Questions tagged [quadratic-forms]
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
182 questions with no upvoted or accepted answers
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Does this variant of a theorem of Hasse (really due to Gauss) have an "elementary" proof?
BACKGROUND
Here are 3 theorems of varying difficulty. Let $M$ be the $Z/2$ subspace of $Z/2[[x]]$ spanned by $f^k$, with the $k>0$ and odd, and $f=x+x^9+x^{25}+x^{49}+\cdots$. For $g$ in $M$, let $...
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16
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0
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588
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The number 1680 and Lagrange's four-square theorem
The number $1680$ has the factorization $2^4\times3\times5\times7$. Rather to my surprise, I found that this number has certain mysterious connection with Lagrange's four-square theorem.
QUESTION: ...
- 15.6k
14
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0
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476
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Cohomological interpretations of quadratic form invariants over rings?
The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local ...
- 141
12
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767
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What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$
This is related to my first MO question and Kevin Buzzard's conjecture at
Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
In December 2010 my question appeared in the M.A.A. Monthly, ...
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11
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0
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158
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Characterization of certain 4-dimensional lattices
Let $\Lambda \subset {\bf Q}^4$ be a lattice, i.e., $\Lambda$ is a free abelian group and $\Lambda \otimes {\bf Q} = {\bf Q^4}$.
The determinants of those dilation-rotations (i.e. linear maps of ${\bf ...
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10
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0
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177
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Matrices whose pairwise products form a basis
Over the vector space of 2x2 matrices, the Pauli matrices $I, X, Y, Z$ form a complete basis. Each of these matrices square to $I$, and with the additional relation that $Z = iXY$, we see that every ...
- 1,203
9
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196
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Leech lattice and rational varieties
Question: Is there a smooth rational variety $X$ of complex dimension $4n$, $n \in \mathbb{N}$; such that the intersection form on $H^{4n}(X,\mathbb{Z})$ is the Leech lattice?
My motivation is mainly ...
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9
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217
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Is there any study about positive definiteness of some matrix space whose matrices don't have to be positive-definite?
--Updated description--
I'm trying to investigate the stability of tensegrity structures, and this question is related to the second order test.
Suppose there is a vector space $W=\operatorname{span}...
- 91
8
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263
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Positive integer solutions of $ab+ac+ad+bc+bd+cd=n$
Consider a quadratic form $Q(a,b,c,d)=ab+ac+ad+bc+bd+cd$ on $\mathbb Z^4$.
For some reason I am interested in the number of solutions $(a,b,c,d)\in\mathbb Z_{> 0}^4$ of $Q(a,b,c,d)=n$ as a function ...
- 5,063
8
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0
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243
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Which continuous function is optimal for sieving?
In 1968, Barban and Vehov considered [1] the problem of determining for which continuous functions $\rho:\mathbb{R}^+\to [0,1]$ satisfying certain properties ($\rho(t)=1$ for $t\leq U_0$, $\rho(t)=0$ ...
- 20.2k
8
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264
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Number of representations of an integer as the dot product of integer vectors
Let $r_k(n)$ denote the number of solutions in positive integers to the equation: $$n = a_1 b_1 + a_2 b_2 + \ldots + a_k b_k.$$
What estimates and/or asymptotics are available for
(1) $...
- 13k
8
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0
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roots of quadratic forms
This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...
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7
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260
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K3 surfaces with no −2 curves
I seem to remember that a K3 surface with big Picard rank always
has smooth rational curves.
This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
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7
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224
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the gaps between values of a positive binary quadratic form at integer points
Suppose that $s$ is a positive irrational number. Consider all possible values of the sum $x^2+sy^2$ where $x$ and $y$ are integers. These values form a set $W=\{0=w_1<w_2<\ldots\}$. Can the ...
7
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Question on some coverings of the euclidean space
Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has ...
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7
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0
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191
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Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$
$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying
$b$ is bilinear,
$b(x,...
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7
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0
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674
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Mock modular forms and (indefinite) quadratic forms
Define the function
$$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$
where $c(n,m,l)$ is defined by
$$ c(n,m,l) =
\begin{cases}
(-1)^{s+l} & \text{if } 4n - m^2 + l^2 = 2s(s+1)\\
0 & \...
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7
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When is the set of numbers represented by certain quaternary quadratic forms completely multiplicative?
Expired by this question
A quadratic form represents all primes except for the primes 2 and 11.
I would like to know some simple sufficient conditions for when the set of numbers integrally ...
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6
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381
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A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
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6
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211
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some problems on sum of two squares
During my experiments with "Mathematica" I arrived to the following observations. My question is that are they interesting, known, solved or not. If they are known could you please give me a reference....
- 841
6
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0
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240
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Positive-definite lattice with O(n,n) Gram matrix generated by minimal vectors
Consider a positive-definite $2n$-dimensional lattice with minimum norm $\mu$. It is sometimes possible to find a generating set of minimal vectors for the lattice such that the Gram matrix takes the ...
- 271
6
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0
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268
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Bound on the determinant of a quadratic form restricted to a subspace
Let $Q\colon \mathbb{Z}^{n}\oplus\mathbb{Z}^m\to\mathbb{R}$ be a real quadratic form, which we denote $Q(x,y)$, $x\in\mathbb{Z}^n$, $y\in\mathbb{Z}^m$. Suppose:
The minimum of $Q(x,y)$ as $y$ varies ...
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5
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175
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Epstein zeta function for non-fundamental discriminant to L-series
Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\...
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5
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162
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Generators of the automorphism group of a quadratic form
Suppose that $q$ is an integral quadratic form, not necessarily unimodular or even non-degenerate, and write $O_q(\mathbb{Z})$ for its automorphism group. According to this question this group is ...
- 19.4k
5
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1
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414
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Binary quadratic forms order four in the form class group not having desired coefficients
I have been looking at binary quadratic forms for a question on MSE, If a binary quadratic form primitively represents $n$
and $n^3$, must it be the identity form?, about forms representing a prime (...
- 25.7k
5
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0
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237
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Linearly independent quadratic forms vanishing on a finite set of points
The question I am interested in can be summed up as follows: given positive integers $n,m,k$, how do we write down $m$ linearly independent quadratic forms $Q_1, \cdots, Q_m \in \mathbb{C}[x_0, \cdots,...
- 26.9k
5
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0
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258
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Sums of two integer squares in arithmetic progressions
Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to ...
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5
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229
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Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?
As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...
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5
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196
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Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?
Recall that the triangular numbers are those integers
$$T_n=n(n+1)/2\ \ \ (n=0,1,2,\ldots).$$
In 1796 Gauss proved that each $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three triangular numbers, ...
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5
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376
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Selmer Group of number fields and Ideal lattices
Let $K$ be a totally real number field of degree $n$ and dicriminant $d$, in this article of F.Lemmermeyer the selmer group of $K$ is defined as
$$\text{Sel}(K):=\{\alpha \in K^{\times}: (\alpha)=...
user115191
5
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0
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primes represented by indefinite quadratic forms
Let $Q$ be an indefinite binary quadratic form with discriminant $D$ and one class per genus (keep the example $x^2 - 2y^2$ in mind). If one asks about the set $P = \{ p : p \text{ prime and } p = Q(...
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5
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308
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Algorithm for solutions to quadratic forms over number fields
Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)?
I am especially interested in the quaternary case. There exist some ...
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5
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0
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596
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Literature on Exponential of a Quadratic Form
Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials
\begin{align}
f(\mathbf{x})=\sum_{i=1}^{L}\operatorname{exp}(-{\mathbf{x}^T\mathbf{A}_i\...
- 1,421
5
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0
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213
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Effect of Covering Radius on Shortest Vector
For "even" integral lattices in dimension at least 4, does a covering radius strictly less than $\sqrt 2$ imply that there is a vector of norm 2, also called a root?
Note that this is simply false in ...
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4
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216
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Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
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On the quadratic equivalence of fields
I have spent the past two years studying abstract Witt rings. These objects are a generalization of "The Witt ring of a field," an algebraic invariant of fields of characteristic not equal to 2. ...
- 141
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Quadrics contained in the (complex) Cayley plane
In the paper
Ilev, Manivel - The Chow ring of the Cayley plane
we can learn, that $CH^8(X)$, with $X := E_6/P_1$, denoting the Cayley plane, has three generators with one of them being the class of ...
- 1,479
4
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Norm variety for n=5, p=2 not isomorphic to a quadric
In the paper "Motivic construction of cohomological invariants", the author displays a list of known norm varieties for several $n,p$ on page $11$. For $p=2, n=5$ it says that a norm variety is given ...
- 1,479
4
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0
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390
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Fully Homomorphic Error Correction?
Consider a field $F$. Suppose we have two vectors $a,b\in F^n$, and an invertible matrix $G\in F^{n\times n}$. Let $c\in F^n$ be the point-wise product of $a$ and $b$, that is, $c_i=a_ib_i$. Let $x=...
- 3,979
4
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Hermitian forms over quaternion algebra
Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let $\sigma=Int(i)\...
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Is the class number of the quadratic field $x^2=3\cdot2^n+1$ $O(n)$?
Numerical evidence suggest that the class number of the quadratic field $x^2=3\cdot2^n+1$ is $O(n)$ while the discriminant is $O(2^n)$.
Here are the class numbers for $n=6 \dots 110$ computed with ...
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3
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0
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160
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Bott periodicity in characteristic p via Clifford algebras
I am currently reading Husemoller's wonderful book on fibre bundles, specifically the section on Clifford algebras. He defines these groups $L_k$ as follows. Let $M_k$ denote the free abelian groups ...
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3
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0
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606
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Proving an optimization problem from continuous input to binary is optimal
Suppose we have a function $f(x,y,z)$ where the inputs are uniform from 0 to 1. The output is either $+1$ or $-1$. And there is a partial symmetry $f(x,y,z) = f(z,y,x)$.
Tell me what the minimum of ...
3
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0
answers
308
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Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?
I seek for very sparse representations of positive integers. Let
$$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$
Recall that a polynomial $P(x,y)$ is integer-valued ...
- 15.6k
3
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0
answers
138
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Is there a method to solve a non-linear quadratic matrix equation?
I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$
Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional ...
- 31
3
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0
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203
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On sums of three squares
Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, any $n\in\mathbb N$ with $n\equiv1,2\pmod4$ can be written as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Clearly, ...
- 15.6k
3
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0
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136
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Can we write each positive integer as $w^2+x^2(1+2y^2+2z^2)$ with $w,x,y,z\in\mathbb Z$ and $x\not=0$?
Lagrange's four-square theorem states that each nonnegative integer is the sum of four squares. Here I ask the following question concerning a refinement of Lagrange's four-square theorem.
QUESTION: ...
- 15.6k
3
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0
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111
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On covering with Idoneal integers
$d\in\Bbb N$ is an idoneal integer if $N\in\Bbb N_{>1}$ can be written uniquely as $N=x^2\pm dy^2$ then $N=2^mp^n$ where $p$ is odd prime and $n\geq0$ and $m\geq0$ holds.
Let the $65$ known ...
user94040
3
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0
answers
171
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Self-dual vertex algebras
Let $(V,Y)$ be a self-dual conformal vertex algebra. For instance, it could be the vertex algebra associated to a positive definite, even, unimodular quadratic form. I look for a formula to compute
$$
...
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3
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0
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292
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Distance between quadratic forms
In notes here http://math.univ-lyon1.fr/homes-www/gille/prenotes/lens.pdf on page $2$ a formulation of distance between two positive quadratic form $[q],[q']$ is given by
$$d([q],[q'])=\frac{\sup_{x\...
user76479