an efficient wave based prediction technique for dynamic plate bending problems with corner stress singularities the finite element method is commonly used for simulations of dynamic plate bending problems a major disadvantage of this method is its practical frequency limitation in that the computational loads become prohibitively large at higher frequencies a newly developed wave based prediction technique aims to relax this frequency limitation through an enhanced computational efficiency this paper discusses the application of the wave based method for the particular case where stress singularities appear in one or more corners of a polygonal plate domain in this case the conventional set of field variable expansion functions is extended with some special purpose functions which incorporate the corner point singularities the beneficial convergence rate of the wave based method as compared with the finite element method is verified for various validation examples

validity of the limp model for porous materials a criterion based on the biot theory the validity of using the limp model for porous materials is addressed in this paper the limp model is derived from the poroelastic biot model assuming that the frame has no bulk stiffness being an equivalent fluid model accounting for the motion of the frame it has fewer limitations than the usual equivalent fluid model assuming a rigid frame a criterion is proposed to identify the porous materials for which the limp model can be used it relies on a new parameter the frame stiffness influence fsi based on porous material properties the critical values of fsi under which the limp model can be used are determined using a one dimensional analytical modeling for two boundary sets absorption of a porous layer backed by a rigid wall and radiation of a vibrating plate covered by a porous layer compared with other criteria the criterion associated with fsi provides information in a wider frequency range and can be used for configurations that include vibrating plates

investigation of the convergence of the mixed displacement pressure formulation for three dimensional poroelastic materials using hierarchical elements recently a mixed pressure displacement u p formulation based on biot s poroelasticity equations has been presented for porous materials this model leads to a reduction of the number of degrees of freedom required for the modeling of three dimensional porous media in comparison to classical displacement displacement u u formulations in this paper an extension of the u p formulation based on hierarchical elements is presented first a variant of the weak integral form of the u p formulation is presented and its numerical implementation using hierarchical elements is detailed together with the application of boundary and loading conditions numerical results are presented to show the accuracy and performance of the present approach in particular the importance of correctly capturing the coupling effects between the two phases is highlighted

an alternative biot s displacement formulation for porous materials this paper proposes an alternative displacement formulation of biot s linear model for poroelastic materials its advantage is a simplification of the formalism without making any additional assumptions the main difference between the method proposed in this paper and the original one is the choice of the generalized coordinates in the present approach the generalized coordinates are chosen in order to simplify the expression of the strain energy which is expressed as the sum of two decoupled terms hence new equations of motion are obtained whose elastic forces are decoupled the simplification of the formalism is extended to biot and willis thought experiments and simpler expressions of the parameters of the three biot waves are also provided a rigorous derivation of equivalent and limp models is then proposed it is finally shown that for the particular case of sound absorbing materials additional simplifications of the formalism can be obtained

a wave based method for the efficient solution of the 2d poroelastic biot equations the vibro acoustic behaviour of poroelastic materials is often formulated as boundary value problems based on the continuum mechanics biot s theory expressed as two coupled partial differential equations this paper presents an extension of the wave based method wbm a numerical technique to solve these vibro acoustic boundary value problems in a computationally efficient manner at present the finite element method fem is the most commonly used prediction technique to deal with these biot equations but suffers from the disadvantage that the system matrices have to be recalculated for each frequency of interest due to the frequency dependent equation parameters this harms the inherent effectiveness of the fem additionally due to the discretisation into a large number of small finite elements and the high number of unknowns per node the computational efforts involved practically restrict the use of fem to low frequency applications the method discussed in this paper is based on an indirect trefftz approach exact solutions of the three coupled waves supported by biot s equations are used as basis functions in a solution expansion to approximate the field variables in a poroelastic boundary value problem this approach leads to smaller systems of equations enabling an efficient solution at higher frequencies

wavelet based numerical homogenization a numerical homogenization procedure for elliptic differential equations is presented it is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions if the operator is in divergence form this is preserved by the homogenization procedure for periodic problems results similar to classical effective coefficient theory are proved the procedure can be applied to problems that are not cell periodic

a wave based method for the axisymmetric dynamic analysis of acoustic and poroelastic problems abstract element based approaches are most commonly applied to predict the harmonic response of coupled acoustic and poroelastic problems a major disadvantage of these techniques is the practical frequency limitation in that the computational efforts become prohibitively large at higher frequencies a recently developed wave based method is an efficient alternative deterministic prediction technique which aims at relaxing this frequency limitation by using exact solutions of the governing equations to approximate the field variables this paper extends the applicability range of this method towards axisymmetric problems novel wave function sets complying with the governing axisymmetric equations are derived and applied to both coupled and uncoupled acoustic and poroelastic problems the beneficial efficiency of the method as compared to standard finite element descriptions is illustrated by means of four numerical validation studies

application of generalized complex modes to the calculation of the forced response of three dimensional poroelastic materials finite element models based on biot s u p formulation for poroelastic materials are widely used to predict the behaviour of structures involving porous media the numerical solution of such problems requires however important computational resources and the solution algorithms are not optimized to improve the solution process a modal approach based on an extension of the complex modes technique has been proposed recently and applied successfully to a simplified mono dimensional problem in this paper this technique is investigated in the case of three dimensional poroelastic problems the technique is first recalled then analytical proof of the stability of the model are given followed by considerations of numerical improvements of the method an energetic interpretation of the generalized complex modes is then given and some numerical results are presented to illustrate the performance of the approach

toward local effective parameter theories using multiresolution decomposition using the recently developed theory of multiresolution decomposition a formulation that governs the response of a linear dynamical system with nonstationary microscale heterogeneities is reduced to two coupled formulations one governing the response smoothed on an arbitrary chosen reference scale with the response fine details as forcing and one governing the response details with the response smooth as a forcing by substituting the formal solution of the latter in the former a new framework specifically tuned to macroscale variations of the response in which the effects of the nonstationary microscale heterogeneity are described via a macroscale effective material operator is obtained localization of across scale couplings as well as the dependence of the response smooth and the effective material operator on the microscale and macroscale geometries are investigated via general asymptotic considerations and specific numerical examples the latter concerns the response of a fluid loaded elastic plate with nonstationary microscale mass heterogeneity

a multi level wave based numerical modelling framework for the steady state dynamic analysis of bounded helmholtz problems with multiple inclusions the wave based method wbm is an alternative numerical prediction method for both interior and exterior steady state dynamic problems which is based on an indirect trefftz approach it applies wave functions which are exact solutions of the governing differential equation to describe the dynamic field variables the smaller system of equations and the absence of pollution errors make the wbm very suitable for the treatment of helmholtz problems in the mid frequency range where element based methods are no longer feasible due to the associated computational costs a sufficient condition for convergence of the method is the convexity of the considered problem domain as a result only problems of moderate geometrical complexity can be considered and some geometrical features cannot be handled at all in this paper these limitations are alleviated through the development of a general modelling framework based on existing wbm methodologies which allows for the efficient introduction of inclusion configurations in bounded wbm models for problems governed by one or more helmholtz equations the feasibility and efficiency of the method is illustrated by means of numerical verification studies in which the methodology is applied to two types of dynamic problems on the one hand a single helmholtz equation associated with the steady state dynamic behaviour of acoustic cavities is studied on the other hand the framework is applied to the solution of the navier system of partial differential equations that describe the elastodynamic response of two dimensional perforated solids

extremal functions for moser s inequality let q be a bounded smooth domain in ri and u x a c1 function with compact support in q moser s inequality states that there is a constant co depending only on the dimension n such that 1 wn1 n j ee n u dx co where iqi is the lebesgue measure of q and wn 1 the surface area of the unit ball in rn we prove in this paper that there are extremal functions for this inequality in other words we show that the sup j nw u dx u e wj 11vuh1n 1 is attained earlier results include carleson chang 1986 q is a ball in any dimension and flucher 1992 q is any domain in 2 dimensions

a generalized macroscopic model for sound absorbing poroelastic media using the homogenization method abstract this paper proposes a new macroscopic model for sound absorbing poroelastic media which is derived by using the homogenization theory based on the method of asymptotic expansions the derivation of the macroscopic properties and governing equations takes into account the multiphysics occurring in poroelastic media for sound absorption including elastic motions of the solid phase compressible viscous fluid flow and the distributions of pressure and temperature in the fluid phase the coupled effects between the elastic solid and the fluid pressure and the temperature and the fluid pressure are also considered in contrast to the conventional biot s model which includes heuristic formulae the proposed method yields a rigorous model that is consistent with the principal governing equations on the microscopic scale utilizing several models that have simple microscopic geometry and comparing the numerical solutions obtained using the proposed method with corresponding analytical solutions we demonstrate that the derived macroscopic governing equations can provide accurate and effective predictions

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

convergence of poroelastic finite elements based on biot displacement formulation the convergence of linear poroelastic elements based on biot displacement formulation is investigated the aim is to determine a mesh criterion that provides reliable results under a given frequency limit the first part deals with 1d applications for which resonance frequencies can be related to biot wavelengths their relative contributions to the motion are given in order to determine if the mesh criteria for monophasic media are suitable for poroelastic media the imposition of six linear elements per wavelength is found for each biot wave as a primary condition for convergence for 3d applications convergence rules are derived from a generic configuration i e a clamped porous layer because of the complex deformation the previous criterion is shown to be insufficient influence of the coupling between the two phases is demonstrated

an efficient finite element scheme for solving the three dimensional poroelasticity problem in acoustics in this paper the finite element method fem is used to solve the three dimensional poroelasticity problem in acoustics based on the isotropic biot allard theory a displacement finite element model is derived using the lagrangian approach together with an analogy with solid elements from this model it is seen that the damping and stiffness matrices of the poroelastic media are complex and frequency dependent this leads to cumbersome calculations for large finite element models and spectral analyses to overcome this difficulty an efficient algorithm is proposed it is based on low frequency approximations of the frequency dependent dissipation mechanisms in poroelastic media this efficient algorithm allows the poroelastic materials to be modeled with classical fem codes also the acoustic poroelastic and the poroelastic poroelastic coupling conditions are presented the proposed model is compared to existing literature for both two dimensional and three dimensional problems excellent compari

characterization of the critical sobolev space on the optimal singularity at the origin in the present paper we investigate the optimal singularity at the origin for the functions belonging to the critical sobolev space hnp p rn 1 p with this purpose we shall show the weighted gagliardo nirenberg type inequality gn u lq rn dx x s c 1n s 1q 1p q1p u lp rn n s pnq n2pu lp rn 1 n s pnq where c depends only on n and p here 0 s n and p q with some p p determined only by n and p additionally in the case n 2 and nn 1 p we can prove the growth orders for s as s n and for q as q are both optimal gn allows us to prove the trudinger type estimate with the homogeneous weight furthermore it is obvious that gn cannot hold with the weight x n itself however with a help of the logarithmic weight of the type log1 x r x n at the origin we cover this critical weight simultaneously we shall give the minimal exponent r q p p so that the continuous embedding can hold

equivalent moser type inequalities in r2 and the zero mass case abstract we first investigate concentration and vanishing phenomena concerning moser type inequalities in the whole plane which involve complete and reduced sobolev norms in particular we show that the critical ruf inequality is equivalent to an improved version of the subcritical adachi tanaka inequality which we prove to be attained then we consider smooth compactly supported functions with respect to the dirichlet norm 2 and we prove an optimal lorentz zygmund type inequality with explicit extremals and from which can be derived classical inequalities in h 1 r 2 such as the adachi tanaka inequality and a version of ruf s inequality

weak anisotropic symmetric formulations of biot s equations for vibro acoustic modelling of porous elastic materials in this paper a fully anisotropic symmetric weak formulation of biot s equations for vibro acoustic modelling of porous elastic materials in the frequency domain is proposed starting from biot s equations in their anisotropic form a mixed displacement pressure formulation is discussed in terms of cartesian tensors the anisotropic equation parameters appearing in the differential equations are derived from material parameters which are possible to determine through experimental testing or micro structural simulations of the fluid and the porous skeleton solutions are obtained by applying the finite element method to the proposed weak form and the results are verified against a weak displacement based formulation for a foam and plate combination

measuring dependence powerfully and equitably given a high dimensional data set we often wish to find the strongest relationships within it a common strategy is to evaluate a measure of dependence on every variable pair and retain the highest scoring pairs for follow up this strategy works well if the statistic used a has good power to detect non trivial relationships and b is equitable meaning that for some measure of noise it assigns similar scores to equally noisy relationships regardless of relationship type e g linear exponential periodic in this paper we define and theoretically characterize two new statistics that together yield an efficient approach for obtaining both power and equitability to do this we first introduce a new population measure of dependence and show three equivalent ways that it can be viewed including as a canonical smoothing of mutual information we then introduce an efficiently computable consistent estimator of our population measure of dependence and we empirically establish its equitability on a large class of noisy functional relationships this new statistic has better bias variance properties and better runtime complexity than a previous heuristic approach next we derive a second related statistic whose computation is a trivial side product of our algorithm and whose goal is powerful independence testing rather than equitability we prove that this statistic yields a consistent independence test and show in simulations that the test has good power against independence taken together our results suggest that these two statistics are a valuable pair of tools for exploratory data analysis

an efficient wave based prediction technique for plate bending vibrations the finite element method is a commonly used prediction method for dynamic simulations of plate bending problems a major disadvantage of this method is its practical frequency limitation as its application involves prohibitively large computational efforts at higher frequencies a newly developed wave based prediction technique pushes up the frequency limit through an enhanced computational efficiency this paper discusses the principles of the wave based method for convex plate problems the beneficial convergence rate of the wave based method as compared with the finite element method is verified for various validation examples which include the most commonly encountered boundary conditions

deriving the equations of motion for porous isotropic media the equations of motion and stress strain relations for the linear dynamics of a two phase fluid solid isotropic porous material have been derived by a direct volume averaging of the equations of motion and stress strain relations known to apply in each phase the equations thus obtained are shown to be consistent with biot s equations of motion and stress strain relations however the effective fluid density in the equation of relative flow has an unambiguous definition in terms of the tractions acting on the pore walls the stress strain relations of the theory correspond to quasistatic stressing i e inertial effects are ignored it is demonstrated that using such quasistatic stress strain relations in the equations of motion is justified whenever the wavelengths are greater than a length characteristic of the averaging volume size

an extension of complex modes for the resolution of finite element poroelastic problems a new modal synthesis technique inspired from complex modes is presented in this paper the purpose of this method is to find an alternative to the weakness of usual modal techniques in order to improve resolution of finite element poroelastic problems firstly the construction of modes and the synthesis for a general forcing is explained secondly the approach is applied to a one dimensional poroelastic problem finally the approach is numerically tested and its performance is investigated by comparison with solutions obtained with a direct resolution of the system

a discontinuous galerkin method with plane waves for sound absorbing materials poro elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications including the aerospace automotive and energy industries more efficient computational models are required to further optimise the use of such materials in this paper we present a discontinuous galerkin method dgm with plane waves for poro elastic materials using the biot theory solved in the frequency domain this approach offers significant gains in computational efficiency and is simple to implement costly numerical quadratures of highly oscillatory integrals are not needed it is shown that the biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave based dgm a key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media this is particularly important when modelling porous materials as they are generally coupled with other media such as the surround fluid or an elastic structure the validation of the method is described first for a simple wave propagating through a porous material and then for the scattering of an acoustic wave by a porous cylinder the accuracy conditioning and computational cost of the method are assessed and comparison with the standard finite element method is included it is found that the benefits of the wave based dgm are fully realised for the biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material

reply to reshef et al falsifiability or bust the term equitability was introduced by reshef et al in ref 1 to describe measures of statistical dependence that give similar scores to equally noisy relationships of different types their paper also introduced a new statistic the maximal information coefficient mic that was said to satisfy this equitability criterion there has since been much interest in mic due primarily to its claimed equitability 2 3 however neither the original paper 1 nor follow up work 4 provided an unambiguous mathematical definition of equitability in particular the types of noise permissible in the noisy relationships used to define equitability were not described

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

functional associations at global brain level during perception of an auditory illusion by applying maximal information coefficient abstract maximal information coefficient mic is a recently introduced information theoretic measure of functional association with a promising potential of application to high dimensional complex data sets here we applied mic to reveal the nature of the functional associations between different brain regions during the perception of binaural beat bb bb is an auditory illusion occurring when two sinusoidal tones of slightly different frequency are presented separately to each ear and an illusory beat at the different frequency is perceived we recorded sixty four channels eeg from two groups of participants musicians and non musicians during the presentation of bb and systematically varied the frequency difference from 1 hz to 48 hz participants were also presented non binuaral beat nbb stimuli in which same frequencies were presented to both ears across groups as compared to nbb i bb conditions produced the most robust changes in the mic values at the whole brain level when the frequency differences were in the classical alpha range 8 12 hz and ii the number of electrode pairs showing nonlinear associations decreased gradually with increasing frequency difference between groups significant effects were found for bbs in the broad gamma frequency range 34 48 hz but such effects were not observed between groups during nbb altogether these results revealed the nature of functional associations at the whole brain level during the binaural beat perception and demonstrated the usefulness of mic in characterizing interregional neural dependencies

expressions of dissipated powers and stored energies in poroelastic media modeled by u u and u p formulations this paper is devoted to the rigorous obtention of the energy balance in porous materials the wave propagation in the porous media is described by biot allard s u u and u p formulations the paper derives the expressions for stored kinetic and strain energies together with dissipated energies it is shown that in the case of mixed formulations these expressions do not correspond to the real and imaginary parts of the variational formulations a quantitative convergence analysis of finite element scheme is then undertaken with the help of these indicators it is shown that the order of convergence of these indicators for linear finite element is one and that they are then well fitted to check the validity of finite element models

application of an efficient wave based prediction technique for the analysis of vibro acoustic radiation problems recently a new numerical prediction technique has been developed for the analysis of vibro acoustic problems in bounded domains this wave based prediction technique adopts an indirect trefftz approach unlike the conventional element based methods the novel technique uses wave functions that exactly satisfy the governing dynamic equations to describe the dynamic variables fine domain discretizations are no longer required and the resulting model sizes are much smaller than with the element based methods this results in a more time efficient prediction technique which also allows handling of mid frequency applications this paper discusses how the wave based prediction technique can be used for vibro acoustic radiation problems in unbounded domains its beneficial characteristics compared to the element based methods are demonstrated through a validation example

enhanced weak integral formulation for the mixed u_ p_ poroelastic equations recently atalla et al j acoust soc am 104 1444 1452 1998 and debergue et al j acoust soc am 106 2383 2390 1999 presented a weak integral formulation and the general boundary conditions for a mixed pressure displacement version of the biot s poroelasticity equations finite element discretization was applied to the formulation to solve 3d vibro acoustic problems involving elastic acoustic and poroelastic domains in this letter an enhancement of the weak integral formulation is proposed to facilitate its finite element implementation it is shown that this formulation simplifies the assembly process of the poroelastic medium the imposition of its boundary conditions and its coupling with elastic and acoustic media

the thresholds of the existence of maximizers for the critical sharp singular moser trudinger inequality under constraints this paper is addressed to study the existence of maximizers for the singular moser trudinger supremum under constraints in the critical case begin aligned mt n a beta sup u in w 1 n mathbb r n vert nabla u vert n a vert u vert n n 1 int mathbb r n phi n left 1 beta n alpha n u frac n n 1 right x beta dx end aligned where a 0 beta in 0 n phi n t e t sum k 0 n 2 frac t k k alpha n n omega n 1 1 n 1 and omega n 1 denotes the surface area of the unit sphere in mathbb r n more precisely we study the effect of the parameter a to the attainability of mt n a beta we will prove that for each beta in 0 n there exist the thresholds a beta and a beta such that mt n a beta is attained for any a in a beta a beta and is not attained for a a beta we also give some qualitative estimates for a beta and a beta our results complete the recent studies on the sharp moser trudinger type inequality under constraints due to do o sani and tarsi commun contemp math 19 27 2016 lam proc am math soc 145 4885 4892 2017 math nachr 291 14 15 2272 2287 2018 and ikoma ishiwata and wadade math ann 373 1 2 831 851 2019

a trefftz based numerical modelling framework for helmholtz problems with complex multiple scatterer configurations the aim of the work presented in this paper is the numerical solution of low and mid frequency time harmonic acoustic multiple scattering problem a novel so called multi level modelling approach is proposed which is applicable to the study of a configuration of well separated obstacles of arbitrary shape on which any type of acoustic boundary condition can be applied the generic character of the method is obtained by embedding the superposition principle for the multiple scattering influence in a state of the art acoustic modelling technique the so called wave based method the resulting approach successfully alleviates the geometrical limitations of the underlying trefftz based method and preserves the method s computational efficiency resulting in a generic multiple scattering modelling framework with a superior computational efficiency in the low as well as the mid frequency range several numerical validation examples show that the proposed approach is as accurate as the classical single scattering wave based method and illustrate the computational efficiency as compared to boundary element methods

reply to murrell et al noise matters the concept of statistical equitability plays a central role in the 2011 paper by reshef et al 1 formalizing equitability first requires formalizing the notion of a noisy functional relationship that is a relationship between two real variables x and y having the form y f x y f x where f is a function and is a noise term whether a dependence measure satisfies equitability strongly depends on what mathematical properties the noise term is allowed to have the narrower one s definition of noise the weaker the equitability criterion becomes unfortunately the allowed mathematical properties of is an issue

boundary conditions for the weak formulation of the mixed u p poroelasticity problem this paper presents the boundary conditions that apply to the weak integral formulation of the biot mixed u p poroelasticity equations these boundary conditions are derived from the classical boundary conditions of the biot displacement u u poroelasticity equations they are applied to the surface integrals of the associated weak form to account for exterior excitations supports and couplings with exterior elastic acoustic poroelastic media and a septum it will be shown that the derived boundary conditions for the u p formulation lead to simpler finite element equations compared to those obtained from the u u formulation finally two numerical examples are presented to validate the poroelastic septum coupling condition and to highlight the limitations of the free edge condition on a poroelastic medium

cleaning up the record on the maximal information coefficient and equitability although we appreciate kinney and atwal s interest in equitability and maximal information coefficient mic we believe they misrepresent our work we highlight a few of our main objections below fig 1 equitability of mic and mutual information under a range of noise models the equitability of mic and mutual information across a subset of noise models analyzed in refs 1 and 4 for each noise model the relationships tested are as in ref 4 in each regarding our original paper 1 kinney and atwal 2 state mic is said to satisfy not just the heuristic notion of equitability but also the mathematical criterion of r2 equitability the latter being their formalization of the heuristic notion that we introduced this statement is simply false we were explicit in our paper that our claims regarding mic s performance were based on large scale simulations we tested mic s equitability through simulations these show that for a large collection of test functions with varied sample sizes noise levels and noise models mic roughly equals the coefficient of determination r2 relative to each respective noiseless function although we mathematically proved several things about mic none of our claims imply that it satisfies kinney and atwal s r2 equitability which would require that mic exactly equal r2 in the infinite data limit thus their proof that no dependence measure can satisfy r2 equitability although interesting does not uncover any error in our work and their suggestion that it does is a gross misrepresentation kinney and atwal seem ready to toss out equitability as a useful criterion based on their theoretical result we argue however that regardless of whether perfect equitability is possible approximate notions of equitability remain the right goal for many data exploration settings just as the theory of np completeness does not suggest we stop thinking about np complete problems but instead that we look for approximations and solutions in restricted cases an impossibility result about perfect equitability provides focus for further research but does not mean that useful solutions are unattainable similarly as others have noted 3 kinney and atwal s proof requires a highly permissive noise model and so the attainability of r2 equitability under more limited noise models such as those in our work remains an open question finally the authors argue that mutual information is more equitable than mic however they provide as justification only a single noise model only at limiting sample sizes n 5 000 as we ve shown in follow up work 4 which they themselves cite but fail to address mic is more equitable than mutual information estimation under many other realistic noise models even at a sample size of 5 000 kinney and atwal have stated it matters how one defines noise 5 and a useful statistic must indeed be robust to a wide range of noise models equally importantly we ve established in both our original and follow up work that at sample size regimes less than 5 000 mic is more equitable than mutual information estimates across all noise models tested mic s superior equitability in these settings is not an artifact we neglected as kinney and atwal suggest but rather a weakness of mutual information estimation and an important consideration for practitioners we expect that the understanding of equitability and mic will improve over time and that better methods may arise however accurate representations of the work thus far will allow researchers in the area to most productively and collectively move forward

equitability mutual information and the maximal information coefficient how should one quantify the strength of association between two random variables without bias for relationships of a specific form despite its conceptual simplicity this notion of statistical equitability has yet to receive a definitive mathematical formalization here we argue that equitability is properly formalized by a self consistency condition closely related to data processing inequality mutual information a fundamental quantity in information theory is shown to satisfy this equitability criterion these findings are at odds with the recent work of reshef et al reshef dn et al 2011 science 334 6062 1518 1524 which proposed an alternative definition of equitability and introduced a new statistic the maximal information coefficient mic said to satisfy equitability in contradistinction to mutual information these conclusions however were supported only with limited simulation evidence not with mathematical arguments upon revisiting these claims we prove that the mathematical definition of equitability proposed by reshef et al cannot be satisfied by any nontrivial dependence measure we also identify artifacts in the reported simulation evidence when these artifacts are removed estimates of mutual information are found to be more equitable than estimates of mic mutual information is also observed to have consistently higher statistical power than mic we conclude that estimating mutual information provides a natural and often practical way to equitably quantify statistical associations in large datasets

on the effect of equivalent constraints on a maximizing problem associated with the sobolev type embeddings in mathbb r n in this paper we consider the attainability of a maximizing problem begin aligned d sup vert u vert h 1 n gamma 1 left vert u vert n n alpha vert u vert p p right end aligned where n ge 2 n 0 0 gamma le n and vert u vert h 1 n gamma left vert u vert n gamma vert nabla u vert n gamma right frac 1 gamma the existence of a maximizer for d is closely related to the exponent gamma in fact we show that the value begin aligned alpha alpha inf vert u vert h 1 n gamma 1 left frac 1 vert u vert n n vert u vert p p right end aligned is a threshold in terms of the attainability of d

r2 equitability is satisfiable kinney and atwal 1 make excellent points about mutual information the maximal information coefficient 2 3 and equitability one of their central claims however is that no nontrivial dependence measure can satisfy r 2 r2 equitability we argue that this is the result of a poorly constructed definition which we quote a dependence measure d x y d x y is r 2 r2 equitable if and only if when evaluated on a joint probability distribution p x y p x y that corresponds to a noisy functional relationship between two real random variables x and y the following relation holds d x y g r 2 f x y d x y g r2 f x y here g is a function that does not depend on p x y p x y and f is the function defining the noisy functional relationship i e y f x y f x

existence and nonexistence of maximizers for variational problems associated with trudinger moser type inequalities in mathbb r n we discuss the existence of a maximizer for a maximizing problem associated with the trudinger moser type inequality in mathbb r n n geq 2 different from the bounded domain case we obtain both of the existence and the nonexistence results the proof requires a careful estimate of the maximizing level with the aid of normalized vanishing sequences

moving average fields macro scale response measures and homogenizing micro scale variation there are two critical issues when deriving a macro scale prediction model starting from a more complete underlying model the first is the precise relationship of the fields predicted by the more complete model and the fields predicted by the macro scale model the second is the manner of solving a closure problem that is invariably encountered in all such derivations the understanding that moving averages of the fields predicted by the more complete model are the fields predicted by the macro scale model is challenged on the grounds that accomplishing a moving average does not eliminate micro scale variation it only appears to do so in one representation of the moving average field the solution of a closure problem by assumption is challenged on the grounds that the most common assumptions are demonstrably invalid even while leading to prediction models that can provide reasonable estimates of the macro scale response in some scenarios in presenting the challenges it is further shown how a multiresolution analysis by an orthogonal wavelet system provides a framework for both precisely defining macro scale response fields i e fields from which all micro scale variation has been eliminated and presenting a formally exact solution for a precisely described closure problem

a multiresolution approach to homogenization and effective modal analysis of complex boundary value problems we apply multiresolution techniques to study the effective properties of bound ary value problems conditions under which boundary values are preserved in the effective ho mogenized formulation are developed and discussed relations between the eigenfunctions and eigenvalues of the generic formulation and those of the effective formulation are explored and their dependence on the micro scale is studied applications to the construction of effective green func tions in a complex lamination are discussed the analytic results are demonstrated via numerical computations

existence and non existence of maximizers for the moser trudinger type inequalities under inhomogeneous constraints in this paper we study the existence and non existence of maximizers for the moser trudinger type inequalities in mathbb r n of the form begin aligned d n alpha a b sup u in w 1 n mathbb r n vert nabla u vert l n mathbb r n a vert u vert l n mathbb r n b 1 int mathbb r n phi n left alpha u n right dx end aligned here n ge 2 n frac n n 1 a b 0 alpha in 0 alpha n and phi n t e t sum j 0 n 2 frac t j j where alpha n n omega n 1 1 n 1 and omega n 1 denotes the surface area of the unit ball in mathbb r n we show the existence of the threshold alpha alpha a b n in 0 alpha n such that d n alpha a b is not attained if alpha in 0 alpha and is attained if alpha in alpha alpha n we also provide the conditions on a b in order that the inequality alpha alpha n holds