4,787 research outputs found
Riemann Hypothesis: a GGC factorisation
A GGC (Generalized Gamma Convolution) representation of Riemann's Xi-function
is constructed
Free energy of a folded semiflexible polymer confined to a nanochannel of various geometries
Monte Carlo simulations are used to study the conformational properties of a
folded semiflexible polymer confined to a long channel. We measure the
variation in the conformational free energy with respect to the end-to-end
distance of the polymer, and from these functions we extract the free energy of
the hairpin fold, as well as the entropic force arising from interactions
between the portions of the polymer that overlap along the channel. We consider
the scaling of the free energies with respect to varying the persistence length
of the polymer, as well as the channel dimensions for confinement in
cylindrical, rectangular and triangular channels. We focus on polymer behaviour
in both the classic Odijk and back folded Odijk regimes. We find the scaling of
the entropic force to be close to that predicted from a scaling argument that
treats interactions between deflection segments at the second virial level. In
addition, the measured hairpin fold free energy is consistent with that
obtained directly from a recent theoretical calculation for cylindrical
channels. It is also consistent with values determined from measurements of the
global persistence length of a polymer in the backfolded Odijk regime in recent
simulation studies
On Hilbert's 8th Problem
A Hadamard factorization of the Riemann Xi-function is constructed to
characterize the zeros of the zeta function
Polymer translocation into and out of an ellipsoidal cavity
Monte Carlo simulations are used to study the translocation of a polymer into
and out of a ellipsoidal cavity through a narrow pore. We measure the polymer
free energy F as a function of a translocation coordinate, s, defined to be the
number of bonds that have entered the cavity. To study polymer insertion, we
consider the case of a driving force acting on monomers inside the pore, as
well as monomer attraction to the cavity wall. We examine the changes to F(s)
upon variation in the shape anisometry and volume of the cavity, the polymer
length, and the strength of the interactions driving the insertion. For
athermal systems, the free energy functions are analyzed using a scaling
approach, where we treat the confined portion of the polymer to be in the
semi-dilute regime. The free energy functions are used with the Fokker-Planck
equation to measure mean translocation times, as well as translocation time
distributions. We find that both polymer ejection and insertion is faster for
ellipsoidal cavities than for spherical cavities. The results are in
qualitative agreement with those of a Langevin dynamics study in the case of
ejection but not for insertion. The discrepancy is likely due to
out-of-equilibrium conformational behaviour that is not accounted for in the FP
approachComment: 11 pages, 11 figure
van Dantzig Pairs, Wald Couples and Hadamard Factorisation
Some consequences of a duality between the Hadamard-Weierstrass factorisation
of an entire function and van Dantzig-Wald couples of random variables are
explored. We demonstrate the methodology on particular functions including the
Riemann zeta and xi-functions, Ramanujan's tau function, L-functions and Gamma
and Hyperbolic functions
Posterior Concentration for Sparse Deep Learning
Spike-and-Slab Deep Learning (SS-DL) is a fully Bayesian alternative to
Dropout for improving generalizability of deep ReLU networks. This new type of
regularization enables provable recovery of smooth input-output maps with
unknown levels of smoothness. Indeed, we show that the posterior distribution
concentrates at the near minimax rate for -H\"older smooth maps,
performing as well as if we knew the smoothness level ahead of time.
Our result sheds light on architecture design for deep neural networks, namely
the choice of depth, width and sparsity level. These network attributes
typically depend on unknown smoothness in order to be optimal. We obviate this
constraint with the fully Bayes construction. As an aside, we show that SS-DL
does not overfit in the sense that the posterior concentrates on smaller
networks with fewer (up to the optimal number of) nodes and links. Our results
provide new theoretical justifications for deep ReLU networks from a Bayesian
point of view
Deep Learning: Computational Aspects
In this article we review computational aspects of Deep Learning (DL). Deep
learning uses network architectures consisting of hierarchical layers of latent
variables to construct predictors for high-dimensional input-output models.
Training a deep learning architecture is computationally intensive, and
efficient linear algebra libraries is the key for training and inference.
Stochastic gradient descent (SGD) optimization and batch sampling are used to
learn from massive data sets
Bayesian Particle Tracking of Traffic Flows
We develop a Bayesian particle filter for tracking traffic flows that is
capable of capturing non-linearities and discontinuities present in flow
dynamics. Our model includes a hidden state variable that captures sudden
regime shifts between traffic free flow, breakdown and recovery. We develop an
efficient particle learning algorithm for real time on-line inference of states
and parameters. This requires a two step approach, first, resampling the
current particles, with a mixture predictive distribution and second,
propagation of states using the conditional posterior distribution. Particle
learning of parameters follows from updating recursions for conditional
sufficient statistics. To illustrate our methodology, we analyze measurements
of daily traffic flow from the Illinois interstate I-55 highway system. We
demonstrate how our filter can be used to inference the change of traffic flow
regime on a highway road segment based on a measurement from freeway
single-loop detectors. Finally, we conclude with directions for future
research
Deep Learning for Short-Term Traffic Flow Prediction
We develop a deep learning model to predict traffic flows. The main
contribution is development of an architecture that combines a linear model
that is fitted using regularization and a sequence of layers.
The challenge of predicting traffic flows are the sharp nonlinearities due to
transitions between free flow, breakdown, recovery and congestion. We show that
deep learning architectures can capture these nonlinear spatio-temporal
effects. The first layer identifies spatio-temporal relations among predictors
and other layers model nonlinear relations. We illustrate our methodology on
road sensor data from Interstate I-55 and predict traffic flows during two
special events; a Chicago Bears football game and an extreme snowstorm event.
Both cases have sharp traffic flow regime changes, occurring very suddenly, and
we show how deep learning provides precise short term traffic flow predictions
Deep Learning: A Bayesian Perspective
Deep learning is a form of machine learning for nonlinear high dimensional
pattern matching and prediction. By taking a Bayesian probabilistic
perspective, we provide a number of insights into more efficient algorithms for
optimisation and hyper-parameter tuning. Traditional high-dimensional data
reduction techniques, such as principal component analysis (PCA), partial least
squares (PLS), reduced rank regression (RRR), projection pursuit regression
(PPR) are all shown to be shallow learners. Their deep learning counterparts
exploit multiple deep layers of data reduction which provide predictive
performance gains. Stochastic gradient descent (SGD) training optimisation and
Dropout (DO) regularization provide estimation and variable selection. Bayesian
regularization is central to finding weights and connections in networks to
optimize the predictive bias-variance trade-off. To illustrate our methodology,
we provide an analysis of international bookings on Airbnb. Finally, we
conclude with directions for future research
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