# Tagged Questions

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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### The Mathematics of Tetris

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...
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### Rotation invariant tensors

It is often claimed that the only tensors invariant under the orthogonal transformations (rotations) are the Kronecker delta $\delta_{ij}$, the Levi-Civita epsilon $\epsilon_{ijk}$ and various ...
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### Prove that if fewer than $n$ students in class are initially infected, the whole class will never be completely infected.

During 6.042, the students are sitting in an $n$ × $n$ grid. A sudden outbreak of beaver flu (a rare variant of bird flu that lasts forever; symptoms include yearning for problem sets and craving for ...
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### three grasshoppers jumping on a plane.

The problem is dead simple: Three grasshoppers sit on a plane not in a line. Every second just one of the grasshoppers hops symmetrically over one of the others. Can they return to the initial ...
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### Proof there is no way to chose signs to make sequential sum $1+2+3+\cdots+10$ even [closed]

I've figured that for the sum $$1+2+3+4+5+6+7+8+9+10=55$$ There is no way to chose the signs of the numbers to get an even sum. I'm really struggling to prove this and would appreciate some ...
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### Invariants of a matrix

I'm teaching a course in physics, and I need a simple and intuitive proof that a matrix ($3\times3$, but it doesn't matter) has exactly 1 invariant which is linear in its entries, 2 that are quadratic,...
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### Using the invariance principle: how to solve $n+d(n)+d(d(n))=m$?

Let $d(n)$ be the digital sum of $n$. How to solve $n+d(n)+d(d(n))=m$, where $n$ and $m$ are natural?
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### Understading the integral form of a conservation law

When I think of a conservation law I think of a continuity equation like the following $$\partial_t \rho = -\nabla \cdot \vec j$$ But now I'm reading a book on electrodynamics (that's honestly a bit ...
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### What should you call a property, like an invariant, but that is reversed instead of preserved?

Suppose $P$ is some property of some objects and $f$ is a function on those objects. If $Px$ implies $Pf(x)$ and $\lnot Px$ implies $\lnot Pf(x)$, then we might say that "$P$ is invariant under $f$". ...
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### Numbers in a sequence

The number sequence $1, 9, 8, 2...$ satisfies the following rule: each element of the sequence starting from the fifth, is equal to the last digit of the sum of the previous four members. Will we ever ...
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### The Invariance Principle

I had come across a problem practicing to get better at approaching different types of problems from different field topics and this one had got me kind of stuck in what direction to go. Not so ...
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### Are Bezier curves invariant under conformal mapping?

I've spent quite a bit of time on google trying to find information on whether or not Bezier curves are invariant under conformal mapping (i.e. a conformal mapping of all points on the curve is the ...
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### Similarity classes of matrices

Let $M_n(K)$ be the set of all $n\times n$ matrices over a field $K$. If $\mathcal{R}$ is the equivalence relation defined by matrix similarity, what does the quotient $M_n(K)/\mathcal{R}$ looks like? ...
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### Creating all strongly connected graphs with given in-degree with equal probability

I am looking for a way to sample uniformly from the space of all strongly connected directed graphs (without self-loops) of $n$ nodes and in-degree $k=(k_1,...,k_n)$ with $1 \leq k_i \leq n-1$. In ...
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### Inertial Frames of Refereence

I am told that in Newtonian mechanics, no coordinate system is "superior" to any other. Also, all inertial frames are in a state of constant, rectilinear motion with respect to one another. So am I ...
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### notation for invariation

Let $\Lambda = \{T \in \operatorname{Her}_2(\mathcal{O}) ; T \ge 0\})$ and $\mathcal{O}$ the maximal order of some quadratic imaginary number field. I write $T[U] := U^* \cdot T \cdot U$ where $U$ is ...
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### Prove that n is divisible by 4 in a cylic sum with variables which have only two possible values

It is known that $a_1, a_2, a_3, ... , a_n \in \left\{-1, 1 \right\}$ and $S = a_1a_2a_3a_4 + a_2a_3a_4a_5 + ... + a_na_1a_2a_3 = 0$ Prove that $n \equiv 0\space(mod\space 4)$ I know this problem ...
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Show that the product of two delta functions $\delta{(x)}$$\delta{(y)} is invariant under rotation around the origin. This is a problem from Zee's textbook on Gravity on page 51. The book was ... 1answer 156 views ### Connection between the Tutte and characteristic polynomials? Both the Tutte polynomial T_G(x,y) and the characteristic polynomial \phi_G(x) encode a great amount of structure of the input graph G. I've read somewhere that the Tutte polynomial has a kind ... 2answers 59 views ### Invariant subspaces Let T be a linear operator on a finite dimensional vector space V over a field F such that every subspace of V is invariant under T then how to prove T is digonalizable ? Is the converse true ? 2answers 76 views ### loop invariant for simple algorithms The following is an algorithm which finds the maximum value in a list of integers, and I want to prove that it is correct by using a loop invariant. ... 3answers 106 views ### Fibonacci Loop Invariants I've taking an Algorithms course. This is non-graded homework. The concept of loop invariants are new to me and it's taking some time to sink in. This was my first attempt at a proof of correctness ... 2answers 60 views ### Constructing the sequence: 0\rightarrow (x-y)^{S_2} \stackrel{f}{\rightarrow} k[x+y,xy]\stackrel{g}{\rightarrow} k[y] Let S_2, a group of two elements, act on k[x,y] by permuting x and y. It is clear that$$ 0\rightarrow (x-y) \rightarrow k[x,y]\rightarrow \dfrac{k[x,y]}{(x-y)}\cong k[y] \rightarrow 0 $$... 1answer 56 views ### Write V=P_2(\mathbb{R}) as a direct sum of V=W_1\oplus W_2 \oplus W_3 So, if I let T:P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R}) and is a linear endomorphism given by T(f(x))=f(x)-f(2x-1). Then I have to writeV=P_2(\mathbb{R}) as a direct sum of V=W_1\oplus W_2 \... 1answer 42 views ### Boundedness of RHS implies existance of invariant cube Consider a system of ODEs of the form$$\dot x_1 = f_1(x_1,x_2)-g_1(x_1,x_2)x_1 \\ \dot x_2 = f_2(x_1,x_2)-g_2(x_1,x_2)x_2,$$where f,g are bounded, Lipschitz-continuous functions. (Then by the ... 1answer 160 views ### Identifying k[x_1,x_2,y_1,y_2]^{\epsilon} with k[x,y]\wedge k[x,y] Suppose the symmetric group S_2 of order 2 acts on k^4=Spec \;k[x_1, x_2, y_1, y_2] by the following: for \sigma\not=e,$$\sigma\circ(x_1, x_2, y_1, y_2)=(x_2,x_1,y_2,y_1).$$That is, the ... 1answer 106 views ### Is there any invariance under the inversion mapping? In geometry, there is a transformation called the inversion mapping which maps nonorthogonal circles into nonorthogonal lines and vice versa.(If I make a mistake, inform me, since I am not very ... 0answers 41 views ### Isospectral transformation of ODE Is there a transformation of coefficients of differential operators w/periodic coefficients on the real line a_n(x+2\pi)=a_n(x), that preserves the eigenvalues of their monodromy matrices?$$Df(x)=\... 0answers 31 views ### Invariant equation If I have the equation:$x=-$tan$x$I can send$x \rightarrow -x$and the equation doesn't change. If I define$x>0$one of the possible ranges$x$can take without solving the equation is$\...
I've been reading about the use of invariants in contest math. I saw the following problem (in my own words): There are $N = 2n$ numbers placed on a circle. Then we increase two any consecutive ...