the effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking stretching sheet the effects of variable viscosity and thermal conductivity on the flow and heat transfer in a laminar liquid film on a horizontal shrinking stretching sheet are analyzed the similarity transformation reduces the time independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations the resulting five parameter problem is solved by the homotopy perturbation method the results are presented graphically to interpret various physical parameters appearing in the problem

valuations on algebras with involution let a be a central simple algebra with involution of the first or second kind let v be a valuation on the fixed part f of z a a special v gauge g on a is a kind of value function on a extending v on f such that g x x 2g x for all x in a it is shown under certain restrictions if the residue characteristic is 2 that if v is henselian then there is a special v gauge g if and only if is anisotropic and g is unique if v is not henselian it is shown that there is a special v gauge g if and only if remains anisotropic after scalar extension from f to the henselization of f with respect to v when this occurs g is the unique invariant v gauge on a

cutoff for the ising model on the lattice introduced in 1963 glauber dynamics is one of the most practiced and extensively studied methods for sampling the ising model on lattices it is well known that at high temperatures the time it takes this chain to mix in l 1 on a system of size n is o logn whether in this regime there is cutoff i e a sharp transition in the l 1 convergence to equilibrium is a fundamental open problem if so as conjectured by peres it would imply that mixing occurs abruptly at c o 1 logn for some fixed c 0 thus providing a rigorous stopping rule for this mcmc sampler however obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple markov chains already for the one dimensional ising model showing cutoff is a longstanding open problem

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

a cutoff phenomenon for quantum markov chains we derive upper and lower bounds on the convergence behavior of certain classes of one parameter quantum dynamical semigroups the classes we consider consist of tensor product channels and of channels with commuting liouvillians we introduce the notion of cutoff phenomenon in the setting of quantum information theory and show how it exemplifies the fact that the convergence of quantum stochastic processes is not solely governed by the spectral gap of the transition map we apply the new methods to show that graph states can be prepared efficiently albeit not in constant time by dissipation and give the exact scaling behavior of the time to stationarity

the defektsatz for central simple algebras let q be a central simple algebra finite dimensional over its center f and let v be a valuation ring of f the v has en extension to q i e there exists a dubrovin valuation ring b of q satisfying v f b generally the number of extensions of v to q is not finite and therefore the so called intersection property of dubrovin valuation rings b 1 b n is introduced this property is defined in terms of the prime ideals and the valuation overrings of the intersection b b n it is shown that there exists a uniquely determined natural number n depending only on v and having the following property if b 1 b k are extensions of v having the intersection property then k n and k n holds if and only if b 1 b k is integral over v

hall current and mhd effects on heat transfer over an unsteady stretching permeable surface with thermal radiation the influence of hall current and thermal radiation on flow and heat transfer characteristics in a viscous fluid over an unsteady stretching permeable surface is studied in this paper the unsteadiness in the flow and temperature fields is because of the time dependent stretching velocity and surface temperature similarity transformations are used to convert the governing time dependent nonlinear boundary layer equations for momentum and thermal energy to a system of nonlinear ordinary differential equations these equations are solved numerically by applying shooting method using runge kutta fehlberg method and analytically by differential transform method with pade approximants dtm pade comparison of the numerical results and analytical solutions based on dtm pade approximant is made with the earlier published results the effects of the unsteadiness parameter thermal radiation hall effects suction injection parameter and non uniform heat source sink parameter on flow and heat transfer characteristics as well as on the local nusselt number are shown graphically

maximal orders over valuation rings abstract in this paper we study maximal orders over commutative valuation rings in central simple algebras we are particularly interested in maximal orders which are either bezout or semihereditary we construct a class of bezout maximal orders and a class of semihereditary maximal orders and show that for any valuation ring v resp v with value group z m any bezout resp semihereditary maximal order over v belongs to the class constructed furthermore we classify all maximal orders in m2 f over a valuation ring with value group z m and in mn f given a mild defectless assumption

a note on valued division algebras let d be a finite dimensional division algebra with center f and let b be a valuation ring of d i e xeb or x 1 e b holds for all nonzero x in d then b is called an extension of v f b let s be the set of all extensions of v to d and let p b be the group of all bijections of s which are induced by the inner automorphisms of d in this note we investigate p b it is shown that p b is solvable and an upper bound for p b is given finally theorem 4 5 gives a necessary and sufficient condition for b such that p b is equal to this bound

exact solution for a class of random walk on the hypercube a class of families of markov chains defined on the vertices of the n dimensional hypercube n 0 1 n is studied the single step transition probabilities pn ij with i j n are given by p n ij frac 1 alpha d ij 2 alpha n where 0 1 and dij is the hamming distance between i and j this corresponds to flip independently each component of the vertex with probability frac 1 alpha 2 alpha the m step transition matrix p n ij m is explicitly computed in a close form the class is proved to exhibit cutoff a model independent result about the vanishing of the first m terms of the expansion in of p n ij m is also proved

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

cutoff phenomena for random walks on random regular graphs the cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite markov chains to equilibrium many natural families of chains are believed to exhibit cutoff and yet establishing this fact is often extremely challenging an important such family of chains is the random walk on g n d a random d regular graph on n vertices it is well known that almost every such graph for d 3 is an expander and even essentially ramanujan implying a mixing time of o logn according to a conjecture of peres the simple random walk on g n d for such d should then exhibit cutoff with high probability whp as a special case of this durrett conjectured that the mixing time of the lazy random walk on a random 3 regular graph is whp 6 o 1 log2n in this work we confirm the above conjectures and establish cutoff in total variation its location and its optimal window both for simple and for non backtracking random walks on g n d namely for any fixed d 3 the simple random walk on g n d whp has cutoff at d d 2 logd 1n with window order logn surprisingly the non backtracking random walk on g n d whp has cutoff already at logd 1n with constant window order we further extend these results to g n d for any d no 1 that grows with n beyond which the mixing time is o 1 where we establish concentration of the mixing time on one of two consecutive integers

value functions on central simple algebras in this paper we study noncommutative valuation rings as defined by dubrovin while there is in general no valuation associated to a dubrovin valuation ring we show that there is a value function associated to any dubrovin valuation ring integral over its center by using value functions we obtain information on dubrovin valuation rings in a tensor product both generalizing and giving a much simpler proof of a result about valued division algebras by being able to work directly with central simple algebras we gain new information about division algebras over henselian fields

convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation absorption abstract we consider a convective flow in a porous medium of an incompressible viscous conducting fluid impinging on a permeable stretching surface with suction and internal heat generation absorption using a similarity transformation the governing equations of the problem are reduced to a coupled third order nonlinear ordinary differential equations we first examine a number of special cases for which we may obtain exact solutions we then obtain analytical solutions by the homotopy analysis method and numerical solutions by a boundary value problem solver in order to further study the behavior of the nonlinear differential equations for various values of the physical parameters our numerical solutions are shown to agree with the available results in the literature we then employ the numerical results to bring out the effects of the suction parameter heat source sink parameter stretching parameter porosity parameter the prandtl number and the free convection parameter on the flow and heat transfer characteristics in the absence of suction and free convection our findings are in agreement with the corresponding numerical results of attia h a attia on the effectiveness of porosity on stagnation point flow towards a stretching surface with heat generation comput mater sci 38 2007 741 745

abrupt convergence and escape behavior for birth and death chains we link two phenomena concerning the asymptotical behavior of stochastic processes i abrupt convergence or cut off phenomenon and ii the escape behavior usually associated to exit from metastability the former is characterized by convergence at asymptotically deterministic times while the convergence times for the latter are exponentially distributed we compare and study both phenomena for discrete time birth and death chains on with drift towards zero in particular this includes energy driven evolutions with energy functions in the form of a single well under suitable drift hypotheses we show that there is both an abrupt convergence towards zero and escape behavior in the other direction furthermore as the evolutions are reversible the law of the final escape trajectory coincides with the time reverse of the law of cut off paths thus for evolutions defined by one dimensional energy wells with sufficiently steep walls cut off and escape behavior are related by time inversion

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

valuation rings in finite dimensional division algebras a subring b of a division algebra d is called a valuation ring of d if x or x 1 is contained in b for every x in d x 0 such a ring is called an extension of the valuation ring b of k the centre of d if b k b let d be a division algebra finite dimensional over its centre k d k n2 b a valuation ring of k and b bi iei the set of all extension of b in d theorem 1 b possesses at most n extensions in d i e b n theorem 2 any two extensions of b in d are conjugate in d theorems 3 and 4 the set t of elements in d which are integral over b is a subring of d if and only if b 1 in this case t bi bi in b

correspondences between valued division algebras and graded division algebras abstract if d is a tame central division algebra over a henselian valued field f then the valuation on d yields an associated graded ring gd which is a graded division ring and is also central and graded simple over gf after proving some properties of graded central simple algebras over a graded field including a cohomological characterization of its graded brauer group it is proved that the map d gd g yields an index preserving isomorphism from the tame part of the brauer group of f to the graded brauer group of gf this isomorphism is shown to be functorial with respect to field extensions and corestrictions and using this it is shown that there is a correspondence between f subalgebras of d with center tame over f and graded gf subalgebras of gd

glauber dynamics for the mean field ising model cut off critical power law and metastability we study the glauber dynamics for the ising model on the complete graph also known as the curie weiss model for 1 we study metastability in particular we show that the glauber dynamics restricted to states of non negative magnetization has mixing time o n log n

value functions and associated graded rings for semisimple algebras we introduce a type of value function y called a gauge on a finite dimensional semisimple algebra a over a field f with valuation v the filtration on a induced by y yields an associated graded ring gr y a which is a graded algebra over the graded field gr v f key requirements for y to be a gauge are that gr y a be graded semisimple and that dim gr v f gr y a dim f a it is shown that gauges behave well with respect to scalar extensions and tensor products when v is henselian and a is central simple over f it is shown that gr y a is simple and graded brauer equivalent to gr w d where d is the division algebra brauer equivalent to a and w is the valuation on d extending v on f the utility of having a good notion of value function for central simple algebras not just division algebras and with good functorial properties is demonstrated by giving new and greatly simplified proofs of some difficult earlier results on valued division algebras

the successive approximation method and pade approximants for solutions the non linear boundary value problem in this paper we suggested an successive approximation method and pade approximants method for the solution of the non linear differential equation first we calculate power series of the given equation system then transform it into pade approximants series form which give an arbitrary order for solving differential equation numerically we compare our results with the result obtained by successive method for the non linear equation

entropy driven cutoff phenomena in this paper we present in the context of diaconis paradigm a general method to detect the cutoff phenomenon we use this method to prove cutoff in a variety of models some already known and others not yet appeared in literature including a non reversible random walk on a cylindrical lattice all the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon in the case of birth and death chains this mechanism is fairly well understood our work is an effort to generalize this picture to more general systems such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one dimensional problem in those situations the drift may be looked for by means of a suitable partitioning of the state space into classes using a statistical mechanics language it is then possible to set up a kind of energy entropy competition between the weight and the size of the classes under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease

cutoff for nonbacktracking random walks on sparse random graphs a finite ergodic markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale discovered in the context of card shuffling aldous diaconis 1986 this phenomenon is now believed to be rather typical among fast mixing markov chains yet establishing it rigorously often requires a challengingly detailed understanding of the underlying chain here we consider non backtracking random walks on random graphs with a given degree sequence under a general sparsity condition we establish the cutoff phenomenon determine its precise window and prove that the suitably rescaled cutoff profile approaches a remarkably simple universal shape

extensions of valuation rings in central simple algebras certain subrings r of simple algebras q finite dimensional over their center k are studied these rings are called q valuation rings since they share many properties with commutative valuation rings let v be a valuation ring of k the center of q and let m be the set of q valuation rings r in q with r n k v then 1s 1 this extension theorem which does not hold if one considers only total valuation rings was proved by n i dubrovin here first a somewhat different proof of this result is given and then information about the set m is obtained theorem the elements in j are conjugate if v has finite rank theorem the elements in m are total valuation rings if m contains one total valuation ring in this case q is a division ring theorem i i 1 if 5 contains an invariant total valuation ring

cutoff at the entropic time for sparse markov chains we study convergence to equilibrium for a large class of markov chains in random environment the chains are sparse in the sense that in every row of the transition matrix p the mass is essentially concentrated on few entries moreover the random environment is such that rows of p are independent and such that the entries are exchangeable within each row this includes various models of random walks on sparse random directed graphs the models are generally non reversible and the equilibrium distribution is itself unknown in this general setting we establish the cutoff phenomenon for the total variation distance to equilibrium with mixing time given by the logarithm of the number of states times the inverse of the average row entropy of p as an application we consider the case where the rows of p are i i d random vectors in the domain of attraction of a poisson dirichlet law with index 0 1 our main results are based on a detailed analysis of the weight of the trajectory followed by the walker this approach offers an interpretation of cutoff as an instance of the concentration of measure phenomenon

quantitative approximations of evolving probability measures and sequential markov chain monte carlo methods we study approximations of evolving probability measures by an interacting particle system the particle system dynamics is a combination of independent markov chain moves and importance sampling resampling steps under global regularity conditions we derive non asymptotic error bounds for the particle system approximation in a few simple examples including high dimensional product measures bounds with explicit constants of feasible size are obtained our main motivation are applications to sequential mcmc methods for monte carlo integral estimation

study of a third grade non newtonian fluid flow between two parallel plates using the multi step differential transform method in this paper the multi step differential transform method mdtm one of the most effective method is implemented to compute an approximate solution of the system of nonlinear differential equations governing the problem it has been attempted to show the reliability and performance of the mdtm in comparison with the numerical method fourth order runge kutta and other analytical methods such as hpm ham and dtm in solving this problem the first differential equation is the plane couette flow equation which serves as a useful model for many interesting problems in engineering the second one is the fully developed plane poiseuille flow equation and finally the third one is the plane couette poiseuille flow

characterization of cutoff for reversible markov chains a sequence of markov chains is said to exhibit total variation cutoff if the convergence to stationarity in total variation distance is abrupt we consider reversible lazy chains we prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of worst in some sense sets of stationary measure at least alpha for some alpha in 0 1 we also give general bounds on the total variation distance of a reversible chain at time t in terms of the probability that some worst set of stationary measure at least alpha was not hit by time t as an application of our techniques we show that a sequence of lazy markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their lazy mixing times tends to infty

total variation cutoff in birth and death chains the cutoff phenomenon describes a case where a markov chain exhibits a sharp transition in its convergence to stationarity diaconis proc natl acad sci usa 93 4 1659 1664 1996 surveyed this phenomenon and asked how one could recognize its occurrence in families of finite ergodic markov chains peres american institute of mathematics aim research workshop palo alto http www aimath org wwn mixingtimes 2004 noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing time and spectral gap tends to infinity and conjectured that in many settings this condition should also be sufficient diaconis and saloff coste ann appl probab 16 4 2098 2122 2006 verified this conjecture for continuous time birth and death chains started at an endpoint with convergence measured in separation it is natural to ask whether the conjecture holds for these chains in the more widely used total variation distance in this work we confirm the above conjecture for all continuous time or lazy discrete time birth and death chains with convergence measured via total variation distance namely if the product of the mixing time and spectral gap tends to infinity the chains exhibit cutoff at the maximal hitting time of the stationary distribution median with a window of at most the geometric mean between the relaxation time and mixing time in addition we show that for any lazy or continuous time birth and death chain with stationary distribution the separation 1 p t x y y is maximized when x y are the endpoints together with the above results this implies that total variation cutoff is equivalent to separation cutoff in any family of such chains

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc