TURBO-STRAIGHTENING FOR DECOMPOSITION INTO STANDARD BASES

1992 ◽  
Vol 02 (03) ◽  
pp. 275-290 ◽  
Author(s):  
CHRISTOPHE CARRE ◽  
ALAIN LASCOUX ◽  
BERNARD LECLERC
Keyword(s):  
Scalar Product ◽  
Young Tableaux ◽  
Symmetric Polynomials ◽  
Harmonic Polynomials ◽  
Technical Tool ◽  
Linear Basis ◽  
Plücker Relations ◽  
By Products ◽  
Main Technical Tool ◽  
The Ideal

Specht and Hodge have shown that the space generated by products of minors of a matrix admits a linear basis in bijection with Young tableaux. The decomposition of any element into this basis is called straightening and corresponds to the iterative use of Plücker relations. Thanks to a well-known isomorphism between the space of harmonic polynomials and the space of polynomials modulo the ideal generated by symmetric polynomials, we can now use as a main technical tool the canonical scalar product on this later space. This leads to a different, and possibly better, algorithm for straightening.

scholarly journals Square-integrable representations of non-unimodular groups

1976 ◽  
Vol 15 (1) ◽  
pp. 1-12 ◽  
Author(s):  
A.L. Carey
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Hilbert Algebra ◽  
Technical Tool ◽  
Integrable Representation ◽  
Hilbert Algebras ◽  
Orthogonality Relations ◽  
Square Integrable Representation ◽  
Main Technical Tool

In the last three years a number of people have investigated the orthogonality relations for square integrable representations of non-unimodular groups, extending the known results for the unimodular case. The results are stated in the language of left (or generalized) Hilbert algebras. This paper is devoted to proving the orthogonality relations without recourse to left Hilbert algebra techniques. Our main technical tool is to realise the square integrable representation in question in a reproducing kernel Hilbert space.

scholarly journals The non-archimedean SYZ fibration

2019 ◽  
Vol 155 (5) ◽  
pp. 953-972 ◽  
Author(s):  
Johannes Nicaise ◽  
Chenyang Xu ◽  
Tony Yue Yu
Keyword(s):  
Explicit Description ◽  
Affine Structure ◽  
Technical Tool ◽  
One Dimensional ◽  
Codimension Two ◽  
Main Technical Tool

We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away from a codimension-two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt (divisorially log terminal) models along one-dimensional strata.

scholarly journals Co-tame polynomial automorphisms

2019 ◽  
Vol 29 (05) ◽  
pp. 803-825 ◽  
Author(s):  
Eric Edo ◽  
Drew Lewis
Keyword(s):  
Characteristic Zero ◽  
Independent Interest ◽  
Technical Tool ◽  
Polynomial Automorphism ◽  
Polynomial Automorphisms ◽  
Negative Results ◽  
Dimension Formula ◽  
Positive Results ◽  
Main Technical Tool

A polynomial automorphism of [Formula: see text] over a field of characteristic zero is called co-tame if, together with the affine subgroup, it generates the entire tame subgroup. We prove some new classes of automorphisms of [Formula: see text], including nonaffine [Formula: see text]-triangular automorphisms, are co-tame. Of particular interest, if [Formula: see text], we show that the statement “Every [Formula: see text]-triangular automorphism is either affine or co-tame” is true if and only if [Formula: see text]; this improves upon positive results of Bodnarchuk (for [Formula: see text], in any dimension [Formula: see text]) and negative results of the authors (for [Formula: see text], [Formula: see text]). The main technical tool we introduce is a class of maps we term translation degenerate automorphisms; we show that all of these are either affine or co-tame, a result that may be of independent interest in the further study of co-tame automorphisms.

scholarly journals Semi-infinite Plücker Relations and Weyl Modules

2018 ◽  
Vol 2020 (14) ◽  
pp. 4357-4394 ◽  
Author(s):  
Evgeny Feigin ◽  
Ievgen Makedonskyi
Keyword(s):  
Type A ◽  
Character Formula ◽  
Flag Varieties ◽  
Coordinate Ring ◽  
Weyl Modules ◽  
Plücker Relations ◽  
Plücker Embedding ◽  
Formal Version ◽  
The Ideal

Abstract The goal of this paper is two-fold. First, we write down the semi-infinite Plücker relations, describing the Drinfeld–Plücker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, that is, the quotient by the ideal generated by the semi-infinite Plücker relations. We establish the isomorphism with the algebra of dual global Weyl modules and derive a new character formula.

scholarly journals Explicit Correlation Amplifiers for Finding Outlier Correlations in Deterministic Subquadratic Time

Algorithmica ◽  
2020 ◽  
Vol 82 (11) ◽  
pp. 3306-3337
Author(s):  
Matti Karppa ◽  
Petteri Kaski ◽  
Jukka Kohonen ◽  
Padraig Ó Catháin
Keyword(s):  
Positive Integer ◽  
Binary Data ◽  
Randomized Algorithm ◽  
Time Algorithm ◽  
High Dimensionality ◽  
Parameter Range ◽  
Technical Tool ◽  
Similarity Join ◽  
Explicit Correlation ◽  
Main Technical Tool

Abstract We derandomize Valiant’s (J ACM 62, Article 13, 2015) subquadratic-time algorithm for finding outlier correlations in binary data. This demonstrates that it is possible to perform a deterministic subquadratic-time similarity join of high dimensionality. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant’s randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders by Reingold et al. (Ann Math 155(1):157–187, 2002). We say that a function $$f:\{-1,1\}^d\rightarrow \{-1,1\}^D$$ f : { - 1 , 1 } d → { - 1 , 1 } D is a correlation amplifier with threshold $$0\le \tau \le 1$$ 0 ≤ τ ≤ 1 , error $$\gamma \ge 1$$ γ ≥ 1 , and strength p an even positive integer if for all pairs of vectors $$x,y\in \{-1,1\}^d$$ x , y ∈ { - 1 , 1 } d it holds that (i) $$|\langle x,y\rangle |<\tau d$$ | ⟨ x , y ⟩ | < τ d implies $$|\langle f(x),f(y)\rangle |\le (\tau \gamma )^pD$$ | ⟨ f ( x ) , f ( y ) ⟩ | ≤ ( τ γ ) p D ; and (ii) $$|\langle x,y\rangle |\ge \tau d$$ | ⟨ x , y ⟩ | ≥ τ d implies $$\left (\frac{\langle x,y\rangle }{\gamma d}\right )^pD \le \langle f(x),f(y)\rangle \le \left (\frac{\gamma \langle x,y\rangle }{d}\right )^pD$$ ⟨ x , y ⟩ γ d p D ≤ ⟨ f ( x ) , f ( y ) ⟩ ≤ γ ⟨ x , y ⟩ d p D .

scholarly journals Uniformity, universality, and computability theory

2017 ◽  
Vol 17 (01) ◽  
pp. 1750003
Author(s):  
Andrew S. Marks
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Technical Tool ◽  
Free Products ◽  
Borel Sets ◽  
Borel Equivalence Relations ◽  
Strengthened Form ◽  
Shift Action ◽  
Natural Classes ◽  
Main Technical Tool

We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.

scholarly journals Effective Density for Inhomogeneous Quadratic Forms I: Generic Forms and Fixed Shifts

2020 ◽  
Author(s):  
Anish Ghosh ◽  
Dubi Kelmer ◽  
Shucheng Yu
Keyword(s):  
Quadratic Forms ◽  
Effective Density ◽  
Independent Interest ◽  
Diophantine Condition ◽  
Technical Tool ◽  
Optimal Density ◽  
Density Result ◽  
Similar Density ◽  
Main Technical Tool

Abstract We establish effective versions of Oppenheim’s conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result, which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the 2nd moment of Siegel transforms on certain congruence quotients of $SL_n(\mathbb{R}),$ which we believe to be of independent interest. In a sequel, we use different techniques to treat the companion problem concerning generic shifts and fixed quadratic forms.

Heat Kernels and Green Functions on Metric Measure Spaces

2014 ◽  
Vol 66 (3) ◽  
pp. 641-699 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Jiaxin Hu
Keyword(s):  
Exit Time ◽  
Metric Measure Spaces ◽  
Doubling Property ◽  
Technical Tool ◽  
Capacity Estimate ◽  
Gaussian Estimate ◽  
Mean Exit Time ◽  
Measure Spaces ◽  
Volume Doubling ◽  
Main Technical Tool

AbstractWe prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling property, the elliptic Harnack inequality, and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball that uses two-sided estimates of a Green function in a ball.

Locally Optimal Tests against Periodic Autoregression: Parametric and Nonparametric Approaches

1996 ◽  
Vol 12 (1) ◽  
pp. 88-112 ◽  
Author(s):  
Mohamed Bentarzi ◽  
Marc Hallin
Keyword(s):  
Lagrange Multiplier ◽  
Periodic Structures ◽  
Theory And Practice ◽  
Multiple Time ◽  
Unpublished Manuscript ◽  
Technical Tool ◽  
Optimal Tests ◽  
Asymptotically Optimal Tests ◽  
Lagrange Multiplier Tests ◽  
Main Technical Tool

Locally asymptotically optimal tests are derived for the null hypothesis of traditional AR dependence, with unspecified AR coefficients and unspecified innovation densities, against an alternative of periodically correlated AR dependence. Parametric and nonparametric rank-based versions are proposed. Local powers and asymptotic relative efficiencies (with respect, e.g., to the corresponding Gaussian Lagrange multiplier tests proposed in Ghysels and Hall [1992, “Lagrange Multiplier Tests for Periodic Structures,” unpublished manuscript, CRDE, Montreal] and Liitkepohl [1991, Introduction to Multiple Time Series Analysis, Berlin: Springer-Verlag; 1991, pp. 243–264, in W.E. Griffiths, H. Liitkepohl, & M.E. Block (eds.), Readings in Econometric Theory and Practice, Amsterdam: North-Holland] are computed explicitly; a rank-based test of the van der Waerden type is proposed, for which this ARE is uniformly larger than 1. The main technical tool is Le Cam's local asymptotic normality property.

scholarly journals THE EXOTIC (HIGHER ORDER LÉVY) LAPLACIANS GENERATE THE MARKOV PROCESSES GIVEN BY DISTRIBUTION DERIVATIVES OF WHITE NOISE

2013 ◽  
Vol 16 (03) ◽  
pp. 1350020 ◽  
Author(s):  
LUIGI ACCARDI ◽  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
White Noise ◽  
Higher Order ◽  
Technical Tool ◽  
Arithmetic Means ◽  
Infinite Dimensional ◽  
Second Derivatives ◽  
Lévy Laplacian ◽  
Order Arithmetic ◽  
Derivatives Of ◽  
Main Technical Tool

We introduce, for each a ∈ ℝ+, the Brownian motion associated to the distribution derivative of order a of white noise. We prove that the generator of this Markov process is the exotic Laplacian of order 2a, given by the Cesàro mean of order 2a of the second derivatives along the elements of an orthonormal basis of a suitable Hilbert space (the Cesàro space of order 2a). In particular, for a = 1/2 one finds the usual Lévy Laplacian, but also in this case the connection with the 1/2-derivative of white noise is new. The main technical tool, used to achieve these goals, is a generalization of a result due to Accardi and Smolyanov5 extending the well-known Cesàro theorem to higher order arithmetic means. These and other estimates allow to prove existence of the heat semi-group associated to any exotic Laplacian of order ≥ 1/2 and to give its explicit expression in terms of infinite dimensional Fourier transform.

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