d _k=1 is the projection pursuit regression (PPR) model ? = h(x) = β₀ + Σ_jβ_jf_j(α^T _jx) where the f_j , are general smooth functions with mean 0 and norm 1, and Σ^d _k=1α² _kj=1. The standard PPR algorithm of Friedman and Stuetzle (1981) estimates the smooth functions f_j using the supersmoother nonparametric scatterplot smoother. Friedman's algorithm constructs a model with M _max linear combinations, then prunes back to a simpler model of size M ≤ M _max, where M and M _max are specified by the user. This article discusses an alternative algorithm in which the smooth functions are estimated using smoothing splines. The direction coefficients α_j, the amount of smoothing in each direction, and the number of terms M and M _max are determined to optimize a single generalized cross-validation measure.     

An Erdös-Ko-Rado theorem for the subcubes of a cube

LetP be that partially ordered set whose elements are vectors x=(x ₁, ...,x _n ) withx _i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx _i =y _i orx _i =0 for alli. LetN _i (P)={x εP : |{j:x _j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN ₁(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f ₀, ...,f _n ) for which there is an Erdös-Ko-Rado familyF inP such that |N _i (P) ∩F|=f _i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.     

Every set is “symmetric” for some function

Let 〈h _n 〉 denote a sequence of positive real numbers. We show that for every set A ? ? there exists a function f: ? → ω such that A = {x ∈ ?: (∈〈h _n 〉)[h _n ↘ 0 & (? _n ∈ ?)(f(x ? h _n ) = f(x + h _n ) = f(x))]}. This solves a problem of K. Ciesielski, K. Muthuvel and A. Nowik.     

Self-adjusting stochastic processes

Let A_t(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that A_t+1(h, k) ? A_t(h, k). If the self-adjustment (due to transition h → kx) is A_{t + 1}(h, j) = λA_t(h, j) + (1 ? λ)δ_jk (0 < λ < 1), then with probability 1 the process is eventually periodic. If A₀(i, j) < 1 for all i, j and if the self-adjustment satisfies A_{t + 1}(h, k) = ?(A_t(h, k)) with ?(x) twice differentiable and increasing, x < ?(x) < 1 for 0 ? x < 1,?(1) = ?′(1) = 1, then, with probability 1, lim A_t does not exist.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Selection principle for pointwise bounded sequences of functions

For a number ? > 0 and a real function f on an interval [a, b], denote by N(?, f, [a, b]) the least upper bound of the set of indices n for which there is a family of disjoint intervals [a _i , b _i ], i = 1, …, n, on [a, b] such that |f(a _i ) ? f(b _i )| > ? for any i = 1, …, n (sup Ø = 0). The following theorem is proved: if {f _j } is a pointwise bounded sequence of real functions on the interval [a, b] such that n(?) ≡ lim sup _j→∞ N(?, f _j , [a, b]) < ∞ for any ? > 0, then the sequence {f _j } contains a subsequence which converges, everywhere on [a, b], to some function f such that N(?, f, [a, b]) ≤ n(?) for any ? > 0. It is proved that the main condition in this theorem related to the upper limit is necessary for any uniformly convergent sequence {f _j } and is “almost” necessary for any everywhere convergent sequence of measurable functions, and many pointwise selection principles generalizing Helly’s classical theorem are consequences of our theorem. Examples are presented which illustrate the sharpness of the theorem.     

Solving polynomial optimization problems via the truncated tangency variety and sums of squares

Let f,g_i,i=1,…,l,h_j,j=1,…,m, be polynomials on Rⁿ and S?{x∈Rⁿ∣g_i(x)=0,i=1,…,l,h_j(x)≥0,j=1,…,m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.