Questions tagged [applied-mathematics]

the branch of mathematics that deals with the mathematical aspects of problems from science and engineering: applied analysis, numerical mathematics, applied statistics etc. (For applications of mathematics in general, cf. also [tag:applications].)

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75 views

Linearized NLS/GP around a soliton and the spectrum of the evolution operator

I apologize if this has been asked before but so far I haven't found it anywhere. Consider the Nonlinear Schrödinger equation with a potential (i.e. Gross- Pitaevskii) in $\mathbb{R}^{d}$ $$i\Psi_{t} =...
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Another question from Villani's monograph “Hypocoercivity”

I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 62 (the first snapshot), he defined a new inner product $((\cdot,\cdot))$ as in (...
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The relation between unit quaternions and the virtual rotation of an angular velocity vector?

I am working with a calculus employed in multi rigid body dynamics problems introduced by professor E. Haug (c.f. this book). Let's define $\boldsymbol{\mathrm{e}} \in \mathcal{R}^4$ as a set of unit ...
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Recursive formula for integral of Chebyshev-type integral

Define $$ I_{m,n}(x,y,r) = \int_a^b T_m(x + r \sin(\gamma)) T_n(y-r \cos(\gamma)) d\gamma $$ where $T_m(x)$ are the Chebyshev polynomials of the first kind, and $a$ and $b$ are constants. Assume that ...
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Is Steven J. Miller's “research” on election fraud sound? And was he paid for it? [closed]

I recently encountered the following piece regarding alleged massive voter fraud in Pennsylvania: https://justthenews.com/sites/default/files/2020-11/...
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5answers
441 views

What are efficient pooling designs for RT-PCR tests?

I realize this is long, but hopefully I think it may be worth the reading for people interested in combinatorics and it might prove important to Covid-19 testing. Slightly reduced in edit. The ...
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119 views

Notions of “completeness” and “sufficiency” of a mathematical model

I'm modelling a real-world problem as having instances $i$ in a set $P$. As a very simple artificial example, consider the problem of choosing a meeting room given a certain number of people. Then $i =...
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2answers
146 views

Examples of applications of hyperbolic conservation laws

I am giving a talk in front of my applied PDE research group on hyperbolic conservation laws, the most basic form of which is the PDE $$ u_t + f(u)_x = 0 $$ where $u$ is the conserved quantity and $f$ ...
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1answer
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Recent uses of applied mathematics in pure mathematics

In this answer Yves de Cornulier mentioned a talk about the possible uses of persistent homology in geometric topology and group theory. Persistent homology is a tool from the area of topological data ...
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SIR model constraint [closed]

During these past months, I've heard a lot about some pandemic modelling techniques, specially the so-called SIR model. Before I begin, I'd like to stress that my interest and question are just a ...
5
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1answer
186 views

Renormalization group strategies

Before introducing block spin transformations in chapter four of Random Walks, Critical Phenomena and Triviality in Quantum Field Theory, the authors state the following: "In this chapter we sketch ...
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67 views

Gradient descent in $U(n)^r$

I have a function $f:U(n)^r\rightarrow \mathbb{R}$ which I would like to minimize. Here, $U(n)$ is the set of unitary matrices, and $r$ should be considered to be much bigger than $n$. For instance, $...
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100 views

Spins in classical statistical mechanics

I'm reading Kupiainen's notes on the renormalization group and also caught my attention. Actually, this is something that often causes my some confusion. On page 43, in the section about Ginzburg-...
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Multiexponential analysis of infection counts with errors

In the past, I have seen some decompositions of sums of exponential decays into components by the Padé-Laplace method: Apply the Laplace transform $${\frak L}(\sum_{i=1}^n a_i e^{k_i t}) = \sum_{i=1}^...
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1answer
244 views

How many persons pass your 1.5 meter neighbourhood during 1 week ? If the distribution is power law what is the exponent?

Consider a graph with vertices being people (in some region), and make an edge if one person pass another closer than say 1.5 meter during say one week. (Such a graph might be thought a kind of ...
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8answers
3k views

Relevant mathematics to the recent coronavirus outbreak

I would like to ask about (old* and new) reliable mathematical literature relevant to various mathematical aspects of the recent coronavirus outbreak: In particular, standard statistical/mathematical ...
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1answer
102 views

Persistent homology stability results (query about Lipschitz functions)

One of the beneficial properties of persistent homology is its stability results (so called robustness to noise). Usually the referenced paper is this paper titled "Lipschitz functions have $L_p$-...
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2answers
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Physical interpretation of the Manifold Hypothesis

Motivation: Most dimensionality reduction algorithms assume that the input data are sampled from a manifold $\mathcal{M}$ whose intrinsic dimension $d$ is much smaller than the ambient dimension $D$. ...
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154 views

Help to understand a limit $\varepsilon\rightarrow 0$ computation on a fluid mechanic paper

In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if $$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
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Physical applications based on mathematical model of non-instantaneous impulsive evolution equations

In this paperhttps://www.researchgate.net/publication/269404928_Periodic_solutions_for_nonlinear_evolution_equations_with_non-instantaneous_impulses authors prove the existence and stability of the ...
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2answers
234 views

A question involving directional derivatives and differential inequalities

This is a follow-up question to A question about copulas and directional derivatives. Since no answer was given, I am going to precise the definition of copula. I am interested in proving (or ...
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7answers
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Applications of mathematics in clinical setting

What are some examples of successful mathematical attempts in clinical setting, specifically at the patient-disease-drug level? To clarify, by patient-disease-drug level, I mean the mathematical work ...
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125 views

The topological complexity of polytopes

Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be ...
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1answer
388 views

Fast Bourgain embedding (or similar embeddings)?

Currently I am working on applications of Bourgain Embedding (or similar embeddings of finite metric spaces to $l_2$) to automatic feature engineering for machine learning/data science ( http://www....
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4answers
951 views

Differential geometry applied to biology

This was originally a question posted here on MathSE. But I'll ask again here to see if I can get some different answers. I'm looking for current areas of research which apply techniques from ...
2
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1answer
68 views

Probability distributions with irregular behaviour

Might there be a probability distribution $\mathcal{D}$ such that if we sample $a_i \sim \mathcal{D}([-N,N])$ where $[-N,N] \subset \mathbb{Z}$ then if we define the asymptotic estimate $f$: \begin{...
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193 views

Research-level blogs on complex networks:

I'm an applied mathematician that has a research interest in complex networks for modelling biological systems and I wondered whether the MathOverflow community might know of research-level blogs that ...
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0answers
83 views

Plethora of variant neural networks?

Since a decade ago when new life was breathed in to neural networks in the form of deep learning a plethora of different architectures have come about. Is there a reference that gives compendium of ...
3
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0answers
60 views

Second order non-instantaneous impulsive evolution equations

The first order linear non-instantaneous impulsive evolution equations is given as; $$u'(t)=Au(t)~~ t\in[s_i,t_{i+1}],\,i\in\mathbb{N_0}:=\{0,1,2,...\}$$ $$u(t_i^+)=(E+B_i)u(t_i^-),\,\,i\in\mathbb{...
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22answers
6k views

Which high-degree derivatives play an essential role?

Q. Which high-degree derivatives play an essential role in applications, or in theorems? Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration), and the ...
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69 views

Use of Asymptotics in Diffusion Maps

Question for brevity: Suppose $\varepsilon >0$ is small and that $$ f(\varepsilon) = f_1(\varepsilon) + \mathcal{O}(\varepsilon^k) $$ where $f_1$ has order $\varepsilon^{-\delta}$ for small fixed ...
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Is the field of q-series 'dead'? [closed]

I had a discussion with my advisor about what am I interested as my future research direction and I said it is special functions and q-series. He laughed and said that the topic is essentially dead ...
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479 views

The collected works of John von Neumann

Might there be an online collection of John von Neumann's collected works in pdf format? I'm particularly interested in his approach to applied mathematics(ex. shockwaves, hydrodynamics). Note: I ...
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0answers
134 views

Doubts related Shifting from Pure to Applied math [closed]

I am a second year (Pure) Math and (Theoretical) Physics undergraduate in India. I want to do a masters in Applied/Computational Science or Math, for which I have apply after next 7 months. I have ...
5
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1answer
98 views

Numerical instability of the axis-angle representation of rotations in 3D

Suppose that I have $1000$ pair of points where each pair consists of a point in $\mathbb{R}^3$ and its image after a rotation in $\mathrm{SO}(3)$ with some noise. I have used RANSAC to find the ...
3
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1answer
132 views

Variation of steepest descent/Laplace methods for non-exponential integrands

I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type $$\int_C f(z) M(\lambda g(z)) dz$$ for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
6
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1answer
167 views

Further Developments of Lieb-Schultz-Mattis theorem in Mathematics

The Lieb-Schultz-Mattis theorem [1] and its higher-dimensional generalizations [2] says that a translation-invariant lattice model of spin-1/2's cannot allow a non-degenerate ground state preserving ...
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1answer
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Is there any paper which summarizes the mathematical foundation of deep learning?

Is there any paper which summarizes the mathematical foundation of deep learning? Now, I am studying about the mathematical background of deep learning. However, unfortunately I cannot know to what ...
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2answers
1k views

On Mathematical Foundations of Football

Football (soccer) is arguably one of the most unpredictable sports. Countless variables play a role in determining the outcome of a certain football match. Due to the high complexity of the entire set ...
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3answers
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On Mathematical Analysis of MathSciNet & MathOverflow

This question has two original motivations: mathematical and social. The mathematical motivation is mainly based on what I have seen about Zipf's law here and there. The Zipf's law simply states ...
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257 views

How to promote a blog?

Math behind might be interesting. Quite recent bloggingg activity might have interesting math model. The point is that bloggers compete for subscribers and at the same time cooperate gaining ...
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3answers
608 views

Are there any books/articles that apply abstract coordinate free differential geometry to basic thermodynamics?

The mathematical structure of thermodynamics by Peter Salamon (pdf) would be an example, but i would like a more abstract natural formulation of application of differential geometry or even geometric ...
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Questions about generalized Polynomial Chaos, book by Dongbin Xiu

I have some questions about Chapter 5 from the book Numerical Methods for Stochastic Computations, by Dongbin Xiu. Theorem 5.7: Let $Y$ be a random variable and $\mathbb{E}[Y^2]<\infty$. Let $Z$ ...
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9answers
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Mathematical conjectures on which applications depend

What are some examples of mathematical conjectures that applied mathematicians assume to be true in applications, despite it being unknown whether or not they are true?
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1answer
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Why is persistent cohomology so much faster than persistent homology

I refer to this paper: de Silva, Vin; Morozov, Dmitriy; Vejdemo-Johansson, Mikael. Dualities in persistent (co)homology. Inverse Problems 27 (2011), no. 12, 124003, 17 pp. (Journal link, arXiv link). ...
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240 views

Industrial research projects on “mathematical modeling and PDEs” [closed]

Apparently there are several companies in a great variety of fields (medical, biological, engineering, etc.) that need "consulting on mathematical modeling and PDEs" from applied mathematicians. I'...
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4answers
4k views

Is there a mathematical and information theoretic explanation for this cube packing phenomenon?

I saw this unintuitive result on dice packing: A jumble of thousands of cubic dice, agitated by an oscillating rotation, can rapidly become completely ordered, a result that is hard to produce ...
7
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1answer
366 views

A game-theoretical question in a political economy model

My research question in a dynamic model of political competition boils down to the following conjecture. I am confident that it holds (all simulations work), but I have not been able to prove it yet. ...
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0answers
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Quantum Optimization as approximating $\mathbb{CP}^{2^n -1}$ with the orbits of a subgroup of SU($2^n$)

For example given a great circle within the sphere, we can think about computing the average distance of a point on the sphere from the great circle. Slightly more generally, given a subgroup $H \...
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3answers
3k views

Do bubbles between plates approximate Voronoi diagrams?

For example, soap bubbles:                   Image from UPenn: "A 2-dimensional foam of wet soap bubbles squashed between glass plates, after 10 hours ...