255 research outputs found
Rigidity and volume preserving deformation on degenerate simplices
Given a degenerate -simplex in a -dimensional space
(Euclidean, spherical or hyperbolic space, and ), for each , , Radon's theorem induces a partition of the set of -faces into two
subsets. We prove that if the vertices of the simplex vary smoothly in
for , and the volumes of -faces in one subset are constrained only to
decrease while in the other subset only to increase, then any sufficiently
small motion must preserve the volumes of all -faces; and this property
still holds in for if an invariant of
the degenerate simplex has the desired sign. This answers a question posed by
the author, and the proof relies on an invariant we discovered
for any -stress on a cell complex in . We introduce a
characteristic polynomial of the degenerate simplex by defining
, and prove that the roots
of are real for the Euclidean case. Some evidence suggests the same
conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr
Confident Predictability: Identifying reliable gene expression patterns for individualized tumor classification using a local minimax kernel algorithm
<p>Abstract</p> <p>Background</p> <p>Molecular classification of tumors can be achieved by global gene expression profiling. Most machine learning classification algorithms furnish global error rates for the entire population. A few algorithms provide an estimate of probability of malignancy for each queried patient but the degree of accuracy of these estimates is unknown. On the other hand <it>local minimax learning </it>provides such probability estimates with best finite sample bounds on expected mean squared error on an individual basis for each queried patient. This allows a significant percentage of the patients to be identified as <it>confidently predictable</it>, a condition that ensures that the machine learning algorithm possesses an error rate below the tolerable level when applied to the confidently predictable patients.</p> <p>Results</p> <p>We devise a new learning method that implements: (i) feature selection using the k-TSP algorithm and (ii) classifier construction by local minimax kernel learning. We test our method on three publicly available gene expression datasets and achieve significantly lower error rate for a substantial identifiable subset of patients. Our final classifiers are simple to interpret and they can make prediction on an individual basis with an individualized confidence level.</p> <p>Conclusions</p> <p>Patients that were predicted confidently by the classifiers as cancer can receive immediate and appropriate treatment whilst patients that were predicted confidently as healthy will be spared from unnecessary treatment. We believe that our method can be a useful tool to translate the gene expression signatures into clinical practice for personalized medicine.</p
Manin matrices and Talalaev's formula
We study special class of matrices with noncommutative entries and
demonstrate their various applications in integrable systems theory. They
appeared in Yu. Manin's works in 87-92 as linear homomorphisms between
polynomial rings; more explicitly they read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). We claim
that such matrices behave almost as well as matrices with commutative elements.
Namely theorems of linear algebra (e.g., a natural definition of the
determinant, the Cayley-Hamilton theorem, the Newton identities and so on and
so forth) holds true for them.
On the other hand, we remark that such matrices are somewhat ubiquitous in
the theory of quantum integrability. For instance, Manin matrices (and their
q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and
the so--called Cartier-Foata matrices. Also, they enter Talalaev's
hep-th/0404153 remarkable formulas: ,
det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show
that theorems of linear algebra, after being established for such matrices,
have various applications to quantum integrable systems and Lie algebras, e.g
in the construction of new generators in (and, in general,
in the construction of quantum conservation laws), in the
Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also
discuss applications to the separation of variables problem, new Capelli
identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints
e.g. in Newton id-s fixed, normal ordering convention turned to standard one,
refs. adde
A quantum isomonodromy equation and its application to N=2 SU(N) gauge theories
We give an explicit differential equation which is expected to determine the
instanton partition function in the presence of the full surface operator in
N=2 SU(N) gauge theory. The differential equation arises as a quantization of a
certain Hamiltonian system of isomonodromy type discovered by Fuji, Suzuki and
Tsuda.Comment: 15 pages, v2: typos corrected and references added, v3: discussion,
appendix and references adde
Integrable Models From Twisted Half Loop Algebras
This paper is devoted to the construction of new integrable quantum
mechanical models based on certain subalgebras of the half loop algebra of
gl(N). Various results about these subalgebras are proven by presenting them in
the notation of the St Petersburg school. These results are then used to
demonstrate the integrability, and find the symmetries, of two types of
physical system: twisted Gaudin magnets, and Calogero-type models of particles
on several half-lines meeting at a point.Comment: 22 pages, 1 figure, Introduction improved, References adde
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