## 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 $$N of $${x_i }$$ are equivalent C at the base of the unit vector in $$l_ {1} ^ {n}$$.

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 $$N leq x_ {1} , it follows that $$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$$(B _ { ell ^ {2}} ^ {+}) = 1$$But I do not know why $$r _ {x} (B _ { ell ^ {2}} ^ {+}) = 1$$. What element in $$B _ { ell ^ {2}} ^ {+}$$ can take to prove that $$r _ {x} (B _ { ell ^ {2}} ^ {+}) = 1$$?