The Experts below are selected from a list of 15 Experts worldwide ranked by ideXlab platform

John R. Stembridge - One of the best experts on this subject based on the ideXlab platform.

  • ERRATUM: EXPLICIT MATRICES FOR IRREDUCIBLE REPRESENTATIONS OF WEYL GROUPS
    2005
    Co-Authors: John R. Stembridge
    Abstract:

    In the paragraph Preceding Lemma 2.4 of [1], we defined an ordering s1,...,sn of the simple reflections to be graceful if for all i<j<k,wehaveiand k adjacent in the Coxeter graph (i.e., si and sk do not commute) only if j and k are also adjacent. We also wrote that it is “easy to check that the standard order we have chosen for each Weyl group is graceful. ” However, this is not correct for the orderings we used for the Weyl groups of type E; in fact, the triple (i, j, k) =(1, 2, 3) clearly violates the condition. One way to fix this error is simply to interchange s2 and s3 in the standard ordering. The remainder of the paper remains completely unaffected; however, this does change the matrices that our algorithm would construct. A better way to fix the error is to improve the definition. We should instead define the ordering s1,...,sn to be graceful if for all i<jsuch that i and j are adjacent in the Coxeter graph, each of si,si+1,...,sj−1 belong to distinct irreducible components of the parabolic subgroup generated by s1,s2,...,sj−1. The orderings we used for the simple reflections in the Weyl groups of type E do satisfy this weaker condition, Lemma 2.4 remains valid as stated, and the first paragraph of the proof now simplifies to the following: “For clarity, we will assume that r = n−3; the general case follows by essentially the same argument. Given that the ordering is graceful, it follows that Wn−1 is a direct product of parabolic subgroups, say WI × WJ, with generating sets I and J that include sn−1 and sn−2, respectively.” No other changes are needed. In particular, the statements and proofs of Proposition 2.5 and Corollary 2.6 are valid in their original forms

  • Addendum to “Explicit matrices for irreducible representations of Weyl groups”
    2004
    Co-Authors: John R. Stembridge
    Abstract:

    In the paragraph Preceding Lemma 2.4 of [1], we defined an ordering s1,..., sn of the simple reflections to be graceful if for all i < j < k, we have i and k adjacent in the Coxeter graph (i.e., si and sk do not commute) only if j and k are also adjacent. We also wrote that it is “easy to check that the standard order we have chosen for each Weyl group is graceful. ” However, this is not correct for the orderings we used for the Weyl groups of type E; in fact, the triple (i, j, k) = (1, 2, 3) clearly violates the condition. One way to fix this error is simply to interchange s2 and s3 in the standard ordering. The remainder of the paper remains completely unaffected; however, this does change the matrices that our algorithm would construct. A better way to fix the error is to improve the definition. We should instead define the ordering s1,..., sn to be graceful if for all i < j such that i and j are adjacent in the Coxeter graph, each of si, si+1,..., sj−1 belong to distinct irreducible components of the parabolic subgroup generated by s1, s2,...,sj−1. The orderings we used for the simple reflections in the Weyl groups of type E do satisfy this weaker condition, Lemma 2.4 remains valid as stated, and the first paragraph of the proof now simplifies to the following: “For clarity, we will assume that r = n − 3; the general case follows by essentially the same argument. Given that the ordering is graceful, it follows that Wn−1 is a direct product of parabolic subgroups, say WI × WJ, with generating sets I and J that include sn−1 and sn−2, respectively.” No other changes are needed. In particular, the statements and proofs of Proposition 2.5 and Corollary 2.6 are valid in their original forms

Le, Nam Q. - One of the best experts on this subject based on the ideXlab platform.

  • Remarks on the Green's function of the linearized Monge-Amp\`ere operator
    2015
    Co-Authors: Le, Nam Q.
    Abstract:

    In this note, we obtain sharp bounds for the Green's function of the linearized Monge-Amp\`ere operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge-Amp\`ere measure satisfying a doubling condition. Our result is an affine invariant version of the classical result of Littman-Stampacchia-Weinberger for uniformly elliptic operators in divergence form. We also obtain the $L^{p}$ integrability for the gradient of the Green's function in two dimensions. As an application, we obtain a removable singularity result for the linearized Monge-Amp\`ere equation.Comment: v2: Fix a missing sentence Preceding Lemma 3.3; To appear in Manuscripta Mat

K. Newey - One of the best experts on this subject based on the ideXlab platform.

  • Econometrica Supplementary Material SUPPLEMENT TO “LOCAL IDENTIFICATION OF NONPARAMETRIC AND SEMIPARAMETRIC MODELS”
    2014
    Co-Authors: Xiahong Chen, Victor Chernozhukov, Sokbae Lee, Whitney Newey, Xiaohong Chen, K. Newey
    Abstract:

    THIS SUPPLEMENTAL MATERIAL GIVES PROOFS for the results of Sections 4 and 5 of the paper as well as some additional results and discussions. S1. DISCUSSION AND AN EXAMPLE FOR Lemma 3 The first item we consider is discussion and examples of the genericity re-sult in Lemma 3. Below, we provide examples for Lemma 3 for the case A = B = L2[01] that highlight the range of algorithms permitted by condi-tions 1 and 2 Preceding Lemma 3, including cases where various restrictions on m ′ are imposed: boundedness, compactness, weak positivity, and density restrictions. Genericity arguments use the idea of randomization, and are of-ten employed in economic theory, functional analysis, and probability the-ory; see, for example, Anderson and Zame (2000), Marcus and Pisier (1981), Ledoux and Talagrand (2011). Andrews (2011) previously used a related no-tion of genericity, called prevalence within bounded sets, to argue that rich classes of operators induced by densities in nonparametric IV are L2-complete. Though inspired in part by Andrews (2011), Lemma 3 of the main articl

Victor Chernozhukov - One of the best experts on this subject based on the ideXlab platform.

  • Econometrica Supplementary Material SUPPLEMENT TO “LOCAL IDENTIFICATION OF NONPARAMETRIC AND SEMIPARAMETRIC MODELS”
    2014
    Co-Authors: Xiahong Chen, Victor Chernozhukov, Sokbae Lee, Whitney Newey, Xiaohong Chen, K. Newey
    Abstract:

    THIS SUPPLEMENTAL MATERIAL GIVES PROOFS for the results of Sections 4 and 5 of the paper as well as some additional results and discussions. S1. DISCUSSION AND AN EXAMPLE FOR Lemma 3 The first item we consider is discussion and examples of the genericity re-sult in Lemma 3. Below, we provide examples for Lemma 3 for the case A = B = L2[01] that highlight the range of algorithms permitted by condi-tions 1 and 2 Preceding Lemma 3, including cases where various restrictions on m ′ are imposed: boundedness, compactness, weak positivity, and density restrictions. Genericity arguments use the idea of randomization, and are of-ten employed in economic theory, functional analysis, and probability the-ory; see, for example, Anderson and Zame (2000), Marcus and Pisier (1981), Ledoux and Talagrand (2011). Andrews (2011) previously used a related no-tion of genericity, called prevalence within bounded sets, to argue that rich classes of operators induced by densities in nonparametric IV are L2-complete. Though inspired in part by Andrews (2011), Lemma 3 of the main articl

Sokbae Lee - One of the best experts on this subject based on the ideXlab platform.

  • Econometrica Supplementary Material SUPPLEMENT TO “LOCAL IDENTIFICATION OF NONPARAMETRIC AND SEMIPARAMETRIC MODELS”
    2014
    Co-Authors: Xiahong Chen, Victor Chernozhukov, Sokbae Lee, Whitney Newey, Xiaohong Chen, K. Newey
    Abstract:

    THIS SUPPLEMENTAL MATERIAL GIVES PROOFS for the results of Sections 4 and 5 of the paper as well as some additional results and discussions. S1. DISCUSSION AND AN EXAMPLE FOR Lemma 3 The first item we consider is discussion and examples of the genericity re-sult in Lemma 3. Below, we provide examples for Lemma 3 for the case A = B = L2[01] that highlight the range of algorithms permitted by condi-tions 1 and 2 Preceding Lemma 3, including cases where various restrictions on m ′ are imposed: boundedness, compactness, weak positivity, and density restrictions. Genericity arguments use the idea of randomization, and are of-ten employed in economic theory, functional analysis, and probability the-ory; see, for example, Anderson and Zame (2000), Marcus and Pisier (1981), Ledoux and Talagrand (2011). Andrews (2011) previously used a related no-tion of genericity, called prevalence within bounded sets, to argue that rich classes of operators induced by densities in nonparametric IV are L2-complete. Though inspired in part by Andrews (2011), Lemma 3 of the main articl