Math RSS Feeds

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math updates on arXiv.org

+ Quasipinning and its relevance for $N$-Fermion quantum states. (arXiv:1409.0019v1 [quant-ph])

Authors: Christian Schilling

Fermionic natural occupation numbers (NON) do not only obey Pauli's famous exclusion principle but are even further restricted to a polytope by the generalized Pauli constraints, conditions which follow from the fermionic exchange statistics. Whenever given NON are pinned to the polytope's boundary the corresponding $N$-fermion quantum state $|\Psi_N\rangle$ simplifies due to a selection rule. We show analytically and numerically for the most relevant settings that this rule is stable for NON close to the boundary, if the NON are non-degenerate. In case of degeneracy a modified selection rule is conjectured and its validity is supported. As a consequence the recently found effect of quasipinning is physically relevant in the sense that its occurrence allows to approximately reconstruct $|\Psi_N\rangle$, its entanglement properties and correlations from 1-particle information. Our finding also provides the basis for a generalized Hartree-Fock method by a variational ansatz determined by the selection rule.

+ Instantons on conical half-flat 6-manifolds. (arXiv:1409.0030v1 [hep-th])

Authors: Severin Bunk, Olaf Lechtenfeld, Alexander D. Popov, Marcus Sperling

We present a general procedure to construct 6-dimensional manifolds with SU(3)-structure from SU(2)-structure 5-manifolds. We thereby obtain half-flat cylinders and sine-cones over 5-manifolds with Sasaki-Einstein SU(2)-structure. They are nearly Kahler in the special case of sine-cones over Sasaki-Einstein 5-manifolds. Both half-flat and nearly Kahler 6-manifolds are prominent in flux compactifications of string theory. Subsequently, we investigate instanton equations for connections on vector bundles over these half-flat manifolds. A suitable ansatz for gauge fields on these 6-manifolds reduces the instanton equation to a set of matrix equations. We finally present some of its solutions and discuss the instanton configurations obtained this way.

+ Tracking Dynamic Point Processes on Networks. (arXiv:1409.0031v1 [stat.ML])

Authors: Eric C. Hall, Rebecca M. Willett

Cascading chains of events are a salient feature of many real-world social, biological, and financial networks. In social networks, social reciprocity accounts for retaliations in gang interactions, proxy wars in nation-state conflicts, or Internet memes shared via social media. Neuron spikes stimulate or inhibit spike activity in other neurons. Stock market shocks can trigger a contagion of volatility throughout a financial network. In these and other examples, only individual events associated with network nodes are observed, usually without knowledge of the underlying dynamic relationships between nodes. This paper addresses the challenge of tracking how events within such networks stimulate or influence future events. The proposed approach is an online learning framework well-suited to streaming data, using a multivariate Hawkes point process model to encapsulate autoregressive features of observed events within the social network. Recent work on online learning in dynamic environments is leveraged not only to exploit the dynamics within the underlying network, but also to track that network structure as it evolves. Regret bounds and experimental results demonstrate that the proposed method performs nearly as well as an oracle or batch algorithm.

+ Noncollision Singularities in a Planar Four-body Problem. (arXiv:1409.0048v1 [math.DS])

Authors: Jinxin Xue

In this paper, we show that there is a Cantor set of initial conditions in a planar four-body problem such that all the four bodies escape to infinity in finite time avoiding collisions. This proves the Painlev\'{e} conjecture for the four-body case, thus settles the conjecture completely.

+ On Asymptotic Normality of the Local Polynomial Regression Estimator with Stochastic Bandwidths. (arXiv:1409.0055v1 [math.ST])

Authors: Carlos Martins-Filho, Paulo Saraiva

Nonparametric density and regression estimators commonly depend on a bandwidth. The asymptotic properties of these estimators have been widely studied when bandwidths are nonstochastic. In practice, however, in order to improve finite sample performance of these estimators, bandwidths are selected by data driven methods, such as cross-validation or plug-in procedures. As a result nonparametric estimators are usually constructed using stochastic bandwidths. In this paper we establish the asymptotic equivalence in probability of local polynomial regression estimators under stochastic and nonstochastic bandwidths. Our result extends previous work by Boente and Fraiman (1995) and Ziegler (2004).

+ Dendriform-Tree Setting for Fully Non-commutative Fliess Operators. (arXiv:1409.0059v1 [math.CO])

Authors: Luis A. Duffaut Espinosa, W. Steven Gray, Kurusch Ebrahimi-Fard

This paper provides a dendriform-tree setting for Fliess operators with matrix-valued inputs. This class of analytic nonlinear input-output systems is convenient, for example, in quantum control. In particular, a description of such Fliess operators is provided using planar binary trees. Sufficient conditions for convergence of the defining series are also given.

+ Lower bound for ranks of invariant forms. (arXiv:1409.0061v1 [math.AG])

Authors: Harm Derksen, Zach Teitler

We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad--Schreyer--Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic symmetric matrices.

+ Spherical and Gaussian Spin Glasses. (arXiv:1409.0062v1 [cond-mat.dis-nn])

Authors: Giuseppe Genovese, Daniele Tantari

We report some results on spin glass systems with soft spin, spherical and Gaussian. We analyse the Legendre variational structure linking the two models, in the spirit of equivalence of ensembles. At the end we discuss the existing link between these spin glasses and the analogical Hopfield model of neural networks.

Mathematics News -- ScienceDaily

+ Spot light on tailor-made multicyclic type of polymers
Scientists have synthesized multicyclic type of polymers for the first time offering insights for tailoring polymer properties as well as the mathematics of complex geometries.
+ Flu outbreak provides rare study material
Five years ago this month, one of the first U.S. outbreaks of the H1N1 virus swept through the Washington State University campus, striking some 2,000 people. A university math and biology professor has used a trove of data gathered at the time to gain insight into how only a few infected people could launch the virus's rapid spread across the university community.
+ Combining math and music to open new possibilities
The power of mathematics to open new possibilities in music has been demonstrated by scientists for years. Modern experiments with computer music are just the most recent example.
+ Learning by watching, toddlers show intuitive understanding of probability
Most people know children learn many skills simply by watching people around them. Without explicit instructions youngsters know to do things like press a button to operate the television and twist a knob to open a door. Now researchers have taken this further, finding that children as young as age 2 intuitively use mathematical concepts such as probability to help make sense of the world around them.
+ Neuroscience and big data: How to find simplicity in the brain
Scientists can now monitor and record the activity of hundreds of neurons concurrently in the brain, and ongoing technology developments promise to increase this number. However, simply recording the neural activity does not automatically lead to a clearer understanding of how the brain works. In a new article, researchers describe the scientific motivations for studying the activity of many neurons together, along with a class of machine learning algorithms for interpreting the activity.
+ How worms crawl: mathematical model challenges traditional view
A new mathematical model for earthworms and insect larvae challenges the traditional view of how these soft bodied animals get around. Researchers say that there is a far greater role for the body's mechanical properties and the local nerves which react to the surface that the animal is traveling across.
+ How children's brains memorize math facts
As children learn basic arithmetic, they gradually switch from solving problems by counting on their fingers to pulling facts from memory. The shift comes more easily for some kids than for others, but no one knows why. Now, new brain-imaging research gives the first evidence drawn from a longitudinal study to explain how the brain reorganizes itself as children learn math facts.
+ Powerful math creates 3-D shapes from simple sketches
A new graphics system that can easily produce complex 3-D shapes from simple professional sketches will be unveiled by computer scientists. The technology has the potential to dramatically simplify how designers and artists develop new product ideas. Converting an idea into a 3-D model using current commercial tools can be a complicated and painstaking process.