Mathematics

Resource-mediated competition between two plant species with different rates of water intake

Speaker: 
Chunyi Gai
Date: 
Fri, Oct 14, 2022
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

We propose an extension of the well-known Klausmeier model of vegetation to two plant species that consume water at different rates. Rather than competing directly, the plants compete through their intake of water, which is a shared resource between them. In semi-arid regions, the Klausmeier model produces vegetation spot patterns. We are interested in how the competition for water affects the co-existence and stability of patches of different plant species. We consider two plant types: a “thirsty” species and a “frugal” species, that only differ by the amount of water they consume per unit growth, while being identical in other aspects. We find that there is a finite range of precipitation rate for which two species can co-exist. Outside of that range (when the rate is either sufficiently low or high), the frugal species outcompetes the thirsty species. As the precipitation rate is decreased, there is a sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. The pattern consisting of only frugal spots is the most resilient. The next-most-resilient pattern consists of all-thirsty patches, with the mixed pattern being less resilient than either of the homogeneous patterns. We also examine numerically what happens for very large precipitation rates. We find that for a sufficiently high rate, the frugal plant takes over the entire range, outcompeting the thirsty plant.

Class: 

Multiplicative functions in short intervals

Speaker: 
Paranedu Darbar
Date: 
Thu, Oct 6, 2022
Location: 
PIMS, University of British Columbia
PIMS, University of Lethbridge
PIMS, University of Northern British Columbia
Zoom
Online
Conference: 
L-Functions in Analytic Number Theory Seminar
Abstract: 

In this talk, we are interested in a general class of multiplicative functions. For a function that belongs to this class, we will relate its “short average” to its “long average”. More precisely, we will compute the variance of such a function over short intervals by using Fourier analysis and by counting rational points on certain binary forms. The discussion is applicable to some interesting multiplicative functions such as

$$
\mu_k(n), \frac{\phi (n)}{n}, \frac{n}{\phi (n)}, \mu^2(n)\frac{\phi(n)}{n},
\sigma_\alpha (n), (-1)^{\#\left\{p: p^k | n \right\}}
$$

and many others and it provides various new results and improvements to the previous result
in the literature. This is a joint work with Mithun Kumar Das.

 

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

Class: 

Joint value distribution of L-functions

Speaker: 
Junxian Li
Date: 
Thu, Sep 22, 2022
Location: 
PIMS, University of British Columbia
PIMS, University of Lethbridge
PIMS, University of Northern British Columbia
Zoom
Online
Conference: 
L-Functions in Analytic Number Theory Seminar
Abstract: 

It is believed that distinct primitive L-functions are “statistically independent”. The independence can be interpreted in many different ways. We are interested in the joint value distributions and their applications in moments and extreme values for distinct L-functions. We discuss some large deviation estimates in Selberg and Bombieri-Hejhal’s central limit theorem for values of several L-functions. On the critical line, values of distinct primitive L-functions behave independently in a strong sense. However, away from the critical line, values of distinct Dirichlet L-functions begin to exhibit some correlations.

This is based on joint works with Shota Inoue.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

Class: 

Rotary Molecular Motors Driven By Transmembrane Ionic Currents

Speaker: 
Charles S Peskin
Date: 
Wed, Sep 28, 2022
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

There are two rotary motors in biology, ATP synthase and the bacterial flagellar motor. Both are driven by transmembrane ionic currents. We consider an idealized model of such a motor, essentially an electrostatic turbine. The model has a rotor and a stator, which are closely fitting cylinders. Attached to the rotor is a fixed density of negative charge, with helical symmetry. Positive ions move longitudinally by drift and diffusion on the stator. A key assumption is local electroneutrality of the combined charge distribution. With this setup we derive explicit formulae for the transmembrane current and the angular velocity of the rotor in terms of the transmembrane electrochemical potential difference of the positive ions and the mechanical torque on the motor. This relationship between "forces" and "fluxes" turns out to be linear, and given by a symmetric positive definite matrix, as anticipated by non-equilibrium thermodynamics, although we do not make any use of that formalism in deriving the result. The equal off-diagonal terms of this 2x2 matrix describe the electromechanical coupling of the motor. Although macroscopic, the model can be used as a foundation for stochastic simulation via the Einstein relation.

Class: 

Gaps in the sequence square root n mod 1

Speaker: 
Keivan Mallahi-Karai
Date: 
Mon, Oct 3, 2022
Location: 
University of Utah
Online
Conference: 
University of Utah Seminar in Ergodic Theory
Abstract: 

In this talk I will present some of aspects of the proof of a theorem of Elkies and McMullen (Duke Math Journal, 2004) on the asymptotic distribution of the gap sizes for the finite sequence ( √n (mod 1) : 1 ≤ n ≤ N ) as N goes to infinity. The proof relies, among other things, on tools from homogenous dynamics and relates the problem to one in the geometry of numbers.

Class: 
Subject: 

Fubini foiled: pathological foliations from symbolic codings

Speaker: 
Aaron Calderon
Date: 
Tue, Sep 13, 2022
Location: 
Online
Conference: 
Online working seminar in Ergodic Theory
University of Utah Seminar in Ergodic Theory
Abstract: 

In this talk, I'll present a counterintuitive construction of A. Katok (exposited by Milnor) which at first glance seems to contradict Fubini's theorem. In particular, one can build a full-measure set E in the unit square and a foliation of the square by smooth curves such that any leaf of the foliation meets E in exactly one point. If time permits, I'll also mention work of Ruelle-Wilkinson and Shub-Wilkinson that shows these sorts of pathological examples are common in non-uniformly hyperbolic dynamics.

Class: 
Subject: 

Agent-based modelling and topological data analysis of zebrafish patterns

Speaker: 
Bjorn Sandstede
Date: 
Wed, Sep 21, 2022
Location: 
PIMS, University of British Columbia
Online
Zoom
Conference: 
Mathematical Biology Seminar
Abstract: 

Patterns are widespread in nature and often form during early development due to the self-organization of cells or other independent agents. One example are zebrafish (Danio rerio): wild-type zebrafish have regular black and gold stripes, while mutants and other fish feature spotty and patchy patterns. Qualitatively, these patterns display impressive consistency and redundancy, yet variability inevitably exists on both microscopic and macroscopic scales. I will first discuss an agent-based model that suggests that both consistency and richness of patterning on zebrafish stems from the presence of redundancy in iridophore interactions. In the second part of my talk, I will focus on how we can quantify features and variability of patterns to facilitate predictive analyses. I will discuss an approach based on topological data analysis for quantifying both agent-level features and global pattern attributes on a large scale. The proposed methodology is able to quantify the differential impact of stochasticity in cell interactions on wild-type and mutant patterns and predicts stripe and spot statistics as a function of varying cellular communication. This is joint work with Alexandria Volkening and Melissa McGuirl.

Class: 

No IET is Mixing

Speaker: 
Gianluca Faraco
Date: 
Tue, Sep 6, 2022
Location: 
University of Utah
Online
Conference: 
Online working seminar in Ergodic Theory
University of Utah Seminar in Ergodic Theory
Abstract: 

In 1980, Katok proved that no interval exchange transformation (IET) is mixing for any Borel invariant measure. The same holds for any special flow constructed by mean of any IET and a roof function of bounded variation. In this talk, I aim to explain to you the proof of these results.

Class: 
Subject: 

Statistical Estimation with Differential Privacy

Speaker: 
Gautam Kamath
Date: 
Thu, Aug 25, 2022
Location: 
PIMS, University of Victoria
Conference: 
Mathematics of Ethical Decision-making Systems
Abstract: 

Naively implemented, statistical procedures are prone to leaking information about their training data, which can be problematic if the data is sensitive. Differential privacy, a rigorous notion of data privacy, offers a principled framework to dealing with these issues. I will survey recent results in differential private statistical estimation, presenting a few vignettes which highlight novel challenges for even the most fundamental problems, and suggesting solutions to address them. Along the way, I’ll mention connections to tools and techniques in a number of fields, including information theory and robust statistics.

Class: 
Subject: 

Furstenberg's topological x2 x3 result

Speaker: 
Barak Weiss
Date: 
Tue, Aug 16, 2022
Location: 
University of Utah
Zoom
Conference: 
Online working seminar in Ergodic Theory
University of Utah Seminar in Ergodic Theory
Abstract: 

In 1967 Furstenberg proved that any closed subset of the one dimensional torus R/Z, invariant under the two maps x -> 2x mod 1, x -> 3x mod 1, is either finite or the entire torus. I will explain a proof of this result due to Boshernitzan (1994). Furstenberg's proof is slightly longer but perhaps more conceptual. I will explain the main steps in Furstenberg's approach and their connection to joinings.

Class: 
Subject: 

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