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LU growth factor applied to LDL of a Positive Semidefinite matrix [on hold]

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
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Local coordinates on (infinite dimensional) Lie groups, factorization of Rieman zeta functions

Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...
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Does this count as a canonical decomposition for non-elementary hyperbolic 3-orbifolds?

Let $\Gamma$ be a Kleinian group and let $\mathbb{H}^3$ be the upper half-space model for hyperbolic 3-space. Then $\mathbb{H}^3/\Gamma$ is an orientable hyperbolic 3-orbifold (with the group action ...
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What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
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Surgery to unlink $S^p$ and $S^q$ in $S^d$

We know that $S^p$ and $S^q$ can be linked in $S^d$ if $p+q<d$. Let us consider the simplest case where both $S^p$ and $S^q$ are un-knotted spheres. I am looking for a surgery to unlink $S^p$ and ...
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Difficult examples of invertible differentiable functions [on hold]

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
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Finding kernel and image [on hold]

X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) What is the dimension of X? Find ker f and im f. Find bases for ker f and im f. Is f a bijection? My attempt: 1. I calculate dim ...
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Minimum number of real multiplications to multiply two quaternions [on hold]

Karatsuba multiplication of two complex numbers can be performed with just three real multiplications (instead of four) as follows: $$(a+bi)(c+di) = (ac-bd) + i ((a+b)(c+d) - ac-bd)$$ We only need the ...
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Can someone help me find a Mathematical documentary that aired on British television within the past 10 years about Leibniz? [on hold]

I have been searching for a documentary that aired on British television between around 2006 and 2012 which was centred around the German Mathematician, Gottfried Leibniz. All that I can remember ...
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This question is on the intuition behind the Caratheodory definition of measurable sets as given in Billingsley. He motivates by saying that we "should" call a set $A$ measurable if $$P^*(A) + ... 0answers 116 views Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism MOTIVATION Nicolas and Serre have analyzed the structure of the space of mod 2 modular forms of level 1, viewed as a "Hecke-module". They show that for each p>2, the operator T_p acting on ... 0answers 51 views Contraction with a vector field and pullback bundle While trying to understand the proof of "smooth" version of Kostant-Hochschild-Rosenberg theorem (which is due to Connes for compact smooth manifolds) I found the following argumentation: one is ... 0answers 61 views Elliptic Curve: Q=nP [on hold] I have a question relating to Elliptic Curve Scalar Multiplication between two points. Given two points on that curve, Q and P, where Q=nP, is it possible to find n if we have an m such that mQ=P? 1answer 85 views Does a Gaussian process shrink under a contraction map Let T \subset \mathbb R^n, and assume it's a finite set if that helps. Consider the symmetric Gaussian process (X_t)_{t\in T} defined by X_t = \langle G, t\rangle, where G is a standard ... 1answer 117 views A question of terminology regarding integer partitions I am wondering if there is a standard notation and name for the following. Let \lambda be a partition \lambda_1\geq \lambda_2\geq\cdots\geq \lambda_r\geq 1 of n into r parts. Then we can ... 0answers 112 views Is there any computation of K^0(X) and K_0(X) for a singular curve X? Let X be a projective curve over an algebraic closed field k which characteristic zero. Define K^0(X) as the Grothendieck group of the derived category Perf(X) and K_0(X) as the Grothendieck ... 0answers 73 views Syzygies in integral domains Let f be a homogeneous irreducible element in a graded commutative Noetherian ring. What is the possible set of elements f_1 and f_2 such that f=f_1g_1+ f_2g_2? Even in very particular cases ... 2answers 89 views Variation of Hodge structures associated to a hermitian symmetric domain Let D be an irreducible hermitian symmetric domain. Then there exists a variation of Hodge structures (h_s)_{s\in D} on a vector space V satisfying specific conditions which depend on D such ... 1answer 107 views Bounding the number of lattice points inside an n-dimensional ellipsoid I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie inside an n-dimensional ellipse. That is, given an n-dimensional ellipsoid ... 4answers 195 views Uniform upper bound for the sum over primes \sum_{p \leq x} p^{-1+\varepsilon} I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click). In equation (27) the authors, apparently, used the following ... 1answer 345 views Can topological cyclic homology compute Picard groups? Let K be a number field, and \mathcal{O}_K its ring of integers. Then there is an isomorphism$$K_0(\mathcal{O}_K) \cong \mathbb{Z} \oplus Pic(\mathcal{O}_K) where $Pic(\mathcal{O}_K)$ is the ...
How to calculate the Inverse Fourier Transform of the following functions: $\dfrac{1}{-1+2\pi i x}$ $\dfrac{1}{(2\pi ix)^2-2 \pi ix +1}$ I don't know how to evaluate the integrals.