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We present the results that are necessary in the ongoing lattice calculations of the gluon parton distribution functions (PDFs) within the pseudo-PDF approach. We identify the two-gluon correlator functions that contain the invariant amplitude determining the gluon PDF in the light-cone $z^2 \to 0$ limit, and perform one-loop calculations in the coordinate representation in an explicitly gauge-invariant form. Ultraviolet (UV) terms, which contain $\ln (-z^2)$-dependence cancel in the reduced Ioffe-time distribution (ITD), and we obtain the matching relation between the reduced ITD and the light-cone ITD. Using a kernel form, we get a direct connection between lattice data for the reduced ITD and the normalized gluon PDF., Comment: Talk given by W. Morris at XXVIII International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS2021)

Two types of dynamics, chaotic and monotone, are compared. It is shown that monotone maps in strongly ordered spaces do not have chaotic attracting sets.

Unless another thing is stated one works in the $C^\infty$ category and manifolds have empty boundary. Let $X$ and $Y$ be vector fields on a manifold $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f\colon M\rightarrow\mathbb R$. A subset $K$ of the zero set ${\mathsf Z}(X)$ is an essential block for $X$ if it is non-empty, compact, open in ${\mathsf Z}(X)$ and its Poincar\'e-Hopf index does not vanishes. One says that $X$ is non-flat at $p$ if its $\infty$-jet at $p$ is non-trivial. A point $p$ of ${\mathsf Z}(X)$ is called a primary singularity of $X$ if any vector field defined about $p$ and tracking $X$ vanishes at $p$. This is our main result: Consider an essential block $K$ of a vector field $X$ defined on a surface $M$. Assume that $X$ is non-flat at every point of $K$. Then $K$ contains a primary singularity of $X$. As a consequence, if $M$ is a compact surface with non-zero characteristic and $X$ is nowhere flat, then there exists a primary singularity of $X$.

Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y > x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic points are dense in X, then F is globally periodic., Comment: Incorrect proof of main theorem

It is proved that a certain type of monotone flow has a global period provided periodic points are dense.

Let X be a subset of R^n whose interior is connected and dense in X, ordered by a polyhedral cone in R^n with nonempty interior. Let T be a monotone homeomorphism of X whose periodic points are dense. Then T is periodic., Comment: Presented at AIDS Conference on Dynamical Systems and Differential Equations, July 2016

On a real ($\mathbb F=\mathbb R$) or complex ($\mathbb F=\mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\it tracks} $X$ if $[Y,X]=fX$ for some continuous function $f\colon M\rightarrow\mathbb F$. Let $K$ be a compact subset of the zero set ${\mathsf Z}(X)$ such that ${\mathsf Z}(X)-K$ is closed, with nonzero Poincar\'e-Hopf index (for example $K={\mathsf Z}(X)$ when $M$ is compact and $\chi(M)\neq 0$) and let $\mathcal G$ be a finite-dimensional Lie algebra of analytic vector fields on $M$. \smallskip {\bf Theorem.} Let $X$ be analytic and nontrivial. If every element of $\mathcal G$ tracks $X$ and, in the complex case when ${\mathsf i}_K (X)$ is positive and even no quotient of $\mathcal G$ is isomorphic to ${\mathfrak {s}}{\mathfrak {l}} (2,\mathbb C)$, then $\mathcal G$ has some zero in $K$. \smallskip {\bf Corollary.} If $Y$ tracks a nontrivial vector field $X$, both of them analytic, then $Y$ vanishes somewhere in $K$. \smallskip Besides fixed point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X is an element of A. The following result is proved: If K is a locally maximal compact set of zeroes of X and the Poincar'e-Hopf index of X at K is nonzero, there is a point in K at which all the elements of A vanish., Comment: 8 pages

Let Y and X denote C^k vector fields on a possibly noncompact surface with empty boundary, k >0. Say that Y tracks X if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of zeroes of X. THEOREM Assume the Poincar'e-Hopf index of X at K is nonzero, and the k-jet of X at each point of K is nontrivial. If g is a supersolvable Lie algebra of C^k vector fields that track X, then the elements of g have a common zero in K. Applications are made to attractors and transformation groups.

Smale's approach \cite{Smale80} to the classical two-players repeated Prisoner's Dilemma game is revisited here for $N$-players and Network games in the framework of Blackwell's approachability, stochastic approximations and differential inclusions.

Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $\varphi$ be the local flow on $M$ induced by a $1$-parameter subgroup. Assume $K$ is a compact set of fixed points of $\varphi$ and $U$ is a neighborhood of $K$ containing no other fixed points. Theorem: If the Dold fixed-point index of $\varphi_t|U$ is nonzero for sufficiently small $t>0$, then ${\rm Fix} (G) \cap K \ne \emptyset$.

Let X be an analytic vector field on a real or complex 2-manifold, and K a compact set of zeros of X whose fixed point index is not zero. Let A denote the Lie algebra of analytic vector fields Y on M such that at every point of M the values of X and [X,Y] are linearly dependent. Then the vector fields in A have a common zero in K. Application: Let G be a connected Lie group having a 1-dimensional normal subgroup. Then every action of G on M has a fixed point., Comment: 22 pages

Let K denote a compact invariant set for a strongly monotone semiflow in an ordered Banach space E, satisfying standard smoothness and compactness assumptions. Suppose the semiflow restricted to K is chain transitive. The main result is that either K is unordered, or else K is contained in totally ordered, compact arc of stationary points; and the latter cannot occur if the semiflow is real analytic and dissipative. As an application, entropy is 0 when E = R^3 . Analogous results are proved for maps. The main tools are results of Mierczynski [27 ] and Terescak [37 ], Comment: Published in Dynamics of Continuous, Discrete and Impulsive Systems, Vol. 5, 1999, 529-543

Necessary or sufficient conditions are presented for the existence of various types of actions of Lie groups and Lie algebras on manifolds.

Questions of the following sort are addressed: Does a given Lie group or Lie algebra act effectively on a given manifold? How smooth can such actions be? What fxed-point sets are possible? What happens under perturbations? Old results are summarized, and new ones presented, including: For every integer n there are solvable (in some cases, nilpotent) Lie algebras g that have effective C-infinity actions on all n-manifolds, but on some (in many cases, all) n-manifolds, g does not have effective analytic actions, Comment: A Celebration of the Mathematical Legacy of Raoul Bott, 69-78. Centre de Recherches Math\'ematiques, U. de Montr\'eal. Published in Proceedings & Lecture Notes Vol. 50, (P. R. Kotiuga, ed.), Amer. Math. Soc. Providence RI 2010. ISBN: 978-0-8218-4777-0

A vector field X on a manifold M with possibly nonempty boundary is inward if it generates a unique local semiflow $\Phi^X$. A compact relatively open set K in the zero set of X is a block. The Poincar\'e-Hopf index is generalized to an index for blocks that may meet the boundary. A block with nonzero index is essential. Let X, Y be inward $C^1$ vector fields on surface M such that $[X,Y]\wedge X=0$ and let K be an essential block of zeros for X. Among the main results are that Y has a zero in K if X and $Y$ are analytic, or Y is $C^2$ and $\Phi^Y$ preserves area. Applications are made to actions of Lie algebras and groups.

Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology., Comment: Submitted to Journal of Biological Dynamics. 13 pages

Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that no directed feedback loops are negative. Several new results for cooperative systems are obtained in the process.

Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or towards the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence. For monotone reaction-diffusion systems with Neumann boundary conditions on convex domains, we show that the set of continuous initial data corresponding to solutions that converge to a spatially homogeneous equilibrium is prevalent. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved., Comment: 18 pages

We give a condition ensuring that the operators in a nilpotent Lie algebra of linear operators on a finite dimensional vector space have a common eigenvector., Comment: Four pages

The main result is that the Jacobian determinant of an analytic open map from Euclidean n-space to itself does not change sign. A corollary of the proof is that the set of branch points has dimension < n-1., Comment: 4 pages

Let p be a saddle fixed point for an orientation-preserving surface diffeomorphism f admitting a homoclinic point q. Let V be an open 2-cell bounded by a simple loop formed by two arcs joining p to q lying respectively in the stable and unstable curves at p. It is shown that f|V has fixed point index 1 or 2 depending only on the geometry of V near p., Comment: Slight correction of published version. 13 pages