All Questions
9,056 questions
1
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Semigroup algebras with one dimensional center
Let $S$ be a finite semigroup and $K$ a field of characteristic 0 (we can assume the complex numbers for simplicity).
Question: Is there a characterization when the center of the semigroup algebra $...
- 26.5k
0
votes
2
answers
545
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Is $S^1=\mathbb{R}/\mathbb{Z}$ a ring? [closed]
Is there a ring structure on $S^1=\mathbb{R}/\mathbb{Z}$?
- 705
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votes
0
answers
322
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Chromatic filtration on stable homotopy
Chapter 5 of Ravenel's green book starts with the sentence “[The chromatic spectral sequence] is a mechanism for organizing the Adams-Novikov $E_2$ term and ultimately $\pi_*(S^0)$ itself." My ...
- 133
29
votes
4
answers
1k
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Which stable homotopy groups are represented by parallelizable manifolds?
The Pontryagin-Thom construction allows one to identify the stable homotopy groups of spheres with bordism classes of stably normally framed manifolds. A stable framing of the stable normal bundle ...
- 27.5k
4
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1
answer
466
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Top local cohomology - recommendations
I need some background in local cohomology to read a certain paper, which exploits the structure of the top local cohomology. Even after acquainting myself to local cohomology, I fail to understand ...
user340793
13
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2
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2k
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Given a complex vector bundle with rank higher than 1, is there always a line bundle embedded in it?
When I read the GTM 082, "The Splitting Principle of the complex vector bundle", I see that in the proof we split off one subbundle at a time by pulling back to the projectivization of a ...
- 157
1
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0
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102
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Continuity of the 1-jet prolongation map $\text{imm}(M,N)\to \text{fimm}(M,N)$
I am studying the article Immersion Theory for Homotopy theorists by Michael Weiss for my bachelor thesis. The main theorem states that the space of immersions and formal immersions between two smooth ...
- 11
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185
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Proving a Result About Pontryagin Numbers Without Forms
I've been reading the book Geometry of Differential Forms by Shigeyuki Morita, and I came across the following theorem on page 226 the other day:
Proposition 5.53 (Pontryagin). Two cobordant closed (...
- 377
36
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3
answers
2k
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Defining $SU(n)$ in HoTT
From a recent answer by Mike Shulman, I read:
"HoTT is (among other things) a foundational theory, on roughly the same ontological level as ZFC, whose basic objects can be regarded as $\infty$-...
- 43.2k
17
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1
answer
898
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Does there exist a closed manifold with vanishing reduced rational cohomology but nonvanishing odd torsion cohomology?
Question: Let $p$ be an odd prime. Does there exist a closed manifold $M$ with $\widetilde H^\ast(M; \mathbb Q) = 0$ but $\widetilde H^\ast(M; \mathbb F_p) \neq 0$?
When $p = 2$, an example is given ...
- 64k
5
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295
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If $\pi_\ast A$ is graded-commutative, then is $A_\ast$ a lax monoidal functor?
Let $A$ be a homotopy ring spectrum. Then the homology theory $A_\ast : Spectra \to GrAb$ lifts to a homology theory valued in $GrMod(\pi_\ast A)$. If $A$ is homotopy commutative, then this functor $...
- 64k
51
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5
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Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 63.1k
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1
answer
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Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
I'm interested in both version of the question in the title, i.e. in the topological category and in the smooth category. By a topological immersion I mean a local embedding. I was asking in ...
- 439
7
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1
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260
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Bordism for oriented triangulable manifolds without smooth differentiable structures
We know the bordism group for oriented smooth differentiable structures such as $\Omega_d^{SO}$ that requires the special orthogonal group structure on the tangent bundle $TM$ of manifold $M$.
$$\...
- 10.5k
17
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3
answers
2k
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Variety of commuting matrices
Let $G=\operatorname{GL}(n,\mathbb{C})$ and $\mathfrak{g}=\operatorname{Mat}(n,\mathbb{C})$ and let us consider the two varieties $X,Y$ defined as $$X=\{(x,y) \in G \times G \ | \ xy=yx\} $$ and $$Y=\{...
- 1,061
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285
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When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations?
Let $\mathcal{M}$ be a
locally finitely presentable model category, cofibrantly generated by
two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial
cofibrations with presentable domain ...
- 1,514
18
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1
answer
1k
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Application of higher categories in algebra
Higher categories and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher ...
4
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2
answers
228
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Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$?
Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective ...
- 771
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0
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360
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on definitions of stacks
There are two ways to define a stack.
The first one is that the presheaf of sets Isom (a,b) is a sheaf and that every descent data is effective.
The second one says that a stack is a homotopy sheaf of ...
- 494
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1
answer
214
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If $A_\ast$ has a Künneth theorem, then is $A$ a module over Morava $K$-theory?
$\newcommand\Spt{\mathit{Spt}}\newcommand\GrAb{\mathit{GrAb}}$Let $A$ be a ring spectrum. Suppose that $A$ has a Künneth theorem — i.e. the homology theory $A_\ast : \Spt \to \GrAb$ is a strong ...
- 64k
1
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1
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442
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Higher homotopy local systems
The concept of a local system in algebraic geometry is often described as a locally constant constructible sheaf on a scheme $X$, which is in essence a sheaf whose stalk at a point $x$ comes equipped ...
- 2,907
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524
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Is there anything special about the Honda formal group?
The "standard" Morava E-theory $E_n$ (at a prime $p$) is typically defined using the so-called "Honda formal group law", the unique FGL $\Gamma_n$ over $\mathbb{F}_{p^n}$ ...
- 1,969
10
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0
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420
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What is the original source for the Goerss-Hopkins-Miller-Lurie theorem on tmf?
The central basic theorem of topological modular forms states that the structure sheaf of $\widehat{\mathcal{M}}_{ell}$ lifts to a sheaf of complex-oriented $E_{\infty}$-rings whose formal groups are ...
- 1,969
11
votes
1
answer
473
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Characteristic classes of non-linear sphere bundles
It is well known that the diffeomorphism group of there sphere $\operatorname{Diff}(S^n)$ has the homotopy type of a product $X:=O(n+1)\times \operatorname{Diff}_{\partial D^n}(D^n)$ of the orthogonal ...
- 6,546
1
vote
1
answer
179
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Goresky-Hingston product on cohomology of the relative free loop space on $S^1$
I'm interested in the computations of the Goresky-Hingston product (defined https://arxiv.org/abs/0707.3486)
on the cohomology of the relative free loop space on the circle (or better yet, their ...
- 113
12
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1
answer
360
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Can the Bousfield class of projective space be computed directly?
Recall that the Bousfield class of a spectrum $E$, written $\langle E\rangle$, is the class of spectra $X$ such that $X\wedge E$ is not contractible. For example the Bousfield class of any of the ...
- 2,052
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173
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Which positive flat stable model structures on (flavors of) spectra have the property that cofibrant operad-algebras forget to cofibrant spectra?
Let $M$ be a monoidal model category and $O$ an operad valued in $M$, and the category of $O$-algebras inherits a model structure from $M$ where a map $f$ is a weak equivalence (resp. fibration) if ...
- 30.3k
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122
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Is there a name for this condition on a monoid?
Suppose we have a commutative monoid ${\mathcal M}=\langle M,\otimes\rangle$ such that the usual divisibility relation $\leq_\otimes$ given by $a\leq_\otimes b\Leftrightarrow \exists c(a\otimes c=b)$ ...
- 1,951
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2
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586
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When are bundles of odd and even differential forms isomorphic?
Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ ...
- 501
14
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1
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562
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Mednykh's formula for 2d manifold with boundaries? How to count principal G-bundles with prescribed holonomies?
Let S be a closed, orientable 2d manifold and G a finite group. Since a principal G-bundle over S is specified by maps $\phi : \pi_1(S) \rightarrow G$ modulo the adjoint action by G, the way to count ...
- 171
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154
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Are the relations of being homeomorphic or being homotopy equivalent on the compact polyhedra definable in the structure of the natural numbers?
Let $ K(\mathbb{N}) $ denote the set of all finite simplicial complexes with vertices in $\mathbb{N}$.
Let $ f\colon \mathbb{N} \to K(\mathbb{N}) $ be a computable bijection.
Let $ R $ = { $ (m, n) $ |...
- 307
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0
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Given a proper submersion $f: X \setminus F_0 \to D \setminus 0$ which extends to $X \to D$, are cycles on $X$ which run around $0$ boundaries in $X$?
I have a proper map of complex manifolds
$$f: X \to D,$$
where $D \subset \mathbb C$ is the unit disc. By assumption, $f$ has connected fibers, is smooth over $D \setminus 0$, and a smooth fiber $F$ ...
- 1,286
9
votes
1
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309
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Comparing cohomology of a total complex with the cohomology of semidirect product
$\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an ...
- 1,759
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3
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1k
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Why it is convenient to be cartesian closed for a category of spaces?
In 1967 Steenrod wrote what later became a quite celebrated paper, A convenient category of topological spaces (Michigan Math. J. 14 (1967) 133–152). The paper conveys the work of many (among the most ...
- 9,111
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1
answer
288
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Explicit generators from Serre spectral sequence
Let $p: E \to B$ be a locally trivial fibration with fiber $F$. If necessary, suppose that $B$ is simply connected. Suppose that the Serre spectral sequence leaves the term $H_p(B, H_q(F, \mathbb{Q}))$...
1
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0
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146
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Cohomology of the classifying space of a semidirect product, and some specific examples with cyclic groups
Let $G$ and $A$ be finite abelian groups and $\rho :G \rightarrow \text{Aut}(G)$ a representation of $G$. We can form the semidirect product $A\rtimes _{\rho} G$. Just to agree on the notation this is ...
- 813
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1
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Easiest proof of computability of homotopy groups of spheres
Has it gotten easier to prove all homotopy groups of spheres are computable? I don’t care if the computation is inefficient, what’s the easiest proof? Are we still stuck doing Postnikov towers?
- 827
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2
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350
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What is an unstable dual-Steenrod comodule?
$\newcommand\Sq{\mathit{Sq}}$Recall that a (graded) module $V^\ast$ over the Steenrod algebra $\mathcal A^\ast$ is said to be unstable if $\Sq^i v = 0$ for $i > |v|$. The motivating example, of ...
- 64k
7
votes
1
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444
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Which monoids have a faithful irreducible representation?
Let $*$ be a binary operation on a set $M$, with an identity element $e\in M$.
A monoid representation of $(M,*,e)$ is a map $\delta:M\to (S\to S)$ for some set $S$, such that $\delta(e)=\mathrm{id}_S$...
- 24.8k
8
votes
3
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543
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Jones' theorem for non-simply-connected spaces?
Let $X$ be a smooth manifold. Jones' theorem says that $H^\bullet(\mathcal{L}X)\cong HH_\bullet(\Omega^\bullet_X)$, where $\mathcal{L}X$ is the free loop space of $X$. Is there a modification of this ...
- 2,788
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3
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887
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Finite CW complex with finite non-abelian fundamental group and higher homologies zero
I want to build a finite CW complex such that $\pi_1$ is non-abelian and $H_i$ are zero for $i\geq 2.$
From Hatcher for a given group G, one can create an example of a 2-complex $X_G$ with $\pi_1(X_G)=...
- 1,177
5
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0
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387
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Calculating étale fundamental groups from the usual fundamental group
$\newcommand{Spec}{\operatorname{Spec}}$Let $X$ be a connected affine smooth variety over $\mathbb{Q}$, with a point $x\in X(\Spec(\mathbb{Q})$.
For any algebraically closed field $K$ of ...
- 541
3
votes
1
answer
211
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Defining the classifying space of a group acting on a set
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}$Recall that $n$-simplices of the classifying space (simplicial set) $BG$ of a group are orbits of the diagonal action by shifts
on $n+1$-...
- 513
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2
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2k
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Has anyone seen a nice map of multiplicative cohomology theories?
I have seen lots of descriptions of this map in the literature but never seen it nicely drawn anywhere.
I could try to do it myself but I really lack expertise, hence am afraid to miss something or ...
- 18.4k
4
votes
1
answer
458
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Pullback of complex vector bundles along a retraction of compact Hausdorff spaces: a direct proof instead?
Consider a pointed compact Hausdorff space $(X,x_0)$ and a closed pointed subspace $i:A\subset X$ such that there exists a continuous map $r:X\rightarrow A$ such that $r|_A=\text{Id}_A$. Set
$$q:(X,...
user490645
11
votes
2
answers
722
views
$ \mathbb{R}P^n $ bundles over the circle
Is every $ \mathbb{R}P^{2n} $ bundle over the circle trivial?
Are there exactly two $ \mathbb{R}P^{2n+1} $ bundles over the circle?
This is a cross-post of (part of) my MSE question
https://math....
- 4,081
2
votes
1
answer
117
views
Primal identity in matrix semigroup
Given a finite set of matrix $\{M_1,M_2,\cdots,M_n\}\subseteq \mathbb{C}^{d\times d}$, we consider the semigroup generated by matrix product.
We call $s_1\cdots s_k$ an identity index if
$M_{s_1}M_{...
- 1,503
3
votes
1
answer
182
views
Simplicial set from all orderings of simplicial complex
Given an abstract simplicial complex $K$ on a set of vertices $V$, we can form a semi-simplicial set by $F(K)$ sending $F(K)_n$ to be the set of ordered $(n+1)$-tuples of vertices in $V$ forming an $n$...
- 2,044
14
votes
1
answer
1k
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Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?
The question is inspired by an answer to The concept of Duality
It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...
- 18.4k
1
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0
answers
293
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Does Thom-Milnor theorem provide a bound for the number of connected components of a zero set of a polynomial system?
I have a system $$f_1(x_1,\dots,x_m) = 0,...,f_p(x_1,\dots,x_m) = 0,$$
where $f_1,\dots,f_p$ are real polynomials and $x_1,\dots,x_m$ are real independent variables. I'm interested in an upper bound ...
- 1,058