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Questions tagged [invariance]

A property of an object is called invariant if, given some steps that alter the object, always remains, no matter what steps are used in what order.

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Is there any quick way to compute/approximate a symmetric, scale-invariant (declining color) gradient around an ellipse?

First the goal is to draw en ellipse with a (grey color) gradient like this: With minimum at the center of the line and symmetrically declining towards the in- and outside. Other than shown in the ...
UncleBob's user avatar
1 vote
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Simplest proof that exactness implies mixing

Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
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Prove that modified RBF function satisfies Mercer conditions.

Suppose that I have a modified RBF kernel function. $k(\mathbf{x},\mathbf{y}) = \exp{(-||\mathbf{x}-P\mathbf{y}||^2 })$ where $\mathbf{x},\mathbf{y}$ represent $d$ dimensional inputs and $P$ is the ...
flammmes's user avatar
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What quantity is “invariant” in a 3-body invariant manifold?

Invariant manifolds are used to calculate low-energy trajectories for spacecraft transiting between Lagrange points. I understand that an invariant manifold is a topological manifold that is invariant ...
Woody's user avatar
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How is the invariance of $ds^2$ in a coordinate change defined?

I will show my question with the help of an exercise I stumbled upon recently because I think it will illustrate my problem more clearly. I'm simplifying the exercise by just showing the steps and not ...
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How to identify the number of degrees of freedom in a system of particles using the Principle of Relativity

So consider the following laws of motion in an inertial system : $$ m_k \partial_{tt} x_k = - \partial_{x_k} V(x_1, .., x_k)$$ where $x_i \in \mathbb{R}^3$ is a particle of mass $m_i$. By principle of ...
Suspicious Fred's user avatar
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What is this value called in algebraic topology?

I have a weak familiarity with homotopies and homologies and I have been thinking about similar constructs on my own in the past. Now I would like to classify these ideas of mine before studying these ...
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'Easy' proof for Invariance of Domain for a 2D ball

In class, Invariance of Domain was proved using lots of machinery, but I came up with the following 'proof' whose truth I am suspicious of. Explicitly statement to prove is : Let $B$ = $\{ \vec{x} \...
Mahammad Yusifov's user avatar
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2 answers
103 views

Help Analyze whether, at a given moment, the following cases are possible....

Question The numbers $4$, $8$, $9$ and $15$ are written on the board. Carla deletes three of them and then writes three more numbers following the rule: if the numbers $a$, $b$, $c$ are deleted, the ...
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Trace Polynomial of $\mathfrak{sl}(2, F)$ — Example in Section 23.3 in Humphreys Lie algebras book

Let L be a Lie algebra, H a CSA of L, and $(V, \phi)$ be an irreducible representation of $L$. For any non-negative integer $k$, define the trace polynomial $f_k:L \rightarrow F$ by $f_k(z) = Tr(\phi(...
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Arithmetic Geometric Mean inequality, Book "Problem Solving Strategies" by Arthur Engel.

I put the images to make it easier to follow. I don't understand why $\cos \alpha_{n+1} = \cos \frac{\alpha_n}{2}$ implies that $\alpha_n = \frac{\alpha_0}{2^n}$. I also don't understand why he ...
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Puzzle of an ant rearranging stacks of seeds in a line [duplicate]

Interesting puzzle that I haven't been able to solve or find a solution to. An ant rearranges a line of stacks of seeds as follows: With each iteration, the ant goes to each stack in order and grabs ...
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Why does the invariant 1-tensor integral gives 0 for any volume?

I was going through David Tong's vectors calculus notes. In Chapter 6.1.3 (Invariant Integrals) he gives the example Here are some examples. First, suppose that we have a 3d integral over the ...
vueenx's user avatar
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Permutational invariants of degree-$m$ supersymmetric tensors

Let $A$ be a super-symmetric $(n \times.....\times n)$ $m$-fold tensor, so that: $$ A\left(i_{1},\dots,i_{m}\right) = A\left(j_{1},\dots,j_{m}\right), $$ when $\left(j_{1},\dots,j_{m}\right)$ is any ...
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Finding a Monovariant of a certain Map

In solving a puzzle related to linear algebra over $\mathbb{Z}/2\mathbb{Z}$, I came across the function $\varphi:\mathbb{Z}^n\rightarrow \mathbb{Z}^n$ given by $$(x_1, x_2, \dots, x_n)\mapsto(|x_1-x_n|...
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Induction Proof of Loop Invariance Given the Invariance

Consider the following program segment i=1 total = 1 while i<n i = i + 1 total = total + i Let p be the proposition "total = $\frac{{i}{(i+1)}}{2}$ &...
Rikiita's user avatar
3 votes
1 answer
144 views

Is it possible to arrange the integers $1, 1, 2, 2,..., 1998, 1998$ such that there are exactly $i − 1$ other numbers between any two $i$'s?

Is it possible to arrange the integers $1, 1, 2, 2,..., 1998, 1998$ such that there are exactly $i − 1$ other numbers between any two $i$'s? This is a problem related to invariants and I'm trying to ...
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When do two integer linear programs yield the same solution?

This question was cross-posted to operations research stack exchange An illustrative example Consider an integer linear program $\min -2x_1 + x_2$ subject to $x_1 - x_2 \leq 3$ and $x_1 + x_2 \leq 10$ ...
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2 answers
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Three grasshoppers jumping on a line

The following problem appears on pp. 86 Terrence Tao's "Solving Mathematical Problems: a personal perspective". Three grasshoppers are on a line. Each second, one (and only one) grasshopper ...
vietajumping's user avatar
1 vote
1 answer
136 views

Verify my proof of Viviani's Theorem: In an equilateral triangle, the sum of distances from an interior point to each side is constant

Please verify and critique my proof of Viviani's Theorem: In an equilateral triangle, the sum of the distances from any point within the triangle to each of its sides equals the altitude of the ...
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Invariant problem starting with the sequence $1,2, ... ,100$

Initially, we are given the sequence $1,2, ... ,100$. Every minute, we erase any two numbers $u$ and $v$ and replace them with the value $uv + u + v$. Clearly, we will be left with just one number ...
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2 votes
2 answers
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How do I solve exercise 2.2.4 Blackburn/de Rijke/Venema’s “Modal Logic”

Here’s a transcript of the original exercise. (There‘s even a hint given by the authors in the textbook as you can see. But precisely this hint confuses me). 2.2.4 Consider the binary until operator $...
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1 vote
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128 views

Projectivity that keeps Euclidean distance invariant

It is known that in projective transformations the distance does not have to remain invariant as it occurs in Euclidean Geometry with isometries. However, I have found an article called Geometries of ...
Pablo Ib's user avatar
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1 answer
36 views

Show that the origin of $\mathbb{R}^3$ is not invariant under spatial translations

It is a known fact that $\mathbb{R}^3$ does not model classical physical space accurately since it includes some non-invariant structure under translations, such as the origin $(0,0,0)$. However, I am ...
Promethèus's user avatar
3 votes
1 answer
111 views

Alice plays a game of choosing numbers and replacing them with some other. [closed]

The following $100$ numbers are written on the board:$$2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1.$$ Alice chooses two numbers $a,b,$ erases them and writes the number $\dfrac{ab - 1}{a+b+2}$ on ...
轻型八神's user avatar
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Does probability distribution satisify translation invariance

According to E(n) Equivariant Normalizing Flows, Translation Invariance Recall that we want the distribution $p_V (V)$ to be translation invariant with respect to the overall location and orientation ...
Owen's user avatar
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2 votes
2 answers
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Transformations in Noether's theorem and point transformations

In lagrangian mechanics, I've come across a proof that point transformations don't change the Euler-Lagrange equations of motion. Suppose our new coordinates are simply $\tilde{q_i}=\tilde{q_i}(q_1,...
Nakshatra Gangopadhay's user avatar
1 vote
2 answers
151 views

Invariance of the Laplacian operator under the action of rotations centred at the origin

I need to solve the following problem: $$\begin{cases} \Delta u(x, y) = 1 \ \ \mathrm{if} \ (x, y) \in D\\ u(x, y) = 0 \ \ \ \ \; \mathrm{if} (x, y) \in D \end{cases}$$ where $D$ is the domain ...
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1 vote
1 answer
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Determine if system is linear, time-invariant and causal from differential equation

Determine if the system described by the differential equation $$\ddot{y}(t) + 2\dot{y}(t)-6y(t) = 2x(t) $$ is linear, time-invariant and causal. $y(t)$ is the system output and $x(t)$ is the system ...
Carl's user avatar
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1 vote
1 answer
82 views

Eventually constant votes. [duplicate]

Let's say that there are $2n+1$ persons sitting at a round table. Each time these people will vote by yes or no. The process starts by an initial vote and in the next vote if at least of person number ...
PNT's user avatar
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1 vote
1 answer
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Can a free module have invariant basis number if its ring doesn't?

Is there a free $R$-module $M$ such that $M$ has invariant basis number (IBN), while $R$ does not have the IBN property? My use of "invariant basis number" here (for modules) doesn't seem to ...
WillG's user avatar
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0 answers
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About running variance in Batch Normalization layer

A batch normalization(BN) layer is normally used to reduce the covariance shit problem in neural networks. Where in a layer, input $x$ will be normalized to something like $x^{\prime}$ = $\frac{x-\mu(...
Dennis Wu's user avatar
1 vote
1 answer
122 views

Calculus of Variations, Noether's Theorem, need help understanding and solving this problem

I have been given the following problem: Investigate whether the functional $I=\int_{t_1}^{t_2}t\dot{x}^2dt$ is invariant under the transformation $\bar{t}=t+\epsilon$ and $\bar{x}=x$, with $\epsilon$ ...
Martin Sieburg's user avatar
6 votes
1 answer
131 views

'Invariants' in a category of modules

I have a commutative unital ring $R$, a full additive subcategory $\mathcal{C}$ of $\text{Mod}_R$ that is closed under isomorphisms and an operation $f \colon \mathrm{Ob}(\mathcal{C}) \to \mathbb{Z}_{\...
user829347's user avatar
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1 vote
1 answer
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Why is $B_\pi$ an invariant bilinear form on $\mathfrak{g}$?

Definition: Let $\mathfrak{g}$ be a Lie algebra and $(\pi ,V)$ be a representation of $\mathfrak{g}$ on $V$. A bilinear form $B$ on $V$ is said to be invariant under $\pi$ if for any $x\in \mathfrak{g}...
Anil Bagchi.'s user avatar
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0 votes
1 answer
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Invariants of modules over principal ideal domains

I have a principal ideal domain $A$, and am trying to show that a variety of $A$-modules are non-isomorphic. So I am looking for 'invariants' of $A$-modules, by which I mean mappings $M\mapsto f(M)\in\...
user829347's user avatar
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1 answer
168 views

Where in PCA does the non-uniqueness of eigenvectors come from?

I tried comparing sklearn.decomposition.KernelPCA with a linear kernel to sklearn.decomposition.PCA on the same data set and got different eigenvectors. My understanding is that these should be ...
Galen's user avatar
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5 votes
2 answers
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A step in a proof that a set is Lebesgue measurable if and only if its translation is

I'm trying to fill in the gaps in the following proof: In the above, $\mu^*$ refers to the Lebesgue outer measure, the Carathéodory definition of measurability states that a set $A$ is Carathéodory ...
Sam's user avatar
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1 vote
1 answer
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Is $y(n) = x(nT)$ causal,time invariant and linear?

If $T>0$ is the sampling period of a device which produces one output signal $Yn$ from a continuous input signal $x(t)$, and n is any integer, would it be causal,time invariant and/or linear? My ...
TheExtraSpicyBeef's user avatar
0 votes
1 answer
87 views

Invariant distribution of a Markov chain - joint probability

I have found the following invariant distribution $\pi$, invariant w.r.t. my transition probability matrix: $$P = \begin{bmatrix} 0.8 & 0.2 \\ 0.5 & 0.5 \end{bmatrix}$$ $\pi = [0.7142857, 0....
yoyo_24's user avatar
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Practical applications of knot concordance

Could someone mention some of the practical applications of knot concordance? I was not able to find any from Google Scholar. Thanks in advance.
Omar Shehab's user avatar
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63 views

2-dimensional invariant subspace using Jordan matrix

If given a matrix which is invertible, and knowing the Jordan canonical form is: $$J = \begin{bmatrix} -2 & 0 & 0 & 0\\ 0&5&0&0\\0&0&-8&0\\0&0&0&8\end{...
Roo4ma's user avatar
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1 answer
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Where can I find information about sets whose subsets produce unique values?

I'm interested in sets whose subsets produce unique values under a given commutative operation. I don't know what this is called. I tried searching for terms like unique commutative invariant, but ...
lmonninger's user avatar
2 votes
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73 views

Why are (co)homology groups algebraic invariants?

Background & definitions: I am studying algebraic topology, in particular (co)homology groups and Mayer-Vietoris sequences. Important terms: Algebraic invariant = property of a topo. space ...
Tereza Tizkova's user avatar
1 vote
1 answer
285 views

distance between two functions by considering invariance

For two functions $f(x): \mathbb R^n \mapsto \mathbb R^m$, $g(x): \mathbb R^n \mapsto \mathbb R^m$. An usual way to define their distance may be $$ \|f-g\| = \sqrt{\int_x \|f(x)-g(x)\|^2 dx}. $$ I ...
Jiaji Huang's user avatar
1 vote
0 answers
119 views

Let $Y\sim F_{\theta}$. When $g(Y, \theta)$ that does not depend on $\theta$?

Let $Y$ be a random vector in $\mathbb{R}^k$, with distribution function belonging to a family $\{F_{\theta}, \theta\in\Theta\}$ is a parametric family of distribuitiuons (e.g. normal with unknown ...
Albert Paradek's user avatar
1 vote
0 answers
31 views

Proving that periodic vector-valued function satisfies $f(\mathbf x)= f(\mathbf x')$ iff $\mathbf x' = \mathbf x + k + \mathbf am$

In order to create a layer for a neural network that uniquely encodes relative relationships between a set of angles, I'm looking for a differentiable function with some specific properties; I'm ...
monkeypuzzle's user avatar
2 votes
0 answers
83 views

What are some examples of beautiful invariance?

I have been recently intrigued by the 2011 IMO Problem 2 which describes a windmill process exemplified in this video by 3Blue1Brown. The question is fundamentally about invariance in that the number ...
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1 vote
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104 views

A question based on Invariance principle

. Suppose the positive integer n is odd. First Al writes the numbers $1, 2,..., 2n$ on the blackboard. Then he picks any two numbers a, b, erases them, and writes, instead, $|a − b|$. Prove that an ...
Death Champion's user avatar
2 votes
1 answer
80 views

Iterating (programmatically) over "all" integer polynomials (how to order them and when to stop to be sure we checked all what we wanted?)

I have some annoyingly stubborn problems that are basically of the form "are all integer polynomials of some kind a sum of a few products of polynomials of some other kind?". For instance I ...
Jakub Kamiński's user avatar

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