Xu : Sagbi bases of Cox-Nagata rings. Journal of the European Mathematical Society , 12 2 , p. Cartwright ; S. Brady ; D. Orlando ; B. Benfey : Reconstructing spatiotemporal gene expression data from partial observations. Bioinformatics , 25 19 , p. Craciun ; A. Dickenstein ; A.

Sturmfels : Toric dynamical systems. Journal of symbolic computation , 44 11 , p. Dueck ; S. Hosten and B. Sturmfels : Normal toric ideals of low codimension. Journal of pure and applied algebra , 8 , p. Gritzmann ; B. Sturmfels and G. Ziegler : Guest editors' foreword : [special issue dedicated to the memory of Victor Klee].

Discrete and computational geometry , 42 2 , p. Herrmann ; A. Jensen ; M. Joswig and B. Sturmfels : How to draw tropical planes. Laubenbacher and B. Sturmfels : Computer algebra in systems biology. The American mathematical monthly , 10 , p. Sturmfels : Computeralgebra in der Systembiologie. Informatik-Spektrum , 32 1 , p.

Lin and B. Sturmfels : Polynomial relations among principal minors of a 4 x 4-matrix. Journal of algebra , 11 , p. Xu : Marginal likelihood integrals for mixtures of independence models. Journal of machine learning research , 10 , p. Morton ; L. Pachter ; A.

Shiu ; B. Sturmfels and O. Wienand : Convex rank tests and semigraphoids. SIAM journal on discrete mathematics , 23 3 , p. Nie and B. Sturmfels : Matrix cubes parameterized by eigenvalues. SIAM journal on matrix analysis and applications , 31 2 , p. Speyer and B.

Sturmfels : Tropical mathematics. Mathematics magazine , 82 3 , p. Bogart ; A. Jensen ; D. Speyer ; B. Thomas : Computing tropical varieties. Ahnert ; H. Edelsbrunner ; T. Fink ; E. Glynn ; G. Hattem ; A. Kudlicki ; Y. Mileyko ; J. Morton ; A. Mushegian ; L. Pachter ; M. Rowicka ; A. Gritzmann and B. Sturmfels : Victor L. Klee Notices of the American Mathematical Society , 55 4 , p. Hemmecke ; J. Wienand : Three counter-examples on semi-graphoids.

Combinatorics, probability and computing , 17 02 , p. Huggins ; B. Sturmfels ; J. Yu and D. Yuster : The hyperdeterminant and triangulations of the 4-cube. Mathematics of computation , 77 , p. Sturmfels and S. Sullivant : Toric geometry of cuts and splits. The Michigan mathematical journal , 57 , p. Sturmfels and J. Tevelev : Elimination theory for tropical varieties. Mathematical research letters , 15 3 , p. Beerenwinkel ; N. Eriksson and B. Sturmfels : Conjunctive Bayesian networks. Bernoulli , 13 4 , p.

Beerenwinkel ; L.

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Pachter and B. Sturmfels : Epistasis and shapes of fitness landscapes. Statistica Sinica , 17 4 , p. Pachter ; B. Sturmfels ; S. Elena and R. Lenski : Analysis of epistatic interactions and fitness landscapes using a new geometric approach. Journal of symbolic computation , 42 , p. Dickenstein ; E. Feichtner and B. Sturmfels : Tropical discriminants. Journal of the American Mathematical Society , 20 4 , p. Drton ; B. Sullivant : Algebraic factor analysis : tetrads, pentads and beyond.

Probability theory and related fields , , p. Holtz and B. Sturmfels : Hyperdeterminantal relations among symmetric principal minors. Journal of algebra , 2 , p. Sturmfels : Computing the integer programming gap. Combinatorica , 27 3 , p. Huggins ; L. Sturmfels : Toward the human genotope. Bulletin of mathematical biology , 69 8 , p. Joswig ; B. Yu : Affine buildings and tropical convexity.

Albanian journal of mathematics , 1 4 , p. Sturmfels : The cyclohedron test for finding periodic genes in time course expression studies. Sturmfels : The mathematics of phylogenomics. SIAM review , 49 1 , p. Tevelev and J. Yu : The Newton polytope of the implicit equation. Moscow mathematical journal , 7 2 , p. Sturmfels : Evolution on distributive lattices. Journal of theoretical biology , 2 , p. Catanese ; S. Hosten ; A. Khetan and B. Sturmfels : The maximum likelihood degree. American journal of mathematics , 3 , p. Jarrah ; R. Sturmfels : Monomial dynamical systems over finite fields.

Complex systems , 16 4 , p. Dewey ; P. Huggins ; K. Woods ; B. Sturmfels and L. Pachter : Parametric alignment of Drosophila genomes. PLoS computational biology , 2 6 , p. Geiger ; C. Meek and B. Sturmfels : On the toric algebra of graphical models. The annals of statistics , 34 3 , p. Sturmfels : Resultants in genetic linkage analysis. Journal of symbolic computation , 41 2 , p. Nie ; J. Demmel and B. Sturmfels : Minimizing polynomials via sum of squares over the gradient ideal.

## Best Math Education Degrees

Mathematical programming , 3 , p. Sullivant : Combinatorial secant varieties. Pure and applied mathematics quarterly , 2 3 , p. Altmann and B. Sturmfels : The graph of monomial ideals. Journal of pure and applied algebra , , p. Sturmfels : Matroid polytopes, nested sets and Bergman fans. Portugaliae mathematica , 62 4 , p. Garcia ; M. Stillman and B. Sturmfels : Algebraic geometry of Bayesian networks.

Journal of symbolic computation , 39 , p. Sturmfels : Solving the likelihood equations. Foundations of computational mathematics , 5 4 , p. Notices of the American Mathematical Society , 52 10 , p. Sturmfels : Can biology lead to new theorems? Annual report of the Clay Mathematics Institute , , p. Sullivant : Toric ideals of phylogenetic invariants.

Journal of computational biology , 12 2 , p. De Loera ; D. Haws ; R. Hemmecke ; P. Yoshida : Short rational functions for toric algebra and applications. Journal of symbolic computation , 38 2 , p. Develin and B. Sturmfels : Tropical convexity. Documenta mathematica , 9 , p. Sturmfels : Erratum for 'Tropical convexity' [Doc. Haiman and B. Sturmfels : Multigraded Hilbert schemes. Journal of algebraic geometry , 13 4 , p. Hosten ; D. Maclagan and B.

Sturmfels : Supernormal vector configurations. Journal of algebraic combinatorics : an international journal , 19 3 , p. Sturmfels : Parametric inference for biological sequence analysis. Sturmfels : Tropical geometry of statistical models. Sturmfels : The tropical Grassmannian. Advances in geometry , 4 3 , p. Yu : Classification of six-point metrics. De Loera and B. Sturmfels : Algebraic unimodular counting. Mathematical programming , 96 2 , p. Santos and B. Sturmfels : Higher Lawrence configurations.

Sturmfels : Alexander duality in subdivisions of Lawrence polytopes. Advances in geometry , 3 2 , p. Cattani ; A. Dickenstein and B. Sturmfels : Binomial residues. Annales de l'Institut Fourier , 52 3 , p. De Loera ; F. Discrete and computational geometry , 27 1 , p. Sturmfels : Elimination theory in codimension 2. Journal of symbolic computation , 34 2 , p. Hausel and B. Documenta mathematica , 7 , p. Novik ; A. Postnikov and B. Sturmfels : Syzygies of oriented matroids.

Duke mathematical journal , 2 , p. Bayer ; S. Popescu and B. Sturmfels : Syzygies of unimodular Lawrence ideals. Sturmfels : Rational hypergeometric functions. Compositio mathematica , 2 , p. Sturmfels : A sagbi basis for the quantum Grassmannian. Miller ; B. Sturmfels and K. Yanagawa : Generic and cogeneric monomial ideals. Journal of symbolic computation , 29 , p. Sturmfels : Four counterexamples in combinatorial algebraic geometry. Journal of algebra , 1 , p. Sturmfels : Solving algebraic equations in terms of A-hypergeometric series.

Discrete mathematics , , p. Onn and B. Sturmfels : Cutting corners. Advances in applied mathematics , 23 1 , p. Saito ; B. Takayama : Hypergeometric polynomials and integer programming. Cattani and A. Dickenstein : The search for rational A-hypergeometric functions. Bayer ; I. Peeva and B. Sturmfels : Monomial resolutions. Mathematical research letters , 5 , p. Bayer and B.

Sturmfels : Cellular resolutions of monomial modules. Sturmfels : Residues and resultants. Journal of mathematical sciences , 5 1 , p. Diaconis and B. Sturmfels : Algebraic algorithms for sampling from conditional distributions. The annals of statistics , 26 1 , p. Eisenbud ; I. Proceedings of the American Mathematical Society , 3 , p. Huber ; F. Sturmfels : Numerical Schubert calculus.

Journal of symbolic computation , 26 6 , p. Peeva ; V. Reiner and B. Sturmfels : How to shell a monoid. Mathematische Annalen , 2 , p. Sturmfels : Syzygies of codimension 2 lattice ideals. Mathematische Zeitschrift , 1 , p. Sturmfels : Generic lattice ideals. Journal of the American Mathematical Society , 11 2 , p. Roos and B. Sturmfels : A toric ring with irrational Poincare-Betti series. Proceedings of the Japan Academy : series A, mathematical sciences , 74 7 , p.

Sturmfels : Polynomial equations and convex polytopes. Fulton and B. Sturmfels : Intersection theory on toric varieties. Topology , 36 2 , p. Guckenheimer ; M. Myers and B. Sturmfels : Computing Hopf bifurcations 1.

SIAM journal on numerical analysis , 34 1 , p. Huber and B. Sturmfels : Bernstein's theorem in affine space. Discrete and computational geometry , 17 2 , p. Takayama : Hypergeometric polynomials and integer programming [Japanese]. Applicable algebra in engineering, communication and computing , 8 4 , p.

Thomas and B. Sturmfels : Variation of cost functions in integer programming. Mathematical programming , 77 2 , p. De Loera ; S. Hosten ; F. Sturmfels : The polytope of all triangulations of a point configuration. Documenta mathematica , 1 , p. Eisenbud and B. Sturmfels : Binomial ideals. Duke mathematical journal , 84 1 , p. Journal of algebra , 3 , p. Combinatorica , 15 3 , p. Fulton ; R. MacPherson ; F. Sturmfels : Intersection theory on spherical varieties.

Journal of algebraic geometry , 4 1 , p. Sturmfels : A polyhedral method for solving sparse polynomial systems. Mathematics of computation , 64 , p. Kalkbrener and B. Teachers in math education enter a strong and robust job market upon graduation. There will never be a shortage of need for teachers, especially for those who teach math. There are more than one million teachers employed in high schools in the US, and high school teaching is expected to see an 8 percent increase in jobs over the next decade.

This translates to almost 80, new jobs, many of which will likely be math and STEM related positions. The modern world cannot function without math. Jobs for math teachers will always exist, and the demand for these educators will only grow as the tech economy becomes even larger. A great degree starts with a strong mathematics department, which will help educators to become incredibly knowledgeable about their subject matter.

On top of this, having a program that introduces students to the foundations of secondary education and contemporary issues in math education will give teachers the best possible training, for the future classroom and the job market. These programs are largely residential, meaning you take classes on campus. College Choice considers a multitude of factors including institutional success, student well-being, and financial return to develop rankings.

Student well-being measures how satisfied current and former students felt with the services and programs offered at their school. In the end, we believe these are the 25 Best Math Education Degrees. As it began enrolling students in , UNC is one of three schools able to claim the title of oldest public school in the United States. The school now has 14 schools, through which it offers more than 70 courses of study; they also opened their own hospital in , UNC Health Care, which focuses on both research and treatment, but has taken on a specialization in cancer care.

In Fall , Chapel Hill had around 30, students and 3, academic staff. UNC offers a Bachelor of Science degree in Mathematics, though students entering the program also have the option of pursuing a Bachelor of Arts. The BS option is considered to be more comprehensive, and better prepares students for future study in the field of mathematics. Students pursuing the major are required to take courses such as:. Overall the degree requires a total of credit-hours. Students also have the option of pursuing a BS in Applied Mathematics, which requires taking courses in the natural sciences.

Students interested in pursuing Pure Mathematics or Mathematical Biology have a pre-arranged set of recommended courses. UC San Diego offers more than undergraduate and graduate degree programs to a student population of 28, undergraduate and 7, postgraduate students. San Diego has a number of different Mathematics majors available to its students.

Sample courses from the Math Department include:. Requirements vary based on the route chosen by the student, but each will have general foundations in calculus and other basic mathematics. There are also available minors in Mathematics and Mathematics Education. The minor in Mathematics Education requires 34 units of coursework, which sometimes include courses that are cross-listed with the education department.

The program is intended to prepare students for pursuing a secondary-school teaching credential. Courses in the program include:. The areas covered in these courses encompass the areas of mathematics that are typically tested by the state of Texas and the TExES secondary mathematics examination. In total the program requires completion of credit-hours. Students are allowed to complete four semesters of ROTC in place of pursuing government general education courses. It currently enrolls about 12, students, including both graduate and undergraduate learners.

Its student to faculty ratio ensures that all students have the opportunity to receive personal attention and guidance, enabling them to get more out of their education. Denver has a Bachelor of Science degree program in Mathematics. Required courses in the BS program are:. Students also have the option of pursuing a Bachelor of Arts in Mathematics instead. The main distinction between the programs is that the BS degree requires further upper-level math courses.

Students are required to complete 52 quarter credit-hours of MATH courses, which includes 24 credits of upper-level math courses. Students are allowed to transfer up to quarter credit-hours, but they are required to complete at least the final 45 credit-hours at the University of Denver itself. The school has an affiliation with Churches of Christ. It was originally founded in as the Childers Classical Institute.

In , they had a total enrollment of 4, students, which included just shy of graduate students. Students must choose between tracks in mathematics, actuarial science, applied mathematics, and pure mathematics. Courses within these specializations include:. The BA degree gives students a strong liberal arts background in mathematics, but also allows studies in other disciplines.

The BS degree is a more robust and deeper dive into mathematical theory. ACU offers a concentration in actuarial science, which it bills as one of the most in-demand and highest-paying job fields in the country. The program requires credit-hours within major, depending on the track chosen. Students must also complete hours of elective courses, which again depend on the track and degree plan that the student chooses. Bernard M. Baruch is a part of the City University of New York public school system.

They have around 1, academic staff members and just over 18, total students, most of whom are undergraduates. Baruch offers three different mathematics-based majors, which include a general mathematics major, a financial mathematics major, and an actuarial science major. Course requirements vary between these majors, but sample courses include:. Some majors have specific courses that are prerequisites before a student may begin or declare the major. Students interested in teaching mathematics are recommended to contact the Center for Advertisement and Orientation.

Students may also choose to complete a mathematics minor, which includes a smaller list of courses from the general mathematics major requirements. Students attempting to enter the mathematics department take the COMPASS test, which helps to ensure a minimum level of mathematics competency. Some students who pass this test may also be required to take a continuing studies course before they would be allowed to begin mathematics major.

It is the oldest and largest public university in the University of Minnesota system. They have more than 50, total students, which makes them one of the largest universities in the United States. They also offer specializations in actuarial science, mathematical biology, computer applications of mathematics, and math education. Sample math courses include:. The college also offers specific support and guidance for students who are planning to pursue further graduate study in mathematics.

There is also an undergraduate honors option for students whose academic performance makes them eligible. Students pursuing the Mathematics Education Specialization end up pursuing a Bachelor of Arts degree. The Department of Mathematics offers a number of departmental scholarships, which majors may apply.

It is the third-largest school in the University of California system, enrolling 30, undergraduates and around 7, postgraduates. Students begin the program by taking introductory courses in calculus and linear algebra, before pursuing courses such as:. UC Davis has a recommended course of study for students looking to pursue this exam. Siena College is a private liberal arts college located in Loudonville, NY. They have 3, full-time students, and around academic staff members.

They have three schools within the college, which includes the schools of business, liberal arts, and science. Siena College offers a number of different options for Mathematics majors, which include BA and BS options in general mathematics. Courses in the math program include:. Graduates from the program have gone on to become data analysts, systems administrators, compliance officers, and teachers. Siena has produced more than mathematics teachers in the past 30 years, and more than 90 percent of those graduates are still teaching. Recent graduates have won New York State scholarships for excellence in teaching.

Mansfield was founded in as the Mansfield Classical Seminary, before transitioning into a normal school in the late nineteenth century. The school became a full university in and now has a student population of around 2, Mansfield has a Bachelor of Science program in Mathematics.

The program requires the completion of credit-hours of coursework, and has three possible concentrations: applied mathematics, computer science, and pure mathematics. Students in the program also build ties to K groups, community colleges, and local industry. Many students from local community colleges are also able to transfer into the program.

Noting, for example, that if 2. Similarly, if 2. In Exercises 2. We break out the proof of Theorem 2. Note that the comparison with cones from above is equivalent to the condition 2. Likewise, comparison with cones from below may be restated as 2. Using 2. Thus W is empty. In showing that comparison with cones implies AML, we only used comparison with cones with nonnegative slopes from above, and com- parison with cones with nonpositive slopes from below.

Show that u enjoys comparison with cones. Hint: the proof above needs only minor tweaking. First we use comparison with cones from above, which implies that, using the form 2. The inequality 2. Rewrite 2. We are going to plug 2. First, on the left of 2. Then the inequality of the extremes in 2. It is set up this way to make the next remark easy. Crandall for y, w near x. See Section 3. The full assertions now follow from Section 3.

Formulate a similar condition which guarantees that u enjoys comparison with cones from below in U and conclude that the u of Exercise 2. Note that this class of functions is not C 2 in general, an example being the distance to an interval on the complement of the interval. Hint: There are a number of ways to do this. Crandall Exercise 2. There are a number of ways to do this, including using Exercise 2. Hint: Review Section 2. The only comparisons used by Savin in [49] are those of Exercise 2. The theory we are discussing splits naturally into two halves.

The reason lies in the very notion of a viscosity solution even if this theory applies very well to the Laplace equation. However, this can be done. See Section 4 of [8], where further equivalences are given beyond what is discussed in these notes. Thus there is no generality lost in working with continuous functions at the outset we could have used upper-semicontinuous functions earlier. We have already seen an example of this in Section 2. We regard 2. The other assertions comprise, together with their consequences in Lemma 4.

Then we may replace u w by u x in 4. Thus if u has a local maximum point, it is constant in a ball around that point. We turn to b. We turn to c. To prove d , let the assumptions of 4. Crandall We turn to e. To prove the converse, recall that if 4. By the the proof of Section 2. By Section 2. Therefore we have the following variant of 2.

To use this fact and the estimate implicit in 2. We turn to d. Note the notational peculiarities. We will want to display the function argument later, and the identities 4. First, via Lemma 4. To establish 4. We next show that 4. Using 4. We use this information in 4. Exercise 4. Perform the integration of 4. Show that 4. Observe that 4. It then follows from 4.

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Here one only needs to check the sign of a derivative. Use 4. See Sections 8 and 9 concerning citations for Exercises 4. Clearly, in Exercise 4. The result allows U to be unbounded and a boundary function b which grows at most linearly. Theorem 5. A self-contained presentation of the proof of [9], which does not require familiarity with viscosity solution theory, is given in [8].

This time, however, we fully reduce the result to standard arguments from viscosity solution theory, and we do not render our discussion self-contained in this regard. We just sketch the new proof, as it applies to the case of a bounded domain. The main point is an approximation result.

Proposition 5. This is consistent with the conventions of [31]. The trickiest point is e , which we discuss below. Full details are available in [28]; see also Barron and Jensen [13], Proposition 5. However, they do not consider subsolutions nor note the approximation property f in this case. After that, we present our proof of the most subtle point, e , of Proposition 5.

This sort of nonlinear change of variables is a standard tool in comparison theory of viscosity solutions. Crandall which implies, using 5. It is straightforward to check that these computations are valid in the viscosity sense. It follows immediately from Theorem 3. We now discuss key elements of the proof of Proposition 5. To establish 5. Using 5. Combining this with 5. We continue. According to Lemma 4. Indeed, then, using 5. Hence 5. Exercise 5. Provide the little continuation argument. Proposition 6. One proof was already indicated in Exercise 2. Here is another. Now use Exercise 6.

Exercise 6. The main result of this section is the tool we will use to prove the converse to the proposition above. We are able, in the general case, to obtain curves with similar properties to those discussed in the exercises. Before proving this result, let us give an application. Theorem 6. Proof of Proposition 6. According to the increasing slope estimate 4. Passing to the limit in the relations 6. The property 6.

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However, it is con- siderably more elegant to use curves instead of the discrete versions. In addition, they do not pay attention to the maximality and there is a oversight in their proof of v we learned to include v in the list of properties from [13]. Hint: Exercise 6. His paper also requires Proposition 7.

Proposition 7. We prove b , which is new, and relegate a , which is known, to Ex- ercise 7. Let the assumptions of b hold. This proves 7. The assertion b now follows from Section 7. Exercise 7. Prove, by similar arguments, a of the proposition. In the case a of Proposition 7.

Show, in case a of Proposition 7. Use a of Proposition 7. It is a straightforward generalization and extension of a proof learned from [3] in the Euclidean case. However, the norm need not be the Euclidean norm. Lemma 7. Rearranging 7. We use 7. Returning to 7. Let K be a closed subset of IRn. It is assumed that the reader has read the introduction, but not the main text of this article. We do include some pointers, often parenthetical, to appropriate parts of the main text.

Aronsson derived, among other things, interesting information about the set on which these two functions coincide Exercise 7. This is a very special circumstance. Moreover, in general, the McShane-Whitney extensions have a variety of un- pleasant properties Exercise 2. Crandall using the McShane-Whitney extensions rather than cones. This required the boundary data to be Lipschitz continuous in contrast with Theorem 5. He could not, however, prove the uniqueness or stability Theorem 5. With the technology of the times, this is about all anyone could have proved.

Moreover, the question of uniqueness of the function whose existence Aronsson proved would be un- settled for 26 years! Aronsson himself made the gap more evident in the paper [4] in which he produced examples of U, b for which he could show that the problem had no C 2 solution. This work also contained a penetrating analysis of classical solutions of the pde.

However, all of these results are false in the generality of viscosity solutions of the equation see below , which appear as the perfecting instrument of the theory. A rich supply of other solutions was provided as well. Evans in in [32]. They are summarized in [31], which contains a detailed history.

In using [31], note that we are thinking here of G being an increasing function of the Hessian, D2 u, not a decreasing function. As mentioned, in R. Thus, after 26 years, the existence of absolutely minimizing functions as- suming given boundary values was known Aronsson and Jensen , and, at last, the uniqueness Jensen. Among other contributions was an lovely new uniqueness proof by G. Barles and J. Roughly speaking, this proof couples some penetrating observations to the standard machinery of viscosity solutions to reach the same conclusions as Jensen, but without obstacle problems or integral estimates.

This was the state of the art until the method of Section 5 was deployed. After existence and uniqueness, one wants to know about regularity. In our Lemma 4.

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Jensen also gave similar arguments to establish related results. See also [19]. Lindqvist and J. Crandall valid at points where Du x exists. Our equation 4. The appropriate Harnack inequal- ity is closely related to regularity issues for classes of elliptic and parabolic equations, which is one of the reasons to be interested in it. This sort of observation is a routine matter in the viscosity solution theory; there is much else in that paper.

One point of concern is the relationship between the notions of viscosity solutions and solutions in the sense of distributions for the p-Laplace equation. See Juutinen, Lindqvist and Manfredi, [39] and Ishii, [35]. When u enjoys both of these, it enjoys comparison with cones.

At last: a result asserting the existence of a derivative at a particular point. Of course, comparison with cones had already been used by Jensen to derive Lipschitz continuity, and was used contemporaneously with [29] by Bhattacharya [19], etc. Blowups are Linear The next piece of evidence in the regularity mystery was provided by Crandall and Evans, [30].

Using tools from [29] and some new arguments, all rather simple. It would be quite interesting to have more information about it. Probably not. Much of this theory was developed by N. Barron and R. Jensen in collaboration with various coauthors. We refer the reader to the review article by Barron [10] for an overview up to the time of its writing, and to Jensen, Barron and Wang [11], [12] for more recent advances.

In particular, [12] is concerned with vector-valued functions u in the set up we explain below for scalar functions u. We take the following point of view in giving selected references here. None of these topics are mentioned in this work of limited aims. So we are selective, sticking to variants of the main thrust of this article. We write the generic arguments of H as H x, r, p. The reader should go to the references given for this, if it is omitted. The technical conditions under which these authors established this are more severe than those given in [27], corresponding to the more transparent proof given in this paper.

Yu [53] proved several things in this direction. Secondly, he provided an example to show that the answer is no in general if H is merely quasi-convex, but otherwise nice enough. This is not an issue of smoothness of H. This is the generality of [8]. Other arguments given herein do not easily generalize to this case. The case in which H x, p is convex in p, along with other technical as- sumptions, Yu [53] uses comparison type arguments to show similar results.

The general case H x, p is treated as well, in full quasi-convex generality and with minimum regularity on H, in Champion and De Pascale, [25]. There are other interesting things in this paper. Here the Lipschitz functional is primary, and the authors show that the associated absolutely minimizing property is equivalent to a straightforward comparison with distance functions property. In particular, it makes clear that the fact that the distance functions do not themselves satisfy a full comparison with distance function property noted in [8] is not an impediment to the full characterization, properly put.

The paper of Wang, currently available on his website, contains a very nice introduction to which we refer for a further overview. In this context, the results explained herein are quite special, correspond- ing, say, to merely deriving the basic properties of harmonic functions via their mean value property, and all sorts of generalizations are treated later by various theories, not using the mean value property Poisson equations, more general elliptic operators, time dependent versions and so on. This same elegant relation plays a role in approximation arguments given in Le Gruyer [41] and Oberman [47].

The results of Exercise 4. We could go on, but it is time to stop. Aronsson, G. III, Ark. Barles, G. Equations 26 , — Barron, E. C Math. Bieske, T. Bhattacharya, T. Torino , Special Issue, 15—68 Champion, T. Crandall, M. Ra- tional Mech. Evans, L. Crandall Gariepy, R. Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers, preprint, Ishii, Hitoshi, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions.

Jensen, R. Juutinen, P. On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. Le Gruyer, E. Lindqvist, P. Madrid 10 — McShane, E. Oberman, A. Peres, Y. Savin, O. Wang, C. To appear in Trans. Whitney, H. Wu, Y. Yu, Y. A nice recent survey is Kaloshin [K]; and see also the survey paper [E6]. Theorem 1.

Assume we can solve 1. We next study the Hamiltonian dynamics in the new variables: Theorem 1. The point is that these dynamics are trivial. This is in general impossible, since our PDE 1. But KAM Kolmogorov-Arnold-Moser theory tells us that we can in fact carry out this procedure for a Hamiltonian that is an appropriate small per- turbation of a Hamiltonian depending only on p.

Evans Generating Functions, Linearization. Owing to 1. Now according to 1. Fourier series Next we use Fourier series to try to build a solution v of 1. Plug 1. The full details of this procedure are very complicated. There are two approaches to these issues: the Lagrangian, dynamical systems methods dis- cussed in this section and the nonlinear PDE methods explained in the next section. Idea of Proof. This is a fairly standard derivative estimate for solutions of the Euler—Lagrange equation E-L. Evans Theorem 2.

We apply Theorem 2. Evans Remark. We work in the tangent bundle for the Lagrangian viewpoint, and in the cotangent bundle for the Hamiltonian viewpoint. The constant from Theorem 2. Evans Idea of Proof. We must now manufacture a measure giving equality above. One goal of weak KAM theory is studying the structure of the Mather set and the related Aubry set , in terms of the underlying Hamiltonian dynamics. See Fathi [F5] for much more. Evans Theorem 3. Uniqueness of c P. Then Proof.

Assume for the rest of the discussion that v is smooth. See my old paper [E2] for what to do when v is not smooth. Idea of proof. Therefore 3. It then follows from 3. Evans 2. In view of 3. Again, see [E-G1] for the details. We propose to study the asymptotic limit of the PDE 4. We will discover that the structure of weak KAM theory appears in the limit. We illustrate some uses of the approximation 4. Indeed, a direct computation cf. Suppose 4. As discussed in Section 1, if we could really change to the action-angle variables X, P according to 1. In view of the nonresonance condition 4.

Observe next from 4. Recall 4. Using formula 4.

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Tn This proves 4. It would be interesting to have some more careful numerical studies here, as for instance in Gomes-Oberman [G-O]. See also Gomes [G1], [G2] for some more developments. One fundamental question is just how, and if, Mather sets can act as global replacements for the classical KAM invariant tori. Some formal relationships are sketched in my expository paper [E5].

The case of Hamiltonians which are coercive, but nonconvex and nonqua- siconvex in p, is completely open. This highly de- generate, highly nonlinear elliptic equation occurs quite naturally from the variational construction in Section 4, but to my knowledge has no interpre- tation in terms of dynamical systems. References [Am] L. Anantharaman, Gibbs measures and semiclassical approximation to action-minimizing measures, Transactions AMS, to appear. Bernard and B. Concordel, Periodic homogenization of Hamilton-Jacobi equations I: additive eigenvalues and variational formula, Indiana Univ.

Edinburgh , — Third printing, Evans and D. Paris Sr. I Math. Fathi, Orbites heteroclines et ensemble de Peierls, C. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. Fathi and A. Forni and J. Graffi, Springer, Goldstein, Classical mechanics 2nd ed , Addison-Wesley, Analysis 35 , — Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity 10 , Gomes and A. Control Optim. Iturriaga and H.

Sanchez-Morgado, On the stochastic Aubry-Mather theory, to appear. Lions, G. Papanicolaou, and S. Varadhan, Homogenization of Hamilton—Jacobi equations, unpublished, circa Marcellini, Regularity for some scalar variational problems under general growth conditions, J Optimization Theory and Applications 90 , — Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Annali Scuola Normale di Pisa 23 , 1— Mascolo and A. Mather, Minimal measures, Comment. Math Helvetici 64 , — Zeitschrift , — Evans [P-R] I.

Percival and D. The two model functionals that we shall consider in the sequel are: the perimeter of a set E in IRn and the Dirichlet integral of a scalar function u. This fact is classical when E is a smooth open set and u is a C 1 function [22], [21]. However, an approximation argument gives no information about the equality case. Let us start by recalling what the Steiner symmetrization of a measurable set E is. For simplicity, and without loss of generality, in the sequel we shall always consider the symmetrization of E in the vertical direction.

Here and in the sequel we denote by Lk the Lebesgue measure in IRk. Moreover, it is not hard to see that the diameter of E de- creases under Steiner symmetrization, i. Denoting by P E the perimeter of a measurable set in IRn , the following result states that the perimeter too decreases under Steiner symmetrization.

Notice also that from 1. In particular, the integration by parts formula 1. Proposition 1. Notice also that Proposition 1. If we translate Theorem 1. In the sequel, we shall denote by Da u the absolutely continuous part of Du with respect to Lebesgue measure Ln. The singular part of Du will be denoted by Ds u. Next result is an essential tool for studying the behavior of Steiner sym- metrization with respect to perimeter. Lemma 1. Inequality 1. Notice that the argument used in the proof of Lemma 1. In fact, in this case, by applying 1.

As we have ob- served in 1. Let GE be a Borel set satisfying 1. From 1. Fusco Remark 1. To prove inequality 2.