rearrangements of functions let g be measurable non negative functions on r and let tf be their equimeasurable symmetric decreasing rearrangements let f r r r be continuous and suppose that the associated rectangle function defined by f r f a c f b d f a d f b c for r x y r2 a x b c y d is non negative then f g d f tf g d where is lebesgue measure the concept of equimeasurable rearrangement is also defined for functions on a more general class of measure spaces and the inequality holds in the general case if f x y x y where is convex and 0 0 then we obtain tf g d g d in particular if x x p 1 p then we find that the operator s tf is a contraction on lp for 1 p

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

the affine polya szego principle equality cases and stability a brothers ziemer type theorem for the affine polya szego principle and a quantitative affine polya szego principle are established

regularized iterative image restoration with ringing reduction linear space invariant image restoration algorithms often introduce ringing effects near sharp intensity transitions it is shown that these artifacts are attributable to the regularization of the ill posed image restoration problem two possible methods to reduce the ringing effects in restored images are proposed the first method incorporates deterministic a priori knowledge about the original image into the restoration algorithm the second method locally regulates the severity of the noise magnification and the ringing phenomenon depending on the edge information in the image a regularized iterative image restoration algorithm is proposed in which both ringing reduction methods are included by making use of the theory of the projections onto convex sets and the concept of norms in a weighted hilbert space both the numerical performance and the visual evaluation of the results are improved by the use of ringing reduction

kernel synthesis for generalized time frequency distributions using the method of alternating projections onto convex sets cohen s generalized time frequency distribution gtfr requires the choice of a two dimensional kernel the kernel directly affects many performance attributes of the gtfr such as time resolution frequency resolution realness and conformity to time and frequency marginals a number of different kernels may suffice for a given performance constraint high frequency resolution for example interestingly most sets of kernels satisfying commonly used performance constraints are convex we describe a method whereby kernels can be designed that satisfy two or more of these constraints if there exists a nonempty intersection among the constraint sets then the theory of alternating projection onto convex sets pocs guarantees convergence to a kernel that satisfies all of the constraints if the constraints can be partitioned into two sets each with a nonempty intersection then pocs guarantees convergence to a kernel that satisfies the inconsistent constraints with minimum mean square error we apply kernels synthesized using pocs to the generation of some example gtfrs and compare their performance to the spectrogram wigner distribution and cone kernel gtfr

stability of polya szegő inequality for log concave functions abstract a quantitative version of polya szeg inequality is proven for log concave functions in the case of steiner and schwarz rearrangements

singular trudinger moser inequality and fractional p laplace equations in rn abstract in this paper we establish a singular trudinger moser inequality in r n applying that result we study the existence of solution to n s fractional laplacian equation l n s s u x v x u n s 2 u f x u x where 0 n 0 s 1 l n s s is a nonlocal fractional operator with singular kernel k r n 0 r f is a continuous function on r n r which does not satisfy the ambrosetti rabinowitz condition by using mountain pass theorem we obtain the existence of solution to above equation furthermore when f satisfies the ambrosetti rabinowitz condition we obtain the existence to solution of that equation with negative energy in our best knowledge it is the first time the existence to solution of above equation is studied

affine orlicz polya szego principle for log concave functions abstract the affine l p polya szego principle significantly strengthens the classical polya szego principle it is an open problem whether there exists an affine orlicz polya szego principle which includes the affine l p polya szego principle as special case in this paper an affine orlicz polya szego principle for log concave functions is established by using functional steiner symmetrizations

symmetry of constrained minimizers of the cahn hilliard energy on the torus abstract we establish sufficient conditions for a function on the torus to be equal to its steiner symmetrization and apply the result to so called volume constrained minimizers of the cahn hilliard energy the resulting connectedness of superlevel sets is used in two dimensions together with the bonnesen inequality to quantitatively estimate the sphericity of minimizers we also show how two point rearrangements can be used to give an alternate proof of symmetry for the constrained minimizers of the cahn hilliard model

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

combined lp and quasi newton methods for minimax optimization we present an algorithm for minimax optimization that combines lp methods and quasi newton methods the quasi newton algorithm is used only if an irregular solution is detected in which case second order derivative information is needed in order to obtain a fast final rate of convergence we prove that the algorithm can converge only to a stationary point and that normally the final rate of convergence will be either quadratic or superlinear the performance is illustrated through some numerical examples

optimal sobolev imbeddings involving rearrangement invariant quasinorms let m and n be positive integers with n 2 and 1 m n 1 we study rearrangement invariant quasinorms r and d on functions f 0 1 view the mathml source such that to each bounded domain in view the mathml sourcen with lebesgue measure there corresponds c c 0 for which one has the sobolev imbedding inequality r u t c d mu t u cm0 involving the nonincreasing rearrangements of u and a certain mth order gradient of u when m 1 we deal in fact with a closely related imbedding inequality of talenti in which d need not be rearrangement invariant r u t c d d dt x view the mathml sourcen u x u t u x dx u c10 in both cases we are especially interested in when the quasinorms are optimal in the sense that r cannot be replaced by an essentially larger quasinorm and d cannot be replaced by an essentially smaller one our results yield best possible refinements of such limiting sobolev inequalities as those of trudinger strichartz hansson brezis and wainger

on the uniqueness and monotonicity of energy minimisers in the homotopy classes of incompressible mappings and related problems the goal of this paper is to prove the existence and uniqueness of the so called energy minimisers in homotopy classes for the variational energy integral f u x z x f x 2 u 2 u 2 2 dx with f c 0 of class c 2 and satisfying suitable conditions and u lying in the sobolev space of weakly differentiable incompressible mappings of a finite open symmetric plane annulus x onto itself specifically lying in a x u w 1 2 x r 2 det u 1 a e in x and u x on x it is well known that the space a x admits a countably infinite homotopy class decomposition a x s ak with k z we prove that the energy integral f has a unique minimiser in each of these homotopy classes ak furthermore we show that each minimiser is a homeomorphic monotone radially symmetric twist mapping of class c 3 x x or as smooth as f allows thereafter whilst also being a local minimiser of f over a x with respect to the l 1 metric to our best knowledge this is the first uniqueness result for minimisers in homotopy classes in the context of incompressible mappings

on the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems it is the purpose of this paper to discuss the equality sign in d and in related inequalities in particular we shall give an answer to the following question does the equality in d imply that u u this question has been raised by g polya g szego in their book on isoperimetric inequalities l l p 186 they state that their method of proving d unfortunately leaves little hope for discussing the sign of equality another comment on this question can be found in 8 where e lieu conjectures that probably a strict version of d is true as it turns out the doubts of g polya g szeg were indeed justified because there are functions u for which equality holds in d while u does not coincide with u in section 1 of this paper i give an example of such a function on the other hand the conjecture of e lieb is also reasonable since generically i can give a positive answer to the question above there is a dense subset of nonnegative functions in wol 2 1 1 for which the equality in d implies

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 spectral element approach for the stability of delay systems we describe a spectral element approach to study the stability and equilibria solutions of delay differential equations ddes in contrast to the prototypical temporal finite element analysis tfea the described spectral element approach admits spectral rates of convergence and allows exploiting hp convergence schemes the described approach also avoids the limitations of analytical integrations in tfea by using highly accurate numerical quadratures enabling the study of more complicated ddes the effectiveness of this new approach is compared with well established methods in the literature using various case studies specifically the stability results are compared with the conventional tfea and legendre collocation methods whereas the equilibria solutions are compared with the numerical simulations and the homotopy perturbation method hpm solutions our results reveal that the presented approach can have higher rates of convergence than both collocation methods and the hpm copyright 2011 john wiley sons ltd

constructing membership functions using statistical data abstract determination of the membership functions is vital in practical applications of the fuzzy set theory this paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function pdf in the universe of discourse the method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf it is show that for any pdf the method is capable of generating membership functions in accordance with the possibility probability consistency principle membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented

second order γ limit for the cahn hilliard functional the goal of this paper is to solve a long standing open problem namely the asymptotic development of order 2 by convergence of the mass constrained cahn hilliard functional

one dimensional unstable eigenfunction and manifold computations in delay differential equations in this paper we present a new numerical technique for computing the unstable eigenfunctions of a saddle periodic orbit in a delay differential equation this is used to obtain the necessary starting data for an established algorithm for computing one dimensional 1d unstable manifolds of an associated saddle fixed point of a suitable poincare map to illustrate our method we investigate an intermittent transition to chaos in a delay system describing a semiconductor laser subject to phase conjugate feedback

limited angle reconstruction using stabilized algorithms the stability of solutions to the limited angle tomography reconstruction problem obtained by using the projections onto convex sets pocs technique are examined although pocs techniques provide a feasible solution to the reconstruction problem the solution is only one sample from the intersection of the closed convex sets that define the solution space a method for evaluating the ensemble of possible solution waveforms that are in the neighborhood of a solution is presented the ensemble characteristics are used to construct an inverse filter which is then applied to the computed solution the results obtained using this method are less sensitive to noise amplification and are less dependent on both starting data and the number of iterations an estimate of the object dependent extrapolation that is possible using either linear or nonlinear constraints is provided

concentration phenomena for a class of fractional kirchhoff equations in rn with general nonlinearities abstract in this paper we study the following class of fractional kirchhoff problems e 2 s m e 2 s n u s 2 s u v x u f u in r n u h s r n u 0 in r n where e 0 is a small parameter s 0 1 n 2 s is the fractional laplacian v r n r is a positive continuous function m 0 r is a kirchhoff function satisfying suitable conditions and f r r fulfills berestycki lions type assumptions of subcritical or critical type using suitable variational arguments we prove the existence of a family of positive solutions u e which concentrates at a local minimum of v as e 0

flute like musical instruments a toy model investigated through numerical continuation abstract self sustained musical instruments bowed string woodwind and brass instruments can be modelled by nonlinear lumped dynamical systems among these instruments flutes and flue organ pipes present the particularity to be modelled as a delay dynamical system in this paper such a system a toy model of flute like instruments is studied using numerical continuation equilibrium and periodic solutions are explored with respect to the blowing pressure with focus on amplitude and frequency evolutions along the different solution branches as well as jumps between periodic solution branches the influence of a second model parameter namely the inharmonicity on the behaviour of the system is addressed it is shown that harmonicity plays a key role in the presence of hysteresis or quasiperiodic regime throughout the paper experimental results on a real instrument are presented to illustrate various phenomena and allow some qualitative comparisons with numerical results

a polya szego inequality in a fermi metric abstract let n k n with n 2 and 1 k n given a positive function c r n k we form the riemannian metric g on r n associated to the differential expression d s 2 d x 2 x 2 d y 2 where we write r n x x y with x r n k and y r k let be a log convex measure on r k with smooth density and the product measure l n k on r n where c r n k is a positive function we obtain a polya szego inequality of the form r n f u j g u d r n f u s j g u s d for sobolev functions u where the operation s refers to the k n steiner symmetrisation with respect to the gradient operator g is associated to the metric g and the mapping j may be seen as interpolating between the tangent space at x and r n the nonnegative integrand f is continuous and convex in the gradient variable and satisfies some additional hypotheses as an application we derive a polya szego inequality in the hyperbolic plane that takes the above form

stability and convergence of the two parameter cubic spline collocation method for delay differential equations in this paper we propose the cubic spline collocation method with two parameters for solving delay differential equations ddes some results of the local truncation error and the convergence of the spline collocation method are given we also obtain some results of the linear stability and the nonlinear stability of the method for ddes in particular we design an algorithm to obtain the ranges of the two parameters a b which are necessary for the p stability of the collocation method some illustrative examples successfully verify our theoretical results

the feasible solution in signal restoration the feasible solution to the signal restoration problem is defined as the one which satisfies all constraints which can be imposed on the true solution a very important set of constraints can be obtained by examining the statistics of the noise these and other constraints can be described as closed convex sets thus projection onto closed convex sets is the numerical method used to obtain a feasible solution examples of this method demonstrate its usefulness in one and two dimensional signal restoration the limitations of the method are discussed

continuation of bifurcations in periodic delay differential equations using characteristic matrices in this paper we describe a method for continuing periodic solution bifurcations in periodic delay differential equations first the notion of characteristic matrices of periodic orbits is introduced and equivalence with the monodromy operator is demonstrated an alternative formulation of the characteristic matrix is given which can be computed efficiently defining systems of bifurcations are constructed in a standard way including the characteristic matrix and its derivatives for following bifurcation curves in two parameters the pseudo arclength method is used combined with newton iteration two test examples an interrupted machining model and a traffic model with driver reaction time conclude the paper the algorithm has been implemented in the software tool pdde cont

sobolev and hardy littlewood sobolev inequalities this paper is devoted to improvements of sobolev and onofri inequalities the additional terms involve the dual counterparts i e hardy littlewood sobolev type inequalities we focus our attention on optimal constants that can be achieved either by completion of the square methods or by using nonlinear flows and provide various new estimates

digital signal restoration using fuzzy sets a new signal restoration method with considerable flexibility in incorporating a priori information is developed the method defines a fuzzy set for each piece of information to restrict the set of acceptable solutions using fuzzy sets makes it possible to model partially defined information as well as exact knowledge the intersection of all the fuzzy sets is the feasibility set the original signal is a member of this set with a high membership value and any high membership valued element of this set is a nonrejectable solution such solutions can be computed by using optimization techniques ideally the feasibility set contains only the original signal the chance of recovering the original signal decreases as the feasibility set gets larger thus the size of the feasibility set gives a quality measure for the solution the method generated successful results in many restorations for which the conventional techniques have failed and may be applied in image coding and tomography

numerical bifurcation analysis of immunological models with time delays in recent years a large number of mathematical models that are described by delay differential equations ddes have appeared in the life sciences to analyze the models dynamics numerical methods are necessary since analytical studies can only give limited results in turn the availability of efficient numerical methods and software packages encourages the use of time delays in mathematical modelling which may lead to more realistic models we outline recently developed numerical methods for bifurcation analysis of ddes and illustrate the use of these methods in the analysis of a mathematical model of human hepatitis b virus infection

the perimeter inequality under steiner symmetrization cases of equality steiner symmetrization is known not to increase perimeter of sets in rn the sets whose perimeter is preserved under this symmetrization are characterized in the present paper

symmetry of solutions to the optimal exit time control problem in this paper we study the solutions to the optimal exit time control problem such a problem tries to find the state feedback control law with a fixed cost that can keep the state of a randomly perturbed system inside a subset of the state space called the safe set for as long as possible on average by formulating the problem as an optimization problem with pde constraints and using symmetrization techniques we show that when the safe set is a ball the optimal feedback control if it exists must be radially symmetric furthermore we show that among all safe sets with a fixed volume the ball is the best in that it yields the most efficient optimal exit time control the proofs make essential use of the general isoperimetric inequality

contribution to application of energy measure of fuzzy sets abstract the works of de luca termini continued by for example knopfmacher loo and gottwald are the most important on the topic of determination of measures of fuzzy sets the matter is to evaluate how fuzzy a fuzzy set is there are two general concepts of measures of fuzzy set i e entropy and energy measures we show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations applications of the use of energy measure discussed in detail include decision making fuzzy process control and prediction in fuzzy systems

solution of time varying delay systems using an adaptive collocation method abstract an adaptive legendre gauss radau collocation method is presented for solving time varying delay systems by dividing the domain of the problem into a uniform mesh the delay system is replaced with a sequence of initial value problems the method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval in order a specified error tolerance is satisfied we investigate some error estimates on the whole domain of interest some numerical examples are given and comparisons with other methods are made to demonstrate the efficiency and accuracy of the proposed method

higher order sobolev embeddings and isoperimetric inequalities abstract optimal higher order sobolev type embeddings are shown to follow via isoperimetric inequalities this establishes a higher order analogue of a well known link between first order sobolev embeddings and isoperimetric inequalities sobolev type inequalities of any order involving arbitrary rearrangement invariant norms on open sets in r n possibly endowed with a measure density are reduced to much simpler one dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets as a consequence the optimal target space in the relevant sobolev embeddings can be determined both in standard and in non standard classes of function spaces and underlying measure spaces in particular our results are applied to any order sobolev embeddings in regular john domains of the euclidean space in maz ya classes of possibly irregular euclidean domains described in terms of their isoperimetric function and in families of product probability spaces of which the gauss space is a classical instance

faber krahn inequalities for the robin laplacian a free discontinuity approach isoperimetric inequalities for the principal eigenvalues of the robin laplacian are interpreted as free discontinuity problems of unusual type we prove a full range of faber krahn inequalities in a nonlinear setting and for non smooth domains including the open case of the torsional rigidity the key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation as a byproduct we obtain the best constants for a class of poincare inequalities with trace terms in mathbb r n

second order derivatives and rearrangements the classical polya szego principle states that the lebesgue norm of the gradient of a weakly differentiable function on r decaying to 0 at infinity decreases under spherically symmetric rearrangement we present extensions of this principle to second order derivatives in two different directions on one hand when dimension n 1 we show that the space of those functions whose second order distributional derivative is a measure with finite total variation is closed under the operation of decreasing rearrangement moreover a polya szego type principle is proved for such a variation on the other hand in any dimension an inequality between the decreasing rearrangement of the second order derivatives of a function and the first order derivative of its decreasing rearrangement is established as a consequence a new unified approach to second order sobolev type inequalities in rearrangement invariant spaces via 1 dimensional hardy type inequalities is derived dipartimento di matematica e applicazioni per l architettura universita di firenze piazza ghiberti 27 50122 firenze italy e mail cianchi cesit1 unifi it

an iterative method for the extrapolation of band limited functions abstract we derive an iterative algorithm to extrapolate band limited functions using the operators of fourier transform and projection on convex subsets the algorithm allows the use of a priori knowledge in the extrapolation procedure

a quantitative isoperimetric inequality for fractional perimeters abstract recently frank and seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from sobolev seminorms of fractional order this isoperimetric inequality is improved here in a quantitative form

stability of elastic columns with periodic retarded follower forces the objective of this work is to present a stability analysis for elastic columns under the influence of periodically varying follower forces whose orientation is retarded i e depends on the position of the system at a previous time one and two degree of freedom dof discretized systems under the simultaneous influence of both parametric excitation and time delay whose effects on such systems have previously been only considered separately are studied by employing an orthogonal polynomial approximation the infinite dimensional floquet transition matrix associated with the time periodic differential delay system is approximated the stability criteria that all the eigenvalues floquet multipliers of this matrix must lie within the unit circle is then applied the stability charts for different combinations of the remaining system parameters are shown and the previously reported results for the special cases where either the parametric excitation or the time delay vanishes are verified two cases when the parametric forcing period is equal to or twice the delay period are taken into consideration in this work for special cases of the single dof system the numerical stability plots are verified by considering the analytical expressions for the corresponding stability boundaries for an analogous delayed mathieu equation

performance characteristics of the fuzzy sign detector in this paper fuzzy statistical techniques are introduced into the optimum signal detection problems the likelihood ratio for fuzzy detection of known signals is obtained as a special case of the fuzzy signal detector a fuzzy set theoretic approach to sign detection of known signals is considered specifically the test statistic of the fuzzy sign detector for known signals is obtained some properties of the fuzzy sign nonlinearity which constitutes the fuzzy sign detector are also described finally the performance characteristics of the fuzzy sign detector are investigated and compared to those of the crisp sign detector

normalized ground states for the nls equation with combined nonlinearities abstract we study existence and properties of ground states for the nonlinear schrodinger equation with combined power nonlinearities u u u q 2 u u p 2 u in r n n 1 having prescribed mass r n u 2 a 2 under different assumptions on q p a 0 and r we prove several existence and stability instability results in particular we consider cases when 2 q 2 4 n p 2 q p i e the two nonlinearities have different character with respect to the l 2 critical exponent these cases present substantial differences with respect to purely subcritical or supercritical situations which were already studied in the literature we also give new criteria for global existence and finite time blow up in the associated dispersive equation

convex symmetrization and polya szego inequality abstract we prove a polya szego inequality involving a convex symmetrization of functions and we investigate the equality case

on starshaped rearrangement and applications a radial symmetrization technique is investigated and new properties are proven the method transforms functions u into new functions u with starshaped level sets and is therefore called starshaped rearrangement this rearrangement is in general not equimeasurable a circumstance with some surprising consequences the method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems in particular the lipschitz continuity of free boundaries can now be easily obtained in a new fashion introduction in 26 g szego introduced the concept of radial symmetrization this is a nonequimeasurable rearrangement which transforms functions u q r q c r2 with starshaped level sets qc x e flu x c into functions u with symmetric starshaped level sets later 2 3 15 16 22 his approach was generalized to transform functions with level sets containing the origin into starshaped and not necessarily symmetric level sets and to higher dimensions we call a set d c rn starshaped with respect to xo if x e d implies tx 1 t xo e d for every t e 0 1 we call a set d starshaped if it is starshaped with respect to the origin therefore a more appropriate name for this rearrangement might be starshaped rearrangement in particular since the notion of radial symmetrization can be easily confused with circular or spherical symmetrization or spherically symmetric rearrangement one of the standard results in the theory of rearrangements is that the capacity of a condenser decreases under rearrangement i e j vu x lpdx 1 vu x ipdx for p 1 where u 1 on ql u 0 on 9q and u o c q c q e 0 0 j ivu p x ipdx for p 1 provided q c rn is a bounded starshaped domain containing ue 0 and u 0 1 u e c0 1 q u 1 in u 0 and u 0 on iq another convenient feature of other rearrangements is their equimeasurability a rearrangement u u is called equimeasurable if the n dimensional lebesgue received by the editors march 26 1984 and in revised form september 24 1985 1980 mathematics subject classification primary 35j20 35r35 35b05 26d20 lthis research was supported by the deutsche forschungsgemeinschaft dfg and by e reid 1986 american mathematical society 0002 9947 86 1 00 25 per page

errors in reprojection methods in computenzed tomography iterative tomographic reconstruction methods have been developed which can enforce various physical constraints on the reconstructed image an integral part of most of these methods is the repro jection of the reconstructed image these estimated projections are compared to the original projection data and modified according to some criteria based on a priori constraints in this paper the errors generated by such reprojection schemes are investigated bounds for these errors are derived under simple signal energy assumptions and using probabilistic assumptions on the distribution of discontinuities these bounds can be used in the enforcement of constraints in the determination of convergence of the iterative methods and in the detection of artifacts

on population resilience to external perturbations we study a spatially explicit harvesting model in periodic or bounded environments the model is governed by a parabolic equation with a spatially dependent nonlinearity of kolmogorov petrovsky piskunov type and a negative external forcing term delta using sub and supersolution methods and the characterization of the first eigenvalue of some linear elliptic operators we obtain existence and nonexistence results as well as results on the number of stationary solutions we also characterize the asymptotic behavior of the evolution equation as a function of the forcing term amplitude in particular we define two critical values delta and delta 2 such that if delta is smaller than delta the population density converges to a significant state which is everywhere above a certain small threshold whereas if delta is larger than delta 2 the population density converges to a remnant state everywhere below this small threshold our results are shown to be useful for studying

spline collocation methods for solving delay differential equations the convergence and stability analysis of seventh c 3 spline collocation methods if applied to delay differential equations ddes will be considered letting the interior collocation points x i 1 c j x i 1 c jh j 1 3 be dependent on three parameters c 1 c 2 c email protected 0 1 it will be shown that these methods for ddes possess convergence rate of order seven if 0 7279115

pointwise symmetrization inequalities for sobolev functions and applications we develop a technique to obtain new symmetrization inequalities that provide a unified framework to study sobolev inequalities concentration inequalities and sharp integrability of solutions of elliptic equations

analysis of milling dynamics for simultaneously engaged cutting teeth this paper investigates the stability of a milling process with simultaneously engaged teeth and contrasts it to prior work for a single tooth in the cut the stability analyses are performed with the chebyshev collocation method and the state space tfea technique these analyses show that a substantially different stability behavior is observed in addition the stability lobes are shown to undergo rapid transitions for relatively small changes in the radial immersion ratio these transitions are explained in terms of the specific cutting force profiles the stable periodic motion of the tool was also investigated using a harmonic balance approach and a dynamic map created with the tfea technique the findings suggest that a large number of harmonics are required for the harmonic balance approach to obtain the correct solution

symmetric decreasing rearrangement can be discontinuous suppose f xl x2 0 is a continuously differentiable function supported in the unit disk in the plane its symmetric decreasing rearrangement is the rotationally invariant function f xl x2 whose level sets are circles enclosing the same area as the level sets of f such rearrangement preserves lp norms but decreases convex gradient integrals e g p p 1 p 0 j 1 2 3 is a sequence of infinitely differentiable functions also supported in the unit disk which converge uniformly together with first derivatives to f the symmetzed functions also converge uniformly the real question is about convergence of the derivatives of the symmetrized functions we announce that the derivatives of the symmetrized functions need not converge strongly e g it can happen that fj f p 0 for every p we further characterize exactly those f s for which convergence is assured and for which it can fail

on a comparison theorem via symmetrisation talenti s theorem allows us to estimate the norms of the solution of an elliptic boundary value problem posed on a bounded open set in terms of those of the symmetrised problem posed on the ball centred at 0 and such that we are interested in the limiting case when equality is achieved between the norms of the two solutions we show then that and that the solution is radially symmetric and decreasing the novelty of this exposition is that the method used is very elementary compared to the usual arguments

functions of bounded variation and rearrangements the action of the symmetric rearrangement of functions of bounded variation is investigated in particular pointwise regularizing effects are established polya szeg type inequalities with a characterization of equality cases are also proved

a quantitative polya szego principle andrea cianchidipartimento di matematica e applicazioni per l architetturapiazza ghiberti 27 50122 firenze italye mail cianchi uni itluca espositodipartimento di ingegneria dell informazione e matematica applicatavia ponte don melillo 84084 fisciano sa italye mail esposito diima unisa itnicola fuscodipartimento di matematica e applicazionivia cintia 80126 napoli italye mail n fusco unina itcristina trombettidipartimento di matematica e applicazionivia cintia 80126 napoli italye mail cristina unina itjune 27 2006

hard constrained inconsistent signal feasibility problems we consider the problem of synthesizing feasible signals in a hilbert space in the presence of inconsistent convex constraints some of which must imperatively be satisfied this problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones the resulting convex optimization problem is analyzed and numerical solution schemes are presented along with convergence results the proposed formalism and its algorithmic framework unify and extend existing approaches to inconsistent signal feasibility problems an application to signal synthesis is demonstrated

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

image restoration by the method of convex projections part 2 applications and numerical results the image restoration theory discussed in a previous paper by youla and webb 1 is applied to a simulated image and the results compared with the well known method known as the gerchberg papoulis algorithm the results show that the method of image restoration by projection onto convex sets by providing a convenient technique for utilizing a priori information performs significantly better than the gerchberg papoulis method

an hp version of the discontinuous galerkin time stepping method for nonlinear second order delay differential equations with vanishing delays abstract we present and analyze an h p version of the discontinuous galerkin dg time stepping method for nonlinear second order delay differential equations with vanishing delays we derive a priori error bounds in the l 2 and l norm that are fully explicit with respect to the local time steps the local approximation orders and the local regularity of the exact solution we further prove that the h p dg scheme based on geometrically refined time steps and on linearly increasing approximation orders attains exponential rates of convergence for analytic solutions with start up singularities moreover we also propose an h p dg scheme for the state dependent delay differential equations we illustrate the theoretical results with a series of numerical experiments

representation of digital image by fuzzy neural network a multilayer feedforward fuzzy neural network fnn by which the predetermined fuzzy system can be realized is constructed to express a given two dimensional 2 d digital image it is shown that such a network is universal approximator the fnn approach provides us with the representation model of the 2 d discrete image in noise environment some given fuzzy numbers are employed to describe gray levels of a digital image and the deviation of corrupted image to its noise free one a class of fuzzy rules for removing impulse noise in degraded image are presented the corresponding fnn may work as a filter which improves the performances of median and rank conditioned rank selection filters especially such a filter may to a greatest extent maintain the fine structure of the image under the minimum mean absolute error criterion the membership functions of fuzzy numbers are adaptively adjusted and an optimal fnn for image representation is derived in noise environment simultaneously some real image examples are systemically analyzed to show that the fnn may result in higher quality global restoration

an abstract framework in the numerical solution of boundary value problems for neutral functional differential equations we consider the numerical solution of boundary value problems for general neutral functional differential equations the problems are restated in an abstract form and then a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed

collocation methods for the computation of periodic solutions of delay differential equations in this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations ddes periodic solutions are found by solving a periodic two point boundary value problem which is an infinite dimensional problem for ddes in contrast to the case of ordinary differential equations we investigate three collocation methods based on piecewise polynomials we discuss computational issues and show numerical orders of convergence using an extensive number of tests we compare our numerical results with known theoretical convergence results for initial value problems for ddes in particular we show how superconvergence at the mesh points can be lost or recovered depending on the dde model under consideration and on the choice of collocation discretization we end with a brief discussion of adaptive mesh selection

computing unstable manifolds of periodic orbits in delay differential equations we present the first algorithm for computing unstable manifolds of saddle type periodic orbits with one unstable floquet multiplier in systems of autonomous delay differential equations ddes with one fixed delay specifically we grow the one dimensional unstable manifold wu q of an associated saddle fixed point q of a poincare map defined by a suitable poincare section starting close to q along the linear approximation to wu q given by the associated eigenfunction our algorithm grows the manifold as a sequence of points where the distance between points is governed by the curvature of the one dimensional intersection curve wu q of wu q with our algorithm makes it possible to study global bifurcations in ddes we illustrate this with the break up of an invariant torus and a subsequent crisis bifurcation to chaos in a dde model of a semiconductor laser with phase conjugate feedback