Expand all Collapse all | Results 1 - 25 of 30 |
1. CJM 2013 (vol 65 pp. 1287)
$K$-theory of Furstenberg Transformation Group $C^*$-algebras The paper studies the $K$-theoretic invariants of the crossed product
$C^{*}$-algebras associated with an important family of homeomorphisms
of the tori $\mathbb{T}^{n}$ called Furstenberg transformations.
Using the Pimsner-Voiculescu theorem, we prove that given $n$, the
$K$-groups of those crossed products, whose corresponding $n\times n$
integer matrices are unipotent of maximal degree, always have the same
rank $a_{n}$. We show using the theory developed here that a claim
made in the literature about the torsion subgroups of these $K$-groups
is false. Using the representation theory of the simple Lie algebra
$\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the
number of ways that $0$ can be represented as a sum of integers
between $-n$ and $n$ (with no repetitions). By adapting an argument
of van Lint (in which he answered a question of ErdÅs), a simple,
explicit formula for the asymptotic behavior of the sequence
$\{a_{n}\}$ is given. Finally, we describe the order structure of the
$K_{0}$-groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification
results of H. Lin and N. C. Phillips.
Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20 |
2. CJM 2012 (vol 65 pp. 1020)
Monotone Hurwitz Numbers in Genus Zero Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.
Keywords:Hurwitz numbers, matrix models, enumerative geometry Categories:05A15, 14E20, 15B52 |
3. CJM 2011 (vol 63 pp. 1364)
The Cubic Dirac Operator for Infinite-Dimensonal Lie Algebras
Let $\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}} \mathfrak{g}_i$ be an infinite-dimensional graded
Lie algebra, with $\dim\mathfrak{g}_i<\infty$, equipped with a non-degenerate
symmetric bilinear form $B$ of degree $0$. The quantum Weil algebra
$\widehat{\mathcal{W}}\mathfrak{g}$ is a completion of the tensor product of the
enveloping and Clifford algebras of $\mathfrak{g}$. Provided that the
Kac-Peterson class of $\mathfrak{g}$ vanishes, one can construct a cubic Dirac
operator $\mathcal{D}\in\widehat{\mathcal{W}}(\mathfrak{g})$, whose square is a quadratic Casimir
element. We show that this condition holds for symmetrizable
Kac-Moody algebras. Extending Kostant's arguments, one obtains
generalized Weyl-Kac character formulas for suitable ``equal rank''
Lie subalgebras of Kac-Moody algebras. These extend the formulas of
G. Landweber for affine Lie algebras.
Categories:22E65, 15A66 |
4. CJM 2010 (vol 63 pp. 413)
Generating Functions for Hecke Algebra Characters
Certain polynomials in $n^2$ variables that serve as generating
functions for symmetric group characters are sometimes called
($S_n$) character immanants.
We point out a close connection between the identities of
Littlewood--Merris--Watkins
and Goulden--Jackson, which relate $S_n$ character immanants
to the determinant, the permanent and MacMahon's Master Theorem.
From these results we obtain a generalization
of Muir's identity.
Working with the quantum polynomial ring and the Hecke algebra
$H_n(q)$, we define quantum immanants that are generating
functions for Hecke algebra characters.
We then prove quantum analogs of the Littlewood--Merris--Watkins identities
and selected Goulden--Jackson identities
that relate $H_n(q)$ character immanants to
the quantum determinant, quantum permanent, and quantum Master Theorem
of Garoufalidis--L\^e--Zeilberger.
We also obtain a generalization of Zhang's quantization of Muir's
identity.
Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring Categories:15A15, 20C08, 81R50 |
5. CJM 2010 (vol 63 pp. 3)
Free Bessel Laws
We introduce and study a remarkable family of real probability
measures $\pi_{st}$ that we call free Bessel laws. These are related
to the free Poisson law $\pi$ via the formulae
$\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our
study includes definition and basic properties, analytic aspects
(supports, atoms, densities), combinatorial aspects (functional
transforms, moments, partitions), and a discussion of the relation
with random matrices and quantum groups.
Keywords:Poisson law, Bessel function, Wishart matrix, quantum group Categories:46L54, 15A52, 16W30 |
6. CJM 2010 (vol 62 pp. 758)
General Preservers of Quasi-Commutativity Let ${ M}_n$ be the algebra of all $n \times n$ matrices over $\mathbb{C}$. We say that $A, B \in { M}_n$ quasi-commute if there exists a nonzero $\xi \in \mathbb{C}$ such that $AB = \xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi \colon M_n \to M_n$ which preserve quasi-commutativity in both directions.
Keywords:general preservers, matrix algebra, quasi-commutativity Categories:15A04, 15A27, 06A99 |
7. CJM 2009 (vol 62 pp. 109)
Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices
with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$.
All possible inertia values, ranks, and multiple eigenvalues
of $A + B$ are determined. Extension of the results to the sum of $k$ matrices
with $k > 2$ and connections of the results to other subjects such
as algebraic combinatorics are also discussed.
Keywords:complex Hermitian matrices, real symmetric matrices, inertia, rank, multiple eigenvalues Categories:15A42, 15A57 |
8. CJM 2008 (vol 60 pp. 1050)
Adjacency Preserving Maps on Hermitian Matrices Hua's fundamental theorem of the geometry of hermitian matrices
characterizes bijective maps on the space of all $n\times n$
hermitian matrices preserving adjacency in both directions.
The problem of possible improvements
has been open for a while. There are three natural problems here.
Do we need the bijectivity assumption? Can we replace the
assumption of preserving adjacency in both directions by the
weaker assumption of preserving adjacency in one direction only?
Can we obtain such a characterization for maps acting between the
spaces of hermitian matrices of different sizes? We answer all
three questions for the complex hermitian matrices, thus obtaining
the optimal structural result for adjacency preserving maps on
hermitian matrices over the complex field.
Keywords:rank, adjacency preserving map, hermitian matrix, geometry of matrices Categories:15A03, 15A04, 15A57, 15A99 |
9. CJM 2008 (vol 60 pp. 1149)
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle We study the geometry, topology and Lebesgue measure of the set of
monic conjugate reciprocal polynomials of fixed degree with all
roots on the unit circle. The set of such polynomials of degree $N$
is naturally associated to a subset of $\R^{N-1}$. We calculate
the volume of this set, prove the set is homeomorphic to the $N-1$
ball and that its isometry group is isomorphic to the dihedral
group of order $2N$.
Categories:11C08, 28A75, 15A52, 54H10, 58D19 |
10. CJM 2008 (vol 60 pp. 923)
Endomorphisms of Kronecker Modules Regulated by Quadratic Algebra Extensions of a Function Field The Kronecker modules $\mathbb{V}(m,h,\alpha)$, where $m$ is a positive integer, $h$ is
a height function, and $\alpha$ is a $K$-linear functional on the
space $K(X)$ of rational functions in one variable $X$ over an
algebraically closed field $K$, are models for the family of all
torsion-free rank-2 modules that are extensions of finite-dimensional
rank-1 modules. Every such module comes with a regulating polynomial
$f$ in $K(X)[Y]$. When the endomorphism algebra of $\mathbb{V}(m,h,\alpha)$ is
commutative and non-trivial, the regulator $f$ must be quadratic in
$Y$. If $f$ has one repeated root in $K(X)$, the endomorphism algebra
is the trivial extension $K\ltimes S$ for some vector space $S$. If
$f$ has distinct roots in $K(X)$, then the endomorphisms form a
structure that we call a bridge. These include the coordinate rings
of some curves. Regardless of the number of roots in the regulator,
those $\End\mathbb{V}(m,h,\alpha)$ that are domains have zero radical. In addition,
each semi-local $\End\mathbb{V}(m,h,\alpha)$ must be either a trivial extension
$K\ltimes S$ or the product $K\times K$.
Categories:16S50, 15A27 |
11. CJM 2008 (vol 60 pp. 520)
Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this
paper, we characterize those $A$ for which $\|A\|_{E, F}$ are
determined by their actions on decreasing sequences, where $E$ and
$F$ are suitable normed Riesz spaces of sequences. In particular,
our results can apply to the following spaces: $\ell_p$, $d(w,p)$,
and $\ell_p(w)$. The results established here generalize
ones given by Bennett; Chen, Luor, and Ou; Jameson; and
Jameson and Lashkaripour.
Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, NÃ¶rlund mean matrices, summability matrices, matrices with row decreasing Categories:15A60, 40G05, 47A30, 47B37, 46B42 |
12. CJM 2007 (vol 59 pp. 1284)
On Effective Witt Decomposition and the Cartan--Dieudonn{Ã© Theorem Let $K$ be a number field, and let $F$ be a symmetric bilinear form in
$2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical
theorem of Witt states that the bilinear space $(Z,F)$ can be
decomposed into an orthogonal sum of hyperbolic planes and singular and
anisotropic components. We prove the existence of such a decomposition
of small height, where all bounds on height are explicit in terms of
heights of $F$ and $Z$. We also prove a special version of Siegel's
lemma for a bilinear space, which provides a small-height orthogonal
decomposition into one-dimensional subspaces. Finally, we prove an
effective version of the Cartan--Dieudonn{\'e} theorem. Namely, we show
that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can
be represented as a product of reflections of bounded heights with an
explicit bound on heights in terms of heights of $F$, $Z$, and
$\sigma$.
Keywords:quadratic form, heights Categories:11E12, 15A63, 11G50 |
13. CJM 2007 (vol 59 pp. 638)
Distance from Idempotents to Nilpotents We give bounds on the distance from a non-zero idempotent to the
set of nilpotents in the set of $n\times n$ matrices in terms of
the norm of the idempotent. We construct explicit idempotents and
nilpotents which achieve these distances, and determine exact
distances in some special cases.
Keywords:operator, matrix, nilpotent, idempotent, projection Categories:47A15, 47D03, 15A30 |
14. CJM 2007 (vol 59 pp. 488)
Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties We consider the $k$-osculating varieties
$O_{k,n.d}$ to the (Veronese) $d$-uple embeddings of $\PP^n$. We
study the dimension of their higher secant varieties via inverse
systems (apolarity). By associating certain 0-dimensional schemes
$Y\subset \PP^n$ to $O^s_{k,n,d}$ and by studying their Hilbert
functions, we are able, in several cases, to determine whether those
secant varieties are defective or not.
Categories:14N15, 15A69 |
15. CJM 2007 (vol 59 pp. 186)
Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields Purely simple Kronecker modules ${\mathcal M}$, built from an algebraically closed field $K$,
arise from a triplet $(m,h,\alpha)$ where $m$ is a positive integer,
$h\colon\ktil\ar \{\infty,0,1,2,3,\dots\}$ is a height function, and
$\alpha$ is a $K$-linear functional on the space $\krx$ of rational
functions in one variable $X$. Every pair $(h,\alpha)$ comes with a
polynomial $f$ in $K(X)[Y]$ called the regulator. When the module
${\mathcal M}$ admits non-trivial endomorphisms, $f$ must be linear or
quadratic in $Y$. In that case ${\mathcal M}$ is purely simple if and
only if $f$ is an irreducible quadratic. Then the $K$-algebra
$\edm\cm$ embeds in the quadratic function field $\krx[Y]/(f)$. For
some height functions $h$ of infinite support $I$, the search for a
functional $\alpha$ for which $(h,\alpha)$ has regulator $0$ comes
down to having functions $\eta\colon I\ar K$ such that no planar curve
intersects the graph of $\eta$ on a cofinite subset. If $K$ has
characterictic not $2$, and the triplet $(m,h,\alpha)$ gives a
purely-simple Kronecker module ${\mathcal M}$ having non-trivial
endomorphisms, then $h$ attains the value $\infty$ at least once on
$\ktil$ and $h$ is finite-valued at least twice on
$\ktil$. Conversely all these $h$ form part of such triplets. The
proof of this result hinges on the fact that a rational function $r$
is a perfect square in $\krx$ if and only if $r$ is a perfect square
in the completions of $\krx$ with respect to all of its valuations.
Keywords:Purely simple Kronecker module, regulating polynomial, Laurent expansions, endomorphism algebra Categories:16S50, 15A27 |
16. CJM 2005 (vol 57 pp. 82)
Jordan Structures of Totally Nonnegative Matrices An $n \times n$ matrix is said to be totally nonnegative if every
minor of $A$ is nonnegative. In this paper we completely
characterize all possible Jordan canonical forms of irreducible
totally nonnegative matrices. Our approach is mostly combinatorial
and is based on the study of weighted planar diagrams associated
with totally nonnegative matrices.
Keywords:totally nonnegative matrices, planar diagrams,, principal rank, Jordan canonical form Categories:15A21, 15A48, 05C38 |
17. CJM 2004 (vol 56 pp. 776)
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices We explicitly describe
the best approximation in
geodesic submanifolds of positive definite matrices
obtained from involutive
congruence transformations on the
Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of
positive definite matrices.
An explicit calculation for the minimal distance
function from the geodesic submanifold
${\mathrm{Sym}}(p,{\mathbb R})^{++}\times
{\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in
${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is
given in terms of metric and
spectral geometric means, Cayley transform, and Schur
complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$
Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform Categories:15A48, 49R50, 15A18, 53C3 |
18. CJM 2004 (vol 56 pp. 134)
Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States |
Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States Every norm $\nu$ on $\mathbf{C}^n$ induces two norm numerical
ranges on the algebra $M_n$ of all $n\times n$ complex matrices,
the spatial numerical range
$$
W(A)= \{x^*Ay : x, y \in \mathbf{C}^n,\nu^D(x) = \nu(y) = x^*y = 1\},
$$
where $\nu^D$ is the norm dual to $\nu$, and the algebra numerical range
$$
V(A) = \{ f(A) : f \in \mathcal{S} \},
$$
where $\mathcal{S}$ is the set of states on the normed algebra
$M_n$ under the operator norm induced by $\nu$. For a symmetric
norm $\nu$, we identify all linear maps on $M_n$ that preserve
either one of the two norm numerical ranges or the set of states or
vector states. We also identify the numerical radius isometries,
{\it i.e.}, linear maps that preserve the (one) numerical radius
induced by either numerical range. In particular, it is shown that
if $\nu$ is not the $\ell_1$, $\ell_2$, or $\ell_\infty$ norms,
then the linear maps that preserve either numerical range or either
set of states are ``inner'', {\it i.e.}, of the form $A\mapsto
Q^*AQ$, where $Q$ is a product of a diagonal unitary matrix and a
permutation matrix and the numerical radius isometries are
unimodular scalar multiples of such inner maps. For the $\ell_1$
and the $\ell_\infty$ norms, the results are quite different.
Keywords:Numerical range, numerical radius, state, isometry Categories:15A60, 15A04, 47A12, 47A30 |
19. CJM 2003 (vol 55 pp. 1000)
Some Convexity Results for the Cartan Decomposition In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$
where $a(g)$ is the abelian part in the Cartan decomposition of
$g$. This is exactly the support of the measure intervening in the
product formula for the spherical functions on symmetric spaces of
noncompact type. We give a simple description of that support in
the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$,
$\mathbf{C}$ or $\mathbf{H}$. In particular, we show that
$\mathcal{S}$ is convex.
We also give an application of our result to the description of
singular values of a product of two arbitrary matrices with
prescribed singular values.
Keywords:convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values Categories:43A90, 53C35, 15A18 |
20. CJM 2003 (vol 55 pp. 91)
Some Convexity Features Associated with Unitary Orbits Let $\mathcal{H}_n$ be the real linear space of $n\times n$ complex
Hermitian matrices. The unitary (similarity) orbit $\mathcal{U}
(C)$ of $C \in \mathcal{H}_n$ is the collection of all matrices
unitarily similar to $C$. We characterize those $C \in \mathcal{H}_n$
such that every matrix in the convex hull of $\mathcal{U}(C)$ can
be written as the average of two matrices in $\mathcal{U}(C)$. The
result is used to study spectral properties of submatrices of
matrices in $\mathcal{U}(C)$, the convexity of images of $\mathcal{U}
(C)$ under linear transformations, and some related questions
concerning the joint $C$-numerical range of Hermitian matrices.
Analogous results on real symmetric matrices are also discussed.
Keywords:Hermitian matrix, unitary orbit, eigenvalue, joint numerical range Categories:15A60, 15A42 |
21. CJM 2002 (vol 54 pp. 571)
Diagonals and Partial Diagonals of Sum of Matrices Given a matrix $A$, let $\mathcal{O}(A)$ denote the orbit of $A$ under a
certain group action such as
\begin{enumerate}[(4)]
\item[(1)] $U(m) \otimes U(n)$ acting on $m \times n$ complex matrices
$A$ by $(U,V)*A = UAV^t$,
\item[(2)] $O(m) \otimes O(n)$ or $\SO(m) \otimes \SO(n)$ acting on $m
\times n$ real matrices $A$ by $(U,V)*A = UAV^t$,
\item[(3)] $U(n)$ acting on $n \times n$ complex symmetric or
skew-symmetric matrices $A$ by $U*A = UAU^t$,
\item[(4)] $O(n)$ or $\SO(n)$ acting on $n \times n$ real symmetric or
skew-symmetric matrices $A$ by $U*A = UAU^t$.
\end{enumerate}
Denote by
$$
\mathcal{O}(A_1,\dots,A_k) = \{X_1 + \cdots + X_k : X_i \in
\mathcal{O}(A_i), i = 1,\dots,k\}
$$
the joint orbit of the matrices $A_1,\dots,A_k$. We study the set of
diagonals or partial diagonals of matrices in $\mathcal{O}(A_1,\dots,A_k)$,
{\it i.e.}, the set of vectors $(d_1,\dots,d_r)$ whose entries lie
in the $(1,j_1),\dots,(r,j_r)$ positions of a matrix in $\mathcal{O}(A_1,
\dots,A_k)$ for some distinct column indices $j_1,\dots,j_r$. In many
cases, complete description of these sets is given in terms of the
inequalities involving the singular values of $A_1,\dots,A_k$. We
also characterize those extreme matrices for which the equality cases
hold. Furthermore, some convexity properties of the joint orbits are
considered. These extend many classical results on matrix
inequalities, and answer some questions by Miranda. Related results
on the joint orbit $\mathcal{O}(A_1,\dots,A_k)$ of complex
Hermitian matrices under the action of unitary similarities are
also discussed.
Keywords:orbit, group actions, unitary, orthogonal, Hermitian, (skew-)symmetric matrices, diagonal, singular values Categories:15A42, 15A18 |
22. CJM 2001 (vol 53 pp. 758)
Inequivalent Transitive Factorizations into Transpositions The question of counting minimal factorizations of permutations into
transpositions that act transitively on a set has been studied extensively
in the geometrical setting of ramified coverings of the sphere and in the
algebraic setting of symmetric functions.
It is natural, however, from a combinatorial point of view to ask how such
results are affected by counting up to equivalence of factorizations, where
two factorizations are equivalent if they differ only by the interchange of
adjacent factors that commute. We obtain an explicit and elegant result for
the number of such factorizations of permutations with precisely two
factors. The approach used is a combinatorial one that rests on two
constructions.
We believe that this approach, and the combinatorial primitives that have
been developed for the ``cut and join'' analysis, will also assist with the
general case.
Keywords:transitive, transposition, factorization, commutation, cut-and-join Categories:05C38, 15A15, 05A15, 15A18 |
23. CJM 2001 (vol 53 pp. 470)
Hyperbolic Polynomials and Convex Analysis A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to
a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$
has all real roots for all vectors $x$. Motivated by partial
differential equations, G{\aa}rding proved in 1951 that the largest
such root is a convex function of $x$, and showed various ways of
constructing new hyperbolic polynomials. We present a powerful new
such construction, and use it to generalize G{\aa}rding's result to
arbitrary symmetric functions of the roots. Many classical and recent
inequalities follow easily. We develop various convex-analytic tools
for such symmetric functions, of interest in interior-point methods
for optimization problems over related cones.
Keywords:convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interior-point method, semidefinite program, singular value, symmetric function Categories:90C25, 15A45, 52A41 |
24. CJM 2000 (vol 52 pp. 141)
Numerical Ranges Arising from Simple Lie Algebras A unified formulation is given to various generalizations of the
classical numerical range including the $c$-numerical range,
congruence numerical range, $q$-numerical range and von Neumann
range. Attention is given to those cases having connections with
classical simple real Lie algebras. Convexity and inclusion
relation involving those generalized numerical ranges are
investigated. The underlying geometry is emphasized.
Keywords:numerical range, convexity, inclusion relation Categories:15A60, 17B20 |
25. CJM 2000 (vol 52 pp. 197)
Sublinearity and Other Spectral Conditions on a Semigroup Subadditivity, sublinearity, submultiplicativity, and other
conditions are considered for spectra of pairs of operators on a
Hilbert space. Sublinearity, for example, is a weakening of the
well-known property~$L$ and means $\sigma(A+\lambda B) \subseteq
\sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The
effect of these conditions is examined on commutativity,
reducibility, and triangularizability of multiplicative semigroups
of operators. A sample result is that sublinearity of spectra
implies simultaneous triangularizability for a semigroup of compact
operators.
Categories:47A15, 47D03, 15A30, 20A20, 47A10, 47B10 |