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UBC cIRcle BIRS Workshop Lecture Videos Logo
Frame theory: Applications and open problems
Banff International Research Station for Mathematical Innovation and Discovery
— 2015-04-07 Faculty Author affiliation: Air Force Institute of Technology Unreviewed Non UBC http://creativecommons.org/licenses/by-nc-nd/2.5/ca/
UBC cIRcle BIRS Workshop Lecture Videos Logo
The fully coupled 3D numerical model of hydraulic fracturing: ways to improve and possible applications
Banff International Research Station for Mathematical Innovation and Discovery
— 2020-12-07 The model and algorithm for the numerical solution of the three-dimensional problem of hydraulic fracture initiation and further propagation will be presented. The model is fully coupled and takes into account three important processes: elastic deformation of the rock, fluid flow in the fracture, and its further propagation in the rock. The mathematical model consists of three groups of equations. Each of them responses for one process defined above. The elasticity equations are solved by the dual boundary element method (DBEM), the lubrication equations by the finite element method (FEM) improved by simple conservative correction. This correction allows us to preserve the total volume of injected fluid on the discrete level. The fracture propagation criterion gives the system of non-linear equations, which is solved by special modification of relaxation method. In the early stages of propagation we need to explicitly consider the fluid lag, which in general varies along fracture front. It essentially increases needed computational resources. One of the ways to overcome this challenge is to use any approximation of behavior of variables near the fracture front (tip). Are the already developed asymptotic solution applicable here The results obtained by the model include the initiation pressure for the real configuration of perforated well, the shape of the fracture, its position and orientation, as well as the possibility of reorientation and the size of the domain where it reorients. The cementing and casing of the well can be taken into account. From the point of oilfield engineer's view, the model can be useful in the understanding of the early stage of hydraulic fracturing when there are many stop cases that sometimes lead to an unsuccessful hydraulic fracturing. Researcher Author affiliation: Institute of Computational Technologies SB RAS Unreviewed Non UBC http://creativecommons.org/licenses/by-nc-nd/4.0/
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A Bayesian modeling approach of multiple-subject fMRI data
Banff International Research Station for Mathematical Innovation and Discovery
— 2019-02-28 We present a Bayesian nonparametric regression model for the analysis of multiple-subject functional magnetic resonance imaging (fMRI) data. Our goal is to provide a joint analytical framework that allows the detection of regions of the brain that activate in response to a stimulus, while simultaneously taking into account the association, or clustering, of spatially remote voxels within and across subjects. The model incorporates information on both the spatial and temporal correlation structures of the data. It also allows for voxel-dependent and subject-specific parameters. The high dimensionality of the data and the large amount of parameters to be estimated pose computational challenges. We employ variational Bayes algorithms as an approximate computational technique and compare efficiency and estimation results with respect to a full Monte Carlo Markov chain algorithm. We explore performances of the proposed model on simulated data and on real fMRI data. Non UBC Unreviewed Faculty Author affiliation: Rice University http://creativecommons.org/licenses/by-nc-nd/4.0/
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Almost-floating bodies
Banff International Research Station for Mathematical Innovation and Discovery
— 2018-09-26 We discuss a new construction of bodies from a given convex body in $\mathbb{R}^n$ which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to $p$-affine surface areas. Non UBC Unreviewed Author affiliation: University of Michigan Postdoctoral http://creativecommons.org/licenses/by-nc-nd/4.0/
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Towards multi-layered multi-area models of cortical networks
Banff International Research Station for Mathematical Innovation and Discovery
— 2016-06-06 Faculty Author affiliation: Jülich Research Centre Unreviewed Non UBC Theoretical research on the local cortical network has mainly been concerned with the study of random networks composed of one excitatory and one inhibitory population of neurons. This led to fundamental insights on the correlation structure of activity. The present contribution discusses next steps towards a more realistic representation of the cortical microcircuit and the brain-scale architecture. The talk first introduces the draft of a full-scale model of the microcircuit at cellular and synaptic resolution [1] comprising about 100,000 neurons and one billion local synapses connecting them. The emerging network activity exhibits fundamental properties of in vivo activity: asynchronous irregular activity, layer-specific spike rates, higher spike rates of inhibitory neurons as compared to excitatory neurons, and a characteristic response to transient input. As the formal executable specification is publicly available, the model can serve as a testbed for theoretical approaches and can iteratively be refined. A key element in the mean-field theory of systems of heterogeneous populations is the transfer function of the individual elements. Recent progress [2] enables insights into the anatomical origin of oscillations in the multi-layered circuitry [3]. Despite these successes, the explanatory power of local models is limited as half of the synapses of each excitatory nerve cell have non-local origins. The second part of the talk therefore argues for the need of brain-scale models to arrive at self-consistent descriptions and addresses the arising technological and theoretical questions: Are simulations of the required size feasible [4]? Are full scale simulations required as opposed to downscaled representatives [5]? How can anatomical and physiological constraints with their respective uncertainty margins be integrated to arrive at a multi-area model with a realistic activity state [6]? www.nest-initiative.org www.csn.fz-juelich.de [1] Potjans TC, Diesmann M Cerebral Cortex 24(3):785-806 (2014) www.opensourcebrain.org [2] Schuecker J, Diesmann M, Helias M Phys Rev E 92:052119 (2015) [3] Bos H, Diesmann M, Helias M arXiv:1510.00642[q-bio.NC] (2015) [4] Kunkel S, Schmidt M, Eppler JM, Plesser HE, Masumoto G, Igarashi J, Ishii S, Fukai T, Morrison A, Diesmann M, Helias M Front Neuroinform 8:78 (2014) [5] van Albada S, Helias M, Diesmann M PLoS Comput Biol 11(9):e1004490 (2015) [6] Schuecker J, Schmidt M, van Albada S, Diesmann M, Helias M arXiv:1509.03162 [q-bio.NC] (2015) http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Simple walk in three quarter plane
Banff International Research Station for Mathematical Innovation and Discovery
— 2018-03-31 In this talk, we consider the simple walk ($\textit{i.e.}$ walk with a set of steps {$\mathcal{S}=\{\text{W, N, E, S}\}$}) in the lattice plane. We constrain the walk to avoid the negative quadrant. The objective is to compute the number of paths $c(i,j;n)$ of length $n$, starting at $(0,0)$ and ending at $(i,j)$, with $\left(i\geq 0 \text{ or } j\geq 0\right)$ and $n\geq 0$. A way to achieve this goal is to cut the three quarters of the plane into two convex symmetric parts which will be three octants of the plane. Non UBC Unreviewed Author affiliation: Université de Tours Graduate http://creativecommons.org/licenses/by-nc-nd/4.0/
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Correlations and dynamics in spatially extended balanced networks
Banff International Research Station for Mathematical Innovation and Discovery
— 2016-06-07 Non UBC Balanced networks offer an appealing theoretical framework for studying neural variability since they produce intrinsically noisy dynamics with some statistical features similar to those observed in cortical recordings. However, previous balanced network models face two critical shortcomings. First, they produce extremely weak spike train correlations, whereas cortical circuits exhibit both moderate and weak correlations depending on cortical area, layer and state. Second, balanced networks exhibit simple mean-field dynamics in which firing rates linearly track feedforward input. Cortical networks implement non-linear functions and produce non-trivial dynamics, for example, to produce motor responses. We propose that these shortcoming of balanced networks are overcome by accounting for the distance dependence of connection probabilities observed in cortex. We generalize the mean-field theory of firing rates, correlations and dynamics in balanced networks to account for distance-dependent connection probabilities. We show that, under this extension, balanced networks can exhibit either weak or moderate spike train correlations, depending on the spatial profile of connections. Networks that produce moderate correlation magnitudes also produce a signature spatial correlation structure. A careful analysis of in vivo primate data reveals this same correlation structure. Finally, we show that spatiotemporal firing rate dynamics can emerge spontaneously in spatially extended balanced networks. Principal component analysis reveals that these dynamics are fundamentally high-dimensional and reliable, suggesting a realistic spiking model for the rich dynamics underlying non-trivial neural computations. Taken together our results show that spatially extended balanced networks offer a parsimonious model of cortical circuits. Unreviewed Author affiliation: University of Notre Dame Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
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Mahler measure, regulators and modular units
Banff International Research Station for Mathematical Innovation and Discovery
— 2016-04-05 Non UBC Unreviewed Faculty Author affiliation: Universite PARIS 6 http://creativecommons.org/licenses/by-nc-nd/4.0/
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Electro-rheological model based on flexoelectric membranes embedded on non-Newtonian fluids with application to outer hair cells
Banff International Research Station for Mathematical Innovation and Discovery
— 2017-03-07 Non UBC Unreviewed Author affiliation: Universidad Nacional Autónoma de México Faculty Liquid crystal flexoelectric actuation uses an imposed electric field to create mem- brane bending and it is used by the Outer Hair Cells (OHC) located in the inner ear, whose role is to amplify sound through generation of mechanical power. Oscillations in the OHC membranes create periodic viscoelastic flows in the contacting fluid media. A key objective of this work on flexoelectric actuation relevant to OHC is to find the relations and impact of the electro-mechanical properties of the membrane, the rheolog- ical properties of the viscoelastic media, and the frequency response of the generated mechanical power output. The model developed and used in this work is based on the integration of: (i) the flexoelectric membrane shape equation applied to a circular mem- brane attached to the inner surface of a circular capillary, and (ii) the coupled capillary flow of contacting viscoelastic phases, such that the membrane flexoelectric oscillations drive periodic viscoelastic capillary flows, as in OHCs. By applying the Fourier transform formalism to the governing equation an analytical expression for the transfer function, associated to the curvature and electrical field, power dissipation elastic storage were found. The integrated flexoelectric/viscoelastic model and the novel findings contribute to the ongoing quest for a fundamental understanding of the functioning of outer hair cells (OHC), especially on the role of membrane deformation in delivering mechanical power through electromotility and its frequency-dependent power conversion efficiency. http://creativecommons.org/licenses/by-nc-nd/4.0/
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Some properties of hyperbolic dynamics from micro-local analysis
Banff International Research Station for Mathematical Innovation and Discovery
— 2020-12-06 Researcher Author affiliation: Univ. Grenoble Unreviewed Non UBC In hyperbolic dynamics (Anosov dynamics) each trajectory is strongly unstable and its behavior is unpredictable. A smooth probability distribution evolves also in a complicated way since it acquires higher and higher oscillations. Nevertheless using micro-local analysis, this evolution is predictable in the sense of distributions. It is similar to a quantum scattering problem in cotangent space as treated by Helffer and Sjöstrand using escape functions in (86'). In this talk we will use wave-packet transform (or FBI transform) and explain how to derive some spectral properties of the dynamics, as the existence of the intrinsic discrete spectrum of Ruelle resonances, a fractal Weyl law, estimates on the wave front set of the resonances, and band structure in the case of geodesic flow. Collaboration with Masato Tsujii. http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Chern-Simons theory and the Thompson group
Banff International Research Station for Mathematical Innovation and Discovery
— 2020-12-07 Other Author affiliation: Imperial College London/CNRS Unreviewed Non UBC I will present work in progress suggesting a conjecturally general correspondence relating quantum (Chern-Simons) invariants of knots in three-space to (Eynard-Orantin) higher genus mirror symmetry, the computation of the former being solved by the latter. The correspondence rests on the identification of a natural group of piecewise linear transformations on both sides of the correspondence. http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Time-dependent hydraulic fracture initiation and propagation
Banff International Research Station for Mathematical Innovation and Discovery
— 2020-12-07 Unreviewed In engineering design for multi-stage HF treatments of horizontal well stimulation, it is ideal to promote simultaneous growth of all fractures in each stage in order to reduce the number of non-producing perforation clusters. While increased attention has been given to studies of multiple HF growth, time dependence is not typically considered as a factor affecting the HF initiation and following growth. A combined experimental and modeling study is carried out to explore the occurrence of the time-dependent initiation of single/multiple hydraulic fracture(s) and their subsequent propagation. By showing the existence of HF initiation at wellbore pressures that are insufficient to induce instantaneous initiation, and explaining that its underlying mechanism is due to the stable growth of the hydraulic fracture under subcritical conditions, this research leads to new insights for promoting more evenly growth of multiple hydraulic fractures in multi-stage HF treatments. Furthermore, our experimental results indicate that the time delay associated with hydraulic fracture initiation can be affected by various factors, such as the fluid viscosity and acidity, and the confining stresses, thereby leading to the practically-relevant outcome that fluid(s) can be chosen in order to promote initiation and growth of multiple hydraulic fractures and/or single hydraulic fractures under conditions where the required wellbore pressure for instantaneous initiation cannot be reached. Non UBC Author affiliation: University of Pittsburgh Postdoctoral http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Bounds on the Riesz means of mixed Steklov problems
Banff International Research Station for Mathematical Innovation and Discovery
— 2020-12-07 Author affiliation: University of Bristol Researcher Unreviewed Non UBC The goal of this talk is to study bounds on the Riesz means of mixed Steklov problems. The Riesz mean is a convex function of eigenvalues and has an important role and connection with other spectral quantities. We recall the results known in this direction for the Laplace eigenvalues. Then we introduce the mixed Steklov problem and state the main results. We also discuss some key ideas of the proof. This is joint work with Ari Laptev. http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Group valued momentum maps
Banff International Research Station for Mathematical Innovation and Discovery
— 2020-12-09 Faculty Author affiliation: Shanghai Jiao Tong University Unreviewed Non UBC http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Tutorial on von Neumann algebras
Banff International Research Station for Mathematical Innovation and Discovery
— 2020-12-09 Faculty Author affiliation: University of California San Diego Unreviewed Non UBC In the first lecture, I will review basic notions and constructions of von Neumann algebras. The second lecture will be devoted to property Gamma and McDuffâ s property for II_1 factors. In the third lecture, I will discuss the isomorphism problem for ultrapowers of II_1 factors. ========== Lecture 1: Define vN algebras and state the bicommutant theorem. Introduce tracial vN algebras and the hyperfinite II_1 factor. Group and group measure space vN algebras. Lecture 2: Define property Gamma and discuss the connection with inner amenability of groups. Define McDuffâ s property. Examples of II_1 factors that are Gamma but not McDuff. Lecture 3: The ultrapower construction for tracial vN algebras. Discuss dependence on the choice of the ultrafilter and examples of II_1 factors with non-isomorphic ultrapowers. http://creativecommons.org/licenses/by-nc-nd/4.0/
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Semidefinite Programming Relaxations of the Traveling Salesman Problem
Banff International Research Station for Mathematical Innovation and Discovery
— 2020-12-06 Graduate Author affiliation: Cornell University Unreviewed Non UBC In this talk, we analyze a semidefinite programming relaxation of the traveling salesman problem due to de Klerk, Pasechnik, and Sotirov. Our main result is that their relaxation has an unbounded integrality gap. In particular, we give a family of instances such that the gap increases linearly with the number of cities on the tour. To obtain this result, we search for feasible solutions within a highly structured class of cost matrices; the problem of finding such solutions reduces to finding feasible solutions for a related linear program, which we do analytically. The solutions we find imply the unbounded integrality gap. Further, they imply several corollaries that help us better understand the semidefinite program and its relationship to other linear and semidefinite relaxations of the traveling salesman problem. Using the same technique, we show that a more general semidefinite program introduced by de Klerk, de Oliveira Filho, and Pasechnik for the k-cycle cover problem also has an unbounded integrality gap. http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Stability and Iterative Random Forests
Banff International Research Station for Mathematical Innovation and Discovery
— 2020-12-07 Faculty Author affiliation: University of Berkeley Non UBC Unreviewed http://creativecommons.org/licenses/by-nc-nd/4.0/

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