# 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 ...
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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 = $\frac{{i}{(i+1)}}{2}$ &...
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### 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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### 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 ...
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### 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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### 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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### 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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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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 ...