three dimensional steady state flow to a well in a randomly heterogeneous bounded aquifer 1 we consider flow in a confined aquifer of uniform thickness due to a well of zero radius that fully penetrates the aquifer and discharges at a constant rate if the lateral extent of the aquifer is infinite a steady state flow regime never develops it is however well known that if the aquifer is additionally uniform a quasi steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time on the expanding surface head is uniform and time invariant inside this surface head at any time is described by a steady state solution a rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three dimensional transient stochastic flow problem in an aquifer of infinite lateral extent here we take a different approach by developing a three dimensional steady solution for mean flow to a well in a randomly heterogeneous aquifer with a cylindrical prescribed head boundary in analogy to the uniform case we expect our solution to approximate a quasi steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger we treat log conductivity as a statistically homogeneous random field characterized by a gaussian spatial covariance function that may have different horizontal and vertical correlation scales our solution consists of analytical expressions for the ensemble mean and variance of head in the aquifer to second order in the standard deviation of log conductivity it is based on recursive approximations of exact nonlocal moment equations that are free of distributional assumptions and so apply to both gaussian and non gaussian log conductivity fields the analytical solution is supported by numerical monte carlo simulations it clarifies the manner in which relationships between the horizontal and vertical scales of the quasi steady state region and those of statistical anisotropy impact the statistical moments of drawdown and the equivalent and apparent hydraulic conductivities of the aquifer both conductivities are shown to exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal integral scales from it

a positivity result in the theory of macdonald polynomials we outline here a proof that a certain rational function c n q t which has come to be known as the q t catalan is in fact a polynomial with positive integer coefficients this has been an open problem since 1994 because c n q t evaluates to the catalan number at t q 1 it has also been an open problem to find a pair of statistics a b on the collection n of dyck paths pi of length 2n yielding c n q t summation operator pi t a pi q b pi our proof is based on a recursion for c n q t suggested by a pair of statistics recently proposed by j haglund one of the byproducts of our results is a proof of the validity of haglund s conjecture

non convex self dual lagrangians new variational principles of symmetric boundary value problems abstract we study the concept and the calculus of non convex self dual nc sd lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems they indeed provide new representations and formulations for the superposition of convex functions and symmetric operators they yield new variational resolutions for large class of hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones this approach seems to offer several useful advantages it associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of euler lagrange functions these potential functions are quite flexible and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems additionally in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard euler lagrange function perhaps most remarkable however are the permanence properties of nc sd lagrangians their calculus is relatively manageable and their applications are quite broad

integrable solutions of a generalized mixed type functional integral equation in this work we prove the existence of integrable solutions for the following generalized mixed type nonlinear functional integral equation x t g t t x t f t 0 t k t s u t s q x s d s t 0 our result is established by means of a krasnosel skii type fixed point theorem proved by taoudi 2009 in the last section we give an example to illustrate our result

fractional equations and generalizations of schaefer s and krasnoselskii s fixed point theorems abstract we study a fractional differential equation of caputo type by first transforming it into an integral equation with an l 1 0 kernel and then applying fixed point theory of banach and schauder type using a weighted norm to avoid stringent compactness conditions it becomes clear that tedious construction of mapping sets and boundedness conditions can be avoided if we use fixed point theorems of schaefer and krasnoselskii type the weighted norm then produces open sets so large that it is difficult to show that mappings are compact this then leads us to generalize both schaefer s and krasnoselskii s fixed point theorems which yield simple and direct qualitative results for the fractional differential equations the weight g yields compactness but it does much more the generalized fixed point theorems now yield growth properties of the solutions of the fractional differential equations

stochastic analysis of well type flows in randomly heterogeneous porous formations 1 well type flow takes place in a heterogeneous porous formation where the transmissivity is modeled as a stationary random space function rsf general expressions for the covariances of the head and flux are obtained and analyzed the second order approximation of the mean radial flux is represented as the product between the solution valid in a homogeneous domain and a distortion term which adjusts according to the medium heterogeneity the spatial dependence of the function is studied in view of the formation identification problem the equivalent t eq and apparent t ap transmissivity are computed the important result is the relationship may be either eq or ap where th and tg represent the harmonic and the geometric means of the transmissivity respectively the position dependent weight is explicitly calculated indeed close to the well it yields which is understandable in view of the fact that the limit is equivalent to i which is the heterogeneity structure of a stratified formation nevertheless the effective transmissivity of a stratified formation is precisely th in contrast far from the well one has with the flow being slowly varying in the mean there it is shown that grows with increasing in the case of t eq the rate of growing is found similar to dagan and lessoff 2007 to be strongly dependent upon the position in the flow domain whereas t ap is a more robust property finally it is shown how the general results can be used for practical applications

sweep maps a continuous family of sorting algorithms abstract we define a family of maps on lattice paths called sweep maps that assign levels to each step in the path and sort steps according to their level surprisingly although sweep maps act by sorting they appear to be bijective in general the sweep maps give concise combinatorial formulas for the q t catalan numbers the higher q t catalan numbers the q t square numbers and many more general polynomials connected to the nabla operator and rational catalan combinatorics we prove that many algorithms that have appeared in the literature including maps studied by andrews egge gorsky haglund hanusa jones killpatrick krattenthaler kremer orsina mazin papi vaille and the present authors are all special cases of the sweep maps or their inverses the sweep maps provide a very simple unifying framework for understanding all of these algorithms we explain how inversion of the sweep map which is an open problem in general can be solved in known special cases by finding a bounce path for the lattice paths under consideration we also define a generalized sweep map acting on words over arbitrary alphabets with arbitrary weights which is also conjectured to be bijective

lattice diagram polynomials and extended pieri rules abstract the lattice cell in the i 1 st row and j 1 st column of the positive quadrant of the plane is denoted i j if is a partition of n 1 we denote by ij the diagram obtained by removing the cell i j from the french ferrers diagram of we set ij det x p j i y q j i n i j 1 where p 1 q 1 p n q n are the cells of ij and let m ij be the linear span of the partial derivatives of ij the bihomogeneity of ij and its alternating nature under the diagonal action of s n gives m ij the structure of a bigraded s n module we conjecture that m ij is always a direct sum of k left regular representations of s n where k is the number of cells that are weakly north and east of i j in we also make a number of conjectures describing the precise nature of the bivariate frobenius characteristic of m ij in terms of the theory of macdonald polynomials on the validity of these conjectures we derive a number of surprising identities in particular we obtain a representation theoretical interpretation of the coefficients appearing in some macdonald pieri rules

topological results on a certain class of functionals and application abstract we study in this paper problems of the type u u p 1 u x bounded r n u 0 i where x is given and where p 1 n 2 n 2 p 1 if n 2 our main result is that i has an infinite number of solutions for a residual set of in h 1 in particular for many n n there exists an open and dense subset of in h 1 such that i has n distinct solutions for such an this result is to be related to the conjecture developed in 1 of the existence of an infinite number of solutions to i the proof relies on a general characterization of level sets for a certain class of functionals when there are no critical value in a large enough interval in addition to the study of problem i we apply this characterization to give another proof using e g brouwer s fixed point theorem for some classical results about even functionals and saddle points

pumping tests in heterogeneous aquifers an analytical study of what can be obtained from their interpretation using jacob s method interpretation of pumping tests to estimate hydraulic parameter values is typically based on the assumption of aquifer homogeneity the applicability of the traditional methods of interpretation in real aquifers can be questioned since the evaluation of the drawdown curves observed at different locations in a single test may not result in one consistent set of hydraulic parameters thus most hydrogeologists tend to look at estimated transmissivities t as some average property of the medium while estimated storativities s are disregarded in some cases particularly when they are obtained from data measured at the pumping well an analytical study of drawdown under radially convergent flow toward a single point in heterogeneous aquifers shows that large time drawdown values form a straight line on a drawdown versus log time plot jacob s method consists of obtaining estimates for t and s from the slope and intercept of this line we find that even in a heterogeneous field these estimates provide valuable information about the aquifer estimated t values for different observation points tend to converge to a single value which corresponds to the effective t derived under parallel flow conditions estimated storativities however display higher variability but the geometric mean of the sest values can be used as an unbiased estimator of the actual s thus it appears that although jacob s method was originally derived for homogeneous media it can provide valuable information in real aquifers

an evaluation of jacob s method for the interpretation of pumping tests in heterogeneous formations most pumping tests are interpreted using the classical theis assumption of large scale homogeneity with various corrections to account for early time behavior of drawdown curves when drawdowns are plotted versus log time late time data often delineate a straight line which is consistent with jacob s approximation of theis solution but may seem surprising in view of the heterogeneity of natural media the aim of our work is to show that jacob s method leads to a good approximation of the effective transmissivity of heterogeneous media when constrained to late time data a review of several multiwell pumping tests demonstrates that when drawdown curves from each observation well are interpreted separately they produce very similar transmissivity t estimates however the corresponding estimates for storativity span a broad range this behavior is verified numerically for several models of formation heterogeneity a very significant finding of the numerical investigation is that t values estimated using simulated drawdown from individual observation wells are all very close to the effective t value for parallel flow this was observed even in nonmultigaussian t fields where high t zones are well connected and where the effective t is larger than the geometric average of point values this implies that jacob s method can be used for estimating effective t values in many if not most formations

correlation structure of flow variables for steady flow toward a well with application to highly anisotropic heterogeneous formations the study a continuation of that of indelman et al 1996 aims at deriving the second order moments of flow variables such as hydraulic head its gradient and the specific discharge for steady flow toward a fully penetrating well in a confined heterogeneous aquifer the log conductivity y ln k is modeled as a three dimensional stationary function of gaussian correlation of anisotropy ratio e by using first order approximations in 2y and e we derive the variance and the vertical integral scale of the piezometric head h of its radial gradient er and of the radial component of the specific discharge qr owing to the nonuniformity of the average flow these quantities are functions of the distance from the well it is shown that the variances of the head 2h and of its gradient 2er as well as the crossvariance e y between er and y vanish at the well whereas the discharge variance 2qr tends to the product between the log conductivity variance 2y and the squared mean discharge qr 2 this behavior pertains to a stratified formation surrounding the well far from the well 75 horizontal y integral scales i the head variance approaches a constant value for r 10i the moments 2er 2qr and ery tend to the corresponding values for uniform flow but with the local mean head gradient replacing the constant one the head vertical integral scale grows indefinitely with r whereas the vertical integral scale of the flux is larger by one log conductivity vertical scale than the one prevailing in uniform flow this latter property is explained by the presence of the source line which increases the correlations in the vertical direction the present results may be used in identifying the log conductivity statistical parameters from flowmeter velocity measurements in piezometers surrounding pumping or injecting wells

krasnoselskii s fixed point theorem for weakly continuous maps abstract we consider the sum a b m x where m is a weakly compact and convex subset of a banach space x a m x is weakly continuous and b l x with bp 1 p 1 an alternative condition is given in order to guarantee the existence of fixed points in m for a b some illustrative applications are given

conjectured statistics for the q t catalan numbers abstract we introduce the distribution function f n q t of a pair of statistics on catalan words and conjecture f n q t equals garsia and haiman s q t catalan sequence c n q t which they defined as a sum of rational functions we show that f n s q t defined as the sum of these statistics restricted to catalan words ending in s ones satisfies a recurrence relation as a corollary we are able to verify that f n q t c n q t when t 1 q we also show the partial symmetry relation f n q 1 f n 1 q by modifying a proof of haiman of a q lagrange inversion formula based on results of garsia and gessel we obtain a q analogue of the general lagrange inversion formula which involves catalan words grouped according to the number of ones at the end of the word

steady flow toward wells in heterogeneous formations mean head and equivalent conductivity we consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw figure 1 the hydraulic conductivity k is modeled as a three dimensional stationary random space function the two point covariance of y in k kg is of axisymmetric anisotropy with i and i the horizontal and vertical integral scales respectively and kg the geometric mean of k unlike previous studies which assumed constant flux the well boundary condition is of given constant head figure 1 the aim of the study is to derive the mean head h and the mean specific discharge q as functions of the radial coordinate r and of the parameters y2 e i i and rw i an approximate solution is obtained at first order in y2 by replacing the well by a line source of strength proportional to k and by assuming ergodicity i e equivalence between space averages over the vertical and h q ensemble means an equivalent conductivity keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge q as the actual one for the given head hw in the well and a given head in a piezometer at distance r from the well this definition corresponds to the transmissivity determined in a pumping test by an observer that measures hw andq the main result of the study is the relationship 19 keq ka 1 kefu where ka is the conductivity arithmetic mean and kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer the weight coefficient 10 has the simple approximate expression ln r i in r rw near the well 0 and keq ka which is easily understood since for rw i 1 the formation behaves locally like a stratified one in contrast far from the well 1 and keq kefu the flow being slowly varying there since ka kefu our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well however the difference between ka and kefu is small for highly anisotropic formations for which e 1 a nonlocal effective conductivity which depends only on the heterogeneous structure is derived in appendix a along the lines of indelman and abramovich 1994

solutions of supercritical semilinear non homogeneous elliptic problems abstract considering a semilinear elliptic equation u u g x u b x in u 0 on in a bounded domain r n with a smooth boundary we apply a new variational principle introduced in momeni 2011 2017 to show the existence of a strong solution where g can have critical growth to be more accurate assuming g x is the primitive of g x and g x is the fenchel dual of g x we shall find a minimum of the functional i defined by i u g x u u b x d x g x u b x u d x over a convex set k consisting of bounded functions in an appropriate sobolev space the symmetric nature of the functional i provided by existence of a function g and its fenchel dual g alleviate the difficulty and shorten the process of showing the existence of solutions for problems with supercritical nonlinearity it also makes it an ideal choice among the other energy functionals including euler lagrange functional

existence results for a generalized nonlinear hammerstein equation on l1 spaces abstract in this paper we are concerned with existence results for a nonlinear hammerstein integral equation acting on l 1 vector valued functions making use of the de blasi measure of weak noncompactness we establish a variant of the topological fixed point theorem of krasnosel skii for the sum of two operators without requiring weak compactness or weak continuity assumptions such a result provides an adequate tool for deriving an existence theory for problem 1 1

on the reduction of krasnoselskii s theorem to schauder s theorem abstract krasnoselskii noted that many problems in analysis can be formulated as a mapping which is the sum of a contraction and compact map he proved a theorem covering such cases which is the union of the contraction mapping principle and schauder s second fixed point theorem in putting the two results together he found it necessary to add a condition which has been difficult to fulfill although a great many problems have been solved using his result and there have been many generalizations and simplifications of his result in this paper we point out that when the mapping is defined by an integral plus a contraction term the integral can generate an equicontinuous map which is independent of the smoothness of the functions because of that it is possible to set up that mapping not as a sum of contraction and compact map but as a continuous map on a compact convex subset of a normed space an application of schauder s first fixed point theorem will then yield a fixed point without any reference to that difficult condition of krasnoselskii finite and infinite intervals are handled separately for the class of problems considered application is parallel to the much simpler brouwer fixed point theorem

krasnosel skii type fixed point theorems under weak topology features abstract in this paper we prove the following krasnosel skii type fixed point theorem let m be a nonempty bounded closed convex subset of a banach space x suppose that a m x and b x x are two weakly sequentially continuous mappings satisfying i a m is relatively weakly compact ii b is a strict contraction iii x b x a y y m x m then a b has at least one fixed point in m this result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping the results presented in this paper encompass several earlier ones in the literature

square boldsymbol q t lattice paths and boldsymbol nabla p_n the combinatorial q t catalan numbers are weighted sums of dyck paths introduced by j haglund and studied extensively by haglund haiman garsia loehr and others the q t catalan numbers besides having many subtle combinatorial properties are intimately connected to symmetric functions algebraic geometry and macdonald polynomials in particular the n th q t catalan number is the hilbert series for the module of diagonal harmonic alternants in 2n variables it is also the coefficient of s 1 n in the schur expansion of e n using q t analogues of labelled dyck paths haglund et al have proposed combinatorial conjectures for the monomial expansion of e n and the hilbert series of the diagonal harmonics modules this article extends the combinatorial constructions of haglund et al to the case of lattice paths contained in squares we define and study several q t analogues of these lattice paths proving combinatorial facts that closely parallel corresponding results for the q t catalan polynomials we also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator in particular we conjecture combinatorial formulas for the monomial expansion of p n the hilbert series p n h 1 n and the sign character p n s 1 n

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

schaefer krasnoselskii fixed point theorems using a usual measure of weak noncompactness abstract we present some extension of a well known fixed point theorem due to burton and kirk t a burton c kirk a fixed point theorem of krasnoselskii schaefer type math nachr 189 1998 423 431 for the sum of two nonlinear operators one of them compact and the other one a strict contraction the novelty of our results is that the involved operators need not to be weakly continuous finally an example is given to illustrate our results

a stochastic analysis of pumping tests in laterally nonuniform media conventional pumping test analysis methodology assumes that aquifer transmissivity is invariant in space the ramifications of this assumption are examined for hypothetical units whose variations in transmissivity are considered to be reasonable representations of variations possible in natural systems the dependence of pumping test transmissivity on spatial angular and radial and temporal location of observations and the method of drawdown analysis is assessed the dependence on the angular position of an observation well appears of little significance dependence on radial position however can be strong the dependence on the interval of time used in the pumping test analysis is only important in highly variable systems methods of drawdown analysis that yield identical estimates in uniform units yield differing estimates in nonuniform ones as a result of a difference in data fitting procedures a reasonable estimate of transmissivity at a regional scale can be obtained from a cooper jacob analysis applied at a large duration of pumpage in general conventional approaches for pumping test analysis should be viable in the nonuniform aquifers of the type considered here

the delta conjecture we conjecture two combinatorial interpretations for the symmetric function eken where f is an eigenoperator for the modified macdonald polynomials defined by bergeron garsia haiman and tesler both interpretations can be seen as generalizations of the shuffle conjecture a statement originally conjectured by haglund haiman remmel loehr and ulyanov and recently proved by carlsson and mellit we show how previous work of the second and third authors on tesler matrices and ordered set partitions can be used to verify several cases of our conjectures furthermore we use a reciprocity identity and llt polynomials to prove another case finally we show how our conjectures inspire 4 variable generalizations of the catalan numbers extending work of garsia haiman and the first author