conic sections – What is the locus of midpoints of the chords of contact of$ x^2+y^2=a^2$ from the points on the $ell x + my + n = 0$

What is the locus of midpoints of the chords of contact of $ x^2+y^2=a^2$ from the points on the $ell x + my + n = 0$

My approach is as follow

$ell x + my + n = 0 Rightarrow my = – ell x – n Rightarrow y = – frac{{ell x}}{m} – frac{n}{m}$

Let us represent the tangent by
$y = px + q Rightarrow p = – frac{ell }{m};q = – frac{n}{m}$

${x^2} + {y^2} = {a^2} Rightarrow {x^2} + {left( {px + q} right)^2} = {a^2} Rightarrow {x^2} + {p^2}{x^2} + {q^2} + 2xpq – {a^2} = 0$

$ Rightarrow left( {1 + {p^2}} right){x^2} + 2xpq + {q^2} – {a^2} = 0 Rightarrow {x^2} + frac{{2xpq}}{{1 + {p^2}}} + frac{{{q^2} – {a^2}}}{{1 + {p^2}}} = 0$

$h = – frac{{pq}}{{1 + {p^2}}} = – frac{{frac{{ell n}}{{{m^2}}}}}{{1 + frac{{{ell ^2}}}{{{m^2}}}}} = – frac{{ell n}}{{{m^2} + {ell ^2}}}$, where $h$ represent the abscissa of the mid-point

${x^2} + {y^2} = {a^2} Rightarrow {left( {frac{{y – q}}{p}} right)^2} + {y^2} = {a^2}$

$ Rightarrow {left( {y – q} right)^2} + {p^2}{y^2} = {p^2}{a^2} Rightarrow {y^2} + {q^2} – 2qy + {p^2}{y^2} = {p^2}{a^2}$

$ Rightarrow left( {1 + {p^2}} right){y^2} – 2qy + {q^2} – {p^2}{a^2} = 0 Rightarrow {y^2} – frac{{2qy}}{{1 + {p^2}}} + frac{{{q^2} – {p^2}{a^2}}}{{1 + {p^2}}} = 0$

$k = frac{q}{{1 + {p^2}}} = – frac{{frac{n}{m}}}{{1 + frac{{{ell ^2}}}{{{m^2}}}}} = – frac{{mn}}{{{m^2} + {ell ^2}}}& h = – frac{{ell n}}{{{m^2} + {ell ^2}}}$ where $k$ represent the ordinate of the mid point

$ Rightarrow frac{k}{n} = – frac{m}{{{m^2} + {ell ^2}}}& frac{h}{n} = – frac{ell }{{{m^2} + {ell ^2}}}$

$ Rightarrow frac{k}{n} = – frac{m}{{{m^2} + {ell ^2}}}& frac{h}{n} = – frac{ell }{{{m^2} + {ell ^2}}}$

$frac{{{h^2}}}{{{n^2}}} + frac{{{k^2}}}{{{n^2}}} = frac{{{ell ^2} + {m^2}}}{{{{left( {{m^2} + {ell ^2}} right)}^2}}} Rightarrow {h^2} + {k^2} = frac{{{n^2}}}{{{m^2} + {ell ^2}}}$

Not able to approach from here , the term $a^2$ is lost in calculation

fa.functional analysis – an infinity of ideals in $ mathcal {L} ( ell_ infty) $?

Looking at this document, if $ Y $ is a Banach space containing a completed subspace $ X $ with the properties of Corollary 1, then $ mathcal {L} (Y) $ contains $ 2 ^ {2 ^ { aleph_0}} $ closed ideals. Can this be used to show that $ mathcal {L} ( ell_ infty) $ also have so many ideals?

Let $ (e_i) $ to be the canonical basis of a space $ X $ satisfying the conditions of Corollary 1, so that there is an operator $ T: X to X $ of standard at most 1 satisfactory:

(a) For some people $ eta> 0 $ and for each $ M in mathbb {N} $ there is a finite dimension subspace $ E $ of $ X $ such as $ d (E)> M $ and $ | T x | geq eta | x | $ for everyone $ x in E $.

b) For some constants $ Gamma $ and all $ m in mathbb {N} $ there is a $ n in mathbb {N} $ so that each $ m $-dimensional subspace $ E $ of $ (e_i) _ {i geq n} $ satisfied $ gamma_2 (T_ {| E}) leq Gamma $.

We can consider $ X $ as a subspace of $ ell_ infty $, so that there is a canonical integration $ theta: X to ell_ infty $. Replacement $ T $ with $ theta T $ does not change (a) or (b). In fact, we can extend $ T $ to an operator on $ ell_ infty $. Considering this extension, we can no longer directly apply Proposition 1, but it seems that the evidence can be changed without too much difficulty to take into account that $ X $ is a subspace of $ ell_ infty $.

Here's what I thought. In the proof of proposition 1, there are projections used $ P_ alpha: X to (G_k) _ {k in alpha} $. But we can expand $ theta P_ alpha $ to an operator $ widehat {P} _ alpha in mathcal {L} ( ell_ infty) $ who agrees with $ P_ alpha $ sure $ X $. We define $ widehat {Q} _k $ and $ widehat {R} _k $ similarly to use in place of $ Q_k $ and $ R_k $. I believe the rest of the evidence passes.

Am I missing something? Is this still an open problem, lol?

temporal complexity – Subset of vectors $ k $ with the shortest sum, compared to the norm $ ell_ infty $

I have a collection of $ n $ vectors $ x_1, …, x_n in mathbb {R} _ { geq 0} ^ {d} $. Given these vectors and an integer $ k $, I want to find the subset of $ k $ vectors whose sum is the shortest compared to the uniform norm. In other words, find the whole (maybe not unique) $ W ^ * subset {x_1, …, x_n } $ such as $ left | W ^ * right | = k $ and

$$ W ^ * = arg min limits_ {W subset {x_1, …, x_n } land left | W right | = k} left lVert sum limits_ {v in W} v right rVert _ { infty} $$

The brute-force solution to this problem takes $ O (dkn ^ k) $ operations – there are $ {n choose k} = O (n ^ k) $ subsets to be tested, and each takes $ O (dk) $ operations to calculate the sum of the vectors, then find the uniform norm (in this case, just the maximum coordinate, because all the vectors are non-negative).

My questions:

  1. Is there a better algorithm than brute force? The approximation algorithms are correct.

One idea that I had was to consider a convex relaxation where we give each vector a fractional weight $ (0, 1) $ and require that the weights total $ k $. The subset resulting from $ mathbb {R} ^ d $ covering all these weighted combinations is indeed convex. However, even if I can find the optimal weight vector, I don't know how to use this set of weights to choose a subset of $ k $ vectors. In other words, which full rounding scheme to use?

I also thought about dynamic programming but I don't know if it would end up being faster in the worst case.

  1. Consider a variation where we want to find the optimal subset for each $ k $ in $ (n) $. Again, is there a better approach than naively solving the problem for each $ k $? I think there should be a way to use race information on size subsets $ k $ to those of size $ (k + 1) $ etc.

  2. Consider the variation where instead of a subset size $ k $, we receive a target standard $ r in mathbb {R} $. The task is to find the largest subset of $ {x_1, …, x_n } $ whose sum has a uniform standard $ leq r $. In principle, we should seek $ O (2 ^ n) $ vector subsets. Are the algorithms changing? Also, is the decision version (for example, we could ask if there is a size subset $ geq k $ whose sum has a uniform standard $ leq r $) of the NP-hard problem?

ag.algebraic geometry – Do Neron-Severi groups of smooth projective unirational manifolds contain a twist $ ell $?

Let $ X $ to be a smooth projective unirational variety on an algebraically closed characteristic field $ p> 0 $, and $ ell neq p $ a first. My question: the Neron-Severi group of $ X $ contain (not null) $ ell $-torsion? This seems to be closely related to the presence of torsion in the etal cohomology group. $ H ^ 2_ {and} (X, mathbb {Z} _l) $.

ag.algebraic geometry – Dimension of $ ell $ -adic Eilenberg-Maclane space

I am currently studying the $ ell $-adic cohomology function, i.e. the functor $$ F: X rightarrow H ^ i_ {ét} (X, mathbb {Q} _ { ell}). $$
In a certain sense, it is a representable functor, that is to say that there is a $ ell $-adic Eilenberg-Maclane space (see Representability of Weil's cohomology theories in the theory of stable motivational homotopy). I would like to know the size of this space. The normal way to calculate the dimension of a module space is to study the deformations. However, since the spread site of a scheme is determined by the underlying reduced structure, we see that
$$ F (X ( epsilon)) = H ^ i_ {ét} (X ( epsilon), mathbb {Q} _ { ell}) cong H ^ i_ {ét} (X, mathbb {Q } _ { ell}) = F (X). $$
However, confusingly, this implies that this analogue of the Eilenberg-Maclane space is of zero dimension, which seems to me to be very false. Where did I go wrong?

functional analysis – Different definitions of the asymptotic – $ ell_ {1} $ Banach spaces?

The concept of asymptotics$ ell_ {1} $ The Banach space was first introduced in this article (bottom of page 7) as follows:

Banach space $ X $ with a standardized base $ {x_i } $ is said to be asymptotic $ l_1 $ if there is a constant $ C $ so that for each $ n $ it exists $ N = N (n) $ so that all successive normalized blocks $ N <z_1 <z_2 <… <z_n $ of $ {x_i } $ are equivalent C at the base of the unit vector in $ l_ {1} ^ {n} $.

Another definition is given in this article (bottom of page 5). It reads:

Banach space $ X $ with a base $ {e_ {i} } $ is asymptotic$ ell_ {1} $ if there is a constant C $> $ 0 so that for all $ N in mathbb {N} $ and all block sequences $ (x_ {i}) _ {i = 1} ^ {N} $ with $ N leq x_ {1} <x_ {2} < ldots <x_ {N} $, it follows that $ frac {1} {C} sum_ {i = 1} ^ {N} | x_ {i} | leq left Vert sum_ {i = 1} ^ {N} x_ {i} droite Vert $.

My question is why are these definitions equivalent? They seem to be different because the latter insists that the length of the sequence is explicitly linked at least support from $ x_ {1} $.

Show that $ (S ^ 1) ^ * = B ( ell ^ 2) $ knowing that $ ( ell ^ 1) ^ * = l ^ infty $

Is there a way to show that two trace class operators, $ S ^ 1 $, is $ B ( ell ^ 2) $, operators bounded on $ ell ^ 2 $, knowing that double $ ell ^ 1 $ is $ ell ^ infty $?

Theory of representation – Equivalence of categories between the looped algebra of $ sl_ {n + 1} $ and the refined Weyl group of $ GL_ ell (C) $

In this article, Theorem 4.9, page 18, Charri and Pressley claim that there is an equivalence of categories between certain categories of Lie algebra $ tilde { mathfrak g} = mathfrak {sl} _ {n + 1} otimes mathbb C (t, t ^ {- 1}) $ and the Weyl group refined from $ GL _ { ell} ( mathbb C) $.

There, they say that the refined group of Weyl $ GL _ { ell} ( mathbb C) $ is the semi-direct product $ S_ ell ltimes mathbb Z _ { ell}, $ where the action by automorphism is given by permutating coordinates. Why? How can I describe this group of Weyl refines more intrinsically? So far I have only heard of a refined Weyl group of affine Lie algebras linked to a bit simple Lie algebras. And would not it be more natural to consider the refined group of Weyl $ mathfrak sl_ {n + 1} $ instead of $ GL _ { ell} ( mathbb C) $? Is there a relationship between them?

Kolmogorov complexity of $ y $ given $ x = yz $ with $ K (x) geq ell (x) – O (1) $

I am trying to solve Exercise 2.2.2 of "An Introduction to the Kolmogorov Complexity and Its Applications" (Li & Vitányi, Vol 3). The exercise is the following (paraphrased):

Let $ x $ satisfied $ K (x) geq n – O (1) $, or $ n = ell (x) $ is the length of $ x $ in a binary encoding. CA watch $ K (y) geq frac {n} {2} – O (1) $ for $ x = yz $ with $ ell (y) = ell (z) $.

I have tried but I have failed to apply the method of incompressibility, exploiting the fact that there are many $ x $ in a way. More direct approaches to trying to find intelligent recursive functions of $ y $ and $ z $ also did not work.

functional analysis – $ B _ { ell ^ {2}} ^ {+} $ with the standard $ Green green cdot Green green _ { sqrt {2}} $ n has no structure par

Consider the space $ ell ^ {2} $ with the standard
begin {align *}
Green x Green _ {2} = left ( sum _ {i = 1} ^ { infty} x _ {i} ^ {2} right) ^ {1/2}
end {align *}

and we define the equivalent standard
begin {align *}
Green green x Green green _ { sqrt {2}} = max { Green x Green _ {2}, sqrt {2} Green x Green _ { infty} } mbox {.}
end {align *}

Define the positive part of the unit ball
begin {align *}
B _ { ell ^ {2}} ^ {+} = lbrace x in ell ^ {2}: ; Green x Green _ {2} leqslant 1, ; x _ {i} geqslant 0 rbrace mbox {.}
end {align *}

I want to show that $ B _ { ell ^ {2}} ^ {+} $ with the standard $ Green green cdot Green green _ { sqrt {2}} $ do not have normal structure, and to show that, I should show that diam$ (B _ { ell ^ {2}} ^ {+}) $ = $ r _ {x} (B _ { ell ^ {2}} ^ {+} $. I've shown this diam$ (B _ { ell ^ {2}} ^ {+}) = $ 1But I do not know why $ r _ {x} (B _ { ell ^ {2}} ^ {+}) = $ 1. What element in $ B _ { ell ^ {2}} ^ {+} $ can take to prove that $ r _ {x} (B _ { ell ^ {2}} ^ {+}) = $ 1?