Questions tagged [positivity]
The positivity tag has no usage guidance.
146
questions
6
votes
0
answers
117
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$\ell^\infty / ces_0$ as an ordered Banach space
Let
$ces_0:=\{\xi\in\ell^\infty : \lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}\xi_k=0\}$ and $q:\ell^\infty \to \ell^\infty/ces_0$ be the usual quotient map. The space $ces_0$ is closed in $\ell^\...
4
votes
0
answers
113
views
Positivity of q-analogs of central binomial coefficients?
With the usual $q-$notations
$[n]_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q},$
$[n]_q!=[1]_q[2]_q\cdots[n]_q$ and
$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$
let
$$b(n,k,r,q)=\det\left(q^{r\...
- 5,572
1
vote
0
answers
135
views
Preserving the strictly total positivity of special bases by using radial basis functions
Let $\left[\alpha,\beta\right]$ be a non-empty interval and consider an $\left(n+1\right)$-dimensional subspace $\mathbb{S}_{n}$ of $C^n\left(\left[\alpha,\beta\right]\right)$ such that $\mathbb{S}_{n}...
- 31
9
votes
1
answer
294
views
Nonnegative coefficients of a product of polynomials
Let $P(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n]$. Does there exist an algorithm to decide whether there is a nonzero polynomial $Q(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n]$ such that the product $...
- 49.5k
3
votes
0
answers
249
views
On the existence of fixed points of a matrix iteration
Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...
- 2,682
6
votes
1
answer
394
views
Positive-definiteness of radial sinc function in three dimensions
In dimension one, it is well known that $\mathcal{F}\chi_{(-1,1)}=\frac{\sin{x}}{x}$. This implies, in particular, that $\frac{\sin{x}}{x}$ is a definite positive function. I wonder if a similar ...
- 923
8
votes
1
answer
490
views
Maps which are both completely positive and positive
Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...
- 42.5k
37
votes
2
answers
3k
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When can a function be made positive by averaging?
Let $f: {\bf Z} \to {\bf R}$ be a finitely supported function on the integers ${\bf Z}$. I am interested in knowing when there exists a finitely supported non-negative function $g: {\bf Z} \to [0,+\...
- 109k
6
votes
0
answers
113
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Positive splitting of Sobolev convergence
Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
- 201
0
votes
1
answer
305
views
Dual of a stable locally free subsheaf is a locally free quotient sheaf
Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.
By [1] definition 1.2:
A line bundle $L$ ...
- 828
1
vote
1
answer
109
views
Completeness of Lowner order in separable Hilbert space
Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the interesection of positive operator and ...
- 1,493
1
vote
1
answer
217
views
positive real matrix-valued function as linear combination of positive-real functions
In my question I am considering $z$ as the complex variable in the $z$-transform $X(z)$ of a discrete-time sequence $x[n]$:
I have $M$ square complex matrices $\mathbf{R}_m$ of size $N\times N$.
I ...
- 11
2
votes
1
answer
356
views
Positive Solutions of second-order ODE
Consider second-order ordinary differential equations of the form
$u''(t)=a(t)u(t)-2$
I'm interested in general criteria on the function $a(t)$, which guarantee respectively rule out the existence ...
- 98
0
votes
1
answer
345
views
Stability of a matrix product
Motivation: I am working on a research problem and have been stuck for a while. I hope someone can help, as it requires only linear algebra. :)
Let $H$ be a real, invertible and positive semi-...
- 103
7
votes
0
answers
182
views
Positivity of certain polynomial coefficients
Consider the rational functions (in fact, polynomials)
$$F_n(q)=\frac1{(1-q)^{2n}}\sum_{k=0}^n(-q)^k\frac{2k+1}{n+k+1}\binom{2n}{n-k}
\prod_{j=0,\,j\neq k}^n\frac{1+q^{2j+1}}{1+q}.$$
The numbers $\...
- 41.8k
0
votes
0
answers
116
views
Spectral projection of a positive operator
Let $(X,K)$ be a partially ordered Banach space where the cone $K$ is generating and normal. Suppose $B$ is a bounded operator on $X$ such that $B(K)\subseteq K$, the spectral radius $\rho(B)=1$ and ...
- 101
0
votes
1
answer
130
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Polynomials $p$ such that $p$ and $p'$ preserve nonnegative numbers
Expanding on a previous post I made recently, let
$$
\mathscr{P}:= \{ p(x) \in \mathbb{R}[x] \mid p(x) \ge 0,~\forall x\ge 0\}.
$$
The Pòlya-Szegö theorem (see Theorem 3.21 here) asserts that $p \in \...
- 1,711
12
votes
2
answers
737
views
Polynomials that preserve nonnegativity
A polynomial $p \in \mathbb{R}[x_1,\dots,x_n]$ is said to be positive on a subset $S$ of $\mathbb{R}^n$ if $p(x) > 0$ for every $x \in S$. The polynomial $p$ is called nonnegative if $p(x) \ge 0$ ...
- 1,711
2
votes
0
answers
114
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Do integral curves on simple abelian surfaces define big line bundles?
Let $A$ be a simple abelian surface over $\mathbb{C}$.
Let $C\subset A$ be an irreducible and reduced one-dimensional closed subscheme. Since $A$ is simple, the normalization of $C$ is of genus at ...
- 57
47
votes
3
answers
4k
views
Is this proof of Perron's theorem correct, and if so is it original?
A few years ago, I came up with this proof of Perron's theorem for a class presentation:
https://pi.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf
I've written an outline of it below ...
- 473
0
votes
0
answers
152
views
Continuity under various topologies for positive linear functionals
It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is ...
- 5,217
7
votes
1
answer
643
views
nef vs. 1-nef vector bundles
Let $X$ be a compact, connected, Kähler manifold, of dimension $d$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.
By [1] definition 3.1.2:
A line bundle $L$ ...
- 828
1
vote
0
answers
31
views
Non Negative Tensor Tucker Decomposition Error Degradation
I have been working on iterative decomposition methods of tensors with non negativity constraints. I have noticed that $\textbf{N}$on negative $\textbf{T}$ensor $\textbf{F}$actorization "NTF" which is ...
- 111
5
votes
0
answers
279
views
When is the strict transform of very ample divisor ample?
Let $X$ be a projective variety and $Y \subset X$ is a very ample divisor on $X$. Let $Z \subset X$ be a regular subvariety (but $X$ need not be regular along $Z$) of codimension $2$ and $\pi:\tilde{X}...
- 2,126
13
votes
1
answer
897
views
Why do we care about Schur Positivity
Some of the most important open problems in Algebraic Combinatorics concern the Schur positivity of classes of symmetric functions. Why is this an important property to have?
4
votes
1
answer
111
views
Specific quaternary quartic that is positive semi-definite but not sum of squares
Does there exist a quaternary quartic $f$ (a form in $\mathbb{R}[x_1,x_2,x_3,x_4]$ of degree $4$), which is positive semi-definite ($f \geq 0$ on $\mathbb{R}^4$) but not a sum of squares, such that ...
- 41
1
vote
1
answer
542
views
Probability of positive definiteness of a random matrix [duplicate]
Given an $n \times n$ symmetric random matrix whose entries have distribution $N(0,1)$, how to calculate the probability of positive definiteness of this matrix?
- 19
9
votes
0
answers
255
views
Can we extend c.p. normal maps on a finite von Neumann algebra $M$ to $L_0(M)_+$?
Suppose that $M$ is a von Neumann algebra with a finite, normal, faithful trace $\tau$. Let $T\colon M\to M$ be a completely positive, normal map.
Can $T$ be extended to a `positively linear map' ...
- 11.3k
29
votes
3
answers
2k
views
Wanted: Positivity certificate for the AM-GM inequality in low dimension
I'm seeking for a Certificate of Positivity for the AM-GM inequality in five variables
$$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$
Can one write the LHS as a sum
$\,\...
- 489
5
votes
2
answers
281
views
Positivity of coefficients of a polynomial derived from Schubert polynomials
Let $W=\bigcup_{n=1}^\infty S_n$ be the union of all symmetric groups $S_n$. For an element $w\in W$, denote by $\mathfrak{S}_w$ the Schubert polynomial associated to $w$, and by $\partial_w$ the ...
- 1,064
12
votes
3
answers
958
views
What techniques are there to prove Schur positivity?
As the title says, what methods exists for proving that a symmetric polynomial (or function) is Schur positive, perhaps involving extra parameters, in which case coefficients should be polynomials in ...
- 15.6k
1
vote
1
answer
194
views
Does every locally positive-definite function have a positive-definite extension?
Let $B$ denote the unit ball in $\mathbb{R}^d$, and suppose $f\colon B\rightarrow\mathbb{C}$ has the property that for every $n\geq1$ and $x_1,\ldots,x_n\in\mathbb{R}^d$ with $\|x_i-x_j\|<1$, the $...
- 7,573
1
vote
0
answers
30
views
Extending the projective action of several positive linear maps to a complex neighbourhood
I am currently reading a paper which, somewhat indirectly, asserts the following result:
Lemma: Let $\Delta \subset \mathbb{R}^d$ denote the simplex $\{(x_1,\ldots,x_d):\sum_{i=1}^d x_i=1\}$, let $...
- 6,186
4
votes
1
answer
239
views
A discrete operator begets even/odd polynomials
Given a function $f(x)$ define the forward shift operator by $Ef(x)=f(x+1)$ and the discrete derivative $\delta f(x)=(E-1)f(x)=f(x+1)-f(x)$.
Given a partition $\lambda=(\lambda_1,\lambda_2,\dots,\...
- 41.8k
8
votes
0
answers
189
views
Hilbert spaces over the semi-field $\mathbb R_+$
Let $\mathbb R_+$ be the semi-field of non-negative real numbers.
Definition (preliminary): A Hilbert space over $\mathbb R_+$ is a pair $(H,P)$, where $H$ is a complex Hilbert space, and $P\subset H$...
- 42.5k
1
vote
1
answer
94
views
Is it true that $B \leq I \rightarrow B^2 \leq I$ (or higher powers) for positive semidefinite $B$, mimicking positive scalars? [closed]
If we know that $B \geq 0 $ (positive semidefinite) and that $I-B \geq 0$, is it necessarily true that $I-B^2 \geq 0$?
- 11
2
votes
1
answer
183
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Perturbation of linear system of equations: Is the solution still non-negative?
Let $A = (a_{ij})_{i,j=1,\dots,n}$ be a matrix such that
$a_{ij} \ge 0$ for all $i,j = 1, \dots, n,$ and
$A$ is positive definite.
Let $I$ be the identity matrix, and $\pmb{1}$ the vector ...
- 267
2
votes
1
answer
151
views
Interplay between the trace operator and the positive part of a Sobolev function
I consider the Sobolev trace $\gamma \colon H^1 (\mathbb{R}^{N+1}_{+}) \to H^{1/2}(\mathbb{R}^N)$. Let $U^{+}$ denote the positive part of a function $U$. It is correct, or only meaningful, to say ...
- 459
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0
answers
93
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Reference request: Positive solution of positive system of linear equations
Let $A \in \mathbb{R}^{n\times n}$ be an invertible matrix with positive entries,
and $b \in \mathbb{R}^n$ a vector with positive entries.
When does $A^{-1}b$ have all positive entries?
I am looking ...
- 267
0
votes
0
answers
85
views
Show that a certain ratio of diagonal entries dominates a certain ratio of singular values
Let $D=[d_{ij}]_{i,j=1,\ldots,4}$ be a $4 \times 4$ “density matrix”, that is, a Hermitian (possibly symmetric) positive definite matrix having trace 1—that is, the (nonnegative) diagonal entries sum ...
- 723
4
votes
2
answers
643
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When does this linear matrix equation have a unique symmetric, positive definite solution?
I encountered the following matrix equation for $A, N, Q \in \mathbb{R}^{n \times n}$ and $A^T=-A$ and $N^T=-N$
$$[X,A]+N^TXN+Q = 0$$
where $Q$ is symmetric, positive definite. My final goal is to ...
- 41
2
votes
1
answer
150
views
induced map on state spaces
A $*$-homomorphism $f:A\to B$ between C*-algebras is called non-degenerate if $f(A)B=B$.
I guess that I can prove that a non-degenerate *-homomorphism always induces a map on state spaces $f^\ast:S(B)...
- 23
2
votes
0
answers
390
views
Detecting positive solutions to an underdetermined linear system [closed]
Given an m by n matrix A with real entries, how can I determine if there exists a solution to the linear equation Ax=0 whose coordinates are all strictly positive?
I am also interested in the variant ...
- 171
6
votes
1
answer
539
views
On a trace condition for positive definite $2\times 2$ block matrices
Consider the following block matrix
$$
X = \begin{bmatrix} A & C \\ C^\top & B\end{bmatrix},
$$
where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{m\times m}$, and $C\in\mathbb{R}^{n\times m}...
- 2,682
6
votes
1
answer
291
views
Restricting a continuous positive-semidefinite function to a finite subset
The following is a toy version of something I've been fiddling with and I thought it might be more efficient to post it as a question here.
Just to fix definitions: for me, a complex-valued function $...
- 25.5k
3
votes
1
answer
311
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One of Poincaré's theorems about positive rational functions
A rational function in $ \mathbb{R}[x_1, x_2] $ is called positive if $f = g/h$ with $g,h \in \mathbb{R}_{\geq 0}[x_1, x_2]$. Are there some references about the following theorem given by Poincare?
...
- 6,121
6
votes
1
answer
612
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A sufficient condition (or not) for positive semidefiniteness of a matrix?
Let $A\in M_{n}(\mathbb{C})$ be a Hermitian matrix. If for all $z_1,...,z_n\in\mathbb{C}$, $$\sum_{i,j=1}^{n}A_{ij}z_{i}\overline{z_{j}}\ge 0$$ then A is positive semidefinite.
I do not think the ...
- 6,997
2
votes
0
answers
186
views
Sums of hermitian squares in free abelian group algebras and real positive semidefinite matrices
A little context for the following question, first. As Noah Stein notes in a comment below, the present question is closely related to the free semialgebraic geometry studied by Helton and his ...
- 6,997
10
votes
1
answer
501
views
Positivity of power of positive PSD matrices
Background: Let $M$ be an $n\times n$ matrix with nonnegative entries. It is immediate that for any integer $k$, $M^k$ has nonnegative entries.
Suppose now that, on top of having nonnegative entries, ...
- 505
21
votes
4
answers
2k
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Why are quantum groups so called?
I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...
- 827