on a definition of random sequences with respect to conditional probability we study a universal martin lof test with respect to a computable probability on a product space then we define random sequences with respect to a conditional probability by using a section of the set of random points of product space we show that 1 our definition is consistent with fubini s theorem and 2 it is equivalent to the relative notion of randomness under a condition this is an extension of lambalgen s theorem 1987 to a correlated probability

a dynamically consistent method to solve nonlinear multidimensional advection reaction equations with fractional diffusion abstract this work is motivated by an extension of both the burgers fisher and the burgers huxley equations in multiple dimensions considering riesz fractional diffusion initial boundary conditions which are positive and bounded are imposed on a closed and bounded set and a finite difference method is proposed to approximate the solutions of the fractional model the methodology is a linear and implicit technique which is based on fractional centered differences we show in this manuscript that the method can be expressed in vector form using a minkowski matrix under suitable conditions the main properties of minkowski matrices are used then to establish the existence and the uniqueness of the solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness additionally we show that the method is a consistent technique which is stable and convergent with first order of convergence in time and second order in space some illustrative simulations show that the scheme is capable of preserving the positivity and the boundedness of the numerical approximations

steiner symmetric extremals in polya szego type inequalities abstract the cases of equality are analyzed in steiner symmetrization inequalities for dirichlet type integrals in particular minimal assumptions are determined under which functions attaining equality are necessarily steiner symmetric

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

combinatorial lower bound arguments for deterministic and nondeterministic turing machines on introduit de nouvelles techniques combinatoires pour etablir des bornes inferieures quadratiques des machines de turing deterministes et non deterministes a une bande

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

computer evaluation of kauffman polynomials by using gauss codes with a skein template algorithm homfly polynomials of knots and links are evaluated on computers by using an algebraic version with gauss codes of the skein template algorithm the computer codes are written with the language c whose structure is particularly suitable to such evaluations jones conway and alexander polynomials may then be readily obtained as special cases

blind identification of an autoregressive system using a nonlinear dynamical approach the problem of identifying an autoregressive ar system with arbitrary driven noise is considered here using an abstract dynamical system to represent both chaotic and stochastic processes in a unified framework a dynamic based complexity measure called phase space volume psv which has its origins in chaos theory can be applied to identify an ar model in chaotic as well as stochastic noise environments it is shown that the psv of the output signal of an inverse filter applied to identify an ar model is always larger than the psv of the input signal of the ar model therefore by minimizing the psv of the inverse filter output one can estimate the coefficients and the order of the ar system a major advantage of this minimum phase space volume mpsv identification technique is that it works like a universal estimator that does not require precise statistical information about the ar input signal because the theoretical psv is so difficult to compute two approximations of psv are also considered the e psv and nearest neighbor psv both approximations are shown to approach the ideal psv asymptotically the identification performance based on these two approximations are evaluated using monte carlo simulations both approximations are found to generate relatively good results in identifying an ar system in various noise environments including chaotic non gaussian and colored noise

upcrossing inequalities for stationary sequences and applications for arrays s i j 1 i j of random variables that are stationary in an appropriate sense we show that the fluctuations of the process s 1 n n 1 can be bounded in terms of a measure of the mean subadditivity of the process s i j 1 i j we derive universal upcrossing inequalities with exponential decay for kingman s subadditive ergodic theorem the shannon macmillan breiman theorem and for the convergence of the kolmogorov complexity of a stationary sample

numerical methods for the fractional laplacian a finite difference quadrature approach the fractional laplacian delta alpha 2 is a nonlocal operator which depends on the parameter alpha and recovers the usual laplacian as alpha to 2 a numerical method for the fractional laplacian is proposed based on the singular integral representation for the operator the method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights the accuracy of the method is shown to be o h 3 alpha convergence of the method is proven the treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method numerical experiments on known exact solutions validate the predicted convergence rates computational examples include exponentially and algebraically decaying solutions with varying regularity the generalization to nonlinear equations involving the operator is discussed the obstacle problem for the fractional laplacian is computed

a well posed problem in singular fickian diffusion we prove that the problem of solving u t u m 1 u x x text for 1 m leqq 0 with initial conditionu x 0 x and flux conditions at infinity mathop lim limits x to infty u m 1 u x f t mathop lim limits x to infty u m 1 u x g t admits a unique solution u in c infty infty x infty 0 t t for every el1 r 0 0 and every pair of nonnegative flux functionsf g e l loc 0 the maximal existence time is given by t sup left t smallint phi x dx int limits 0 t f s g s ds right this mixed problem is ill posed for anym outside the above specified range

kolmogorov s contributions to information theory and algorithmic complexity briefly information theory says a random object x p x has complexity entropy h xp x log p x with the attendant interpretation that h bits are sufficient to describe x on the average algorithmic complexity says an object x has a complexity k x equal to the length of the shortest binary program that describes x it is a beautiful fact that these ideas are much the same in fact it is roughly true that ek x h moreover if we let pu x pr u prints x be the probability that a given computer u prints x when given a random program it can be shown that log 1 pu x k x for all x thus establishing a vital link between the universal probability measure pu and the universal complexity k more on this later the relationship of these ideas to probability theory was summed up in kolmogorov s 1983 paper which was based on his 1970 talk in nice perhaps only

distortion program size complexity with respect to a fidelity criterion and rate distortion function a novel concept of distortion kolmogorov chaitin complexity is proposed bearing analogy to ordinary kolmogorov chaitin complexity that represents the information content of a single message distortion kolmogorov chaitin complexity can be regarded as the amount of information about a single message that must be conveyed in order to reproduce it with a bounded distortion this explanation is justified by the properties of distortion kolmogorov chaitin complexity and the equivalences between distortion kolmogorov chaitin complexity and the rate distortion function which are analogous to those between ordinary kolmogorov chaitin complexity and the shannon entropy some implications for universal almost sure data compression are also discussed

new polya szego type inequalities and an alternative approach to comparison results for pde s abstract we prove some polya szego type inequalities which involve couples of functions and their rearrangements our inequalities reduce to the classical polya szego principle when the two functions coincide as an application we give a different proof of a comparison result for solutions to dirichlet boundary value problems for laplacian equations proved in 1

a pde approach to fractional diffusion a space fractional wave equation we study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers of order s in 0 1 of symmetric coercive linear elliptic second order operators in bounded domains varomega we realize fractional diffusion as the dirichlet to neumann map for a nonuniformly elliptic problem posed on the semi infinite cylinder mathcal c varomega times 0 infty we thus rewrite our evolution problem as a quasi stationary elliptic problem with a dynamic boundary condition and derive space time and space time regularity estimates for its solution the latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation we propose and analyze two fully discrete schemes the discretization in time is based on finite difference discretization techniques the trapezoidal and leapfrog schemes the discretization in space relies on the tensorization of a first degree fem in varomega with a suitable hp fem in the extended variable for both schemes we derive stability and error estimates we consider a first degree fem in varomega with mesh refinement near corners and the aforementioned hp fem in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open bounded polytopal but not necessarily convex domains varomega subset mathbb r 2 we discuss implementation details and report several numerical examples

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

the linking number and the writhe of uniform random walks and polygons in confined spaces random walks and polygons are used to model polymers in this paper we consider the extension of the writhe self linking number and linking number to open chains we then study the average writhe self linking and linking number of random walks and polygons over the space of configurations as a function of their length we show that the mean squared linking number the mean squared writhe and the mean squared self linking number of oriented uniform random walks or polygons of length n in a convex confined space are of the form o n2 moreover for a fixed simple closed curve in a convex confined space we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form our numerical studies confirm those results they also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons of n edges each is of the form o n equilateral random walks and polygons are used to model polymers in conditions we use numerical simulations to investigate how the self linking and linking number of equilateral random walks scale with their length

the linking number in systems with periodic boundary conditions periodic boundary conditions pbc are often used for the simulation of complex physical systems using the gauss linking number we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing pbc in the case of closed chains in pbc the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system for open chains the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system in this paper we give a rigorous proof that the periodic linking number is defined for the infinite system i e that it converges for one two and three pbc models it gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains similarly for a single oriented chain we define the periodic self linking number and prove that it also is defined for open chains in addition we define the cell periodic linking and self linking numbers giving localizations of the periodic linking numbers these can be used to give good estimates of the periodic linking numbers in infinite systems we also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers finally we study and compare these measures when applied to a pbc model of polyethylene melts

the average crossing number of equilateral random polygons in this paper we study the average crossing number of equilateral random walks and polygons we show that the mean average crossing number acn of all equilateral random walks of length n is of the form a similar result holds for equilateral random polygons these results are confirmed by our numerical studies furthermore our numerical studies indicate that when random polygons of length n are divided into individual knot types the for each knot type can be described by a function of the form where a b and c are constants depending on and n0 is the minimal number of segments required to form the profiles diverge from each other with more complex knots showing higher than less complex knots moreover the profiles intersect with the acn profile of all closed walks these points of intersection define the equilibrium length of i e the chain length at which a statistical ensemble of configurations with given knot type upon cutting equilibration and reclosure to a new knot type does not show a tendency to increase or decrease this concept of equilibrium length seems to be universal and applies also to other length dependent observables for random knots such as the mean radius of gyration rg

non stochastic infinite and finite sequences abstract combining outcomes of coin tossing and transducer algorithms it is possible to generate with probability close to 1 very pathological sequences for which computable probabilistic forecasting is impossible these sequences are not random with respect to any reasonable probability distribution a natural consequence from the definition of such sequences is that each simple measure of the set of all such sequences is equal to 0 it was kolmogorov s and levin s idea to estimate the probability of generating of such sequences in the combinations of probabilistic and algorithmic processes 8 14 21 we collect several results in this direction for infinite sequences and asymptotic results for finite sequences including estimation of space and time of losing randomness for time bounded forecasting systems a correction to 22

comparison and regularity results for the fractional laplacian via symmetrization methods abstract in this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional laplacian this allows to get sharp estimates for the solutions obtained by comparing them with solutions of suitable radial problems furthermore we use such result to prove a priori estimates for solutions in terms of the data providing several regularity results which extend the well known ones for the classical laplacian

a general fractional porous medium equation we develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion u t 2 u m 1 u 0 x r n t 0 u x 0 f x x r n we consider data f l 1 r n and all exponents 0 0 existence and uniqueness of a weak solution is established for m m n n giving rise to an l 1 contraction semigroup in addition we obtain the main qualitative properties of these solutions in the lower range 0 m m existence and uniqueness of solutions with good properties happen under some restrictions and the properties are different from the case above m we also study the dependence of solutions on f m and moreover we consider the above questions for the problem posed in a bounded domain

entropic measures markov information sources and complexity the concept of entropy plays a major part in communication theory the shannon entropy is a measure of uncertainty with respect to a priori probability distribution in algorithmic information theory the information content of a message is measured in terms of the size in bits of the smallest program for computing that message this paper discusses the classical entropy and entropy rate for discrete or continuous markov sources with finite or continuous alphabets and their relations to program size complexity and algorithmic probability the accent is on ideas constructions and results no proofs will be given

the growth of the mean average crossing number of equilateral polygons in confinement the physical and biological properties of collapsed long polymer chains as well as of highly condensed biopolymers such as dna in all organisms are known to be determined at least in part by their topological and geometrical properties with this purpose of characterizing the topological properties of such condensed systems equilateral random polygons restricted to confined volumes are often used however very few analytical results are known in this paper we investigate the effect of volume confinement on the mean average crossing number acn of equilateral random polygons the mean acn of knots and links under confinement provides a simple alternative measurement for the topological complexity of knots and links in the statistical sense for an equilateral random polygon of n segments without any volume confinement constrain it is known that its mean acn acn is of the order here we model the confining volume as a simple sphere of radius r we provide an analytical argument which shows that acn of an equilateral random polygon of n segments under extreme confinement meaning r n grows as o n2 we propose to model the growth of acn as a r n2 b r nln n under a less extreme confinement condition where a r and b r are functions of r with r being the radius of the confining sphere computer simulations performed show a fairly good fit using this model

on the laplace equation with non linear dynamical boundary conditions the main part of the paper deals with local existence and global existence versus blow up for solutions of the laplace equation in bounded domains with a non linear dynamical boundary condition more precisely we study the problem consisting in 1 the laplace equation in 0 infty times omega 2 a homogeneous dirichlet condition 0 infty times gamma 0 3 the dynamical boundary condition frac partial u partial nu u t m 2 u t u p 2 u on 0 infty times gamma 1 4 the initial condition u 0 x u 0 x on partial omega here omega is a regular and bounded domain in mathbb r n with n ge 1 and gamma 0 and gamma 1 endow a measurable partition of partial omega moreover m 1 2 le p r where r 2 n 1 n 2 when n ge 3 r infty when n 1 2 and u 0 in h 1 2 partial omega u 0 0 on gamma 0 the final part of the paper deals with a refinement of a global non existence result by levine park and serrin which is applied to the previous problem

divergence free fields energy and asymptotic crossing number in more than one context the equations of classical physics lead to a closed 2 form co on 3 dimensional space which is a conserved quantity in the sense that it pulls back under time evolution in the physics literature co is written as a divergence free vector field w appendix a describes two settings in which a divergence free w is conserved or frozen one is the magnetic field in ideal magneto hydrodynamics and the other is curl velocity in incompressible inviscid fluid flow if ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho id wt denotes the field in the fluid at time t and x0 is the space variable at time t 0 then

the strong ergodic theorem for densities generalized shannon mcmillan breiman theorem let xi x2 be a stationary process with probability densities f x1 x2 xn with respect to lebesgue measure or with respect to a markov measure with a stationary transition measure it is shown that the sequence of relative entropy densities 1 n log f xi x2 x0 converges almost surely this long conjectured result extends the l convergence obtained by moy perez and kieffer and generalizes the shannon mcmillanbreiman theorem to nondiscrete processes the heart of the proof is a new martingale inequality which shows that logarithms of densities are l dominated

anomalous diffusion in view of einstein s 1905 theory of brownian motion abstract einstein s theory of brownian motion is revisited in order to formulate a generalized kinetic theory of anomalous diffusion it is shown that if the assumptions of analyticity and the existence of the second moment of the displacement distribution are relaxed the fractional derivative naturally appears in the diffusion equation this is the first demonstration of the physical origin of the fractional derivative in marked contrast to the usual phenomenological introduction of it it is also examined if einstein s approach can be generalized to nonlinear kinetic theory to derive a generalization of the porous medium type equation by the appropriate use of the escort distribution

on a time depending monge ampere type equation abstract in this paper we prove a comparison result between a solution u x t x r 2 t 0 t of a time depending equation involving the monge ampere operator in the plane and the solution of a conveniently symmetrized parabolic equation to this aim we prove a derivation formula for the integral of a smooth function g x t over sublevel sets of u x u x t r having the same perimeter in r 2

on the mean and variance of the writhe of random polygons we here address two problems concerning the writhe of random polygons first we study the behavior of the mean writhe as a function length second we study the variance of the writhe suppose that we are dealing with a set of random polygons with the same length and knot type which could be the model of some circular dna with the same topological property in general a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons if the mean writhe is non zero then the knot is chiral how accurate is this method for example if for a specific knot type k the mean writhe decreased to zero as the length of the polygons increased then this method would be limited in the case of long polygons furthermore we conjecture that the sign of the mean writhe is a topological invariant of chiral knots this sign appears to be the same as that of an ideal conformation of the knot we provide numerical evidence to support these claims and we propose a new nomenclature of knots based on the sign of their expected writhes this nomenclature can be of particular interest to applied scientists the second part of our study focuses on the variance of the writhe a problem that has not received much attention in the past in this case we focused on the equilateral random polygons we give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons of length n behaves as a linear function of the length of the equilateral random polygon

fractional diffusion limit for collisional kinetic equations this paper is devoted to diffusion limits of linear boltzmann equations when the equilibrium distribution function is a maxwellian distribution it is well known that for an appropriate time scale the small mean free path limit gives rise to a diffusion equation in this paper we consider situations in which the equilibrium distribution function is a heavy tailed distribution with infinite variance we then show that for an appropriate time scale the small mean free path limit gives rise to a fractional diffusion equation

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

the writhe of polygons on the face centred cubic lattice for polygons on the simple cubic lattice there is an important theorem due to lacher and sumners which shows that the writhe of a polygon is the average of the linking numbers of the polygon and its pushoffs in four particular directions this implies that the writhe of a polygon on the simple cubic lattice is always rational we prove a related theorem for the face centred cubic fcc lattice but show that on this lattice the probability that a polygon of length n has irrational writhe tends to unity as n tends to infinity in addition we show that the expectation of the absolute value of the writhe increases at least as fast as n 1 2 for large n

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

natural classification of knots the principal objective of the knot theory is to provide a simple way of classifying and ordering all the knot types here we propose a natural classification of knots based on their intrinsic position in the knot space that is defined by the set of knots to which a given knot can be converted by individual intersegmental passages in addition we characterize various knots using a set of simple quantum numbers that can be determined upon inspection of minimal crossing diagram of a knot these numbers include crossing number average three dimensional writhe number of topological domains and the average relaxation value

decay estimates for evolution equations with classical and fractional time derivatives abstract using energy methods we prove some power law and exponential decay estimates for classical and nonlocal evolutionary equations the results obtained are framed into a general setting which comprise among the others equations involving both standard and caputo time derivative complex valued magnetic operators fractional porous media equations and nonlocal kirchhoff operators both local and fractional space diffusion are taken into account possibly in a nonlinear setting the different quantitative behaviours which distinguish polynomial decays from exponential ones depend heavily on the structure of the time derivative involved in the equation

nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps we prove an invariance principle for a tagged particle in a simple exclusion process out of equilibrium the scaling limit is a time inhomogeneous process of independent increments related to the solution of a fractional heat equation

kolmogorov s structure functions and model selection in 1974 kolmogorov proposed a nonprobabilistic approach to statistics and model selection let data be finite binary strings and models be finite sets of binary strings consider model classes consisting of models of given maximal kolmogorov complexity the structure function of the given data expresses the relation between the complexity level constraint on a model class and the least log cardinality of a model in the class containing the data we show that the structure function determines all stochastic properties of the data for every constrained model class it determines the individual best fitting model in the class irrespective of whether the true model is in the model class considered or not in this setting this happens with certainty rather than with high probability as is in the classical case we precisely quantify the goodness of fit of an individual model with respect to individual data we show that within the obvious constraints every graph is realized by the structure function of some data we determine the un computability properties of the various functions contemplated and of the algorithmic minimal sufficient statistic

a fractional porous medium equation abstract we develop a theory of existence uniqueness and regularity for the following porous medium equation with fractional diffusion u t 1 2 u m 1 u 0 x r n t 0 u x 0 f x x r n with m m n 1 n n 1 and f l 1 r n an l 1 contraction semigroup is constructed and the continuous dependence on data and exponent is established nonnegative solutions are proved to be continuous and strictly positive for all x r n t 0

bayesianism an algorithmic analysis abstract the bayesian program in statistics starts from the assumption that an individual can always ascribe a definite probability to any event it will be demonstrated that this assumption is incompatible with the natural requirement that the individual s subjective probability distribution should be computable we shall construct a probabilistic algorithm producing with probability extremely close to 1 an infinite binary sequence which is not random with respect to any computable probability distribution we use dawid s notion of randomness computable calibration but the results hold for other widely known notions of randomness as well since the bayesian knows the algorithm he must believe that this sequence will be noncalibrable on the other hand it seems that the bayesian must believe that the sequence is random with respect to his own probability distribution we hope that the discussion of this apparent paradox will clarify the foundations of bayesian statistics we analyse also the time of computation and the place of losing randomness we show that we need only polynomial time and space to demonstrate non calibration effects on finite sequences

multivalued fixed point index and nonlocal problems involving relative rearrangement abstract we apply the fixed point index theory for multivalued problems to nonlocal problems involving relative rearrangement mappings these applications need an extension of the almgren lieb theorem and an improvement of the strong continuity of the relative rearrangement mapping on arbitrary open sets bounded or not the proofs of these results are based on some pointwise equations relating the relative rearrangement of the gradient and the weighted de giorgi perimeter

the bfacf algorithm and knotted polygons the bfacf algorithm applied to polygons involves sampling on a markov chain whose state space is the set of all polygons in three dimensions for the simple cubic lattice the authors prove that the ergodic classes of this markov chain are the knot classes of the polygons

continuity of the temperature in boundary heat control problems abstract motivated by the boundary heat control problems formulated in the book of duvaut and lions we study a boundary stefan problem and a boundary porous media problem we prove continuity of the solution with the appropriate modulus we also extend the results to the fractional order case and to the anomalous diffusion problems

elliptic equations and steiner symmetrization we present a new proof of comparison results via steiner symmetrization for solutions of elliptic equations this proof relies upon a level sets argument

minimum description length induction bayesianism and kolmogorov complexity the relationship between the bayesian approach and the minimum description length approach is established we sharpen and clarify the general modeling principles minimum description length mdl and minimum message length mml abstracted as the ideal mdl principle and defined from bayes s rule by means of kolmogorov complexity the basic condition under which the ideal principle should be applied is encapsulated as the fundamental inequality which in broad terms states that the principle is valid when the data are random relative to every contemplated hypothesis and also these hypotheses are random relative to the universal prior the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized if we restrict the model class to finite sets then application of the ideal principle turns into kolmogorov s minimal sufficient statistic in general we show that data compression is almost always the best strategy both in model selection and prediction

minimum complexity density estimation the authors introduce an index of resolvability that is proved to bound the rate of convergence of minimum complexity density estimators as well as the information theoretic redundancy of the corresponding total description length the results on the index of resolvability demonstrate the statistical effectiveness of the minimum description length principle as a method of inference the minimum complexity estimator converges to true density nearly as fast as an estimator based on prior knowledge of the true subclass of densities interpretations and basic properties of minimum complexity estimators are discussed some regression and classification problems that can be examined from the minimum description length framework are considered

writhe and mutual entanglement combine to give the entanglement length we propose a method to estimate n e the entanglement length that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions this estimate uses the writhe of the chains the writhe of the primitive paths and the number of kinks in the chains in a melt an advantage of this method is that it works for both linear and ring chains works under all periodic boundary conditions does not require knowing the contour length of the primitive paths and does not rely on a smooth set of data we apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies

existence and uniqueness results for a time fractional nonlinear diffusion equation abstract in this work we consider a nonlinear ordinary integro differential equation which arises in the studies of time fractional porous medium equation the nonlocality of the resulting free boundary problem is governed by the erdelyi kober operator which requires using other than classical proof techniques to prove the existence and uniqueness of a compactly supported solution we reduce the free boundary case to the initial value problem next we use the sub and supersolution technique to show that there exists a globally defined unique solution as a side product some estimates on the exact solution are found

second order accuracy finite difference methods for space fractional partial differential equations abstract space fractional partial differential equations are used for simulations of for example diffusion of radioactive materials and financial and other models which are characterized by heavy tailed distributions a number of first order accuracy finite difference methods have been proposed in the present paper we introduce second order accuracy finite difference methods with dirichlet boundary conditions these methods have a parameter in these schemes and the parameter stabilizes the schemes this means that there exist various schemes with second order accuracy but the stability of each scheme is different in the present paper we introduce the most stable scheme for any fractional calculus order by choosing the optimal parameter in addition we describe a phenomenon whereby the expected accuracy cannot be obtained if the analytical solution can be expanded to a series having less than second order around boundaries this also happens in both existing methods and the proposed methods in the present paper we develop the stability conditions for the proposed schemes and numerical examples of second order accuracy and accuracy decay are shown

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

does snooping help one of the most efficient way of improving the performance of learning algorithms is snooping i e using some information about the data to be predicted for choosing the parameters of the learning algorithm or the learning algorithm itself allowing different degrees of snooping makes it possible to attain a better performance lossp x of a prediction strategy p on the given data set x we study the snooping curves lx infk p lossp x where k p is the kolmogorov complexity of the prediction strategy p we prove that every non increasing function can be approximated with arbitrary precision by some snooping function lx our framework is that of on line prediction for simplicity we assume that sequences x are binary

bounds for the minimum step number of knots in the simple cubic lattice knots are found in dna as well as in proteins and they have been shown to be good tools for structural analysis of these molecules an important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot here we address this problem by characterizing both analytically and numerically the minimum length also called minimum step number needed to form a particular knot in the simple cubic lattice our analytical work is based on improvement of a method introduced by diao to enumerate conformations of a given knot type for a fixed length this method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 which is 24 to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34 respectively using an independent method based on the bfacf algorithm we provide a complete list of numerical estimates upper bounds of the minimum step numbers for prime knots up to ten crossings which are improvements over current published numerical results we enumerate all minimum lattice knots of a given type and partition them into classes defined by bfacf type 0 moves

the writhe of a self avoiding walk the writhe of a self avoiding walk in a three dimensional space is the average over all projections onto a plane of the sum of the signed crossings we compute this number using a monte carlo simulation our results suggest that the average of the absolute value of the writhe of self avoiding walks increases as nalpha where n is the length of the walks and alpha approximately 0 5 the mean crossing number of walks is also computed and found to have a power law dependence on the length of the walks in addition we consider the effects of solvent quality on the writhe and mean crossing number of walks

computing the writhe on lattices given a polygonal closed curve on a lattice or space group we describe a method for computing the writhe of the curve as the average of weighted projected writhing numbers of the polygon in a few directions these directions are determined by the lattice geometry the weights are determined by areas of regions on the unit 2 sphere and the regions are formed by the tangent indicatrix to the polygonal curve we give a new formula for the writhe of polygons on the face centred cubic lattice and prove that the writhe of polygons on the body centred cubic lattice the hexagonal simple lattice and the diamond space group is always a rational number and discuss applications to ring polymers

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

curvature of random walks and random polygons in confinement the purpose of this paper is to study the curvature of equilateral random walks and polygons that are confined in a sphere curvature is one of several basic geometric properties that can be used to describe random walks and polygons we show that confinement affects curvature quite strongly and in the limit case where the confinement diameter equals the edge length the unconfined expected curvature value doubles from 2 to to study curvature a simple model of an equilateral random walk in spherical confinement in dimensions 2 and 3 is introduced for this simple model we derive explicit integral expressions for the expected value of the total curvature in both dimensions these expressions are functions that depend only on the radius r of the confinement sphere we then show that the values obtained by numeric integration of these expressions agrees with numerical average curvature estimates obtained from simulations of random walks finally we compare the confinement effect on curvature of random walks with random polygons

a topological invariant to predict the three dimensional writhe of ideal configurations of knots and links we present herein a topological invariant of oriented alternating knots and links that predicts the three dimensional 3d writhe of the ideal geometrical configuration of the considered knot link the fact that we can correlate a geometrical property of a given configuration with a topological invariant supports the notion that the ideal configuration contains important information about knots and links the importance of the concept of ideal configuration was already suggested by the good correlation between the 3d writhe of ideal knot configurations and the ensemble average of the 3d writhe of random configurations of the considered knots the values of the new invariant are quantized multiples of 4 7 for links with an odd number of components including knots and 2 7 plus multiples of 4 7 for links with an even number of components

minimum complexity pursuit for universal compressed sensing the nascent field of compressed sensing is founded on the fact that high dimensional signals with simple structure can be recovered accurately from just a small number of randomized samples several specific kinds of structures have been explored in the literature from sparsity and group sparsity to low rankness however two fundamental questions have been left unanswered what are the general abstract meanings of structure and simplicity do there exist universal algorithms for recovering such simple structured objects from fewer samples than their ambient dimension in this paper we address these two questions using algorithmic information theory tools such as the kolmogorov complexity we provide a unified definition of structure and simplicity leveraging this new definition we develop and analyze an abstract algorithm for signal recovery motivated by occam s razor minimum complexity pursuit mcp requires approximately 2 randomized samples to recover a signal of complexity and ambient dimension n we also discuss the performance of the mcp in the presence of measurement noise and with approximately simple signals

symmetrization for fractional neumann problems abstract in this paper we complement the program concerning the application of symmetrization methods to nonlocal pdes by providing new estimates in the sense of mass concentration comparison for solutions to linear fractional elliptic and parabolic pdes with neumann boundary conditions these results are achieved by employing suitable symmetrization arguments to the stinga torrea local extension problems corresponding to the fractional boundary value problems considered sharp estimates are obtained first for elliptic equations and a certain number of consequences in terms of regularity estimates is then established finally a parabolic symmetrization result is covered as an application of the elliptic concentration estimates in the implicit time discretization scheme

a fast ergodic algorithm for generating ensembles of equilateral random polygons knotted structures are commonly found in circular dna and along the backbone of certain proteins in order to properly estimate properties of these three dimensional structures it is often necessary to generate large ensembles of simulated closed chains i e polygons of equal edge lengths such polygons are called equilateral random polygons however finding efficient algorithms that properly sample the space of equilateral random polygons is a difficult problem currently there are no proven algorithms that generate equilateral random polygons with its theoretical distribution in this paper we propose a method that generates equilateral random polygons in a step wise uniform way we prove that this method is ergodic in the sense that any given equilateral random polygon can be generated by this method and we show that the time needed to generate an equilateral random polygon of length n is linear in terms of n these two properties make this algorithm a big improvement over the existing generating methods detailed numerical comparisons of our algorithm with other widely used algorithms are provided

the writhe of a self avoiding polygon we discuss the writhe of a self avoiding polygon on a lattice as a geometrical measure of its entanglement complexity we prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon and use monte carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils

regularizing effects for ut aϑ u 0 in l1 abstract various initial boundary value problems and cauchy problems can be written in the form du dt a u 0 where r r is nondecreasing and a is the linear generator of strongly continuous nonexpansive semigroup e ta in an l1 space for example if a subject perhaps to suitable boundary conditions we obtain equations arising in flow in a porous medium or plasma physics depending on the choice of while if a x acting in l1 r we have a scalar conservation law in this paper we show that if m m 0 and m 2 m 2 where 1 1 then roughly speaking the norm of t du dt may be estimated in terms of the initial data u0 in l1 such estimates give information about the regularity of solutions asymptotic behaviour etc in applications side issues such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator a are also dealt with these considerations are of independent interest

the mean squared writhe of alternating random knot diagrams the writhe of a knot diagram is a simple geometric measure of the complexity of the knot diagram it plays an important role not only in knot theory itself but also in various applications of knot theory to fields such as molecular biology and polymer physics the mean squared writhe of any sample of knot diagrams with n crossings is n when for each diagram at each crossing one of the two strands is chosen as the overpass at random with probability one half however such a diagram is usually not minimal if we restrict ourselves to a minimal knot diagram then the choice of which strand is the over or under strand at each crossing is no longer independent of the neighboring crossings and a larger mean squared writhe is expected for minimal diagrams this paper explores the effect on the correlation between the mean squared writhe and the diagrams imposed by the condition that diagrams are minimal by studying the writhe of classes of reduced alternating knot diagrams we demonstrate that the behavior of the mean squared writhe heavily depends on the underlying space of diagram templates in particular this is true when the sample space contains only diagrams of a special structure when the sample space is large enough to contain not only diagrams of a special type then the mean squared writhe for n crossing diagrams tends to grow linearly with n but at a faster rate than n indicating an intrinsic property of alternating knot diagrams studying the mean squared writhe of alternating random knot diagrams also provides some insight into the properties of the diagram generating methods used which is an important area of study in the applications of random knot theory