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.
522
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Positive subspaces of quadratic forms
here's my question:
Let $V$ be a k-dimensional vector space over $\mathbb{R}$ and $q$ a quadratic form on $V$ of signature $(m,n)$ , $m+n=k$.
We have $W\subset V$ a positive (with respect to the ...
1
vote
0
answers
113
views
Tensor product with $\mathbb{R}$ of an even unimodular lattice
Let $\Lambda$ be an unimodular even lattice of signature $(m,n)$.
By a classifying theorem by Milnor, $\Lambda$ must be of the form $U^k\oplus E_8(\pm 1)^l$, where $U$ is the hyperbolic plane.
Now ...
3
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0
answers
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pavings and quadratic forms
Hi,
let $L$ be a lattice isomorphic to $\mathbb{Z}^r$ for some positive integer $r$ and $E=L\otimes \mathbb{R}$.
An integral paving in $E$ is a set $\Sigma$ of integral polytopes (the vertices are ...
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Did Smith correctly state the mass formula?
Did Smith correctly state the mass formula?
H.J.S. "normal form" Smith was the first, in 1867, to state the mass formula for integral quadratic forms in a genus of 4 or more variables. This was ...
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1
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Question about Gauss composition law over PID.
Let $m$ be a square free integer, $\mathbb{Q}(\sqrt{m})$ a quadratic field extension of $\mathbb{Q}$, $\Delta$ is its discriminant and $O_{\mathbb{Q}(\sqrt{m})/\mathbb{Q}}$ its ring of integers. We ...
- 433
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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\...
- 1,371
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votes
1
answer
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Filling in a rational orthogonal matrix given one row
Quick version: given natural $n$ and a row of $n$ integers such that the sum of the squares is another square, call it $m^2.$ For $n=5,6,7$ is it always possible to fill in the rest of an $n$ by $n$ ...
- 25.4k
8
votes
1
answer
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A spectral inequality for positive-definite matrices
Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues
$$
\lambda_1 \leq \cdots \leq \lambda_n ,
$$
is there a sharp upper bound for the product $\lambda_2 \cdots \...
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Would a closed universe with special relativity violate causality? Does the universe have to be simply connected?
This question may be more appropriate for physics.stackexchange.com, but it would be helpful to get feedback from experts in Minkowski geometry.
The classic twin paradox is a false thought experiment ...
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Quadratic forms and $p$-adic integers
I want to prove a result on equivalences of quadratic forms over $\mathbb{Q}_p$, with a control on the height of the change-of-basis matrix.
(I am more generally interested in hermitian forms over ...
- 1,500
1
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0
answers
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Orthogonal transformations with trivial spinor norm as product of reflections $r_w$ with $(w,w)=-2$
I'm trying to prove that, for a standard unimodular even lattice $\Lambda$ (by standard I mean that it is direct sum of copies of the hyperbolic plane $U$ and $E_8$) every element of $O^+(\Lambda)$, i....
- 83
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votes
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answer
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Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4?
Given a quadratic form $F$ in $n$ variables, there is an associated theta function $\theta_F(z) = \sum_{m \in \mathbb{Z}} q^{F(m)}$, which is a modular form of weight $n/2$. Letting $F(m) = m^2$ ...
- 585
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Solutions to a quadratic congruence
Fix an odd prime $p$. Let $\alpha = (\alpha_0,\dots,\alpha_k)$ be a solution to the congruence $\sum_{i=0}^{k} \alpha_i^2 \equiv x \mod p$. Now consider the number $N_\alpha$ of solutions to the ...
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2
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Quadratic subextension of the function field of quadric.
Assume $F$ is a field of characteristic $\neq 2$. Let $(V,q)$ be a quadratic space such that $\rm dim~ q\geq 3$. When $q$ is irreducible it is known that
there exist a purely transcendental field ...
- 57
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0
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Solution Existence of a System of Complex Quadratic Equations
Consider $ {x_k}_{k = 1}^K \in \mathbb{C}^{N \times 1}$ a set of $K$ complex vector variables of length $N$. I am interested in finding the existence of a solution of the following
quadratic set of ...
- 332
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0
answers
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Non-negative Quadratic forms with Exterior Forms
Hello All,
I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.
Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{...
7
votes
0
answers
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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 & \...
- 1,288
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votes
2
answers
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On the positive definiteness of a linear combination of matrices
In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
QUESTION:
Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
- 895
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votes
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762
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genus and spinor genus over a number field
Let $F$ be a number field with ring of integers $\mathfrak{o}$. Let $(V,Q)$ be a quadratic space of dimension $n$ over $F$, and let $L$ be a free lattice in $V$ (i.e. $L\cong\mathfrak{o}^n$). If the ...
7
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3
answers
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Realizing proper pure octonions as conjugates
Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ
We have a standard conjugation operator with $\bar{1} = 1$ and $\bar{e_i}= - e_i,$ extend by ...
- 25.4k
2
votes
2
answers
278
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Convex optimization problem to QPP
Briefly, have the following problem:
\begin{equation}
\sum_{i = 0}^n a_i \ (max [ F_i( \bar x ), 0 ] )^2 \rightarrow min, \\\\
s.t.\\\\
A \bar x \leq b
\end{equation}
where $ F( \bar x ) $ is a ...
- 121
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votes
1
answer
240
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two non-degenerate quadratic forms on $GF(2)^2r$
I know this:
There are two non-degenerate quadratic forms on $GF(2)^2r$. The hyperbolic form may be taken to be
$Q^+(x)=x_0 x_1 + \cdots +x_{2r-2}x_{2r-1}$ ,
and the elliptic form to be
$Q^-(x)=x^...
- 179
1
vote
1
answer
252
views
''Local-global-principle'' for certain isometries of lattices
Hi everybody.
I am trying to understand a proof of Kneser. the assertion is on a ''weak version'' of the local-global principle certain isometries: It is Satz (30.9) in kneser book ''Quadratische ...
- 103
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1
answer
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if $\Pi_1$ and $\Pi_2$ be elliptic planes then $\Pi_1 \oplus \Pi_2 $ is still elliptic?
Let $\Omega \in\Lambda^{4}\big(V^{ \star }\big)$ be volume form. Define symplectic bilinear form
$q: \Pi \oplus \Pi \rightarrow R $
$\big( \alpha ,\beta \big) \longrightarrow \alpha \...
user21574
2
votes
1
answer
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Gram matrix modulo 4
Suppose we have a full rank, integer sublattice $L$ of the integer lattice $\mathbb Z^d$, where we fix the dimension $d$. Consider the Gram matrix $M$ of $L$, relative to some basis for $L$, and ...
- 613
2
votes
2
answers
432
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The quadratic form $x^2+ny^2$ via prime factors
Elementary algebra shows that the product of two numbers in the form $x^2 + ny^2$ again has the same form, since if $p = (a^2 + nb^2)$ and $q = (c^2 + nd^2)$,
$$pq = (a^2 + nb^2)(c^2 + nd^2) = (ac \...
- 161
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votes
1
answer
390
views
Can an ellipsoid be moved freely inside another ellipsoid?
An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid equivalent if one can be ...
- 87
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3
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if Y-X is positive semi-definite, are the eigenvalues of Y bigger?
So $X$ and $Y$ are Hermitian matrices (or just symmetric real) of size $n$ by $n$ and suppose $Y\succeq X$, namely $Y-X$ is positive-semidefinite. Now write the eigenvalues of $Y$ as $\alpha_1\leq\...
- 87
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1
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Weyl asymptotics vs. form perturbations
Consider Hilbert spaces $V$,$H$; a closed quadratic form $a$ with domain $V$; and its associated operator $A$ on $H$. (If necessary, the form can be assumed to be coercive.) For the sake of simplicity,...
- 3,328
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0
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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 $...
- 5,412
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Over which fields is the Sylvester law of inertia valid?
Short version:
Over which fields is the (appropriate version of the) "Sylvester law of inertia" valid?
Long version:
Let $V$ be a finite dimensional vector space over the field $\Bbbk$ of ...
- 22.7k
12
votes
0
answers
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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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2
votes
2
answers
296
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Involution of $E_{8}$ lattice
Let $L$ be a lattice associate to the Dykin matrix of type $E_{8}$. I would like to understand involutions of $L$ and their invariant $L^{+}$ and coinvariant lattice $L^-$ (I think they are isomorphic)...
- 21
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votes
1
answer
2k
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Diagonalization of a quadratic form in integers
Hello,
Recently I've been studying the problem of quadratic form diagonalization. Suppose that we have a form $F(x,y,z)$ with corresponding symmetric matrix $M$. This form is equivalent to another ...
- 1,573
2
votes
1
answer
845
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Simultaneous quadratic equations
I have a set of 3 quadratic equtions:
$A_1µ_1^2 + B_1µ_1µ_2 + C_1µ_2^2 + D_1µ_1 + E_1µ_2 + F_1 = 0$
$A_2µ_2^2 + B_2µ_2µ_3 + C_2µ_3^2 + D_2µ_2 + E_2µ_3 + F_2 = 0$
$A_3µ_3^2 + B_3µ_3µ_1 + C_3µ_1^2 + ...
- 23
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votes
3
answers
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Characterising semi-definite positiveness on vectors with non-negative entries
My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this ...
- 1,268
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votes
1
answer
2k
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Number of Totally Isotropic Subspaces
First I want to review some concept from quadratic form.
Let $V$ be quadratic space over finite field $F$ and $char(F)\neq 2$ with quadratic form $q$. For exmaple
$q:V\rightarrow V$ and $|F|=q$ and $\...
- 193
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1
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probably Lagrange or Legendre, Pell variant
Evidently Legendre showed that, for positive primes, if $p \equiv 3 \pmod 8$ there is an integral solution to $x^2 - p y^2 = -2.$ Next, if $q \equiv 7 \pmod 8$ there is an integral solution to $x^2 -...
- 25.4k
3
votes
2
answers
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Invariant for group actions
Hello everybody!
Define the action of $SL_4({\mathbb{Z}})$ on alternating 2-forms or simply skew-symmetric matrices of degree 4 according to the following:
For $B \in SL_4({\mathbb{Z}})$ and an ...
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2
answers
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Primes and $x^2+2y^2+4z^2$
A few months ago, I have asked a question about primes represented by ternary quadratic forms. I got two wonderful answers, which showed me how the theory was way richer and more complex that I ...
- 25.7k
1
vote
1
answer
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Lorentz quotient and orientation
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right) ,
$$
Given a real oriented vector space $V$ with inner product, form Lorentzian $L = V \oplus U....
- 25.4k
5
votes
2
answers
302
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Matrix version of number theoretic integral lattice claim
I know this not, everybody else seems to. This is from page 215 of Cassels, Rational Quadratic Forms, formula 4.1, or SPLAG, page 389 formula (36). quote:
If $f$ and $g$ are forms of
determinant $...
- 25.4k
5
votes
0
answers
206
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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 ...
- 25.4k
3
votes
0
answers
150
views
Quadratic forms over symmetric $3\times3$ matrices
This question is motivated by a technique called compensated compactness (CC), which was elaborated by L. Tartar and F. Murat for the analysis of PDEs. The foreword of CC is a problem about quadratic ...
- 51.6k
2
votes
2
answers
255
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on the determination of a quadratic form from its isotropy group in char. 2
So this question is a continuation of the following one
[1] On the determination of a quadratic form from its isotropy group
For some motivations and relevant backgrounds related to this question ...
- 7,521
10
votes
1
answer
359
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Identifying lattices
I wrote a program that numerically searches for lattices in $\mathbb{R}^d$ with high sphere packing densities. As I have been running the program, it has been able to find, in addition to well-known ...
- 5,926
11
votes
1
answer
821
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Primes $ 1 + x^2 + y^2$
EDIT, Saturday 11:32 am, March 24: a complete answer for $4 + x^2 + y^2$ and generally $4 m^2 + x^2 + y^2$ for fixed $m$ is given in Friedlander and Iwaniec page 282, Theorem 14.8. We might also ...
- 25.4k
7
votes
3
answers
4k
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quadratic forms over fields of characteristic 2
I was wondering if anyone knows any good sources for the theory of quadratic forms over fields of characteristic 2 which are written in English?
- 2,005
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votes
1
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2k
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Sums of three non-zero squares
It is a well-known result of Legendre that a positive integer is sum of three squares unless it is of the form $4^a(8b+7)$.
In
Grosswald, E.; Calloway, A.; Calloway, J. The representation of ...
- 32.2k
3
votes
1
answer
399
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Faux Mordell equation and positive binary quadratic forms
This is about the frequency of integral solutions to
$$ b^2 - 4 a^3 = \Delta, $$
when $\Delta < 0$ is a discriminant of positive binary quadratic forms such that the class number is divisible by ...
- 25.4k