random attractors for stochastic porous media equations perturbed by space time linear multiplicative noise unique existence of solutions to porous media equations driven by continuous linear multiplicative space time rough signals is proven for initial data in l 1 o on bounded domains o the generation of a continuous order preserving random dynamical system rds on l 1 o and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained the random attractor is shown to be compact and attracting in l o norm uniform l bounds and uniform space time continuity of the solutions is shown general noise including fractional brownian motion for all hurst parameters is treated a pathwise wong zakai result for driving noise given by a continuous semimartingale is obtained for fast diffusion equations driven by continuous linear multiplicative space time rough signals existence of solutions is proven for initial data in l m 1 o

yosida approximations for multivalued stochastic partial differential equations driven by levy noise on a gelfand triple abstract we prove the existence and uniqueness of solutions for a class of multivalued stochastic partial differential equations with maximal monotone drift on banach space driven by multiplicative levy noise we also establish the strong convergence result for solutions of the approximating equations where the maximal monotone drift operator is replaced by its yosida approximation as an application the existence and uniqueness of solutions for multivalued stochastic porous medium equations is obtained

weierstrass functions with random phases consider the function f te x sum n 0 infty b n al g b nx te n where b 1 0 al 1 and g is a non constant 1 periodic lipschitz function the phases te n are chosen independently with respect to the uniform probability measure on 0 1 we prove that with probability one we can choose a sequence of scales delta k searrow 0 such that for every interval i of length abs i delta k the oscillation of f te satisfies osc f te i geq c abs i al moreover the inequality osc f te i geq c abs i al varepsilon is almost surely true at every scale when b is a transcendental number these results can be improved the minoration osc f te i geq c abs i al is true for every choice of the phases te n and at every scale

self organized criticality with and without conservation abstract the self organized sandpile models lose criticality if dissipation is introduced recently christensen et al have shown that dissipative automata based on the burridge knopoff earthquake model exhibit critical behavior criticality is qualitatively different for the cases with and without conservation a new characteristic length appears for the dissipative case which diverges slower than the system size for all dissipative models we have found a characteristic frequency in the power spectrum of the released energy which is absent for the conservative case the exponents describing criticality change continuously as a function of the strength of dissipation and crossover phenomena occur in the vicinity of conservation disorder is irrelevant if conservation is present while it destroys criticality in the dissipative case

random attractors for singular stochastic evolution equations abstract the existence of random attractors for singular stochastic evolution equations see perturbed by general additive noise is proven the drift is only assumed to satisfy the standard assumptions of the variational approach to see with compact embeddings in the gelfand triple and singular coercivity for ergodic monotone contractive random dynamical systems it is proven that the random attractor consists of a single random point in case of real linear multiplicative noise finite time extinction is obtained applications include stochastic generalized fast diffusion equations and stochastic generalized singular p laplace equations perturbed by levy noise with jump measure having finite first and second moments

hausdorff and packing dimension of fibers and graphs of prevalent continuous maps abstract the notions of shyness and prevalence generalize the property of being zero and full haar measure to arbitrary not necessarily locally compact polish groups the main goal of the paper is to answer the following question what can we say about the hausdorff and packing dimension of the fibers of prevalent continuous maps let k be an uncountable compact metric space we prove that a prevalent f c k r d has many fibers with almost maximal hausdorff dimension this generalizes a theorem of dougherty and yields that a prevalent f c k r d has graph of maximal hausdorff dimension generalizing a result of bayart and heurteaux we obtain similar results for the packing dimension we show that for a prevalent f c 0 1 m r d the set of y f 0 1 m for which dim h f 1 y m contains a dense open set having full measure with respect to the occupation measure m f 1 where dim h and m denote the hausdorff dimension and the m dimensional lebesgue measure respectively we also prove an analogous result when 0 1 m is replaced by any self similar set satisfying the open set condition we cannot replace the occupation measure with lebesgue measure in the above statement we show that the functions f c 0 1 for which positively many level sets are singletons form a non shy set in c 0 1 in order to do so we generalize a theorem of antunovic burdzy peres and ruscher as a complementary result we prove that the functions f c 0 1 for which dim h f 1 y 1 for all y min f max f form a non shy set in c 0 1 we also prove sharper results in which large hausdorff dimension is replaced by positive measure with respect to generalized hausdorff measures which answers a problem of fraser and hyde

the stochastic obstacle problem for the harmonic oscillator with damping abstract we study the kolmogorov equation associated with a second order stochastic variational inequality related to the harmonic oscillator

on the hausdorff dimension of some graphs consider the functions wb x b cn 1 bnx on 1 0n n oo where b 1 0 0 such that if b is large enough then the hausdorff dimension of the graph of wb is bounded below by 2a c ln b we also show that if a function f is convex lipschitz of order a then the graph of f has a finite measure with respect to hausdorff s measure in dimension 2 a the convex lipschitz functions of order a include zygmund s class a our analysis shows that the graph of the classical van der waerdentagaki nowhere differentiable function has a finite measure with respect to h t t in 1 t we consider the hausdorff dimension of the graphs of various continuous functions we introduce a new geometric property of a function convex lipschitz of some order and obtain an upper bound on the dimension of a graph with this property in particular our analysis includes functions of the form

fractal random series generated by poisson voronoi tessellations in this paper we construct a new family of random series defined on r d indexed by one scaling parameter and two hurst like exponents the model is close to takagi knopp functions save for the fact that the underlying partitions of r d are not the usual dyadic meshes but random voronoi tessellations generated by poisson point processes this approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and hausdorff dimensions the proof of this main result is based on several new distributional properties of the poisson voronoi tessellation on the one hand an estimate of the oscillations of the function coupled with an application of a frostman type lemma on the other hand finally we introduce two related models and provide in particular a box dimension calculation for a derived deterministic takagi knopp series with hexagonal bases

weak solutions to the stochastic porous media equation via kolmogorov equations the degenerate case a stochastic version of the porous medium equation with coloured noise is studied the corresponding kolmogorov equation is solved in the space l 2 h where is an infinitesimally excessive measure then a weak solution is constructed

stochastic porous media equations and self organized criticality the existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and wiener forcing term is proven the finite time extinction of solutions with high probability is also proven in 1 d the results are relevant for self organized criticality behavior of stochastic nonlinear diffusion equations with critical states

similarity and fractality in the modelling of roughness by a multilevel profile with hierarchical structure some modern scaling and fractal approaches to the characterization of surface roughness are under consideration a new mathematical model of a multilevel profile suitable for flexible simulation of roughness is presented as a development from these approaches the iterative construction of the model is described it is shown that the profile is a hierarchical structure with self affine and fractal features its geometrical properties are studied both analytically and numerically and some of them are proved to be of the renormalization type the profile contour length can be finite or infinite depending on the chosen values of the structural parameters some characteristics including the bearing curve and the upper volume function of the profile and the corresponding prestructures are calculated

on a random scaled porous media equation abstract it is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well posed in l in the fast diffusion case we show existence in l p

spde in hilbert space with locally monotone coefficients the aim of this paper is to extend the usual framework of spde with monotone coecients to include a large class of cases with merely locally monotone coecients this new framework is conceptually not more involved than the classical one but includes many more fundamental examples not included previously thus our main result can be applied to various types of spdes such as stochastic reaction diusion equations stochastic burgers type equation stochastic 2 d navier stokes equation stochastic p laplace equation and stochastic porous media equation with non monotone perturbations

random attractors for a class of stochastic partial differential equations driven by general additive noise abstract the existence of random attractors for a large class of stochastic partial differential equations spde driven by general additive noise is established the main results are applied to various types of spde as e g stochastic reaction diffusion equations the stochastic p laplace equation and stochastic porous media equations besides classical brownian motion we also include space time fractional brownian motion and space time levy noise as admissible random perturbations moreover cases where the attractor consists of a single point are also investigated and bounds for the speed of attraction are obtained

a new fractal dimension the topological hausdorff dimension we introduce a new concept of dimension for metric spaces the so called topological hausdorff dimension it is defined by a very natural combination of the definitions of the topological dimension and the hausdorff dimension the value of the topological hausdorff dimension is always between the topological dimension and the hausdorff dimension in particular this new dimension is a non trivial lower estimate for the hausdorff dimension we examine the basic properties of this new notion of dimension compare it to other well known notions determine its value for some classical fractals such as the sierpinski carpet the von koch snowflake curve kakeya sets the trail of the brownian motion etc as our first application we generalize the celebrated result of chayes chayes and durrett about the phase transition of the connectedness of the limit set of mandelbrot s fractal percolation process they proved that certain curves show up in the limit set when passing a critical probability and we prove that actually thick families of curves show up where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large hausdorff dimension the proof of this is basically a lower estimate of the topological hausdorff dimension of the limit set for the sake of completeness we also give an upper estimate and conclude that in the non trivial cases the topological hausdorff dimension is almost surely strictly below the hausdorff dimension finally as our second application we show that the topological hausdorff dimension is precisely the right notion to describe the hausdorff dimension of the level sets of the generic continuous function in the sense of baire category defined on a compact metric space

a new generalization of the takagi function abstract we consider a one parameter family of functions f t x t on 0 1 and partial derivatives t k f t x with respect to the parameter t each function of the class is defined by a certain pair of two square matrices of order two the class includes the lebesgue singular functions and other singular functions our approach to the takagi function is similar to hata and yamaguti the class of partial derivatives t k f t x includes the original takagi function and some generalizations we consider real analytic properties of t k f t x as a function of x specifically differentiability the hausdorff dimension of the graph the asymptotics around dyadic rationals variation a question of local monotonicity and a local modulus of continuity our results are extensions of some results for the original takagi function and some generalizations

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise in this work we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation rm d x t rm div left frac nabla x t nabla x t right rm d t x t rm d w t rm in 0 infty times mathcal o where mathcal o is a bounded and open domain in mathbb r n n geqq 1 and w t is a wiener process of the form w t sum infty k 1 mu k e k beta k t e k in c 2 overline mathcal o cap h 1 0 mathcal o and beta k k in mathbb n are independent brownian motions this is a stochastic diffusion equation with a highly singular diffusivity term one main result established here is that for all initial conditions in l 2 mathcal o it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality thus one obtains a stochastic version of the minimal total variation flow the new approach developed here also allows us to prove the finite time extinction of solutions in dimensions 1 leqq n leqq3 which is another main result of this work

a trotter type result for the stochastic porous media equations abstract this paper proves the continuous dependence with respect to diffusivity of the solutions to stochastic porous media equations with noncoercive monotone diffusivity functions and multiplicative noise

self organization and anomalous diffusion abstract the self organizing process is investigated in the continuum limit of the cellular automaton model introduced by bak tang and wiesenfeld an anomalous diffusion equation is proposed for the description of this process and an analytical method for the solutionis presented in one dimension based on an adiabatic approximation

takagi functions and approximate midconvexity abstract let v be a convex subset of a normed space and let e 0 p 0 be given constants a function f v r is called e p midconvex if f x y 2 f x f y 2 e x y p for all x y v we consider the case p 1 2 and investigate the relations between continuous e p midconvex functions and takagi like functions given by p x k 0 2 2 k p dist 2 k x z for x r it occurs that functions p are optimal 1 p midconvex functions this gives us sharp estimations for continuous e p midconvex functions we also compute the maximum of the function p for a certain set of parameter values

escape rate for 2 dimensional brownian motion conditioned to be transient with application to zygmund functions the escape rate of a 2 dimensional brownian motion conditioned to be transient is determined to be p x t f t i o as t t oo 0 or 1 according as ezre n log f ee or oo the result is used to construct a complex valued zygmund function as a lacunary series whose graph does not have a finite linear hausdorff measure this contrasts the result of mauldin and williams that the graphs of all real valued zygmund functions have a finite linear hausdorff measure

strong solutions for stochastic partial differential equations of gradient type unique existence of analytically strong solutions to stochastic partial differential equations spde with drift given by the subdifferential of a quasi convex function and with general multiplicative noise is proven the proof applies a genuinely new method of weighted galerkin approximations based on the distance defined by the quasi convex function spatial regularization of the initial condition analogous to the deterministic case is obtained the results yield a unified framework which is applied to stochastic generalized porous media equations stochastic generalized reaction diffusion equations and stochastic generalized degenerated p laplace equations in particular higher regularity for solutions of such spde is obtained

finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise abstract in this paper one surveys and sharpens some recent results obtained by the authors on the finite time extinction property of solutions to stochastic diffusion equations of the form d x x m d t x d w and d x sgn x d t x d w where 0 m 1 0 these equations arise as models for nonlinear diffusion processes in porous media plasma and self organized criticality under stochastic gaussian perturbation these equations can be also viewed as control systems governed by fast diffusion porous media equations with a stochastic feedback controller x d w

probabilistic representation for solutions of an irregular porous media type equation the degenerate case summary we consider a possibly degenerate porous media type equation over all of r d with d 1 with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion this equation is motivated by some singular behaviour arising in complex self organized critical systems the main idea consists in approximating the equation by equations with monotone non degenerate coefficients and deriving some new analytical properties of the solution

stochastic porous media equations and self organized criticality convergence to the critical state in all dimensions if x x t is the solution to the stochastic porous media equation in mathcal o subset mathbf r d 1 le d le 3 modelling the self organized criticality barbu et al in commun math phys 285 901 923 2009 and x c is the critical state then it is proved that int infty 0m mathcal o setminus mathcal o t 0 dt infty mathbb p hbox a s and lim t to infty int mathcal o x t x c d xi ell infty mathbb p hbox a s here m is the lebesgue measure and mathcal o t c is the critical region xi in mathcal o x t xi x c xi and x c x 0 a e xi in mathcal o if the stochastic gaussian perturbation has only finitely many modes but is still function valued lim t to infty int k x t x c d xi 0 exponentially fast for all compact k subset mathcal o with probability one if the noise is sufficiently strong we also recover that in the deterministic case l 0

existence of strong solutions for stochastic porous media equation under general monotonicity conditions this paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dx x dt b x dw t in bounded domains of r d with dirichlet boundary conditions here is a maximal monotone graph in r r possibly multivalued with

dynamical properties of the zhang model of self organized criticality critical exponents of the infinitely slowly driven zhang model of self organized criticality are computed for d52 and 3 with particular emphasis devoted to the various roughening exponents besides confirming recent estimates of some exponents new quantities are monitored and their critical exponents computed among other results it is shown that the three dimensional exponents do not coincide with the bak tang wiesenfeld phys rev lett 59 381 1987 phys rev a 38 364 1988 abelian model and that the dynamical exponent as computed from the correlation length and from the roughness of the energy profile do not necessarily coincide as is usually implicitly assumed an explanation for this is provided the possibility of comparing these results with those obtained from renormalization group arguments is also briefly addressed s1063 651x 98 06107 8 pacs number s 64 60 lx

fractal dimensions and singularities of the weierstrass type functions a new type of fractal measures k s 1 s 2 defined on the subsets of the graph of a continuous function is introduced the k dimension defined by this measure is closer to the hausdorff dimension than the other fractal dimensions in recent literatures for the weierstrass type functions defined by w x 0 i g i x where 1 0 1 and g is an almost periodic lipschitz function of order greater than it is shown that the k dimension of the graph of w equals to 2 this conclusion is also equivalent to certain rate of the local oscillation of the function some problems on the knot points and the nondifferentiability of w are also discussed

stochastic nonlinear schrodinger equations abstract this paper is devoted to the well posedness of stochastic nonlinear schrodinger equations in the energy space h 1 r d and is a continuation of our recent work barbu et al 2014 we allow general complex coefficients in the noise thus covering both the conservative and non conservative case the latter being important to include quantum measurement effects in the underlying physical model we consider both focusing and defocusing nonlinearities and prove global well posedness in h 1 r d including also the pathwise continuous dependence on initial conditions with exponents of the nonlinearity in exactly the same sharp range as in the deterministic case in particular in the special conservative case this work improves earlier results in de bouard and debussche 2003 where the critical exponents could not be reached for d 6 in the defocusing and for d 4 in the focusing case respectively moreover the local existence uniqueness and blowup alternative are also established for the energy critical case the approach presented here is mainly based on the rescaling approach already used in barbu et al 2014 to study the l 2 case and also on the strichartz estimates established in marzuola et al 2008 for large perturbations of the laplacian

robust stability under structured and unstructured perturbations the problem of robust stability for linear time invariant single output control systems subject to both structured parametric and unstructured h sub infinity perturbations is studied a generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time invariant dynamic system under perturbations of mixed type is presented the solution involves calculating the h sub infinity norm of a finite number of extremal plants the problem of calculating the exact structured and unstructured stability margins is then constructively solved a feedback control system containing a linear time invariant plant which is subject to both structured and unstructured perturbations is considered the case where the system to be controlled is interval is treated and a nonconservative easily verifiable necessary and sufficient condition for robust stability is given the solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered

the limited rademacher functions and bernoulli convolutions associated with pisot numbers abstract in this paper we give a systematical study of the local structures and fractal indices of the limited rademacher functions and bernoulli convolutions associated with pisot numbers for a given pisot number in the interval 1 2 we construct a finite family of non negative matrices maybe non square such that the corresponding fractal indices can be re expressed as some limits in terms of products of these non negative matrices we are especially interested in the case that the associated pisot number is a simple pisot number i e the unique positive root of the polynomial x k x k 1 x 1 k 2 3 in this case the corresponding products of matrices can be decomposed into the products of scalars based on which the precise formulas of fractal indices as well as the multifractal formalism are obtained

cantor boundary behavior of analytic functions let mathbb d be the open unit disc and let partial mathbb d be the boundary of mathbb d for f z analytic in mathbb d and continuous on overline mathbb d it follows from the open mapping theorem that partial f mathbb d subset f partial mathbb d these two sets have very rich and intrigue geometric properties when f z is univalent then they are equal and there is a large literature to study their boundary behaviors our interest is on the class of analytic functions f z for which the image curves f partial mathbb d form infinitely many loops everywhere they are not univalent of course we formulate this as the cantor boundary behavior we give sufficient conditions for such property making use of the distribution of the zeros of f and the mean growth rate of f examples includes the complex weierstrass functions and the cauchy transform of the canonical hausdorff measure on the sierpiski gasket

the graph and range singularity spectra of b adic independent cascade functions abstract with the iso holder sets of a function we naturally associate subsets of the graph and the range of the function we compute the hausdorff dimension of these subsets for a class of statistically self similar multifractal functions namely the b adic independent cascade functions

stability conditions for polytopes of polynomials in considering the stability or performance of linear systems with uncertain parameters one is led to consider the simultaneous stability of families of polynomials for a general stability region in the complex plane the author gives a stability criterion for polytopes of polynomials in terms of the stability of a minimal number of corners and edges of the polytope the testing set of edges and corners depends entirely on the edge directions of the polytope hence the results are particularly applicable to problems with fixed directions but varying lengths and positions of the edges applications to robust pole placement of uncertain systems and computation of worst case h sub infinity norms are demonstrated

on the dimension of the graph of the classical weierstrass function abstract this paper examines dimension of the graph of the famous weierstrass non differentiable function w b x n 0 n cos 2 b n x for an integer b 2 and 1 b 1 we prove that for every b there exists explicitly given b 1 b 1 such that the hausdorff dimension of the graph of w b is equal to d 2 log log b for every b 1 we also show that the dimension is equal to d for almost every on some larger interval this partially solves a well known thirty year old conjecture furthermore we prove that the hausdorff dimension of the graph of the function f x n 0 n b n x for an integer b 2 and 1 b 1 is equal to d for a typical z periodic c 3 function

graphs of continuous functions and packing dimension abstract we prove that for a typical continuous function f c x over an uncountable compact metric space x the packing dimension of its graph g f x f x x x is dim p x 1 we also consider decompositions of functions in c 0 1 in terms of upper box dimension as well as packing dimension which are quite different from the case of hausdorff dimension

a remark on decomposition of continuous functions abstract in this note we show that given a real 1 2 any real valued continuous function on the unit interval can be decomposed as a sum of two real valued continuous functions whose graphs have hausdorff dimension which answers affirmatively a question raised by bayart and heurteaux 2012 2

fractional derivatives of weierstrass type functions for a special type of fractional differentiation formulas for the weierstrass and the weierstrass mandelbrot functions are shown for integer valued parameters a well conditioned numerical procedure for computing the derivatives in the mean of these functions is derived and used to compute some values

some discrete time counterparts to kharitonov s stability criterion for uncertain systems in 1 kharitonov gave an elegant and simple stability criterion for continuous time systems this note reports on similar results for discrete time systems

finite time extinction for stochastic sign fast diffusion and self organized criticality we prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space time noise corresponding to the bak tang wiesenfeld model for self organized criticality this solves a problem posed and left open in several works barbu methods appl sci 36 1726 1733 2013 rockner and wang j lond math soc 2 87 545 560 2013 barbu et al j math anal appl 389 147 164 2012 barbu and rockner comm math phys 311 539 555 2012 barbu et al comm math phys 285 901 923 2009 c r math acad sci paris 347 1 2 81 84 2009 the highly singular degenerate nature of the drift in interplay with the stochastic perturbation causes the need for new methods in the analysis of mass diffusion and several new estimates and techniques are introduced

hurwitz testing sets for parallel polytopes of polynomials in considering robustness of linear systems with uncertain paramenters one is lead to consider simultaneous stability of families of polynomials efficient hurwitz stability tests for polytopes of polynomials have earlier been developed using evaluations on the imaginary axis this paper gives a stability criterion for parallel polytopes in terms of hurwitz stability of a number of corners and edges the testing set of edges and corners depends entirely on the edge directions of the polytope hence the results are particularly applicable in simultaneous analysis of several polytopes with equal edge directions it follows as a consequence that kharitonov s four polynomial test for independent coefficient uncertainties is replaced by a test of 2q polynomials when the stability region is a sector eiv 0 r q v and r q is a rational number

a convex parameterization of robustly stabilizing controllers this paper treats synthesis of robust controllers for linear time invariant systems uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial the main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations with the new parameterization certain robust performance problems can be stated in terms of quasi convex optimization

existence and uniqueness of the solution for stochastic super fast diffusion equations with multiplicative noise abstract in this paper we prove an existence and uniqueness result for the stochastic porous media equation with very singular diffusion and multiplicative noise by using monotonicity techniques the multiplicative gaussian noise is essential in the proof of existence

the two phase stochastic stefan problem this work is concerned with existence for a stochastic free boundary problem arising in phase transition the stefan two phase problem the existence of an invariant ergodic measure associated with the corresponding transition semigroup p t is also proved together with an integration by parts formula for the generator of p t

the hausdorff dimension of the level sets of takagi s function abstract recently maddock 2006 12 has conjectured that the hausdorff dimension of each level set of takagi s function is at most 1 2 we prove this conjecture using the self affinity of the function of takagi and the existing relationship between the hausdorff and box counting dimensions

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients

on the box dimensions of graphs of typical continuous functions abstract let x r be a bounded set we emphasize that we are not assuming that x is compact or borel we prove that for a typical in the sense of baire uniformly continuous function f on x the lower box dimension of the graph of f is as small as possible and the upper box dimension of the graph of f is as big as possible we also prove a local version of this result namely we prove that for a typical uniformly continuous function f on x the lower local box dimension of the graph of f at all points x x is as small as possible and the upper local box dimension of the graph of f at all points x x is as big as possible