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.)
172 questions
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Ramp and Cliff solutions to the $b$-family
This question is related to that question that I asked a while ago.
It was shown numerically by Holm and Staley [1] and [2] the the family of PDEs (they are peakon equations called the $b$-family, but ...
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Examples of "Cell divisions" in mathematics?
I stumbled upon a mathematical structure, which I would describe as a cell division from biology, while researching prime factorization trees:
The image show a cell division:
blue = Growth of classes,...
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How to justify $\mathbf{S}_{x}= \frac{E_{s}}{N}I_{N\times N}$ as the best choice when the channel matrix is unknown?
I am a graduate student working in Wireless Communication, studying random matrix theory and its applications. In the context of determining channel capacity, I encountered the following generalized ...
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Price embeddings in inefficient markets
I am studying a discrete-time, multi-agent financial market model in which $N$ heterogeneous traders interact via a limit order book to produce a price process $\{P_t\}$. Each trader $i$ has an ...
user551082
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Where does one learn about the weather?
I have recently been curious about the mathematics of meteorology, and weather forecasting in general. Are there any reference books one can use to learn this topic? I have a standard graduate ...
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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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Under what circumstances Is a symmetric matrix representable as a Coulomb matrix?
Question:
I am exploring a neural network architecture inspired by physical interactions, where each neuron has associated "mass" and "position" vectors. The weight matrix between ...
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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 \...
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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 ...
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Equivalence of Wind Forces: Intensity vs. Duration [closed]
The strongest tornado in the world happened recently in Greenfield Iowa with winds over 318 mph: https://www.facebook.com/watch/?v=2176728102678237&vanity=reedtimmer2.0
I am curious, are less ...
user531026
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On the relationship between graph isomorphism and equivalence in ETL workflow dependency graphs
$\newcommand{\inn}{\mathrm{in}}\newcommand{\out}{\mathrm{out}}$Let $G = (V, E)$ and $G' = (V', E')$ be two DAGs representing dependency graphs of ETL workflows. Each node $v \in V$ (or $v' \in V'$) ...
user530362
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Domino equation derivation
Can the functional form of $G$ in the expression $\frac{V}{\sqrt{gH}} = G\left(\frac{d}{H}\right)$ be rigorously derived from first principles, where $V$ is the limiting wave speed of a line of ...
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Spectral theory: a key to unlocking efficient insights in network datasets
In the context of directed or undirected graphs, matrices such as adjacency and Laplacian matrices are commonly used. The eigenbasis of these matrices addresses some practical implications, such as ...
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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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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 ...
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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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What is this three dimensional curve that looks like an infinity sign called?
What is this three dimensional curve that looks like an infinity sign called? (Is there a known parametric equation for it?)
It was generated with this Sagemath - script, where you can zoom in 3d in ...
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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}
$$
...
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Application of Yamabe and Liouville type equation
Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs):
The Yamabe Type Equation (for $n>2$):
\begin{equation}
-\...
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Do the exceptional root systems arise in the real world?
I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a ...
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Reference book for stochastic processes
I am looking for a good reference book for properties of stochastic processes for applied research. What I would like the reference to have is a collection of results on a large list of stochastic ...
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Gaussian white noise model in application
I am interested in applications (to data) of non-parametric statistics, and my question concerned the Gaussian white noise model defined by,
$$
X_{t_1, \ldots, t_d}=f\left(t_1, \ldots, t_d\right) d ...
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Does there exist a Python package that samples random special unitary matrices such that the matrices are parameterized
For reference, the linked paper is Composite parameterization and Haar measure for all unitary and special unitary groups by Christoph Spengler, Marcus Huber and Beatrix C. Hiesmayr (J. Math. Phys. 53,...
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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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Mach's principle, Newton's law and Hilbert sphere?
(This question has originally been posted on reddit, but I thought, that the question raised in the post above, might fit as well here on MO.)
I wanted to share with you something I stumbled upon ...
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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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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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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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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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What are the right mathematical tools / language to analyse complex networks over time?
In this article about human physiology as a complex network the authors say that:
"Lacking adequate analytic tools and a theoretical framework to probe
interactions within and among diverse ...
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A functional inequality which calculates the limitation of human eyes
Find all pair of function $f^-,f^+:[0,1]\rightarrow[0,1]$ such that:
(1)$f^-(x)\leq x\leq f^+(x)$.
(2)$f^-(x)+f^+(1-x)=1$.
(3)$f^-(x)f^-(y)\leq f^-(xy)\leq f^-(x)f^+(y)$.
(4)$f^+(x)f^-(y)\leq f^+(xy)\...
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Question on pure mathematics helping climate change research
While I am a pure mathematics tenured professor, still at a relatively young age, and fairly passionate about my area of research, I cannot help but feel that it may be more useful to humanity if I ...
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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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Mathematicians learning from applications to other fields
Once upon a time a speaker at the weekly Applied Mathematics Colloquium at MIT (one of two weekly colloquia in the math department (but the other one is not called "pure")) said researchers ...
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What are partial differential equations with fast reaction terms?
I know $u_t(t,x)=\Delta u^m(t,x),\;\; (t,x)\in (0,\infty)\times \mathbb{R}$ is the fast-diffusion equation when $m\in (0,1).$
But how are PDEs with fast reaction terms defined in general? I also wish ...
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Is it possible to obtain orthogonal (but not normalized) vectors after QR factorization?
After QR decomposition of a matrix, $M$, the columns of Q are orthonormal. Is it possible after obtaining Q, we recover unnormalized column vectors from $Q$? For example, the matrix M has the ...
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What tools should I use for this problem?
Suppose we have $d$ cylindrical metal bars, with radius $l$, attached orthogonal to a support in random places:
Now we have to attach bars with radius $k$ EVENLY SPACED, with distance $p$ between ...
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General strategy of error bound of matrix exponential
I want to ask General strategy of the error bound of the matrix exponential.
For example, suppose, $A, B$ are finite dimension $n \times n$ matrices with complex coefficients. Using Baker–Campbell–...
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Listing applications of the SVD
The SVD (singular value decomposition) is taught in many linear algebra courses. It's taken for granted that it's important. I have helped teach a linear algebra course before, and I feel like I need ...
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Importance of integral equations
Differential equations are at the heart of applied mathematics - they are used to great success in fields from physics to economics. Certainly, they are very useful in modelling a wide range of ...
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Reference request: probabilistic models on climate (change)
I am looking for probabilistic models to address climate change. Are they known in the existing literature?
I have found the post Math behind climate modeling. concerning PDE models.
Many thanks for ...
user128095
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Explain seemingly non-random figures which arise from random Poisson points with normalization
Context Working with some biological datasets it was puzzling to see the patterns like Figure 2 (right) below. The first feeling was, that it corresponds to some biological effects like correlations ...
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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 “...
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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 ...
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Applied Topology/Topological Data Analysis conferences and journals
Can someone point out links to Applied Topology/Topological Data Analysis conferences and journals?
Thank you!
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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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Ramp and Cliff Solutions to the Viscous Burgers Equation: Explicit Formula?
I read an article in which the authors describe an observed phenomenon as being related to the "classical ramp and cliff Burgers solutions''. Those are described as Burgers solutions that behave ...
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I want to enter graduate school in pure math. Is doing REU in “mathematical modeling” a good idea? Is it an essential skill to learn?
(please let me know if this question is not suitable here)
Hello! I'm an undergraduate rising senior majoring in mathematics and it seems that I got rejected by an REU that is held in my university ...
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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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