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 the [applications] tag.)
49 questions with no upvoted or accepted answers
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Useful applications of applied category theory
Led by John Baez, applied category theory (e.g. [1]) seems to accumulate much popularity. As someone who has noticed the importance of category theory in pure mathematics (e.g. homotopy theory, tqfts, ...
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0
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686
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The function space defined by deep neural nets
Given a deep net graph and the activation functions on the hidden vertices do we have a description of the function space spanned by it? (even if for some specific architectures and activation ...
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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 \...
user112852
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Non-Noetherian (classical) algebraic geometry
My starting point for this question is that, in a very classical sense, algebraic geometry is the study of solution spaces of systems of polynomial equations over an algebraically closed field. It is ...
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Game theory of writing multiple choice tests
Here is a model which seems pretty close to my experience of writing multiple choice tests.
Let's view the answer $t$ to each question as a binary string in $S:=\{ 0,1 \}^k$, all equally likely. The ...
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3
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First time reviewing an applied mathematics paper: how to evaluate it?
I am in the pure math camp but was invited to referee an applied/interdisciplinary paper because I'm a specialist in the underlying mathematical tool. I want to ask for general guidance about ...
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3
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Sheaf theory in TDA
I was wondering wether anyone had any examples as to why it more useful to consider a sheaf theory approach to TDA problems.
I am familiar with some of the benefits of using cellular cosheaves to ...
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How to construct lattice points in bounded symmetric domain?
Consider the Hermitian bounded symmetric domain for $k \leq m$:
$$
C_{k, m} = \{ Z \in \mathbb{C}^{m\times k} \,|\, Z^*Z < I_k \}
$$
where $I_k$ is the $k\times k$ unit matrix. If I am not mistaken,...
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3
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0
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What is known about discrete versions of the spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities?
Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of ...
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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 $\mathbb{R}^{d}$
$$i\Psi_{t} =...
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Another question from Villani's monograph "Hypocoercivity"
I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 62 (the first snapshot), he defined a new inner product $((\cdot,\cdot))$ as in (...
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3
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Notions of "completeness" and "sufficiency" of a mathematical model
I'm modelling a real-world problem as having instances $i$ in a set $P$. As a very simple artificial example, consider the problem of choosing a meeting room given a certain number of people. Then $i =...
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Second order non-instantaneous impulsive evolution equations
The first order linear non-instantaneous impulsive evolution equations is given as;
$$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{...
- 59
3
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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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2
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answers
54
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Ashkin-Teller Model
Consider the two-dimensional Ashkin-Teller model on the square lattice $\mathbb{Z}^2$ with Hamiltonian:
$$ H = - \sum_{\langle i,j \rangle} \left[ K \sigma_i \sigma_j + K \tau_i \tau_j + k \sigma_i \...
- 21
2
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Reference book for stochastic processes
I am looking for a good reference book for properties of stochastic processes for applied research. What I would like the reference to have is a collection of results on a large list of stochastic ...
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Is there a fractional derivative that preserves the composition of the one-parameter Mittag-Leffler function with $x\mapsto x^{\alpha}$?
Let $\alpha\in (0,1)$. The Riemann-Liouville fractional derivative of order $\alpha$ is defined by
$$ \sideset{_0^R}{}{D^{\alpha}f(t)}
=\frac{1}{\Gamma{(1-\alpha)}}
\frac{d}{dt}\left(\int_{0}^{t}
\...
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answers
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Literature on Lyndon words and the Lie commutator
Since I lost my paper notes in a domestic conflagration in Japan some ten years ago, I've occasionally tried to recall one particular author who wrote in the 1900s about Lyndon words / strings, or ...
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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 $${\frak L}(\sum_{i=1}^n a_i e^{k_i t}) = \sum_{i=1}^...
- 28k
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The topological complexity of polytopes
Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be ...
- 3,871
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83
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How analyze the following fully nonlinear equation
Now I want to consider the following pde
$u_t(x,t)=\sigma(x,t)(1+|D_xu(x)|^2)^{1/2}$, with initial condition $u(x,0)=g(x)$ which is analytic, and on domain $D\times \mathbf{R}^{+}$, $D\subset \mathbf{...
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Is the stationary distribution of this Markov chain uniform?
First, a little bit of background: Since 2012, Canada has decided to phase out the penny for its coinage system. Product prices may still use arbitrary cents, especially since prices do not typically ...
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Where to read about this kind of "measure of irredundancy" of a set from a family of sets?
Studying a very practical problem from psychometrics, I encountered the following construction.
Let $(X,\mu)$ be a measure space; if preferred, you can presume $\mu$ is a probability measure. In any ...
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2
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0
answers
215
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Number of breakpoints in parametric maximum flow problems
The parametric maximum flow problem can be formulated as
$$f(\lambda) = \min_{x\in\{0,1\}^n} \left( \sum_{i}(a_i + b_i\lambda)x_i + \sum_{i,j}c_{ij}x_ix_j \right),
$$
where all $c_{ij}<0$ (so that ...
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1
vote
0
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46
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Regression models as local sections of a chain complex
Let's say we find some regression equation $\ell$ (best fit / linear / whatever words you need to put here) for a sample $D$, subset of population $P$. This equation/model can be thought of as a ...
- 1,611
1
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0
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Maximizing the integral of a transformation that depends on a neighborhood of values of the original function
I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one. Also, any suggestion on changes that might make the problem better are welcome.
...
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Which real functions benefit from the Fundamental Theorem of Interval Analysis?
I'm reading Introduction to Interval Analysis, by Moore, Baker & Cloud and complementing it with Global Optimization using Interval Analysis, by Hansen & Walster.
Theorem 5.1 - Fundamental ...
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0
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A metric geometry problem which calculates the limitation of human eyes
This is the update version of this question A functional inequality which calculates the limitation of human eyes
Let an Euclidean space $M$ (or a path connected metric space) be partitioned into ...
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1
vote
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Recursive formula for integral of Chebyshev-type integral
Define
$$
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 ...
- 11
1
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83
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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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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-...
- 3,515
1
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0
answers
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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 ...
- 59
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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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0
answers
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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 ...
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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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connectedness of coincidence set
Consider the following obstacle problem in the whole domain $\mathbb{R}^n$
min{$\Delta u$, $u$-$\phi$}=0
with prescribed boundary value $\lim_{|x|\rightarrow\infty}u(x)=0$ and $\phi$ (can be assumed ...
- 41
1
vote
0
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501
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Distribution of random vectors
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...
- 143
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vote
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Use Lie Sub-Groups of GL(3, R) for elastic deformation ?
I'm interested in representing elastic deformations (e.g. stretching) using Lie groups. There are a few references to using $GL(3,\mathbf{R})$ but I'm wondering if possible to use subgroups of $GL(3,\...
- 157
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Practical applications of dandelin spheres
I know that dandelin spheres can be used to prove the focal properties of conic sections, but I heard that they can be used to help track the orbits of planets. All the sources I looked up only said ...
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Conjugate gradient-like algorithm with multiple search directions
I am solving an $n*n$ system $Ax=b$ in CUDA where $A$ is a sparse matrix. Currently I am solving it using the conjugate gradient algorithm.
I have noticed that $Ax$ where $x$ is $n*1$ has roughly the ...
- 71
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Interpolation polynomials with constraints
Lets consider a collection of $n$ points $\{Z_i\}_{i=1}^n\subseteq \mathbb{D}^k(1)=\{(z_1,\ldots,z_k)\in\mathbb{C}^k:\forall j\leq k, |z_j|\leq 1\}$. Let $h: \{Z_i\}_{i=1}^n \to \mathbb{D}^1(1)$ be a ...
- 502
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Numerical method for mixed system of equations and nonlinear inequalities
I am currently encountering challenges in determining the solution method for the following system of equations and inequalities:
$$
\begin{aligned}
&F(x) = 0\\
&G(x) < 0\\
\end{aligned}
$$
...
- 1
0
votes
0
answers
49
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Gaussian white noise model in application
I am interested in applications (to data) of non-parametric statistics, and my question concerned the Gaussian white noise model defined by,
$$
X_{t_1, \ldots, t_d}=f\left(t_1, \ldots, t_d\right) d ...
- 73
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241
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How to measure perceived note similarity in music / simplicity of ratios?
I have discovered a method to measure the similarity of two successive musical notes which I wanted to share with a question:
It is known in music theory that two successive pitches $a,b$ which sound “...
- 3,064
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A new method for processing music scores?
I have developed a method and python script:
https://github.com/githubuser1983/algorithmic_python_music
which allows the user to input a midi file and then chose a few numbers as parameters, and the ...
- 3,064
0
votes
0
answers
162
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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} \...
- 383
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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 ...
- 3,871
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Job sites for applied/interdisciplinary Mathematics?
I was wondering whether there're job sites that post jobs in applied/interdisciplinary mathematics, more specially, say postdocs or higher positions in mathematics and medical imaging, mathematics and ...
- 1,467
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0
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608
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Orthogonal Projections in Lie Theory
I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...
- 157