a variation embedding theorem and applications fractional sobolev spaces also known as besov or slobodetski spaces arise in many areas of analysis stochastic analysis in particular we prove an embedding into certain q variation spaces applications include a new route to a regularity result by kusuoka for stochastic differential equations integration against besov paths a regularity criterion for rough paths and a new regularity result for cameron martin paths associated to fractional brownian motion

common subexpression algorithms for space complexity reduction of gaussian normal basis multiplication the use of normal bases for representing elements in a binary field is attractive in some applications because it is easy to perform squaring operations in hardware in such cases the costs of implementing the multiplication operation become a primary concern we present new algorithms for reducing the space complexity of gaussian normal basis multipliers over binary fields gf 2 m where m is odd compared with previous results our approach incurs no additional costs in time complexity and achieves improvements in space complexity over a wide range of finite fields and digit sizes for the binary fields specified in the nist fips 186 3 elliptic curve digital signature algorithm standards document our algorithms reduce by 16 respectively 27 the number of xor gates needed for the implementation of a digit level parallel input parallel output multiplier over a 163 bit respectively 409 bit binary field

erratum time reversed dynamical entropy and irreversibility in markovian random processes a concept of time reversed entropy per unit time is introduced in analogy with the entropy per unit time by shannon kolmogorov and sinai this time reversed entropy per unit time characterizes the dynamical randomness of a stochastic process backward in time while the standard entropy per unit time characterizes the dynamical randomness forward in time the difference between the time reversed and standard entropies per unit time is shown to give the entropy production of markovian processes in nonequilibrium steady states

predictive information in a nonequilibrium critical model we propose predictive information that is information between a long past of duration t and the entire infinitely long future of a time series as a general order parameter to study phase transitions in physical systems independently of the underlying dynamics it can be used in particular to study nonequilibrium transitions and other exotic transitions where a simpler order parameter cannot be identified using traditional symmetry arguments as an example we calculate predictive information for a stochastic nonequilibrium dynamics problem that forms an absorbing state under a continuous change of a parameter the information at the transition point diverges as logt and we calculate the expression for a smooth crossover to t0 away from the transition

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

smoothness of densities for area like processes of fractional brownian motion we consider a process given by a n dimensional fractional brownian motion with hurst parameter frac 1 4 h frac 1 2 along with an associated levy area like process and prove the smoothness of the density for this process with respect to lebesgue measure

general rough integration levy rough paths and a levy kintchine type formula we consider rough paths with jumps in particular the analogue of lyons extension theorem and rough integration are established in a jump setting offering a pathwise view on stochastic integration against cadlag processes a class of levy rough paths is introduced and characterized by a sub ellipticity condition on the left invariant diffusion vector fields and a certain integrability property of the carnot caratheodory norm with respect to the levy measure on the group using hunt s framework of lie group valued levy processes examples of levy rough paths include a standard multi dimensional levy process enhanced with a stochastic area as constructed by d williams the pure area poisson process and brownian motion in a magnetic field an explicit formula for the expected signature is given

an algorithm to design finite field multipliers using a self dual normal basis the concept of using a self dual normal basis to design the massey omura finite field multiplier is presented an algorithm is given to locate a self dual normal basis for gf 2 sup m for odd m a method to construct the product function for designing the massey omura multiplier is developed it is shown that the construction of the product function based on a self dual basis is simpler than that based on an arbitrary normal basis

low complexity multiplier architectures for single and hybrid double multiplications in gaussian normal bases the extensive rise in the number of resource constrained wireless devices and the needs for secure communications with the servers imply fast and efficient cryptographic computations for both parties efficient hardware implementation of arithmetic operations over finite field using gaussian normal basis is attractive for public key cryptography as it provides free squarings in this paper we first present two low complexity digit level multiplier architectures it is shown that the proposed multipliers outperform the existing gaussian normal basis gnb multiplier structures available in the literature then for the first time using these two architectures we propose a new digit level hybrid multiplier which performs two successive multiplications with the same latency as the one for one multiplication we have studied the efficiency of the proposed hybrid architecture in terms of area and time delay for different digit sizes the main advantage of this new hybrid architecture is to speed up exponentiation and point multiplication whenever double multiplication is required and the traditional schemes fail due to the data dependencies we have investigated the applicability of the proposed hybrid structure to reduce the latency of exponentiation based cryptosystems our analysis and timing results show that the expected acceleration in double exponentiation is considerable prototypes of the presented low complexity multiplier architectures and the proposed hybrid architecture are implemented and experimental results are presented

smoothness of the density for solutions to gaussian rough differential equations we consider stochastic differential equations driven by a multi dimensional gaussian process under the assumption that the vector fields satisfy hormander s bracket condition we demonstrate that the solution admits a smooth density for any strictly positive time t provided the driving noise satisfies certain non degeneracy assumptions our analysis relies on an interplay of rough path theory malliavin calculus and the theory of gaussian processes our result applies to a broad range of examples including fractional brownian motion with hurst parameter greater than 1 4 the ornstein uhlenbeck process and the brownian bridge returning after time t

fast normal basis multiplication using general purpose processors for cryptographic applications normal bases have received considerable attention especially for hardware implementation we consider fast software algorithms for normal basis multiplication over the extended binary field gf 2 sup m we present a vector level algorithm which essentially eliminates the bit wise inner products needed in the conventional approach to the normal basis multiplication we then present another algorithm which significantly reduces the dynamic instruction counts both algorithms utilize the full width of the data path of the general purpose processor on which the software is to be executed we also consider composite fields and present an algorithm which can provide further speed ups and an added flexibility toward hardware software codesign of processors for very large finite fields

differential equations driven by rough paths with jumps abstract we develop the rough path counterpart of ito stochastic integration and differential equations driven by general semimartingales this significantly enlarges the classes of ito forward stochastic differential equations treatable with pathwise methods a number of applications are discussed

a matrix decomposition method for optimal normal basis multiplication we introduce a matrix decomposition method and prove that multiplication in gf 2 k with a type 1 optimal normal basis for can be performed using k 2 1 xor gates irrespective of the choice of the irreducible polynomial generating the field the previous results achieved this bound only with special irreducible polynomials furthermore the decomposition method performs the multiplication operation using 1 5k k 1 xor gates for type 2a and 2b optimal normal bases which matches previous bounds

densities for rough differential equations under hormander s condition we consider stochastic differential equations dy v y dx driven by a multidimensional gaussian process x in the rough path sense t lyons rev mat iberoamericana 14 1998 215 310 using malliavin calculus we show that y t admits a density for t 0 t provided i the vector fields v v 1 v d satisfy hormander s condition and ii the gaussian driving signal x satisfies certain conditions examples of driving signals include fractional brownian motion with hurst parameter h 1 4 the brownian bridge returning to zero after time t and the ornstein uhlenbeck process

efficient algorithms and architectures for field multiplication using gaussian normal bases recently implementations of normal basis multiplication over the extended binary field gf 2 sup m have received considerable attention a class of low complexity normal bases called gaussian normal bases has been included in a number of standards such as ieee and nist for an elliptic curve digital signature algorithm the multiplication algorithms presented there are slow in software since they rely on bit wise inner product operations in this paper we present two vector level software algorithms which essentially eliminate such bit wise operations for gaussian normal bases our analysis and timing results show that the software implementation of the proposed algorithm is faster than previously reported normal basis multiplication algorithms the proposed algorithm is also more memory efficient compared with its look up table based counterpart moreover two new digit level multiplier architectures are proposed and it is shown that they outperform the existing normal basis multiplier structures as compared with similar digit level normal basis multipliers the proposed multiplier with serial output requires the fewest number of xor gates and the one with parallel output is the fastest multiplier

field theoretic treatment of an effective action for a model of catalyzed autoamplification reaction diffusion models can exhibit continuous phase transitions in behaviors and their dynamics at criticality often exhibit scalings with key parameters that can be characterized by exponents while models with only a single field that transitions between absorbing and nonabsorbing states are well characterized and typically fall in the directed percolation universality class the effects of coupling multiple fields remain poorly understood we recently introduced a model which has three fields one of which relaxes exponentially one of which displays critical behavior and one of which has purely diffusive dynamics but exerts an influence on the critical field tchernookov j chem phys 130 134906 2009 simulations suggested that this model is in a universality class distinct from other reaction diffusion systems studied previously although the three fields give rise to interesting physics they complicate analysis of the model with renormalization group methods here we show how to systematically simplify the action for this model such that analytical expressions for the exponents of this universality class can be obtained by standard means we expect the approach taken here to be of general applicability in reaction diffusion systems with coupled order parameters that display qualitatively different behaviors close to criticality

rough differential equations driven by signals in besov spaces rough differential equations are solved for signals in general besov spaces unifying in particular the known results in holder and p variation topology to this end the paracontrolled distribution approach which has been introduced by gubinelli imkeller and perkowski 24 to analyze singular stochastic pdes is extended from holder to besov spaces as an application we solve stochastic differential equations driven by random functions in besov spaces and gaussian processes in a pathwise sense

functional renormalization group and the field theory of disordered elastic systems we study elastic systems such as interfaces or lattices pinned by quenched disorder to escape triviality as a result of dimensional reduction we use the functional renormalization group difficulties arise in the calculation of the renormalization group functions beyond one loop order even worse observables such as the two point correlation function exhibit the same problem already at one loop order these difficulties are due to the nonanalyticity of the renormalized disorder correlator at zero temperature which is inherent to the physics beyond the larkin length characterized by many metastable states as a result two loop diagrams which involve derivatives of the disorder correlator at the nonanalytic point are naively ambiguous we examine several routes out of this dilemma which lead to a unique renormalizable field theory at two loop order it is also the only theory consistent with the potentiality of the problem the beta function differs from previous work and the one at depinning by novel anomalous terms for interfaces and random bond disorder we find a roughness exponent zeta 0 208 298 04epsilon 0 006 858epsilon 2 epsilon 4 d for random field disorder we find zeta epsilon 3 and compute universal amplitudes to order o epsilon 2 for periodic systems we evaluate the universal amplitude of the two point function we also clarify the dependence of universal amplitudes on the boundary conditions at large scale all predictions are in good agreement with numerical and exact results and are an improvement over one loop finally we calculate higher correlation functions which turn out to be equivalent to those at depinning to leading order in epsilon

complexity through nonextensivity abstract the problem of defining and studying complexity of a time series has interested people for years in the context of dynamical systems grassberger has suggested that a slow approach of the entropy to its extensive asymptotic limit is a sign of complexity we investigate this idea further by information theoretic and statistical mechanics techniques and show that these arguments can be made precise and that they generalize many previous approaches to complexity in particular unifying ideas from the physics literature with ideas from learning and coding theory there are even connections of this statistical approach to algorithmic or kolmogorov complexity moreover a set of simple axioms similar to those used by shannon in his development of information theory allows us to prove that the divergent part of the subextensive component of the entropy is a unique complexity measure we classify time series by their complexities and demonstrate that beyond the logarithmic complexity classes widely anticipated in the literature there are qualitatively more complex power law classes which deserve more attention

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

malliavin calculus for regularity structures the case of gpam malliavin calculus is implemented in the context of hairer 2014 16 this involves some constructions of independent interest notably an extension of the structure which accommodates a robust and purely deterministic translation operator in l2l2 directions between models in the concrete context of the generalized parabolic anderson model in 2d one of the singular spdes discussed in the afore mentioned article we establish existence of a density at positive times

the jain monrad criterion for rough paths and applications to random fourier series and non markovian hormander theory we discuss stochastic calculus for large classes of gaussian processes based on rough path analysis our key condition is a covariance measure structure combined with a classical criterion due to jain and monrad ann probab 11 1983 46 57 this condition is verified in many examples even in absence of explicit expressions for the covariance or volterra kernels of special interest are random fourier series with covariance given as fourier series itself and we formulate conditions directly in terms of the fourier coefficients we also establish convergence and rates of convergence in rough path metrics of approximations to such random fourier series an application to spde is given our criterion also leads to an embedding result for cameron martin paths and complementary young regularity cyr of the cameron martin space and gaussian sample paths cyr is known to imply malliavin regularity and also ito like probabilistic estimates for stochastic integrals resp stochastic differential equations despite their rough pathwise construction at last we give an application in the context of non markovian hormander theory

non equilibrium scaling in the schlogl model from the reaction diffusion master equation formulation of the schlogl model the authors develop a comprehensive description of the non equilibrium dynamics using the poisson transformation and the dynamical renormalisation group they firstly show how the equilibrium limit is recovered and secondly describe the low frequency low momenta scaling behaviour of the concentration fluctuations the approach extends simplifies or corrects previous studies and is easily generalised to treat more complicated systems