potential theory for the sum of the 1 laplacian and p laplacian abstract the variational problem of minimizing the functional u d u 1 p d u p a u on a domain r n under zero boundary values which appears for example in the theory of bingham fluids shows interesting phenomena such as the formation of a subset of with positive measure where the minimizer is constant a plateau the corresponding euler lagrange equation should be 1 u p u a but the 1 laplacian is not defined in points where d u 0 we show that it is nevertheless possible to define subsolutions and supersolutions for this equation we show that the comparison principle is valid and we begin developing a potential theory analogous to what is known about the p laplacian in addition we apply those notions to rotationally invariant solutions and we discuss properties of the plateau and a possible connection to the theory

fractal correlation in heterogeneous systems abstract noisy or irregular signals are usually thought of as being composed of or driven by random processes however some signals which appear noisy such as fractal signals are correlated and the fractal dimensions d provides a measure of the degree of correlation between elements of signals densities intensities of a property voltages etc over space or time to arrive at this correlation methods of analysis using the dispersion of a signal such as the relative dispersion or the hurst rescaled range analysis give measures of the fractal dimension d and of the two point correlation between adjacent segments of the signal we have derived expressions for the two point correlation over an extended range of observations of non adjacent neighbors the result allows estimation of the fractal dimension d from the autocorrelation function directly we use the result as a test for self similarity to determine whether or not the correlation fall off is the same for different sized groupings of the data this means that the estimate of the fractal dimension d and the correlation requires observations at one unit size to estimate the range extended correlation behavior with verification at one other unit size the correlation coefficient between n th neighbor units is r n 0 5 n 1 2 h 2 n 2 h n 1 2 h where the hurst coefficient h 2 d for one dimensional signals asymptotically the ratio of successive values of r n is r n r n 1 n n 1 2 h 2 the extended range fractal correlation technique is therefore both efficient and relatively robust

higher holder regularity for the fractional p laplacian in the superquadratic case abstract we prove higher holder regularity for solutions of equations involving the fractional p laplacian of order s when p 2 and 0 s 1 in particular we provide an explicit holder exponent for solutions of the non homogeneous equation with data in l q and q n s p which is almost sharp whenever s p p 1 n q the result is new already for the homogeneous equation

establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes stochastic fractal signals can be characterized by the hurst coefficient h which is related to the exponents of various power law statistics characteristic of these processes two techniques widely used to estimate h are spectral analysis and detrended fluctuation analysis dfa this paper examines the analytical link between these two measures and shows that they are related through an integral transform numerical simulations confirm this relationship for ideal synthesized fractal signals their performance as estimators of h is compared based on a mean square error criterion and found to be similar dfa measures are derived for physiological signals of heartbeat r ensuremath r intervals through the integral transform of a spectral density estimate these agree with directly calculated dfa estimates indicating that the relationship holds for signals with nonideal fractal properties it is concluded that dfa and spectral measures provide equivalent characterizations of stochastic signals with long term correlation

quaternionic cayley transform revisited abstract a cayley type transform in the context of quaternions previously introduced by the author is revisited a simplified definition is given and shorter proofs are exhibited as new results the range of this transform restricted to a class of unbounded normal operators is described and an application to a moment problem with constraints is given the extension problem to normal operators in this new context is also revisited

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti

regularity of quasi minimizers for non uniformly elliptic integrals abstract in this paper we consider a class of non uniformly elliptic integral functionals and we prove the local boundedness of the quasi minimizers our approach is based on a suitable adaptation of the celebrated de giorgi proof and it relies on an appropriate caccioppoli type inequality

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

blocks adjustment reduction of bias and variance of detrended fluctuation analysis using monte carlo simulation abstract the length of minimal and maximal blocks equally distant on log log scale versus fluctuation function considerably influences bias and variance of dfa through a number of extensive monte carlo simulations and different fractional brownian motion fractional gaussian noise generators we found the pair of minimal and maximal blocks that minimizes the sum of mean squared error of estimated hurst exponents for the series of length n 2 p p 7 15 sensitivity of dfa to sort range correlations was examined using arfima p d q generator due to the bias of the estimator for anti persistent processes we narrowed down the range of hurst exponent to 1 2 h 1

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

nonlocal harnack inequalities abstract we state and prove a general harnack inequality for minimizers of nonlocal possibly degenerate integro differential operators whose model is the fractional p laplacian

local operator algebras fractional positivity and the quantum moment problem in the present paper we introduce quantum measures as a concept of quantum functional analysis and develop the fractional space technique in the quantum or local operator space framework we prove that each local operator algebra or quantum algebra has a fractional space realization this approach allows us to formulate and prove a noncommutative albrechtvasilescu extension theorem which in turn solves the quantum moment problem

interior c0 γ regularity for vector valued minimizers of quasilinear functionals abstract we consider weak minimizers of variational integrals whose integrands are quasilinear under suitable conditions on the integrand we obtain results on interior everywhere h lder regularity for the weak minimizers more precise results on the partial regularity can be deduced from our main result too some illustrating examples are given at the end of the paper

minimizing non convex multiple integrals a density result we consider variational problems of the form where is a bounded open set in r n f r n r is a possibly non convex lower semicontinuous function with p growth at infinity for some 1 p u 0 is any function in w 1 p assuming that the convex envelope f of f is affine on each connected component of the set f f we prove the existence of solutions to p for every continuous function g such that i g has no strict local minima and ii every convergent sequence of extremum points of g eventually belongs to an interval where g is constant thus showing that the set of continuous functions g that yield existence to p is dense in the space of continuous functions on r

local regularity for solutions of anisotropic obstacle problems abstract this paper deals with anisotropic obstacle problems for the a harmonic equation div a x u x 0 a local regularity result is given under suitable assumptions

on the nash point equilibria in the calculus of variations abstract the existence and partial regularity of the nash point equilibria for a pair of multiple integrals j u v f x u v u v dx k u v g x u v u v dx are studied the conditions as well as the results are similar to those for local minima obtained by acerbi fusco and giaquinta giusti

unbounded extensions and operator moment problems abstract extension results expressed in terms of complete boundedness leading to necessary and sufficient conditions for the solvability of power moment problems with unbounded operator data are given as an application necessary and sufficient conditions for the existence of selfadjoint and normal extensions for some classes of commuting tuples of unbounded linear operators are obtained

quaternionic cayley transform abstract in this paper we define a quaternionic cayley transform for some linear operators acting in h h where h is a hilbert space which permits the joint investigation of the pairs of symmetric operators in particular this leads to new criteria for the existence of commuting self adjoint extensions of certain pairs of symmetric operators

evaluating scaled windowed variance methods for estimating the hurst coefficient of time series three scaled windowed variance methods standard linear regression detrended and brdge detrended for estimating the hurst coefficient h are evaluated the hurst coefficient with 0 h 1 characterizes self similar decay in the time series autocorrelation function the scaled windowed variance methods estimate h for fractional brownian motion fbm signals which are cumulative sums of fractional gaussian noise fgn signals for all three methods both the bias and standard deviation of estimates are less than 0 05 for series having n 2 9 points estimates for short series n 2 8 are unreliable to have a 0 95 probability of distinguishing between two signals with true h differing by 0 1 more than 2 15 points are needed all three methods proved more reliable based on bias and variance of estimates than hurst s rescaled range analysis periodogram analysis and autocorrelation analysis and as reliable as dispersional analysis the latter methods can only be applied to fgn or differences of fbm while the scaled windowed variance methods must be applied to fbm or cumulative sums of fgn

rotation invariant moment problems an important theorem of marcel riesz cf 14 states that the polynomials are dense in l2 x when x is a determinate measure on the real line in the indeterminate case riesz also characterized the measures x for which the polynomials are dense in l2 they are the so called nevanlinna extremal measures introduced in nevanlinna 11 it does not seem to be known whether the polynomials are dense in l2 g when x is a determinate measure on r d d 1 cf the expository paper by fuglede 7 as well as the research problems book 8 p 529 where devinatz poses the problem as question 1 and ascribes it to the physicist john challifour 1978 in this paper we shall settle the question in the negative there exist rotation invariant measures on r d d 1 which are determinate but for which the polynomials are not dense in l2 p such measures x are necessarily of the following very special form

stability for the solutions of div u p 2 u f with varying p abstract elliptic partial differential equations with principal part div u p 2 u are applied in physics e g for the description of phenomena in glaceology the objective of our note is a natural question of stability for solutions as p varies

partial regularity of p x harmonic maps let g x and hij u be uniformly elliptic symmetric matrices and assume that hij u and p x 2 are sufficiently smooth we prove partial regularity of minimizers for the functional f u z

existence and boundedness of a minimiser for a constrained minimisation problem on rn with limiting exponent we use the concentration compactness principle to study the existence of a minimiser of the minimisation problem where u u 1 u n we also prove the boundedness of the minimiser of l 1 by using the reverse holder inequality

partial lipschitz regularity of minimizers of asymptotically convex functionals with general growth conditions abstract we consider the partial lipschitz continuity of minimizers of functionals of the form v f x v d v d x where r n is open and bounded the map x u f x u appearing in the above functional is assumed to be holder continuous with respect to its first two arguments of class c 2 in its third argument and asymptotically related to the more regular map x u a x u g as the main novelty is the allowance that even as the map f may retain full dependence on the minimizer u not only its gradient du and the spatial coordinate x

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

on the blow up of heat flow for conformal 3 harmonic maps using a comparison theorem chang ding and ye 1992 proved a finite time derivative blow up for the heat flow of harmonic maps from d 2 a unit ball in r 2 to s 2 a unit sphere in r 3 under certain initial and boundary conditions we generalize this result to the case of 3 harmonic map heat flow from d 3 to s 3 in contrast to the previous case our governing parabolic equation is quasilinear and degenerate technical issues such as the development of a new comparison theorem have to be resolved

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

harnack inequalities for double phase functionals we prove a harnack inequality for minimisers of a class of non autonomous functionals with non standard growth conditions they are characterised by the fact that their energy density switches between two types of different degenerate phases

liouville theorems partial regularity and holder continuity of weak solutions to quasilinear elliptic systems this paper describes the connections between liouville type theorems and interior regularity results for bounded weak solutions of quasilinear elliptic systems with quadratic growth it is shown that equivalence does in general hold only in some restricted sense a complete correspondence can be established in certain cases e g for small solutions and for minima of quadratic variational integrals

removability of singular sets of harmonic maps it is proved that a harmonic map with small energy and the monotonicity property is smooth if its singular set is rectifiable and has a finite uniform density moreover the monotonicity property holds if the singular set has a lower dimension or its gradient has higher integrability this work generalizes the results in cl df lg12 which were proved under the assumption that the singular sets are isolated points or smooth submanifolds

scalar minimizers with fractal singular sets lack of regularity of local minimizers for convex functionals with non standard growth conditions is considered it is shown that for every 0 there exists a function ac such that the functional admits a local minimizer uw1 p whose set of non lebesgue points is a closed set with dim n p and where 1 p n n q

an algorithm for the characterization of time series based on local regularity abstract the pointwise holder exponent describes the local regularity of a function this description is particularly applicable to complex time series representing natural phenomena that exhibit statistical self similarity where the scaling exponent fluctuates from point to point in this work an algorithm is proposed for the estimation of both global and local scaling exponents of time series the technique is based on scaled window variance method and its practical implementation relies on the convolution operation the proposed algorithm is an attractive alternative to existing techniques because of its simplicity and computational efficiency the accuracy of the algorithm is verified by application on synthetic signals with a prescribed time evolution of holder exponents case studies are presented in the fields of fluid dynamics room acoustics and machinery diagnostics

evaluating maximum likelihood estimation methods to determine the hurst coefficient a maximum likelihood estimation method implemented in s plus s mle to estimate the hurst coefficient h is evaluated the hurst coefficient with 0 5 h 1 characterizes long memory time series by quantifying the rate of decay of the autocorrelation function s mle was developed to estimate h for fractionally differenced fd processes however in practice it is difficult to distinguish between fd processes and fractional gaussian noise fgn processes thus the method is evaluated for estimating h for both fd and fgn processes s mle gave biased results of h for fgn processes of any length and for fd processes of lengths less than 210 a modified method is proposed to correct for this bias it gives reliable estimates of h for both fd and fgn processes of length greater than or equal to 211

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

on interior regularity of minimizers of p x energy functionals abstract the aim of the paper is to obtain everywhere c 1 regularity in the interior for a minimizer of the p x energy functional e v g x h i j v d v i x d v j x p x 2 d x defined for maps v r m r n under the so called one sided condition supposing the lipschitz continuity assumption on the exponent p x 2

evenly spacing in detrended fluctuation analysis abstract detrended fluctuation analysis is a widely used method which aims at assessing the level of self similarity in time series this method analyzes the diffusion properties of the signal by computing the linear regression slope in the diffusion plot representing in log log coordinates the relationship between the variability of the signal and the length of the intervals over which this variability is computed we compare in this paper the results obtained with logarithmically spaced and evenly spaced diffusion plots the study shows the substantial benefits of evenly spacing especially in the reduction of the variability of estimation

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

subelliptic harmonic maps we study a nonlinear harmonic map type system of subelliptic pde in particular we solve the dirichlet problem with image contained in a convex ball

on square positive extensions and cubature formulas we consider linear functionals ld defined over pd p r x1 xn and study how to extend ld to a square positive l p r i e to a linear functional l with l p2 0 for all p p we use the connection of square positive functionals to real ideals to orthogonal polynomials and to moment matrices finally we report how such extension techniques are used to construct and analyse cubature formulas

deriving dispersional and scaled windowed variance analyses using the correlation function of discrete fractional gaussian noise methods for estimating the fractal dimension d or the related hurst coefficient h for a one dimensional fractal series include hurst s method of rescaled range analysis spectral analysis dispersional analysis and scaled windowed variance analysis which is related to detrended fluctuation analysis dispersional analysis estimates h by using the variance of the grouped means of discrete fractional gaussian noise series dfgn scaled windowed variance analysis estimates h using the mean of grouped variances of discrete fractional brownian motion dfbm series both dispersional analysis and scaled windowed variance analysis have small bias and variance in their estimates of the hurst coefficient this study demonstrates that both methods derive their accuracy from their strict mathematical relationship to the expected value of the correlation function of dfgn the expected values of the variance of the grouped means for dispersional analysis on dfgn and the mean of the grouped variance for scaled windowed variance analysis on dfbm are calculated an improved formulation for scaled windowed variance analysis is given the expected values using these analyses on the wrong kind of series dispersional analysis on dfbm and scaled windowed variance analysis on dfgn are also calculated

interior w2 p x regularity for elliptic equations of nondivergence form abstract in this paper we consider the following elliptic equations of nondivergence form i j 1 n a i j 2 u x i x j f in r n where the coefficients a i j are real valued bounded measurable functions defined in r n belong to vmo loc see definition 2 5 and satisfy the uniformly elliptic condition under the assumption that p satisfies the log holder continuity condition we will establish the interior w 2 p x regularity for strong solutions to the elliptic equations of nondivergence form

global minimizers for a p ginzburg landau type energy in r 2 given a p 2 we prove existence of global minimizers for a p ginzburg landau type energy over maps on r2 with degree d 1 at infinity for the analogous problem on the half plane we prove existence of a global minimizer when p is close to 2 the key ingredient of our proof is the degree reduction argument that allows us to construct a map of degree d 1 from an arbitrary map of degree d 1 without increasing the p ginzburg landau energy

detrending moving average algorithm a closed form approximation of the scaling law abstract the hurst exponent h of long range correlated series can be estimated by means of the detrending moving average dma method the computational tool on which the algorithm is based is the generalized variance dma 2 1 n n i n n y i y n i 2 with y n i 1 n k 0 n y i k being the average over the moving window n and n the dimension of the stochastic series y i the ability to yield h relies on the property of dma 2 to vary as n 2 h over a wide range of scales e alessio a carbone g castelli v frappietro eur j phys b 27 2002 197 here we give a closed form proof that dma 2 is equivalent to c h n 2 h and provide an explicit expression for c h we furthermore compare the values of c h with those obtained by applying the dma algorithm to artificial self similar signals

some regularity results for p harmonic mappings between riemannian manifolds abstract let m be a c 2 smooth riemannian manifold with boundary and n a complete c 2 smooth riemannian manifold we show that each stationary p harmonic mapping u m n whose image lies in a compact subset of n is locally c 1 for some 0 1 provided that n is simply connected and has non positive sectional curvature we also prove similar results for minimizing p harmonic mappings with image being contained in a regular geodesic ball moreover when m has non negative ricci curvature and n is simply connected with non positive sectional curvature we deduce a gradient estimate for c 1 smooth weakly p harmonic mappings from which follows a liouville type theorem in the same setting

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

structural conditions guaranteeing c1 γ regularity of weak solutions to nonlinear second order elliptic systems abstract the interior c 1 regularity is proved for weak solutions to a class of nonlinear second order elliptic systems satisfying conditions on the modulus of continuity of coefficient s gradients

the regularity of harmonic maps into spheres and applications to bernstein problems we show the regularity of and derive a priori estimates for weakly harmonic maps from a riemannian manifold into a euclidean sphere under the assumption that the image avoids some neighborhood of a half equator the proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems we apply these results to the spherical and euclidean bernstein problems for minimal hypersurfaces obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in euclidean spaces are trivial

analyzing exact fractal time series evaluating dispersional analysis and rescaled range methods precise reference signals are required to evaluate methods for characterizing a fractal time series here we use fgp fractional gaussian process to generate exact fractional gaussian noise fgn reference signals for one dimensional time series the average autocorrelation of multiple realizations of fgn converges to the theoretically expected autocorrelation two methods commonly used to generate fractal time series an approximate spectral synthesis ssm method and the successive random addition sra method do not give the correct correlation structures and should be abandoned time series from fgp were used to test how well several versions of rescaled range analysis r s and dispersional analysis disp estimate the hurst coefficient 0 0 9 r s detrended overestimates h for time series with h 0 7 estimates of h from all versions of disp usually have lower bias and variance than those from r s all versions of dispersional analysis disp now tested on fgp are better than we previously thought and are recommended for evaluating time series as long memory processes

harnack s inequality for quasiminimizers with nonstandard growth conditions we prove harnack inequalities for quasiminimizers of the variable exponent dirichlet energy integral by employing the de giorgi method

mixed states for an allen cahn type equation this paper continues the recent study of an allen cahn model pde 1 by eliminating a strong spatial reversibility condition and by weakening certain nondegeneracy conditions on families of basic heteroclinic solutions enabling us to obtain multibump solutions in a much more general setting as in 1 novel minimization arguments play a key role in finding solutions

vanishing viscosity limit for homogeneous axisymmetric no swirl solutions of stationary navier stokes equations abstract 1 homogeneous axisymmetric no swirl solutions of three dimensional incompressible stationary navier stokes equations which are smooth on the unit sphere minus the north and south poles have been classified in this paper we study the vanishing viscosity limit of sequences of these solutions as the viscosity tends to zero some sequences of solutions c l o c m converge to solutions of euler equations on the sphere minus the poles while for other sequences of solutions transition layer behaviors occur for every latitude circle there are sequences which c l o c m converge respectively to different solutions of the euler equations on the spherical caps above and below the latitude circle we give detailed analysis of these convergence and transition layer behaviors

holder continuity of weak solutions of p laplacian pdes with vmo coefficients abstract we consider solutions u w 1 p r n of the p laplacian pde a x d u p 2 d u 0 for x r n where is open and bounded more generally we consider solutions of the elliptic system a x g a x d u d u d u 0 x as well as minimizers of the functional g a x d u d x in each case the coefficient map a r is only assumed to be of class v m o l which means that it may be discontinuous without assuming that x a x has any weak differentiability we show that u c loc 0 for each 0 1 the preceding results are in fact a corollary of a much more general result which applies to the functional f x u d u d x in case f is only asymptotically convex