small fast universal turing machines we present deterministic polynomial time universal turing machines utms with state symbol pairs of 3 11 5 7 6 6 7 5 and 8 4 these are the smallest known utms that simulate turing machines in polynomial time

small universal turing machines abstract let utm m n be the class of universal turing machine with m states and n symbols universal turing machines are proved to exist in the following classes utm 24 2 utm 10 3 utm 7 4 utm 5 5 utm 4 6 utm 3 10 and utm 2 18

an almost full embedding of the category of graphs into the category of groups abstract we construct a functor f g raphs g roups which is faithful and almost full in the sense that every nontrivial group homomorphism f x f y is a composition of an inner automorphism of fy and a homomorphism of the form ff for a unique map of graphs f x y when f is composed with the eilenberg mac lane space construction k f x 1 we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null homotopic maps we provide several applications of this construction to localizations i e idempotent functors we show that the questions 1 is every orthogonality class reflective 2 is every orthogonality class a small orthogonality class have the same answers in the category of groups as in the category of graphs in other words they depend on set theory 1 is equivalent to weak vop nka s principle and 2 to vop nka s principle additionally the second question considered in the homotopy category is also equivalent to vop nka s principle

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

some mathematical results on a system of transport equations with an algebraic constraint describing fixed bed adsorption of gases this paper deals with a system of two equations which describes heatless adsorption of a gaseous mixture with two species when one of the components is inert we obtain an existence result of a weak solution satisfying some entropy condition under some simplifying assumptions the proposed method makes use of a godunov type scheme uniqueness is proved in the class of piecewise c 1 functions

approximation of the solution to a system modeling heatless adsorption of gases we present both time and space time explicit approximation schemes for a system of first order nonlinear transport pdes used as a model in chemical engineering for the time discretization using a splitting we introduce an intermediate spatial regularization step to obtain some bounded variation bv estimates for the approximate solutions with bv initial data and we get an o delta t 1 2 l 1 convergence rate with respect to the fully discrete finite volume scheme we show that it is possible to get such estimates without any regularization step because of the dissipative effect of the upwind scheme used to treat the transport part of the equations this scheme is convergent

on small reduced and fast universal accepting networks of splicing processors in this paper we show that accepting networks of splicing processors ansps of size 2 are computationally complete since by definition an ansp needs at least two nodes to perform non trivial computations this completely settles the question of designing complete ansps of minimal size also we derive from this result the fact that all the languages in pspace can be accepted by ansps of size 2 having polynomial length complexity the ansp complexity measure for the space used in a computation however the construction that we propose although efficient from the descriptional complexity and space complexity points of view does not seem to have good properties from the time complexity point of view in this respect we prove that ansps of size three can decide all languages in np in polynomial time the previous lower bound on size for both completeness and efficient acceptance of np languages was seven we also consider ansps with restricted features proving the following normal forms for any ansp there exists an equivalent ansp without input filters and one without output filters finally we show how to construct a small universal ansp and make several considerations on the computational efficiency of universal ansps

on endomorphism rings of non separable abelian p groups there have been spectacular advances in recent years in the so called realization problem for certain classes of abelian groups and modules the advances have derived mainly from powerful combinatorial set theoretic tools pioneered by shelah in the case of separable abelian p groups these results appear e g in 1 4 51 for non separable abelian p groups the most significant contribution has been 2 our objective in this paper is to extend the results in 2 and derive analogous results to those obtained in 4 51 it is perhaps worth pointing out that our methods are entirely algebraic the necessary set theoretical work has been carried out in the separable case and no further set theoretical arguments are needed in this the work is reminiscent of s all our set theory with one exception is in zfc and this is not acknowledged in the statement of individual results in the exceptional case where we work in godel s constructible universe v l this is acknowledged by appending v l after the statement number the principal realization result can be stated as

every endomorphism of a local warfield module of finite torsion free rank is the sum of two automorphisms abstract a well known result of p hill s 1969 trans amer math soc 141 99 105 says that any endomorphism of a totally projective abelian p group for any odd prime is the sum of two automorphisms we will extend this result to local warfield modules of finite torsion free rank over odd primes