density of states and dynamic scaling on the vicsek snowflake fractal real space rescaling techniques have been employed to obtain recursion relations for the parameters characterising the dynamics of both the heisenberg ferromagnet and antiferromagnet on the infinite vicsek snowflake fractal the authors have calculated the associated dynamic exponents zf and za by linearising about the respective fixed points of the ferromagnetic antiferromagnetic recursion relations and find them to satisfy the relation za zf 2 in addition using a functional integral method both the ferromagnetic and antiferromagnetic density of states rho f omega and rho a omega have been calculated exactly and the resulting spectra were found to satisfy the scaling equation rho alpha omega approximately bz alpha df rho lambda alpha omega near their respective fixed points alpha a f df denoting the fractal dimension finally the amplitudes galpha omega appearing in the solution to the scaling equation rho alpha omega approximately galpha omega omega dfz alpha 1 for the density of states rho alpha omega were found to be periodic functions of ln omega with period ln lambda alpha where lambda alpha is the eigenvalue associated with ferromagnetic alpha f or antiferromagnetic alpha a fixed point

some rigorous results for yang mills theories on a cylinder using an approach based on the canonical formalism the yang mills theories on a cylinder are rigorously analyzed in this way the moduli space a g can be explicitly described with a being the space of connections and g the group of gauge transformations in particular a g0 g0 being the group of the pointed gauge transformations is diffeomorphic to the structure group of the theory g whereas a g is g modulo the group of inner automorphisms it is also proven that a g is a principal fiber bundle with structure group g0

regularized continuum yang mills process and feynman kac functional integral giving an ultraviolet regularization and volume cut off we construct a nuclear riemannian structure on the hilbert manifold mathfrak m of gauge orbits this permits us to define a regularized laplace beltrami operator on mathfrak m and an associated global diffusion in mathfrak m governed by this enables us to define via a feynman kac integral a euclidean continuum regularized yang mills process corresponding to a suitable regularization of the kinetic term of the classical yang mills lagrangian ont mathfrak m

neumann domination for the yang mills heat equation long time existence and uniqueness of solutions to the yang mills heat equation have been proven over a compact 3 manifold with boundary for initial data of finite energy in the present paper we improve on previous estimates by using a neumann domination technique that allows us to get much better pointwise bounds on the magnetic field as in the earlier work we focus on dirichlet neumann and marini boundary conditions in addition we show that the wilson loop functions gauge invariantly regularized converge as the parabolic time goes to infinity

classifications of bundle connection pairs by parallel translation and lassos let m be a connected manifold and g be a closed lie subgroup of gl v the general linear group on a finite dimensional vector space v denote by m the space of h1 loops in m starting at a fixed point m let m be the set of p c m g modulo conjugation by an element of g such that p p p for m is the concatenation of the loops and p p if is reparameterization of and the differential of p satisfies a locality condition it is shown that the bundle connection pairs e up to equivalence with structure group g and fiber v are in one to one correspondence with m a similar result has been announced by kobayashi the correspondence is induced by the parallel translation operators of connections furthermore if the manifold m is simply connected then the space of bundle connection pairs can be classified by a collection of lie algebra valued 1 forms on the manifold m called integrated lassos these 1 forms are related to the differentials of elements of m this last result generalizes weil s characterization of u 1 line bundle connection pairs by the curvature 2 form it is also a generalization of gross results to base manifolds m other than rn

ym2 continuum expectations lattice convergence and lassos the two dimensional yang mills theory ym2 is analyzed in both the continuum and the lattice in the complete axial gauge the continuum theory may be defined in terms of a lie algebra valued white noise and parallel translation may be defined by stochastic differential equations this machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesc the expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group the time parameters of the heat kernels are determined by the areas enclosed by the collectionc and the arguments are determined by the crossing topologies of the curves inc the expectations for the wilson lattice models have a similar structure and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations it is also shown that the lasso variables advocated by l gross 36 exist and are sufficient to generate all the measurable functions on the ym2 measure space

a quotient universal digital topology we study a special topology on z 2 and show that both the marcus topology and the khalimsky topology and also one more digital topology on z 2 may be obtained as three of its quotient topologies a quotient closure operator of the topology studied is discussed too and an analogue of the jordan curve theorem for this closure operator is proved

graphs with a path partition for structuring digital spaces we introduce the concept of graphs with a path partition and define a special type of connectedness in these graphs the connectedness is shown to have certain properties suitable for using graphs with a path partition as convenient background structures on digital spaces for the study of digital images we introduce a family of such graphs on z 2 and present a jordan curve theorem for them

border and surfacetracing theoretical foundations in this paper we define and study digital manifolds of arbitrary dimension and provide in particular a general theoretical basis for curve or surface tracing in picture analysis the studies involve properties such as the one dimensionality of digital curves and n 1 dimensionality of digital hypersurfaces that makes them discrete analogs of corresponding notions in continuous topology the presented approach is fully based on the concept of adjacency relation and complements the concept of dimension as common in combinatorial topology this work appears to be the first one on digital manifolds based on a graph theoretical definition of dimension in particular in the n dimensional digital space a digital curve is a one dimensional object and a digital hypersurface is an n 1 dimensional object as it is in the case of curves and hypersurfaces in the euclidean space relying on the obtained properties of digital hypersurfaces we propose a uniform approach for studying good pairs defined by separations and obtain a classification of good pairs in arbitrary dimension we also discuss possible applications of the presented definitions and results

finite dimensional approximations to wiener measure and path integral formulas on manifolds abstract certain natural geometric approximation schemes are developed for wiener measure on a compact riemannian manifold these approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi group on riemannian manifolds the path space is approximated by finite dimensional manifolds h p m consisting of piecewise geodesic paths adapted to partitions p of 0 1 the finite dimensional manifolds h p m carry both an h 1 and a l 2 type riemannian structures g 1 p and g 0 p respectively it is proved that 1 z i p e 1 2 e d vol g i p i d as mesh p 0 where e is the energy of the piecewise geodesic path h p m and for i 0 and 1 z i p is a normalization constant vol g i p is the riemannian volume form relative to g i p and is wiener measure on paths on m here 1 1 and 0 exp 1 6 1 0 scal s ds where scal is the scalar curvature of m these results are also shown to imply the well known integration by parts formula for the wiener measure

a functional integral approaches to the makeenko migdal equations makeenko and migdal phys lett b 88 1 135 137 1979 gave heuristic identities involving the expectation of the product of two wilson loop functionals associated to splitting a single loop at a self intersection point kazakov and kostov nucl phys b 176 1 199 215 1980 reformulated the makeenko migdal equations in the plane case into a form which made rigorous sense nevertheless the first rigorous proof of these equations and their generalizations was not given until the fundamental paper of levy 2017 subsequently driver kemp and hall commun math phys 351 2 741 774 2017 gave a simplified proof of levy s result and then with driver gabriel kemp and hall commun math phys 352 3 967 978 2017 we showed that these simplified proofs extend to the yang mills measure over arbitrary compact surfaces all of the proofs to date are elementary but tricky exercises in finite dimensional integration by parts the goal of this article is to give a rigorous functional integral proof of the makeenko migdal equations guided by the original heuristic machinery invented by makeenko and migdal although this stochastic proof is technically more difficult it is conceptually clearer and explains why the makeenko migdal equations are true it is hoped that this paper will also serve as an introduction to some of the problems involved in making sense of quantizing yang mill s fields

convergence of u 1 3 lattice gauge theory to its continuum limit it is shown that in three space time dimensions the pure u 1 lattice gauge theory with villain action and fixed coupling constant converges to the free electromagnetic field as the lattice spacing approaches zero the same holds for the wilson action on the electric sector

regularity and decay of lattice green s functions in the paper we study a class of lattice covariant laplace operators with external gauge fields we prove that these operators are positive and that their green s functions decay exponentially they also have regularity properties similar to continuous space green s functions all the bounds are uniform in the lattice spacing

large deviations for the yang mills measure on a compact surface we prove the first mathematical result relating the yang mills measure on a compact surface and the yang mills energy we show that at the small volume limit the scaled yang mills measures satisfy a large deviation principle with a rate function which is expressed in a simple and natural way in terms of the yang mills energy

on the bundle of connections and the gauge orbit manifold in yang mills theory in an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations in yang mills theory is ac principal fibration the underlying quotient space the gauge orbit space is seen explicitly to be ac manifold modelled on a hilbert space

dynamic scaling on fractals with sublattices dynamic scaling techniques are developed for antiferromagnets and multiatomic lattices the methods are applied to antiferromagnets and triatomic lattices on sierpinski gasket fractals and to ferro and antiferromagnets on berker hierarchical lattices relationships between the frequencies of ferro and antiferromagnets on the same fractal and of mono and multiatomic fractal systems are given in particular the dynamic exponents za and zf of antiferromagnetic and ferromagnetic sierpinski gaskets or berker lattice fractals are shown to satisfy za 1 2zf

convergence of theu 1 4 lattice gauge theory to its continuum limit it is shown that in four space time dimensions the compactu 1 lattice gauge theory with general energy function converges to a renormalized free electromagnetic field on the current sector as the lattice spacing approaches zero provided the coupling constant is sufficiently large for the wilson energy function it is possible by judicious choice of the gibbs state to get convergence for arbitrary coupling strengths furthermore for all but a countable number of values of the coupling constant the limit exists and is independent of the particular state chosen to define the lattice model

studies of the spectral dimension of the vicsek snowflake fractal by an exact real space renormalization group method the authors obtain the spectral dimension of the vicsek snowflake fractal the result seems to cast some doubt on the range of validity of the einstein relation on the deterministic fractal

a poincare lemma for connection forms denote by p the space of piecewise smooth curves in rn beginning at the origin a path 2 form is a function h on p such that for each element in p h is a 2 form at the endpoint of with values in a lie algebra g for example if a is a smooth g valued connection form on rn with curvature f and parallel translation operator p then the equation la p 1 f 1 p defines la as a path 2 form a necessary and sufficient condition is given to characterize those path 2 forms which arise in this way by way of application it is shown that the birula mandelstam generalization of maxwell s equations to nonabelian gauge fields is equivalent to the yang mills equation

does a generic connection depend continuously on its curvature for a principal bundle with semi simple structure group over a smooth four dimensional base manifold the set of connections gauge potentials a which are uniquely determined by their curvature field or field strength f is generic in the set of all potentials endowed with the whitneyc topology however the operator taking each such fieldf to its potentiala is not continuous partial negative results are given concerning the existence of a smaller generic set on which this operator is continuous