Questions tagged [inner-product]
The inner-product tag has no usage guidance.
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Hash functions and inner product
As part of a research project on derandomization of linear threshold functions I am working on, I am trying to understand the following problem:
Is there a small (polynomial rather than exponential)...
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What is the exponential map from the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\ltimes_\textrm{ad}\mathfrak{sl}(2,\mathbb{C})$ to its Lie group?
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
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Independent inner functions on the unit disk
This problem cropped up in a paper that I am writing and I have thought about it for too long to no avail: let $\mathbb{D}$ be the open unit disk in the complex plane and suppose $\varphi:\mathbb{D}\...
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Triple Petersson Inner Products With Theta Functions
Our current work requires us to bound triple products of the form
$$\langle \Im(*)^{\frac{3}{2}}|\theta^3|^2,\mu_j \rangle,$$
where $\langle \cdot, \cdot \rangle$
is the Petersson inner product, ...
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Unitary operators with the same inner product as vectors
Suppose we have a set of real unit vectors $v_1,\ldots,v_m \in \mathbb{R}^n$. We can always find a set of unitary operators $U_1,\ldots,U_m$ acting on $\mathbb{C}^N$ (for $N$ that is possibly much ...
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The existence of a variety generated in low degree that is epsilon-close to a fixed variety
Let $\mathbb{C}^n$ be the $n$-dimensional complex vector space endowed with the standard Hermitian inner product, let $X \subseteq \mathbb{C}^n$ be an algebraic set that forms a cone, and let $1>\...
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Inner Product on tensor product of Hilbert spaces is unique?
Given two Hilbert Spaces $H$ and $K$, a natural inner product on $H\otimes K$(= vector space tensor product of $H$ and $K$) is given by
$\hspace{.5in}\langle h_1\otimes k_1,h_2\otimes k_2\rangle=\...
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Probability of orthogonal vectors?
Denote $\mathcal I_m=\{-m,-m+1,\dots,0,\dots,m-1,m\}$ where $m\geq0$ is an integer.
Pick a uniformly random vectors $$\hat a=(a_n,a_{n-1},\dots,a_1)\in(\mathcal I_{m_1}+\sqrt{-1}\cdot\mathcal I_{m_2}...
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What are alternative or equivalent definitions of a positive-definite function on a group?
The standard definition of a positive-definite function on a group goes as follows:
Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
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Unimodular eigenvalue of a H-self-adjoint matrix (indefinite innerproduct)
Let $A,H \in \mathbb{C}^{n \times n}$ be such that $H$ is Hermitian and invertible and $A = H^{-1} A^* H$. In this case, $A$ is said to be $H$-self-adjoint. This is due to the fact that if $\langle \...
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Question about finite dimensional representations of a semi-simple Lie group
I have posted a question in MSE
https://math.stackexchange.com/questions/4468138/question-about-finite-dimensional-representations-of-a-semi-simple-lie-group but didn't receive any comment or answer.
...
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Kernels with finite dimensional feature spaces
Suppose $x,y \in \mathbb{R}^n$ for some given fixed n.
Consider a kernel $K(x,y) = f(\langle x, y \rangle)$, I'd like to know which functions $f$ admit a finite dimensional feature map. In other words,...
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Is there a name for this "inner product" on projective space?
$\newcommand{\proj}{\mathbb{P}}\newcommand{\complex}{\mathbb{C}}\newcommand{\ip}[2]{\langle #1 , #2\rangle}\newcommand{\abs}[1]{\lvert #1 \rvert}$There is a natural bijection $\phi: \proj(\complex^n)\...
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Do involutions always stabilize some transverse lagrangians?
Let $V$ be a vector space of dimension $2n\geq 4$ over a field $F$ of characteristic distinct from $2$. Assume that $V$ is equipped with a nondegenerate alternating form $b$. Let $Sp(V)$ denote the ...
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Terminology for research on distributions of inner products
Consider a set of vectors $M$ from an inner product space $V$. The ordered set of inner products of all pairs of elements in $M$ uniquely characterizes $M$ up to isomorphism.
Suppose now that $V$ is ...
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Invariance signature in infinite dimension
Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that:
$g_0$ is positive-definite
$g_t$ is non-degenerate for ...
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What is lost after RKHS embedding of the L1 space?
We know that every distribution or $L^1$ function $f$ over space $\mathcal{X}$ (e.g., $R^d$) can be embedded to an RKHS $\mathcal{H}$ with a $1$-bounded kernel $\mathcal{K}$ (e.g., the RBF kernel) ...
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General bivariate functions that satisfy Cauchy-Schwarz
Have there been any study of general bivariate functions $f:X \times X \to \mathbb{R}$ that satisfy $f(x,y)^2 \leq f(x,x)f(y,y)$. This comes up as a function I'm working with satisfies the asymmetric ...
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Can an orderless set of inner product between N vectors determine unique structure of the vectors?
Suppose we have n vectors {a1,a2,a3,...,an} such that the sum of them is zero vector
a1+a2+a3+...+an=0
Now, we compute the inner product of each two vectors of them, i.e. we compute the Gramian matrix ...
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Complex conjugate and unitary complex conjugate
Definition: Let V be complex finite dimensional inner product space
Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$
Definition: Anti-unitary Complex ...
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Functional Analysis Generalizations: indeterminated inner product and functions over manifolds
There are books or articles that deal with generalizations of functional analysis in the sense that the inner product need not be positive-definite or that works with functions over manifolds?
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How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?
Let $n,d_1,\ldots,d_n > 1$ be integers, and $V_1, \ldots, V_n$ be inner product spaces over $\mathbb C$, having dimensions $d_1, \ldots, d_n$ respectively. We consider the ways in which we may ...
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The discrete orthogonal polynomials
I want a document or something that explains the following proposition:
The discrete orthogonal polynomials are the polynomial solutions of the given diference equation:
$$
\sigma(x)\Delta\nabla P_n(...
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Uniform distribution on pairs of unitary matrices
This question has two parts.
In Part 1, I would like to know if the following distribution on pairs of $d$-dimensional unitary matrices has popped up in the literature:
"Uniform distribution on ...
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Inner products on abelian groups and general modules
Where can I find a discussion about inner products on something more general than vector spaces, and to which extent should one attempt such a generalization?
My particular interest is in abelian ...
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