correction to the stable geometric dimension of vector bundles over real projective spaces trans amer math soc 268 1981 no 1 39 61 mr0628445 83c 55006

gravity coupled with matter and the foundation of non commutative geometry we first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds its unitary representations correspond to riemannian metrics and spin structure whileds is the dirac propagatords x x d 1 whered is the dirac operator we extend these simple relations to the non commutative case using tomita s involutionj we then write a spectral action the trace of a function of the length element which when applied to the non commutative geometry of the standard model will be shown cc to give the sm lagrangian coupled to gravity the internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of sm with all correct quantum numbers in this slightly non commutative case the group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group

linear quadratic mean field stochastic zero sum differential games abstract in this paper we consider the two player linear quadratic mean field stochastic differential game lq mf szsdg where the expected values of state and players control variables are included in the corresponding mean field stochastic system and the objective functional we characterize the explicit condition under which the feedback saddle point equilibrium for the lq mf szsdg exists the results are obtained by using the completion of squares method and the ordered interchangeability property of multiple saddle points we show that the corresponding saddle point equilibrium is linear in state which is characterized by the two coupled riccati differential equations crdes we also discuss the solvability conditions of the crdes numerical results are presented to verify the theoretical results of the paper

cooperative equilibria in differential games abstract this paper deals with a class of equilibria for differential games in which memory strategies are allowed this class permits the inclusion of a threat in a cooperative strategy the threat can be used by one player as soon as the other player does not control the system according to the agreed upon cooperative control we characterize the set of such equilibria

the stable geometric dimension of vector bundles over real projective spaces an elementary argument shows that the geometric dimension of any vector bundle of order 2e over rp depends only on e and the residue of n mod 8 for n sufficiently large in this paper we calculate this geometric dimension which is approximately 2e the nonlifting results are easily obtained using the spectrum bj the lifting results require bo resolutions half of the paper is devoted to proving mahowald s theorem that beginning with the second stage bo resolutions act almost like k z2 resolutions

stable geometric dimension of vector bundles over even dimensional real projective spaces in 6 the geometric dimension of all stable vector bundles over real projective spacep n was determined ifn is even and sufficiently large with respect to the order 2 e of the bundle in f

a fractal like algebraic splitting of the classifying space for vector bundles the connected covers of the classifying space bo induce a decreasing filtration b of h b o 212 by sub hopf algebras over the steenrod algebra a we describe a multiplicative grading on h b o 212 inducing a direct sum splitting of b over a where a is the usual increasing filtration of a the pieces in the splittings are finite and the grading extends that of h r2s3 which splits it into brown gitler modules we also apply the grading to the thomifications m of b where it induces splittings of the corresponding cobordism modules over the entire steenrod algebra these generalize algebraically the previously known topological splittings of the connective cobordism spectra m o m s o and mspin introduction the classifying space for vector bundles bo is of longstanding interest in topology we will describe a splitting of the mod 2 homology algebra of bo having applications to connective cobordism thom spectra the splitting will be multiplicative in other words it will be fully compatible with whitney sums of vector bundles it differs from other familiar splittings in topology in the way it interacts with the connected covers of bo and the steenrod algebra a of cohomology operations we will explain how this interaction is analogous to the geometric properties of the boundary of the fractal mandelbrot set or m set pr the boundary of the m set has two attributes first patterns become more elaborate upon magnification second patterns visible at one level of magnification actually reappear under further magnification self similarity the second property is mandelbrot s idea of a fractal structure mi while the first is an additional feature of certain fractals like the boundary of the m set our results about certain subalgebras of the algebra h bo over the hopf algebra a reveal precisely these two features specifically consider the decreasing algebra filtration b of h bo provided by the images of the connected covers and the standard increasing hopf algebra filtration a of a where a is generated by the first 2 steenrod squares the analogy to the geometric properties of the boundary of the m set is now made precise by interpreting pattern to mean a multiplicative direct sum splitting of an algebra b over the hopf algebra a magnification as descending in the filtration b and more elaborate as ascending in the filtration a received by the editors may 15 1987 1980 mathematics subject classification 1985 revision primary 55r40 55r45 571390 57t05

brief paper soft constrained stochastic nash games for weakly coupled large scale systems in this paper we discuss infinite horizon soft constrained stochastic nash games involving state dependent noise in weakly coupled large scale systems first we formulate linear quadratic differential games in which robustness is attained against model uncertainty it is noteworthy that this is the first time conditions for the existence of robust equilibria have been derived based on the solutions of sets of cross coupled stochastic algebraic riccati equations csares after establishing an asymptotic structure with positive definiteness for csares solutions we derive a recursive algorithm by means of newton s method so that it can be used to obtain solutions for csares as another important feature we propose a high order approximate nash strategy based on iterative solutions finally we provide a numerical example to verify the efficiency of the proposed algorithms

a c algebraic approach to the principal symbol ii we introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of sukochev and zanin j oper theory 2019 and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry we provide a variant of connes trace theorem which applies to certain noncommutative settings with a minimum of technical preliminaries our approach allows us to consider operators with non smooth symbols and we demonstrate the power of our approach by extending connes trace theorem to operators with non smooth symbols in three examples the lie group mathrm su 2 noncommutative tori and moyal planes

stochastic differential games and viscosity solutions of hamilton jacobi bellman isaacs equations in this paper we study zero sum two player stochastic differential games with the help of the theory of backward stochastic differential equations bsdes more precisely we generalize the results of the pioneering work of fleming and souganidis indiana univ math j 38 1989 pp 293 314 by considering cost functionals defined by controlled bsdes and by allowing the admissible control processes to depend on events occurring before the beginning of the game this extension of the class of admissible control processes has the consequence that the cost functionals become random variables however by making use of a girsanov transformation argument which is new in this context we prove that the upper and the lower value functions of the game remain deterministic apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information its combination with bsde methods in particular that of the notion of stochastic backward semigroups introduced by peng bsde and stochastic optimizations in topics in stochastic analysis science press beijing 1997 allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way the upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower hamilton jacobi bellman isaacs equations respectively for this peng s bsde method is extended from the framework of stochastic control theory into that of stochastic differential games

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

v1 and v2 periodicity in stable homotopy theory this paper we construct some self maps related to the elements v1 e 7r2 bp and v2 e 76 bp and use them to obtain families in the 2 primary stable homotopy of spheres in particular we obtain nonzero elements in the 48k 10 stem for all k 0 here bp denotes the brown peterson spectrum associated to p 2 although our results have convenient interpretations in the bp adamsnovikov spectral sequence 251 we make hardly any use of this spectral sequence except in the final step of one argument the only knowledge of bp required here is its homotopy ring our methods are those of the classical k z2 adams spectral sequence ass as we shall be working entirely in the stable category we will denote by x y the abelian group of stable homotopy classes of maps x to y when x is a finite cw complex and y a spectrum we will identify x y with eix y for any integer i and we will not distinguish between a map and its stable homotopy class if v e e x x vk denotes

applications of connes geodesic flow to trace formulas in noncommutative geometry the trace formula of chazarain duistermaat and guillemin expresses that the singularities of the distribution trace of the wave group on a compact riemannian manifoldxis included in the set of periods of the geodesic flow restricted tos x most of the objects involved in this trace formula have analogues in connes noncommutative geometry this paper shows on several significant examples of noncommutative geometry that connes definition of geodesic flow leads to statements analogous to the classical trace formula of chazarain duistermaat and guillemin

maximum principle for nonzero sum stochastic differential game with delays in this technical note we discuss a nonzero sum stochastic differential game with delays not only the state variable but also control variables of players involve delays this kind of games are motivated by some interesting problems arising from economics and finance using anticipated backward stochastic differential equations we establish a necessary condition and a sufficient condition of maximum principle for the delayed game problem to explain theoretical results we apply them to an economic problem

relationship between maximum principle and dynamic programming for stochastic differential games of jump diffusions this paper is concerned with the relationship between maximum principle and dynamic programming for zero sum stochastic differential games of jump diffusions under the assumption that the value function is smooth enough relations among the adjoint processes the generalised hamiltonian function and the value function are given a portfolio optimisation problem under model uncertainty in an incomplete financial market is discussed to show the applications of our result

value in mixed strategies for zero sum stochastic differential games without isaacs condition in the present work we consider 2 person zero sum stochastic differential games with a nonlinear pay off functional which is defined through a backward stochastic differential equation our main objective is to study for such a game the problem of the existence of a value without isaacs condition not surprising this requires a suitable concept of mixed strategies which to the authors best knowledge was not known in the context of stochastic differential games for this we consider nonanticipative strategies with a delay defined through a partition pi of the time interval 0 t the underlying stochastic controls for the both players are randomized along pi by a hazard which is independent of the governing brownian motion and knowing the information available at the left time point t j 1 of the subintervals generated by pi the controls of players 1 and 2 are conditionally independent over t j 1 t j it is shown that the associated lower and upper value functions w pi and u pi converge uniformly on compacts to a function v the so called value in mixed strategies as the mesh of pi tends to zero this function v is characterized as the unique viscosity solution of the associated hamilton jacobi bellman isaacs equation

the image of j in the ehp sequence the ehp sequence is based on a result of james j who showed that there is h a map h such that s q sn s2n 1 is a fibration when localized at 2 the map sn sn is usually labeled e the boundary homomorphism in the homotopy sequence is usually labeled p for our purposes it will be most convenient to combine all the ehp sequences into one system this gives the following filtration of o201 s q s