Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I
Douglas N.
Hoover;
Edwin
Perkins
1-36
Abstract: R. M. Anderson has developed a nonstandard approach to Itô integration in which the Itô integral is interpreted as an internal Riemann-Stieltjes sum. In this paper we extend this approach to integration with respect to semimartingales. Lifting and pushing down theorems are proved for local martingales, semimartingales and other right-continuous processes on a Loeb space.
Nonstandard construction of the stochastic integral and applications to stochastic differential equations. II
Douglas N.
Hoover;
Edwin
Perkins
37-58
Abstract: H. J. Keisler has recently used a nonstandard theory of Itô integration (due to R. M. Anderson) to construct solutions of Itô integral equations by solving an associated internal difference equation. In this paper we use the same general approach to find solutions $y(t)$ of semimartingale integral equations of the form $\displaystyle y(t,\omega) = h(t,\omega) + \int_0^t {f(s,\omega ,y(\cdot ,\omega))\,dz(s)}$ , where $z$ is a given semimartingale, $ h$ is a right-continuous process and $ f(s,\omega , \cdot)$ is continuous on the space of right-continuous functions with left limits, with the topology of uniform convergence on compacts. In addition, we generalize Keisler's continuity theorem and give necessary and sufficient conditions for an internal martingale to be $ S$-continuous.
Tauberian $L\sp{1}$-convergence classes of Fourier series. I
William O.
Bray;
Časlav V.
Stanojević
59-69
Abstract: It is shown that the Stanojević [2] necessary and sufficient conditions for ${L^1}$-convergence of Fourier series of $f \in {L^1}(T)$ can be reduced to the classical form. A number of corollaries of a recent Tauberian theorem are obtained for the subclasses of the class of Fourier coefficients satisfying $ {n^\alpha }\vert\Delta \hat{f}(n)\vert = o(l)\,(n \to \infty)$ for some $0 < \alpha \leqslant \frac{1}{2}$. For Fourier series with coefficients asymptotically even with respect to a sequence $\{{l_n}\} ,{l_n} = o(n)\,(n \to \infty)$, and satisfying $\displaystyle l_n^{ - 1/q}{\left({\sum\limits_{k = n}^{n + [n/{l_n}]} {{k^{p - ... ...} {\vert^p}} \right)^{1/p}} = o(1)\, \quad (n \to \infty), \quad 1/p + 1/q = 1,$ necessary and sufficient conditions for ${L^1}$-convergence are obtained. In particular for ${l_n} = [\parallel {\sigma _n}(f) - f{\parallel ^{ - 1}}]$, an important corollary is obtained which connects smoothness of $f$ with smoothness of $\{\hat f(n)\}$.
Mixed Hodge structures
Fouad
El Zein
71-106
Abstract: The theory of Mixed Hodge Structures (M.H.S.) on the cohomology of an algebraic variety $X$ over complex numbers was found by Deligne in 1970. The case when $X$ is a Normal Crossing Divisor is fundamental. When the variety $X$ is embedded in a smooth ambient space we get the Mixed Hodge Structure using standard exact sequences in topology. This technique uses resolution of singularities one time for a complete variety and $ 2$ times for a quasi-projective one. As applications to the study of local cohomology we give the spectral sequence to the Mixed Hodge Structure on cohomology with support on a subspace $Y$.
Quadratic forms permitting triple composition
Kevin
McCrimmon
107-130
Abstract: In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product, $ Q\left(\{{xyz} \} \right) = Q(x)Q(y)Q(z)$. In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples $\{xyz \} = (xy)z$ or $x(yz)$ determined by a composition algebra, with the quadratic form $Q$ the usual norm form. For any fixed $ Q$ this leads to $ 1$ isotopy class in dimensions $1$ and $2$, $3$ classes in the dimension $4$ quaternion case, and $6$ classes in the dimension $8$ octonion case.
Markov processes with identical hitting probabilities
Joseph
Glover
131-142
Abstract: Let $(X(t),{P^x})$ and $ (Y(t),{Q^x})$ be transient Hunt processes on a state space $E$ satisfying the hypothesis of absolute continuity (Meyer's hypothesis (L)). Let $ T(K)$ be the first entrance time into a set $K$, and assume ${P^x}(T(K) < \infty) = {Q^x}(T(K) < \infty)$ for all compact sets $K \subseteq E$. There exists a strictly increasing continuous additive functional of $X(t),A(t)$, so that if $T(t) = {\text{inf}}\{s:A(s) > t\}$, then $(X(T(t)),{P^x})$ and $ (Y(t),{Q^x})$ have the same joint distributions. An analogous result is stated if $X$ and $Y$ are right processes (with an additional hypothesis). These theorems generalize the Blumenthal-Getoor-McKean Theorem and have interpretations in terms of potential theory.
Dynamical systems and extensions of states on $C\sp{\ast} $-algebras
Nghiem
Dang-Ngoc
143-152
Abstract: Let $(A,G,\tau)$ be a noncommutative dynamical system, i.e. $A$ is a $ {C^{\ast} }$-algebra, $ G$ a topological group and $ \tau$ an action of $ G$ on $A$ by $^{\ast}$-automorphisms, and let $({M_\alpha })$ be an $M$-net on $G$. We characterize the set of $a$ in $A$ such that $ {M_\alpha }a$ converges in norm. We show that this set is intimately related to the problem of extensions of pure states of R. V. Kadison and I. M. Singer: if $B$ is a maximal abelian subalgebra of $A$, we can associate a dynamical system $(A,G,\tau)$ such that $ {M_\alpha }a$ converges in norm if and only if all extensions to $A$, of a homomorphism of $ B$, coincide on $ a$. This result allows us to construct different examples of a ${C^{\ast} }$-algebra $A$ with maximal abelian subalgebra $B$ (isomorphic to $ C({\mathbf{R}}/{\mathbf{Z}})$ or ${L^\infty }[0,1])$ for which the property of unique pure state extension of homomorphisms is or is not verified.
Locally strange hyperbolic sets
Lowell
Jones
153-162
Abstract: The purpose of this paper is to present a very general method of constructing basic sets having complicated local homeomorphism types.
Normal subgroups of ${\rm Diff}\sp{\Omega }({\bf R}\sp{n})$
Francisca
Mascaró
163-173
Abstract: Let $\Omega$ be a volume element on ${{\mathbf{R}}^n}$. $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$ is the group of $\Omega $-preserving diffeomorphisms of $ {{\mathbf{R}}^n}$. $ {\text{Diff}}_W^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements whose set of nonfixed points has finite $\Omega $-volume. $ {\text{Diff}}_f^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements whose support has finite $\Omega$-volume. ${\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements with compact support. $ {\text{Diff}}_{{\text{co}}}^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements compactly $\Omega$-isotopic to the identity. We prove, in the case $ {\text{vo}}{{\text{l}}_{\Omega }}{{\mathbf{R}}^n} < \infty$ and for $ {\text{n}} \geqslant {\text{3}}$ that any subgroup of $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$, $N$, is normal if and only if $ {\text{Diff}}_{{\text{co}}}^\Omega ({{\mathbf{R}}^n}) \subset N \subset {\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$. If $ {\text{vo}}{{\text{l}}_{\Omega }}{{\mathbf{R}}^n} = \infty$, any subgroup of $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$, $N$, satisfying $ {\text{Diff}}_{{\text{co}}}^\Omega ({{\mathbf{R}}^n}) \subset N \subset {\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$ is normal, for $ n \geqslant {\text{3}}$, there are no normal subgroups between $ {\text{Diff}}_W^\Omega ({{\mathbf{R}}^n})$ and $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$ and for $n \geqslant 4$ there are no normal subgroups between ${\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$ and ${\text{Diff}}_f^\Omega ({{\mathbf{R}}^n})$.
Arithmetic equivalent of essential simplicity of zeta zeros
Julia
Mueller
175-183
Abstract: Let $R(x)$ and $S(t)$ be the remainder terms in the prime number theorem and the Riemann-von Mangoldt formula respectively, that is $ \psi (x) = x + R(x)$ and $ N(t) = (1/2\pi)\int_0^t {\log (\tau /2\pi)\,d\tau + S(t) + 7/8 + O(1/t)}$. We are interested in the following integrals: $J(T,\beta) = \int_1^{{T^\beta }} {{{(R(x + x/T) - R(x))}^2}dx/{x^2}} $ and $I(T,\alpha) = \int_1^T {{{(S(t + \alpha /L) - S(t))}^2}dt}$, where $L = {(2\pi)^{ - 1}}\log \,T$. Furthermore, denote by $N(T,\alpha)(N^{\ast}(T))$ the number of pairs of zeros $\frac{1} {2} + i\Upsilon ,\frac{1} {2} + i\Upsilon^{\prime}$ with $ 0 < \Upsilon \leqslant T$ and $0 < (\Upsilon^{\prime} - \Upsilon)L \leqslant \alpha \,((\Upsilon^{\prime} - \Upsilon)L = 0)$--i.e., off-diagonal and diagonal pairs. Theorem. Assume the Riemann hypothesis. The following three hypotheses (A), (B) and $ ({{\text{C}}_1},{{\text{C}}_2})$ are equivalent: for $\beta \to \infty$ and $\alpha \to 0$ as $ T \to \infty$ we have (A) $J(T,\beta) \sim \beta {T^{ - 1}}{\log ^2}T$, (B) $ I(T,\alpha) \sim \alpha T$ and $ ({{\text{C}}_1})\;N^{\ast}(T) \sim TL,({{\text{C}}_2})N(T,\alpha) = o(TL)$. Hypothesis $ ({{\text{C}}_1},{{\text{C}}_2})$ is called the essential simplicity hypothesis.
On strongly indefinite functionals with applications
Helmut
Hofer
185-214
Abstract: Recently, in their remarkable paper Critical point theory for indefinite functionals, V. Benci and P. Rabinowitz gave a direct approach--avoiding finite-dimensional approximations--to the existence theory for critical points of indefinite functionals. In this paper we develop under weaker hypotheses a simpler but more general theory for such problems. In the second part of the paper the abstract results are applied to a class of resonance problems of the Landesman and Lazer type, and moreover they are illustrated by an application to a wave equation problem.
On the absence of positive eigenvalues of Schr\"odinger operators with long range potentials
Hubert
Kalf;
V.
Krishna Kumar
215-229
Abstract: In this paper we consider the problem of obtaining upper bounds for the positive bound states associated with the Schrödinger operators with long range potentials. We have extended the size of the class of long range potentials for which one can establish the nonexistence of positive eigenvalues, improving upon the recent results of G. B. Khosrovshahi, H. A. Levine and L. E. Payne (Trans. Amer. Math. Soc. 253 (1979), 211-228).
Dimension of stratifiable spaces
Shinpei
Oka
231-243
Abstract: We define a subclass, denoted by $E{M_3}$, of the class of stratifiable spaces, and obtain several dimension theoretical results for $ E{M_3}$ including the coincidence theorem for dim and Ind. The class $ E{M_3}$ is countably productive, hereditary, preserved under closed maps and, furthermore, the largest subclass of stratifiable spaces in which a harmonious dimension theory can be established.
Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators
Lawrence C.
Evans
245-255
Abstract: We prove under appropriate hypotheses that the Hamilton-JacobiBellman dynamic programming equation with uniformly elliptic operators, ${\max _{1 \leqslant k \leqslant m}}\{{L^k}u - {f^k}\} = 0$, has a classical solution $u \in {C^{2,\beta }}$, for some (small) Hölder exponent $\beta > 0$.
The index of harmonic foliations on spheres
Franz W.
Kamber;
Philippe
Tondeur
257-263
Abstract: For foliations on a compact oriented manifold there is a natural energy functional, defined with respect to a Riemannian metric. Harmonic Riemannian foliations are then the critical foliations for this functional under an appropriate class of special variations. The index of the title is the index of the Hessian of the energy functional at a critical, i.e., harmonic foliation. It is a finite number. In this note it is shown that for a harmonic Riemannian foliation $ \mathcal{F}$ of codimension $q$ on the $n$-sphere ($n > 2$) this index is greater or equal to $q + 1$. Thus $\mathcal{F}$ is unstable. Moreover the given bound is best possible.
Convergence acceleration for continued fractions $K(a\sb{n}/1)$
Lisa
Jacobsen
265-285
Abstract: A known method for convergence acceleration of limit periodic continued fractions $K({a_n}/1),{a_n} \to a$, is to replace the approximants ${S_n}(0)$ by "modified approximants" ${S_n}({f^{\ast}})$, where $f^{\ast} = K(a/1)$. The present paper extends this idea to a larger class of converging continued fractions. The "modified approximants" will then be $ {S_n}({f^{(n)^{\prime}}})$, where $ K({a^{\prime}_n}/1)$ is a converging continued fraction whose tails ${f^{(n)\prime}}$ are all known, and where ${a_n} - a_n^\prime \to 0$. As a measure for the improvement obtained by this method, upper bounds for the ratio of the two truncation errors are found.
Monotone decompositions of $\theta $-continua
E. E.
Grace
287-295
Abstract: A $\theta $-continuum ( ${\theta _n}$-continuum) is a compact, connected, metric space that is not separated into infinitely many (more than $n$) components by any subcontinuum. The following results are among those proved. The first generalizes earlier joint work with E. J. Vought for ${\theta _n}$-continua, and the second generalizes earlier work by Vought for ${\theta _1}$-continua. A $\theta $-continuum $X$ admits a monotone, upper semicontinuous decomposition $ \mathcal{D}$ such that the elements of $ \mathcal{D}$ have void interiors and the quotient space $X/\mathcal{D}$ is a finite graph, if and only if, for each nowhere dense subcontinuum $H$ of $X$, the continuum $T(H) = \{x \in X\vert$ if $K$ is a subcontinuum of $X$ and $x$ is in the interior of $K$, then $K \cap H \ne \emptyset \} $ is nowhere dense. Also, if $X$ satisfies this condition, then $ X$ is in fact a ${\theta _n}$-continuum, for some natural number $ n$, and, for each natural number $m$, $X$ is a $ {\theta _m}$-continuum, if and only if $ X/\mathcal{D}$ is a ${\theta _m}$-continuum.
Evolution generated by semilinear dissipative plus compact operators
Eric
Schechter
297-308
Abstract: Existence results and sharp continuous dependence results are given for an evolution equation in an arbitrary Banach space. The right-hand side of the equation consists of a linear dissipative term plus a continuous dissipative term plus a compact term.
Linking numbers and the elementary ideals of links
Lorenzo
Traldi
309-318
Abstract: Let $L = {K_1} \cup \, \cdots \cup {K_\mu } \subseteq {S^3}$ be a tame link of $ \mu \geqslant 2$ components, and $H$ the abelianization of $G = {\pi _1}({S^3} - L)$. Let $\mathcal{L} = ({\mathcal{L}_{ij}})$ be the $\mu \times \mu $ matrix with entries in $\mathbf{Z}H$ given by $\mathcal{L}{_{ii}} = \sum\nolimits_{k \ne i} {l({K_i},{K_k}) \cdot ({t_k} - 1)}$ and for $i \ne j\,{\mathcal{L}_{ij}} = l({K_i},{K_j}) \cdot (1 - {t_i})$. Then if $0 < k < \mu$ $\displaystyle \sum\limits_{i = 0}^{k - 1} {{E_{\mu - k + i}}(L) \cdot {{(IH)}^{... ...{k - 1} {{E_{\mu - k + i}}(\mathcal{L}) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}}} }$ Various consequences of this equality are derived, including its application to the reduced elementary ideals. These results are used to give several different characterizations of links in which all the linking numbers are zero.
Reality of the zeros of an entire function and its derivatives
Simon
Hellerstein;
Li Chien
Shen;
Jack
Williamson
319-331
Abstract: In 1914 Pólya raised the problem of classifying the entire functions which together with all their derivatives have only real zeros. In earlier work Hellerstein and Williamson settled this problem for entire functions which are real on the real axis. We complete the classification in all cases and show that it is sufficient to consider the function and its first two derivatives.
On the triangulation of stratified sets and singular varieties
F. E. A.
Johnson
333-343
Abstract: We show that every compact stratified set in the sense of Thom can be triangulated as a simplicial complex. The proof uses that author's description of a stratified set as the geometric realisation of a certain type of diagram of smooth fibre bundles and smooth imbeddings, and the triangulability of smooth fibre bundles. As a consequence, we obtain proofs of the classical triangulation theorems for analytic and subanalytic sets, and a correct proof of Yang's theorem that the orbit space of a smooth compact transformation group is triangulable.
Some canonical cohomology classes on groups of volume preserving diffeomorphisms
Dusa
McDuff
345-356
Abstract: We discuss some canonical cohomology classes on the space $\bar B\mathcal{D}iff_{\omega 0}^cM$, where $\mathcal{D}iff_{\omega 0}^cM$ is the identity component of the group of compactly supported diffeomorphisms of the manifold $M$ which preserve the volume form $\omega$. We first look at some classes ${c_k}(M),1 \leqslant k \leqslant n = {\text{dim}}\,M$, which are defined for all $M$, and show that the top class ${c_n}(M) \in \,{H^n}(\bar B\mathcal{D}iff_{\omega 0}^cM;{\mathbf{R}})$ is nonzero for $M = {S^n},n$ odd, and is zero for $M = {S^n},n$ even. When $ H_c^i(M;{\mathbf{R}}{\text{) = 0}}$ for $ 0 \leqslant i < n$, the classes ${c_k}(M)$ all vanish and a secondary class $s(M) \in \,{H^{n - 1}}(\bar B\mathcal{D}iff_{\omega 0}^cM; {\mathbf{R}})$ may be defined. This is trivially zero when $n$ is odd, and is twice the Calabi invariant for symplectic manifolds when $n = 2$. We prove that $s({{\mathbf{R}}^n}) \ne 0$ when $n$ is even by showing that it is one of a set of nonzero classes which were defined by Hurder in [7].
Weak-star convergence in the dual of the continuous functions on the $n$-cube, $1\leq n\leq \infty $
Richard B.
Darst;
Zorabi
Honargohar
357-372
Abstract: Let $n$ be a positive integer and let $J = \times _{j = 1}^n{[0,1]_j}$ denote the $n$-cube. Let $\mathbf{C} = \mathbf{C}(J)$ denote the (sup norm) space of continuous (real-valued) functions defined on $J$, and let $ \mathfrak{M}$ denote the (variation norm) space of (real-valued) signed Borel measures defined on the Borel subsets of $J$. Let $\left\langle {{\mu _l}} \right\rangle$ be a sequence of elements of $ \mathfrak{M}$. Necessary and sufficient conditions are given in order that $ {\text{li}}{{\text{m}}_l}\int f \,d{\mu _l}$ exists for every $f \in \mathbf{C}$. After considering a finite dimensional case, the infinite dimensional case is entertained.
Nonlinear mappings that are globally equivalent to a projection
Roy
Plastock
373-380
Abstract: The Rank theorem gives conditions for a nonlinear Fredholm map of positive index to be locally equivalent to a projection. In this paper we wish to find conditions which guarantee that such a map is globally equivalent to a projection. The problem is approached through the method of line lifting. This requires the existence of a locally Lipschitz right inverse, ${F^ \downarrow }(x)$, to the derivative map ${F^\prime }(x)$ and a global solution to the differential equation ${P^\prime }(t) = {F^ \downarrow }(P(t))(y - {y_0})$. Both these problems are solved and the generalized Hadamard-Levy criterion $\displaystyle \int_0^\infty {\mathop {\inf }\limits_{\vert x\vert < s} \left({1/\vert{F^ \downarrow }(x)\vert} \right)\,ds = \infty }$ is shown to be sufficient for $F$ to be globally equivalent to a projection map (Theorem 3.2). The relation to fiber bundle mappings is explored in §4.
On generalized Peano derivatives
Cheng Ming
Lee
381-396
Abstract: A function $ F$ is said to have a generalized $n$th Peano derivative at $x$ if $F$ is continuous in a neighborhood of $ x$ and if there exists a positive integer $k$ such that a $k$th primitive of $F$ in the neighborhood has the $(k + n)$th Peano derivative at $ x$; and in this case this $ (k + n)$th Peano derivative at $x$ is proved to be independent of the integer $ k$ and the $k$th primitives, and is called the generalized $n$th Peano derivative of $F$ at $x$ which is denoted as $ {F_{[n]}}(x)$. If ${F_{[n]}}(x)$ exists and is finite for all $ x$ in an interval, then it is shown that ${F_{[n]}}$ shares many interesting properties that are known for the ordinary Peano derivatives. Using the generalized Peano derivatives, a notion called absolute generalized Peano derivative is studied. It is proved that on a compact interval, the absolute generalized Peano derivatives are just the generalized Peano derivatives. In particular, Laczkovich's absolute (ordinary) Peano derivatives are generalized Peano derivatives.
Orthogonal geodesic and minimal distributions
Irl
Bivens
397-408
Abstract: Let $\mathfrak{F}$ be a smooth distribution on a Riemannian manifold $M$ with $ \mathfrak{H}$ the orthogonal distribution. We say that $ \mathfrak{F}$ is geodesic provided $ \mathfrak{F}$ is integrable with leaves which are totally geodesic submanifolds of $M$. The notion of minimality of a submanifold of $M$ may be defined in terms of a criterion involving any orthonormal frame field tangent to the given submanifold. If this criterion is satisfied by any orthonormal frame field tangent to $\mathfrak{H}$ then we say $ \mathfrak{H}$ is minimal. Suppose that $ \mathfrak{F}$ and $\mathfrak{H}$ are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of $ \mathfrak{F}$ is also a submanifold of Euclidean space with mean curvature normal vector field $\eta$. We show that the integral of $\vert\eta {\vert^2}$ over $M$ is bounded below by an intrinsic constant and give necessary and sufficient conditions for equality to hold. We study the relationships between the geometry of $M$ and the integrability of $\mathfrak{H}$. For example, if $\mathfrak{F}$ and $ \mathfrak{H}$ are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then $\mathfrak{H}$ is integrable iff $\mathfrak{F}$ and $ \mathfrak{H}$ are parallel distributions. Similarly if ${M^n}$ has constant negative sectional curvature and dim $ \mathfrak{H} = 2 < n$ then $\mathfrak{H}$ is not integrable. If $\mathfrak{F}$ is geodesic and $\mathfrak{H}$ is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of $ \mathfrak{H}$ are flat submanifolds of $M$ with parallel second fundamental form.
The spectrum of a Riemannian manifold with a unit Killing vector field
David D.
Bleecker
409-416
Abstract: Let $(P,g)$ be a compact, connected, ${C^\infty }$ Riemannian $(n + 1)$-manifold $ (n \geqslant 1)$ with a unit Killing vector field with dual $1$-form $\eta$. For $t > 0$, let ${g_{t}} = {t^{ - 1}}g + (t^{n}-t^{-1})\eta \otimes \eta$, a family of metrics of fixed volume element on $P$. Let $ {\lambda _1}(t)$ be the first nonzero eigenvalue of the Laplace operator on ${C^\infty }(P)$ of the metric ${g_t}$. We prove that if $d\eta$ is nowhere zero, then ${\lambda _1}(t) \to \infty$ as $t \to \infty$. Using this construction, we find that, for every dimension greater than two, there are infinitely many topologically distinct compact manifolds for which $ {\lambda _1}$ is unbounded on the space of fixed-volume metrics.
Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution
J. J.
Duistermaat
417-429
Abstract: The Kostant convexity theorem for real flag manifolds is generalized to a Hamiltonian framework. More precisely, it is proved that if $f$ is the momentum mapping for a Hamiltonian torus action on a symplectic manifold $M$ and $Q$ is the fixed point set of an antisymplectic involution of $M$ leaving $f$ invariant, then $ f(Q) = f(M) =$ a convex polytope. Also it is proved that the coordinate functions of $f$ are tight, using "half-turn" involutions of $ Q$.