scholarly journals Haas' theorem revisited

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
Benoît Bertrand ◽  
Erwan Brugallé ◽  
Arthur Renaudineau
Keyword(s):  
Vector Space ◽  
Algebraic Curve ◽  
Affine Space ◽  
Linear Subspace ◽  
Vector Subspace ◽  
Original Result ◽  
Affine Subspaces ◽  
Trivalent Graph ◽  
Real Algebraic Curve ◽  
Topological Surface

Haas' theorem describes all partchworkings of a given non-singular plane tropical curve $C$ giving rise to a maximal real algebraic curve. The space of such patchworkings is naturally a linear subspace $W_C$ of the $\mathbb{Z}/2\mathbb{Z}$-vector space $\overrightarrow \Pi_C$ generated by the bounded edges of $C$, and whose origin is the Harnack patchworking. The aim of this note is to provide an interpretation of affine subspaces of $\overrightarrow \Pi_C $ parallel to $W_C$. To this purpose, we work in the setting of abstract graphs rather than plane tropical curves. We introduce a topological surface $S_\Gamma$ above a trivalent graph $\Gamma$, and consider a suitable affine space $\Pi_\Gamma$ of real structures on $S_\Gamma$ compatible with $\Gamma$. We characterise $W_\Gamma$ as the vector subspace of $\overrightarrow \Pi_\Gamma$ whose associated involutions induce the same action on $H_1(S_\Gamma,\mathbb{Z}/2\mathbb{Z})$. We then deduce from this statement another proof of Haas' original result. Comment: 22 pages, 14 figures

Super Real Algebraic Curve

2004 ◽  
pp. 396-396
Author(s):  
Yolanda Lozano ◽  
Steven Duplij ◽  
Malte Henkel ◽  
Malte Henkel ◽  
Euro Spallucci ◽  
...  
Keyword(s):  
Algebraic Curve ◽  
Real Algebraic Curve

EMBEDDINGS OF FREE TOPOLOGICAL VECTOR SPACES

2019 ◽  
Vol 101 (2) ◽  
pp. 311-324
Author(s):  
ARKADY LEIDERMAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Dimensional Subspace ◽  
Vector Subspace ◽  
Compact Metric Space ◽  
Topological Vector Spaces ◽  
Finite Dimensional ◽  
Bernstein Set ◽  
Dimensional Cube ◽  
Open Question

It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$, where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

scholarly journals The topological configuration of a real algebraic curve

1991 ◽  
Vol 43 (1) ◽  
pp. 37-50 ◽  
Author(s):  
Takis Sakkalis
Keyword(s):  
Morse Theory ◽  
Algebraic Curve ◽  
Running Time ◽  
Real Algebraic Curve ◽  
Maximum Coefficient ◽  
Topological Configuration

This paper presents an algorithm, motivated by Morse Theory, for the topological configuration of the components of a real algebraic curve {f(x, y) = 0}. The running time of the algorithm is O(n12 (d + log n)2 log n), where n, d are the degree and maximum coefficient size of f(x, y).

SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

2017 ◽  
Vol 97 (1) ◽  
pp. 110-118 ◽  
Author(s):  
SAAK S. GABRIYELYAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Vector Space ◽  
Topological Vector Space ◽  
Unit Interval ◽  
Vector Subspace ◽  
Locally Compact ◽  
Tychonoff Space ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Usual Topology

For a Tychonoff space $X$, let $\mathbb{V}(X)$ be the free topological vector space over $X$, $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$, then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

Range-compatible homomorphisms over the field with two elements

2018 ◽  
Vol 34 ◽  
pp. 71-114
Author(s):  
Clément De Seguins Pazzis
Keyword(s):  
Linear Subspace ◽  
Linear Operators ◽  
Vector Spaces ◽  
Operator Spaces ◽  
Affine Subspaces ◽  
Finite Dimensional ◽  
Reflexive Operator ◽  
Recent Classification ◽  
Special Case

Let U and V be finite-dimensional vector spaces over a field K, and S be a linear subspace of the space L(U, V ) of all linear operators from U to V. A map F : S → V is called range-compatible when F(s) ∈ Im s for all s ∈ S. Previous work has classified all the range-compatible group homomorphisms provided that codimL(U,V )S ≤ 2 dim V − 3, except in the special case when K has only two elements and codimL(U,V )S = 2 dim V − 3. This article gives a thorough treatment of that special case. The results are partly based upon the recent classification of vector spaces of matrices with rank at most 2 over F2. As an application, the 2-dimensional non-reflexive operator spaces are classified over any field, and so do the affine subspaces of Mn,p(K) with lower-rank at least 2 and codimension 3.

scholarly journals The moduli space of stable vector bundles over a real algebraic curve

2009 ◽  
Vol 347 (1) ◽  
pp. 201-233 ◽  
Author(s):  
Indranil Biswas ◽  
Johannes Huisman ◽  
Jacques Hurtubise
Keyword(s):  
Moduli Space ◽  
Algebraic Curve ◽  
Vector Bundles ◽  
Stable Vector Bundles ◽  
Real Algebraic Curve

scholarly journals Degenerate flag varieties in network coding

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ghislain Fourier ◽  
Gabriele Nebe
Keyword(s):  
Network Coding ◽  
Affine Space ◽  
Linear Subspace ◽  
Flag Varieties ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Rank Metric ◽  
Rank Metric Codes ◽  
Information Set ◽  
Degenerate Flag Varieties

<p style='text-indent:20px;'>Building upon the application of flags to network coding introduced in [<xref ref-type="bibr" rid="b6">6</xref>], we develop a variant of this coding technique that uses degenerate flags. The information set is a metric affine space isometric to the space of upper triangular matrices endowed with the flag rank metric. This suggests the development of a theory for flag rank metric codes in analogy to the rank metric codes used in linear subspace coding.</p>

scholarly journals Principal bundles over a real algebraic curve

2012 ◽  
Vol 20 (5) ◽  
pp. 957-988 ◽  
Author(s):  
Indranil Biswas ◽  
Jacques Hurtubise
Keyword(s):  
Algebraic Curve ◽  
Principal Bundles ◽  
Real Algebraic Curve
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