quantitative diophantine approximation on affine subspaces recently adiceam et al adv math 302 231 279 2016 proved a quantitative version of the convergence case of the khintchine groshev theorem for nondegenerate manifolds motivated by applications to interference alignment in the present paper we obtain analogues of their results for affine subspaces

a note on measuring in p we revisit the problem of generalising lutz s resource bounded measure rbm to small complexity classes and propose a definition of a random based rbm on p k n dtime o nk which we argue as being a good generalisation to p of lutz s rbm we cannot unconditionally prove the existence of such a measure but we give sufficient and necessary conditions for its existence we also revisit an rbm for p defined by strauss inform comput 136 1 1997 1 and correct an erroneous claim concerning the relations between and random sets a correction to this mistake is then proposed which is a less powerful but accurate relation between and random sets in order to obtain these results we introduce a mathematical structure called a measuring system which is a general setting that can be used to compare different rbms on any fixed complexity class through a partial ordering relation

the index of the scattering operator on the positive spectral subspace we construct the scattering operator for a spinor field in a time dependent background by the dyson expansion then we show that the restriction of the scattering operator to the positive spectral subspace with respect to a reference hamiltonian is fredholm the computation of the index of this restriction is reduced to the index computation for an elliptic pseudodifferential operator of order zero we obtain the index in terms of a cohomological formula by means of the atiyah singer index theorem

energy gap for yang mills connections ii arbitrary closed riemannian manifolds abstract we prove an l d 2 energy gap result for yang mills connections on principal g bundles p over arbitrary closed riemannian smooth manifolds of dimension d 2 we apply our version of the ojasiewicz simon gradient inequality 16 19 to remove a positivity constraint on a combination of the ricci and riemannian curvatures in a previous l d 2 energy gap result due to gerhardt 23 theorem 1 2 and a previous l energy gap result due to bourguignon lawson and simons 10 theorem c 11 theorem 5 3 as well as an l 2 energy gap result due to nakajima 42 corollary 1 2 for a yang mills connection over the sphere s d but with an arbitrary riemannian metric

weak completeness in e and e 2 abstract the notions of weak m p completeness for the complexity classes e dtime 2 linear and e 2 dtime 2 polynomial are compared an element c of one of these classes is weakly m p complete for the class if the set p m c consisting of all languages a m p c does not have measure 0 in the class the following two results are proven 1 i every problem that is weakly m p complete for e is weakly m p complete for e 2 2 ii there is a problem in e that is weakly m p complete for e 2 but not for e

nontriviality for exponential time w r t weak reducibilities a set a is nontrivial for the linear exponential time class e dtime 2 l i n if email protected e and the sets from e which can be reduced to a are not from a single level dtime 2 k n of the linear exponential hierarchy similarly a set a is nontrivial for the polynomial exponential time class exp dtime 2 p o l y if email protected exp and the sets from exp which can be reduced to a are not from a single level dtime 2 n k of the polynomial exponential hierarchy see 2 here we compare the strength of the nontriviality notions with respect to the underlying reducibilities where we consider the polynomial time variants of many one bounded truth table truth table and turing reducibilities surprisingly the results obtained for e and exp differ while the above reducibilities yield a proper hierarchy of nontriviality notions for e nontriviality for exp under many one reducibility and truth table reducibility coincides

a direct method for minimizing the yang mills functional over 4 manifolds a direct method is employed to minimize the yang mills functional over a 4 dimensional manifold the limiting connection is shown to be yang mills but in a possibly new bundle we show that a topological invariant of the bundle is preserved by the minimizing process this implies the existence of an absolute minimum of the yang mills functional in a wide class of bundles

dirichlet and neumann boundary value problems for yang mills connections the yang mills equations have been extensively studied on compact riemannian manifolds without boundary a substantial amount of work has also been done on non compact examples with a boundary component at infinity see 5 10 11 14 15 here we define and describe a solution of dirichlet and neumann boundary value problems for yang mills connections on a compact four dimensional manifold with smooth boundary the equations in the interior are the full yang mills equations we do not restrict ourselves to self dual or anti self dual solutions in the case where the structure group in the gauge theory is abelian our results are contained in the hodge de rham theory for manifolds with boundary see 6 9 4 3 in the general case our results describe a non linear version of this theory the base manifold m of the theory is a compact orientable four dimensional manifold with smooth boundary m we assume that m is endowed with a riemannian metric we consider a principal fiber bundle p over m with structure group g a compact lie group which will have a metric invariant under the adjoint action in the usual example this comes from trace the unknown in the problem is the connection a i e a differential form in p ad g where g is the lie algebra of g there are two natural boundary value problems one of them requires fixing the tangential part of the connection a on the bundle restricted to m if m is locally described as x 0 this amounts to prescribing a1 x dx a2 x dx 2 a3 x dx 3 on m the connection is fixed in the sense that a particular choice is fixed in the gauge equivalence class of a on the boundary the yang mills equations are satisfied in the interior this corresponds to the dirichlet problem in the other boundary value problem we consider a is free at the boundary but the normal part of the curvature is set equal to zero we call this the neumann problem for the dirichlet problem we consider a sequence of l1 connections that minimizes the yang mills functional with prescribed admissible tangential boundary data we show that there exists a weak limiting connection a by finding local good gauges the limiting connection is only locally defined we show that a solves the yang mills equations and satisfies the boundary conditions up to

the massive feynman propagator on asymptotically minkowski spacetimes we consider the massive klein gordon equation on asymptoti cally minkowski spacetimes in the sense that the manifold is r 1 d and the metric approaches that of minkowski space at infinity in a short range way jointly in time and space variables in this setup we define feynman and anti feynman scattering data and prove the fredholm property of the klein gordon operator with the associated atiyah patodi singer type boundary conditions at infinite times we then construct a parametrix with compact remainder terms for the fredholm problem and prove that it is also a feynman parametrix in the sense of duistermaat and hormander

almost complete sets we show that there is a set that is almost complete but not complete under polynomial time many one p m reductions for the class e of sets computable in deterministic time 2lin here a set in a complexity class c is almost complete for c under some given reducibility if the class of the problems in c that do not reduce to this set has measure 0 in c in the sense of lutz s resource bounded measure theory we also show that the almost complete sets for e under polynomial time bounded length increasing one one reductions and truth table reductions of norm 1 coincide with the almost p m complete sets for e moreover we obtain similar results for the class exp of sets computable in deterministic time 2poly

weakly hard problems a weak completeness phenomenon is investigated in the complexity class rm e rm dtime 2 rm linear according to standard terminology a language h is leq rm p m hard for e if the set rm p m h consisting of all languages a leq rm p m h contains the entire class e a language c is leq rm p m complete for e if it is leq rm p m hard for e and is also an element of e generalizing this a language h is weakly leq rm p m hard for e if the set rm p m h does not have measure 0 in e a language c is weakly leq rm p m complete for e if it is weakly leq rm p m hard for e and is also an element of e the main result of this paper is the construction of a language that is weakly leq rm p m complete but not leq rm p m complete for e the existence of such languages implies that previously known strong lower bounds on the complexity of weakly leq rm p m hard problems for e given by work of lutz mayordomo and juedes are indeed more general than the corresponding bounds for leq rm p m hard problems for e the proof of this result introduces a new diagonalization method called martingale diagonalization using this method one simultaneously develops an infinite family of polynomial time computable martingales betting strategies and a corresponding family of languages that defeat these martingales prevent them from winning too much money while also pursuing another agenda martingale diagonalization may be useful for a variety of applications

an index theorem for lorentzian manifolds with compact spacelike cauchy boundary we show that the dirac operator on a compact globally hyperbolic lorentzian spacetime with spacelike cauchy boundary is a fredholm operator if appropriate boundary conditions are imposed we prove that the index of this operator is given by the same expression as in the index formula of atiyah patodi singer for riemannian manifolds with boundary the index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary in case the metric is of product type near the boundary a feynman parametrix is constructed

the degrees of freedom of compute and forward we analyze the asymptotic behavior of compute and forward relay networks in the regime of high signal to noise ratios we consider a section of such a network consisting of k transmitters and k relays the aim of the relays is to reliably decode an invertible function of the messages sent by the transmitters an upper bound on the capacity of this system can be obtained by allowing full cooperation among the transmitters and among the relays transforming the network into a k k multiple input multiple output mimo channel the number of degrees of freedom of compute and forward is hence at most k in this paper we analyze the degrees of freedom achieved by the lattice coding implementation of compute and forward proposed recently by nazer and gastpar we show that this lattice implementation achieves at most 2 1 1 k 2 degrees of freedom thus exhibiting a very different asymptotic behavior than the mimo upper bound this raises the question if this gap of the lattice implementation to the mimo upper bound is inherent to compute and forward in general we answer this question in the negative by proposing a novel compute and forward implementation achieving k degrees of freedom

the index of scattering operators of dirac equations ii abstract the index formula of scattering operators of the previous paper of the author t matsui the index of scattering operators of dirac equations commun math phys 110 1987 553 571 is shown to hold for a wider class of potentials which contains both gauge potentials with compactly supported energy and instantontype potentials

diophantine approximation and applications in interference alignment abstract this paper is motivated by recent applications of diophantine approximation in electronics in particular in the rapidly developing area of interference alignment some remarkable advances in this area give substantial credit to the fundamental khintchine groshev theorem and in particular to its far reaching generalisation for submanifolds of a euclidean space with a view towards the aforementioned applications here we introduce and prove quantitative explicit generalisations of the khintchine groshev theorem for non degenerate submanifolds of r n the importance of such quantitative statements is explicitly discussed in jafar s monograph 12 4 7 1

explicit bounds for rational points near planar curves and metric diophantine approximation abstract the primary goal of this paper is to complete the theory of metric diophantine approximation initially developed in beresnevich et al 2007 10 for c 3 non degenerate planar curves with this goal in mind here for the first time we obtain fully explicit bounds for the number of rational points near planar curves further introducing a perturbational approach we bring the smoothness condition imposed on the curves down to c 1 lowest possible this way we broaden the notion of non degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of diophantine approximation in summary our findings improve and complete the main theorems of beresnevich et al 2007 10 and extend the celebrated theorem of kleinbock and margulis 1998 20 in dimension 2 beyond the notion of non degeneracy

the analysis of partial differential operators i hormander springer 1985 vol 1 391 pp vol 2 390 pp vol 3 525 pp vol 4 351 pp

integrality of the monopole number in su 2 yang mills higgs theory on ℝ3 we prove that in classical su 2 yang mills higgs theories on 3 with a higgs field in the adjoint representation an integer valued monopole number magnetic charge is canonically defined for any finite actionl1 loc2 configuration in particular the result is true for smooth configurations the monopole number is shown to decompose the configuration space into path components

multiple antenna interference channels with real interference alignment and receive antenna joint processing based on simultaneous diophantine approximation in this paper the degrees of freedom dof regions of constant coefficient multiple antenna interference channels are investigated first we consider a k user gaussian interference channel with mk antennas at transmitter k 1 k k and n j antennas at receiver j 1 j k denoted as a k m k n j channel relying on a result of simultaneous diophantine approximation a real interference alignment scheme with joint receive antenna processing is developed the scheme is used to obtain an achievable dof region the proposed dof region includes two previously known results as special cases namely 1 the total dof of k n n interference channel with k users and n antennas at each node is nk 2 and 2 the total dof of a k m n channel is at least kmn m n we next explore constant coefficient interference networks with k transmitters and j receivers all having n antennas each transmitter emits an independent message and each receiver requests an arbitrary subset of the messages employing the novel joint receive antenna processing the dof region for this setup is obtained we finally consider wireless x networks where each node is allowed to have an arbitrary number of antennas it is shown that the joint receive antenna processing can be used to establish an achievable dof region which is larger than what is possible with antenna splitting as a special case of the derived achievable dof region for constant coefficient x network the total dof of wireless x networks with the same number of antennas at all nodes and with joint antenna processing is tight while the best inner bound based on antenna splitting cannot meet the outer bound finally we obtain a dof region outer bound based on the technique of transmitter grouping

asymptotic analysis of fundamental solutions of dirac operators on even dimensional euclidean spaces abstract we analyze the short distance asymptotic behavior of some quantities formed out of fundamental solutions of dirac operators minimally coupled to finite dimensional matrix valued gauge potentials on even dimensional euclidean spaces the analysis gives a mathematically rigorous proof as well as a mathematically self consistent description to the so called abelian chiral anomaly in quantum gauge field theories

the chern classes of sobolev connections assumef is the curvature field of a connection potential onr4 with finitel2 norm left int limits r 4 left f right 2 dx infty right we show the chern number c 2 1 mathord left vphantom 1 8 right kern nulldelimiterspace 8 pi 2 int limits r 4 f wedge f topological quantum number is an integer this generalizes previous results which showed that the integrality holds forf satisfying the yang mills equations we actually prove the natural general result in all even dimensions larger than 2

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

baire categories on small complexity classes and meager comeager laws we introduce two resource bounded baire category notions on small complexity classes such as p quasipoly subexp and pspace and on probabilistic classes such as bpp which differ on how the corresponding finite extension strategies are computed we give an alternative characterization of small sets via resource bounded banach mazur games as an application of the first notion we show that for almost every language a i e all except a meager class computable in subexponential time p a bpp a we also show that almost all languages in pspace do not have small nonuniform complexity we then switch to the second baire category notion called locally computable and show that the class sparse is meager in p we show that in contrast to the resource bounded measure case meager comeager laws can be obtained for many standard complexity classes relative to locally computable baire category on bpp and pspace another topic where locally computable baire categories differ from resource bounded measure is regarding weak completeness we show that there is no weak completeness notion in p based on locally computable baire categories i e every p weakly complete set is complete for p we also prove that the class of complete sets for p under turing logspace reductions is meager in p if p is not equal to dspace logn and that the same holds unconditionally for quasipoly finally we observe that locally computable baire categories are incomparable with all existing resource bounded measure notions on small complexity classes which might explain why those two settings seem to differ so fundamentally

a chern number for gauge fields on r4 let f be the curvature of some connection on some principal bundle over r4 i show that if f decays as fast as r2lnr 1 as r tends to infinity then 1 8 2 trf2 is an integer if f decays like r 2 any real value is possible there is an analogous statement for r2n n 2 although it fails for r2

weakly useful sequences an infinite binary sequence x is defined to be i strongly useful if there is a computable time bound within which every decidable sequence is turing reducible to x and ii weakly useful if there is a computable time bound within which all the sequences in a non measure 0 subset of the set of decidable sequences are turing reducible to x juedes lathrop and lutz theorectical computer science 132 1994 37 proved that every weakly useful sequence is strongly deep in the sense of bennett the universal turing machine a half century survey 1988 227 and asked whether there are sequences that are weakly useful but not strongly useful the present paper answers this question affirmatively the proof is a direct construction that combines the martingale diagonalization technique of lutz siam journal on computing 24 1995 1170 with a new technique namely the construction of a sequence that is computably deep with respect to an arbitrary given uniform reducibility the abundance of such computably deep sequences is also proven and used to show that every weakly useful sequence is computably deep with respect to every uniform reducibility

diophantine approximation on planar curves and the distribution of rational points let c be a nondegenerate planar curve and for a real positive decreasing function let c denote the set of simultaneously approximable points lying on c we show that c is of khintchine type for divergence i e if a certain sum diverges then the one dimensional lebesgue measure on c of c is full we also obtain the hausdorff measure analogue of the divergent khintchine type result in the case that c is a rational quadric the convergence counterparts of the divergent results are also obtained furthermore for functions with lower order in a critical range we determine a general exact formula for the hausdorff dimension of c these results constitute the first precise and general results in the theory of simultaneous diophantine approximation on manifolds

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

compressibility and resource bounded measure we give a new definition of resource bounded measure based on compressibility of infinite binary strings we prove that the new definition is equivalent to the one commonly used this new characterization offers us a different way to look at resource bounded measure shedding more light on the meaning of measure zero results and providing one more tool to prove such results the main contribution of the paper is the new definition and the proofs leading to the equivalence result we then show how this new characterization can be used to prove that the class of linear autoreducible sets has p measure 0 we also prove that the class of sets that are truth table reducible to a p selective set has p measure 0 and that the class of sets that turing reduce to a subpolynomial dense set has p measure 0 this strengthens various results

information dimension and the degrees of freedom of the interference channel the degrees of freedom dofs of the k user gaussian interference channel determine the asymptotic growth of the maximal sum rate as a function of the signal to noise ratio subject to a very general sufficient condition on the cross channel gains we give a formula for the dofs of the scalar interference channel as a function of the deterministic channel matrix which involves maximization of a sum of information dimensions over k scalar input distributions known special cases are recovered and even generalized in certain cases with unified proofs

the resolution of the yang mills plateau problem in super critical dimensions abstract we study the minimization problem for the yang mills energy under fixed boundary connection in supercritical dimension n 5 we define the natural function space a g in which to formulate this problem in analogy to the space of integral currents used for the classical plateau problem the space a g can be also interpreted as a space of weak connections on a real measure theoretic version of reflexive sheaves from complex geometry we prove the existence of weak solutions to the yang mills plateau problem in the space a g we then prove the optimal regularity result for solutions of this plateau problem on the way to prove this result we establish a coulomb gauge extraction theorem for weak curvatures with small yang mills density this generalizes to the general framework of weak l 2 curvatures previous works of meyer riviere and tao tian in which respectively a strong approximability property and an admissibility property were assumed in addition

holonomy and skyrme s model in this paper we consider two generalizations of the skyrme model one is a variational problem for maps from a compact 3 manifold to a compact lie group the other is a variational problem for flat connections we describe the path components of the configuration spaces of smooth fields for each of the variational problems we prove that the invariants separating the path components are well defined for not necessarily smooth fields with finite skyrme energy we prove that for every possible value of these invariants there exists a minimizer of the skyrme functional throughout the paper we emphasize the importance of holonomy in the skyrme model some of the results may be useful in other contexts in particular we define the holonomy of a distributionally flat l 2 loc connection the local developing maps for such connections need not be continuous

diophantine approximation on planar curves the convergence theory the convergence theory for the set of simultaneously approximable points lying on a planar curve is established our results complement the divergence theory developed in 1 and thereby completes the general metric theory for planar curves

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

integer topological charges for finite energy fields in the o 3 σ model in the 2 1 dimensional classical o 3 model all finite energy fields have integer topological charges regardless of their asymptotic behavior at infinity the topological charge is conserved for the fields with the finite euclidean action without further assumptions on the field equations or asymptotics the fields with continuous first derivatives as well as fields in sobolev like spaces analogical to w1 2 are considered

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we