All Questions
1,966 questions with no upvoted or accepted answers
6
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220
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Is $\mathrm{End}-\{0\}=\mathrm{Aut}$ for derivation Lie algebra?
Is it true that every nonzero endomorphism of Lie $\mathbb{C}$-algebra $\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\partial_{x_n}$ is an automorphism?
As I ...
- 291
6
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205
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Automorphisms of $\mathbb{C}[x, y, z]$ over $\mathbb C[x]$
What are the automorphisms of $\mathbb{C}[x, y, z]$ fixing $\mathbb{C}[x]$? I.e. those automorphisms $\phi:\mathbb{C}[x, y, z]\to\mathbb{C}[x, y, z]$ s.t. $\phi(x) = x$. I am interested in a complete ...
- 329
6
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0
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89
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Maximal number of commuting functions of a finite set
Let $S$ be a finite set with $n$ elements and let $F_S$ denote the set of functions from $S$ to $S$. I wonder whether anything is known about the maximal cardinality of a commuting subset of $F_S$? A ...
- 23.2k
6
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114
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Hilbert series of special linear sections of Grassmannian $Gr(2,n)$
Consider the Grassmannian $\operatorname{Gr}(2,n)$. I want to know Hilbert series of $H_1 \cap H_2 \dots \cap H_m \cap \operatorname{Gr}(2,n)$ in the Plücker embedding of $\operatorname{Gr}(2,n)$, ...
- 641
6
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382
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Is there a Dedekind domain which has infinite class group and is free of finite rank over a finite quotient PID?
Is there a Dedekind domain $B$ satisfying the following two conditions:
$B$ is an algebra over a finite quotient PID $A$ such that $B$ is free of finite rank as a module over $A$;
$B$ has infinite ...
- 3,813
6
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964
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Constructive contents of “the support of a sheaf is closed” or “the flat locus is open”
Out of curiosity I started to go through an introductory course on schemes and to convince myself that I could prove every result (after an appropriate reformulation if necessary) constructively, i.e. ...
- 1,153
6
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132
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Generalization of pseudogroups
Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup
One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
- 181
6
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266
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Classification of one dimensional (non-commutative) formal group laws over $k[\epsilon]/(\epsilon^n)$
It's well-known that any one dimensional formal group law over a $\mathbb Q$-algebra or a reduced ring is commutative, but there are one dimensioal non-commutative formal group laws over rings like $k[...
- 6,622
6
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407
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Homogeneous regular sequence
Consider a $\mathbb{Z}$-graded polynomial ring $R = k[x_1,\cdots,x_n]$ over a field $k$, where the elements $x_i$ are homogeneous (they may have negative degree). Let $I$ be a homogeneous ideal that ...
- 71
6
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111
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Relation between $\mathrm{Proj}(A[tI])$ and $\mathrm{Proj}(A[tJ])$ for ideals $I$ and $J$ of $A$ with $I^2 \subset J \subset I$
Let $k$ be a field and $A$ a noetherian local $k$-algebra.
Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$.
Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \...
- 523
6
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answers
47
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Special monomorphism to encode the inclusion of topological submonoids
Consider the category $\mathrm{TopMon}$ of topological monoids and continuous monoid homomorphisms.
Consider the inclusion $i:\Bbb{R}_{\ge 0}\hookrightarrow \Bbb{R}$, where the spaces are taken with ...
- 2,129
6
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114
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A question about the span of a sequence of polynomials satisfying a linear recurrence
Let F be a finite field and A(n) in F[t], n in N, be defined by a linear recurrence with coefficients in F[t], together with initial conditions. Is there a decision procedure for determining whether ...
- 5,422
6
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285
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On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian
All rings below are commutative with unity.
If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...
- 412
6
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867
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How to extend Ritt's theorem on elementary invertible bijective elementary functions?
The elementary functions according to Liouville and Ritt are the functions of a complex variable built up by applying exponentiation, logarithms and/or algebraic operations finitely often. That means, ...
- 1,053
6
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233
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Morphism that is surjective on PID points is surjective on every Dedekind domain?
Let $f: X=\Bbb A^n \rightarrow Y=\Bbb A^m $ be a morphism between affine spaces over an algebraically closed field $k$. Assume $f: X(R) \rightarrow Y(R)$ is surjective for any PID $R$ over $k$, under ...
- 6,622
6
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357
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Ideals of orders in number fields
Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{O}$ be an order of $K$ and $A$ a proper ideal of $\mathcal{O}$, by which I mean that $\mathcal{O} = \{\alpha \in K : \...
6
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204
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Counting exceptional divisors
Suppose that I blow up an ideal sheaf $J$ in $\mathbb A^2$ via a map $\pi : X \to \mathbb A^2$. I'd like to compute, from the ideal, how many exceptional divisors there are for $\pi$, and be able to ...
- 83
6
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340
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Would you like a subject class for semigroup theory on the arXiv?
After contacting the arxiv recently about possibly adding semigroup theory as a subject class, they suggested I canvas the research community to establish whether such a subject class would be used ...
- 77
6
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117
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Closedness of the partial order in complete Hausdorff semitopological semilattices
First some definitions.
A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the ...
- 41.9k
6
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133
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What is the monoid of skew-symmetric trilinear forms on finite abelian groups?
I am interested in triple cup product operations on the cohomology ring $H^*(Y;\Bbb Z/p^r)$ of 3-manifolds. Trying to extract the algebra, I am led to the following question.
Let's fix a prime power $...
- 9,580
6
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145
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An ideal that there exists a unique ideal maximal with respect to not containing it
Let $I$ be a non-zero ideal of a commutative ring with identity. Is there any equivalent condition to the property that there exists a unique ideal maximal with respect to not containing $I$?
For ...
6
votes
0
answers
81
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Representing meet-semilattices with vector spaces of specified dimensions
Take $K$ to be a field and take $L$ to be a finite meet-semilattice. I'm interested in the set of functions $n: L \rightarrow \mathbb{Z}^{\ge 0}$ such that there is some function $V$ from $L$ to ...
6
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0
answers
120
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How does $\mathcal{O}_{C_K} / p \mathcal{O}_{C_K}$ looks like?
I am reading stuff about Fontaine's periods rings. Let $K$ be a $p$-adic field and $C_K = \widehat{\overline{K}}$. Then $\mathcal{O}_{C_K}/p\mathcal{O}_{C_K}$ isn't perfect, otherwise $R = \...
- 133
6
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868
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Picard group of non reduced scheme
Let X be a non reduced scheme. $X_{red}$ be the reduced scheme. Is it true that Picard group of $X_{red}$=Picard group of X?
Is this map surjective $Pic (X)\rightarrow Pic(X_{red})$?
Is there a ...
user111251
6
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190
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Constructive approach to complete intersections
Let $K$ be a field (probably of positive characteristic) and consider the ring $R=K[\![x_1,\dotsc,x_n]\!]$. Suppose we have an ideal $I=(f_1,\dotsc,f_n)$ (with the same $n$ as before). Suppose we ...
- 56.9k
6
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266
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Universal property of $A_{\mathrm{cris}}/p^n$
It is well known that the ring $A_{\mathrm{cris}}$ of Fontaine is the universal $p$-adically complete divided power thickening of $\mathcal{O}_{\mathbb{C}_p}$ over $\mathbb{Z}_p$; in fact, this is one ...
- 825
6
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171
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Name for property of mixed characteristic DVR: admits regular local homomorphism from DVR with finite residue field
Does anybody happen to know if there is already a name in the literature for the following property of a mixed characteristic DVR: that there exists a local homomorphism that is regular into the ...
- 4,111
6
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224
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Book or survey on Dedekind-finite rings
I'm seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring $R$ is Dedekind-finite ...
- 10.5k
6
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215
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Matrix semigroups in which a weighted average of eigenvalues is multiplicative
A problem in fractal geometry requires me to consider matrix semigroups with the following curious property. For a $d \times d$ real matrix $A$ let $\lambda_1(A) \geq \lambda_2(A) \geq \cdots \geq \...
- 6,206
6
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129
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Center Picard group non-commutative algebra
I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra.
Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by $\mathrm{Pic}(A)$...
- 7,300
6
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126
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Relative variants of the Jacobson radical
Let $B$ be a commutative ring (with 1). The Jacobson radical can be defined as
$$ J(B) = \{b \in B \mid \forall a \in B \colon \quad 1 + a\cdot b \text{ is a unit in } B \} $$ or $$ J(B) =\{ b \in B ...
- 401
6
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0
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672
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Flat + locally of finite presentation + monomorphism = open immersion
It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$:
Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then $...
- 15.9k
6
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0
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293
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Constructing the normal sheaf for the plucker embedding in MAGMA (or a similar programming language)
How would one construct the normal sheaf $N_{G(2,6)/\mathbb P^{14}}$ to the plucker embedding of the grassmannian $G(2,6) \rightarrow \mathbb P^{14}$ as a sheaf in MAGMA (or another programming ...
- 1,308
6
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294
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Laurent and power series over the field with one element?
Question. Is there a suitable notion of the Laurent series ring $\mathbb{F}_1((t))$ and power series ring $\mathbb{F}_1[[t]]$ in some framework for the field with one element $\mathbb{F}_1$?
For ...
user1437
6
votes
0
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607
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On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes
Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
- 3,101
6
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310
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Algebraic closedness in field of fractions
If $A\subseteq B$ are affine domains over an algebraically closed field $k$ of characteristic zero, such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...
6
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321
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Does integral closure commute with pushforward
Suppose that $\pi : Y \to X$ is a proper birational morphism between normal varieties (schemes, whatever). Suppose that $I$ is an ideal sheaf on $Y$.
One can form $\pi_* I$ and construct an ideal ...
- 20.5k
6
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468
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Find a polynomial not in any ideal generated by polynomials of total degree $o(n)$
Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ ...
- 651
6
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181
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Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra
The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating.
A real finite-dimensional vector space $V$ defines the ...
- 3,513
6
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0
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355
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Integral group rings on which stably free modules are free
Let $G$ be a torsion-free group and $ZG$ the integral group rings. Recall that a projective module $P$ over $ZG$ is stably free if there is an isomorphism $P \oplus ZG^n \cong ZG^m$. Are there known ...
- 1,373
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712
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Spectral sequences and Koszul complexes in Deformation Theory
I'm reading this paper of A. Vistoli and I have some questions about the discussion in page 5. This is the context (If you don't want to download the paper):
Let $A'$ be a noetherian local ring with ...
- 987
6
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0
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572
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Computing the pro-solvable closure of a finitely generated subgroup of a free group
The pro-solvable topology on a group $G$ is the unique group topology such that the set of normal subgroups $N\lhd G$ with $G/N$ a finite solvable group is a fundamental system of neighborhoods of the ...
- 38.6k
6
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338
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Are monomorphisms between algebraic spaces representable?
The question in the title can be reformulated as follows. Let $f : Y \to X$ be a monomorphism of algebraic spaces where $X$ is a scheme. Is it true that $Y$ is a scheme?
If $f$ is locally of finite ...
user45795
6
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0
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533
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A question on Castelnuovo-Mumford regularity
Consider a short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ of coherent sheaves on $\mathbb{P}^n$. Assume that $\mathcal{F}''$ (resp. $\mathcal{F}$) is $m-1$ (resp. $m$...
- 61
6
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106
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Irreducibility testing and factoring
It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...
- 96.4k
6
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1k
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Localisation of injectives
When working with injective modules, one bad thing is that they do not necessarily behave well with respect to localisation. Consider a commutative ring $R$ and have a look at the following properties:...
- 6,700
6
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329
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Non-crystallographic cluster algebras
Background
Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed $(\mathbf{...
- 2,600
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243
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I. Kaplansky, Going up in polynomial rings, unpublished manuscript, 1972
Anyone got a copy of this article?
- 1,388
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0
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4k
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Prime ideals in polynomial rings over integers
Im trying to find a characterization of the prime ideals in the polynomial ring $R = \mathbb Z[X,Y]$ in two variables over the integers.
Actually I need to find the maximal ideals in quotient rings $...
6
votes
0
answers
522
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Does existence of an isolated solution imply the Jacobian determinant is non-zero?
Let $f_1,\dots,f_n$ be formal power series in $\mathbb{C}[[x_1,\dots,x_n]]$ whose constant terms are all zero (i.e. $f_1,\dots f_n$ are not units in the ring). Suppose further that the radical of the ...
- 635