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"Theorem"
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What's your angle, Pythagoras? : a math adventure
In ancient Greece, young Pythagoras discovers a special number pattern (the Pythagorean theorem) and uses it to solve problems involving right triangles.
Book
On the local converse theorem for p-adic GLn
In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of ${\\rm GL}_n(F)$, where $F$ is a non-archimedean local field.
Journal Article
The Pythagorean theorem : a 4,000-year history
The author presents a complex history of the Pythagorean Theorem, examining the earliest evidence of knowledge of the theorem to Einstein's theory of relativity.
Book
On the Theorem of Univalence on the Boundary
We give several generalizations of a known theorem from complex analysis, namely the univalence on the boundary theorem. Starting from a purely topological result (Theorems 1 and 11), we obtain univalence conditions for Sobolev mappings.
Journal Article
Pythagoras and the ratios
An ancient Greek boy, Pythagoras, helps his cousins produce pleasant music when he adjusts the mathematical ratios between the part of their pipes and lyres, knowledge he would later use to become a famous philosopher.
Book
ON THE LONG SEQUENCE AND THE HUREWICZ THEOREM FOR THE PERSISTENT HOMOTOPY GROUPS OF A PAIR
In this paper, we show that the persistent homotopy groups fit into a long sequence which is exact of order 2. We prove a persistent version of the Hurewicz theorem for a pair of spaces (X, A). We also show that this version implies the persistent version of the Hurewicz theorem for the space X.
Journal Article
(\\textsf {AD}^{+}\\) implies \\( \\omega _{1}\\) is a club \\( \\Theta \\)-Berkeley cardinal
Following [1], given cardinals \\(\\kappa <\\lambda \\), we say \\(\\kappa \\) is a club \\(\\lambda \\)-Berkeley cardinal if for every transitive set N of size \\(<\\lambda \\) such that \\(\\kappa \\subseteq N\\), there is a club \\(C\\subseteq \\kappa \\) with the property that for every \\(\\eta \\in C\\), there is an elementary embedding \\(j: N\\rightarrow N\\) with \\(\\mathrm {crit }(j)=\\eta \\). We say \\(\\kappa \\) is \\(\\nu \\)-club \\(\\lambda \\)-Berkeley if \\(C\\subseteq \\kappa \\) as above is a \\(\\nu \\)-club. We say \\(\\kappa \\) is \\(\\lambda \\)-Berkeley if C is unbounded in \\(\\kappa \\). We show that under \\(\\textsf {AD}^{+}\\), (1) every regular Suslin cardinal is \\(\\omega \\)-club \\(\\Theta \\)-Berkeley (see Theorem 7.1), (2) \\(\\omega _1\\) is club \\(\\Theta \\)-Berkeley (see Theorem 3.1 and Theorem 7.1), and (3) the ’s are \\(\\Theta \\)-Berkeley – in particular, \\(\\omega _2\\) is \\(\\Theta \\)-Berkeley (see Remark 7.5).Along the way, we represent regular Suslin cardinals in direct limits as cutpoint cardinals (see Theorem 5.1). This topic has been studied in [31] and [4], albeit from a different point of view. We also show that, assuming \\(V=L({\\mathbb {R}})+{\\textsf {AD}}\\), \\(\\omega _1\\) is not \\(\\Theta ^+\\)-Berkeley, so the result stated in the title is optimal (see Theorem 9.14 and Theorem 9.19).
Journal Article
Gödel's theorem : a very short introduction
When Kurt Gödel published his celebrated theorem, showing that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic, it had a profound impact on mathematical ideas and philosophical thought. Adrian Moore places the theorem in its intellectual and historical context, explaining the key concepts and misunderstandings.
Book
textit{h}$ -MINIMUM SPANNING LENGTHS AND AN EXTENSION TO BURNSIDE’S THEOREM ON IRREDUCIBILITY
We introduce the
$\\textbf{h}$
-minimum spanning length of a family
${\\mathcal A}$
of
$n\\times n$
matrices over a field
$\\mathbb F$
, where
$\\textbf{h}$
is a p-tuple of positive integers, each no more than n. For an algebraically closed field
$\\mathbb F$
, Burnside’s theorem on irreducibility is essentially that the
$(n,n,\\ldots ,n)$
-minimum spanning length of
${\\mathcal A}$
exists if
${\\mathcal A}$
is irreducible. We show that the
$\\textbf{h}$
-minimum spanning length of
${\\mathcal A}$
exists for every
$\\textbf{h}=(h_1,h_2,\\ldots , h_p)$
if
${\\mathcal A}$
is an irreducible family of invertible matrices with at least three elements. The
$(1,1, \\ldots ,1)$
-minimum spanning length is at most
$4n\\log _{2} 2n+8n-3$
. Several examples are given, including one giving a complete calculation of the
$(p,q)$
-minimum spanning length of the ordered pair
$(J^*,J)$
, where J is the Jordan matrix.
Journal Article