## geometry ag.algebraica – An equivalence \$ -mathbb {A} _1 \$ -homotopy is an equivalence \$ \$ mathbb {A} _1 \$ -weak

Leave $$Spc_S = L_ {Nis} sPre (Sm_S)$$ Y $$Spc_S ^ { mathbb {A} _1} = L _ { mathbb {A} _1} L_ {Nis} sPre (Sm_S)$$, the location of two Bousfield's $$sPre (Sm_S)$$ To obtain the category of motivated homotopy. $$Spc_S ^ { mathbb {A} _1}$$.
Here is the definition of $$mathbb {A} _1$$-homotopia

A map $$f: F a G$$ is a $$mathbb {A} _1$$-difference of equivalence if it is a $$mathbb {A} _1$$– local weak equivalence in $$Spc_S ^ { mathbb {A} _1}$$, which means that for any fibrant object $$X$$, there is a weak equivalence of simplicial sets.

$$map (F, X) a map (G, Y)$$

where $$map (X, Y)$$ is the mapping space in $$sPre (Sm_S)$$.

I'm not sure how to show that $$mathbb {A} _1$$equivalence -homotopy is a $$mathbb {A} _1$$– Weak equivalence What is the relationship between $$F times mathbb {A} _1$$ Y $$F times Delta_1$$ where $$Delta_1$$ is the constant simplicial presemigo in $$Delta_1$$?

## real analysis: differentiability of \$ G (x) = int _ { mathbb R} e ^ {tx} f (t) dt \$ on \$ [0,1] \$

Leave $$f colon mathbb R to mathbb R$$ be a non-negative and measurable function, and assume that both $$int _ { mathbb R} f (t) dt < infty$$ Y $$int _ { mathbb R} e ^ tf (t) < infty$$.

Show that the integral $$G (x) = int _ { mathbb R} e ^ {tx} f (t) dt$$ it is finite when $$0 le x le 1$$. Then prove that the function $$G (x)$$ is continuous in $$0 le x le 1$$, and differentiable in $$0 .

My intent:

It is trivial to prove that $$G (x) < infty$$ when $$0 le x le 1$$. To show that $$G (x)$$ is continuous in $$0 le x le 1$$, we have to consider the difference:
begin {align} G (x_2) -G (x_1) & = int _ { mathbb R} e ^ {tx_2} f (t) dt- int _ { mathbb R} e ^ {tx_1} f (t) dt \ & = int _ { mathbb R} (e ^ {tx_2} -e ^ {tx_1}) f (t) dt end {align}
where $$x_1, x_2 in [0,1]$$.

Note that $$| (e ^ {tx_2} -e ^ {tx_1}) f (t) | le max {2f (t), 2e ^ tf (t), f (t) + e ^ tf (t) } in L ^ 1 ( mathbb R),$$ it turns out that
$$lim_ {x_2 a x_1} [G(x_2)-G(x_1)]= int _ { mathbb R} 0 cdot f (t) dt = 0$$
that is to say., $$G (x)$$ is continuous in $$0 le x le 1$$.

Next, we study the differentiability of $$G (x)$$ in $$[0,1]$$.

We have
begin {align} frac {G (x_2) -G (x_1)} {x_2-x_1} = int _ { mathbb R} frac {e ^ {tx_2} -e ^ {tx_1}} {x_2-x_1} f (t ) dt end {align}
Y $$frac {e ^ {tx_2} -e ^ {tx_1}} {x_2-x_1} f (t) = frac {e ^ {tx_2} -e ^ {tx_1}} {tx_2-tx_1} tf (t ).$$
If we leave $$x_2 a x_1$$, so $$lim_ {x_2 a x_1} frac {e ^ {tx_2} -e ^ {tx_1}} {x_2-x_1} f (t) = e ^ {tx_1} tf (t).$$
But this time, I can not find a built-in integrable function $$g (t)$$ such that $$| e ^ {tx_1} tf (t) | in $$mathbb R$$. So, how to prove the differentiability of $$G (x)$$ in $$[0,1]$$?

## co.combinatorics – Binary search extension to determine a hyperplane by dividing a set of points into \$ mathbb {R} ^ d \$

They give us a set $$S$$ of $$n$$ points in $$mathbb {R} ^ d$$ and a vector (hidden) $$mathbf {w} in mathbb {R} ^ d$$, where each point $$mathbf {v} in S$$ is associated with a binary tag equal to the sign of $$mathbf {w} ^ { top} mathbf {v}$$. Set $$S$$ It is known but both the labeling assignment of its points and the vector. $$mathbf {w}$$ they are initially unknown. Sequentially, at each step of the time, we can select a point of $$S$$ and ask for its label, so that its label is revealed.

Question: Which is the minimum number $$q$$ (expressed in terms of $$n$$ Y $$d$$) of the necessary consultations to determine the labels of everyone points in $$S$$, finished everyone possible sets $$S in mathbb {R} ^ d$$?

Example: Yes $$d = 1$$ clearly $$q$$ it is logarithmic in $$n$$.

## Each projective module \$ mathbb {Z} / p ^ n mathbb {Z} \$ – is a free module \$ mathbb {Z} / p ^ n mathbb {Z} \$ – if p is a prime number

If p is a prime number, how to test each project? $$mathbb {Z} / p ^ n mathbb {Z}$$-module is a free $$mathbb {Z} / p ^ n mathbb {Z}$$-module?

## number theory: is there a maximum of \$ m in mathbb {N} \$ such that for some cousins ​​\$ p \$, \$ p_r = p + displaystyle sum_ {n = 1} ^ r 2 ^ n \$ is prime for \$ r = 1,2, …, m \$?

Is there a maximum? $$m en mathbb {N}$$ such that for some prime $$p$$, $$p_r = p + displaystyle sum_ {n = 1} ^ r 2 ^ n$$ is a cousin to $$r = 1,2, …, m$$?

There is probably no simple (or known) answer to this question, but I thought it was worth asking.

So far, I have concluded that $$p$$ must necessarily have $$7$$ as its first digit, that is, $$p = 7 , ( text {mod} , 10)$$. Yes $$p = x , ( text {mod} , 10)$$ for some $$x in {1,3,9 }$$, the process is cut in some $$p_k = 5 , ( text {mod} , 10)$$.

For example, yes $$p = 1 , ( text {mod} , 10)$$, begin {align} p & = 1 , ( text {mod} , 10) \ p_1 & = 3 , ( text {mod} , 10) \ p_2 & = 5 , ( text {mod} , 10) Rightarrow p_2 notin mathbb {P} end {align}

by $$p = 7 , ( text {mod} , 10)$$, we have
begin {align} p_ {4k} & = 7 , ( text {mod} , 10) \ p_ {4k + 1} & = 9 , ( text {mod} , 10) \ p_ {4k + 2} & = 3 , ( text {mod} , 10) \ p_ {4k + 3} & = 1 , ( text {mod} , 10) end {align}

for $$k in mathbb {N} cup {0 }$$ (Let's suppose for simplicity that $$p_0 = p$$).

Consider these two statements:
begin {align} existence p in mathbb {P} &: ; p_r = p + sum_ {n = 1} ^ r 2 ^ n in mathbb {P} ; ; forall r in mathbb {N} tag {1} \ exists m in mathbb {N} &: ; text {for some p in mathbb {P} }, ; p_r = p + sum_ {n = 1} ^ r 2 ^ n in mathbb {P} ; ; forall r in {1,2, …, m } tag {2} \ & ; ; ; ; , text {and} ; forall p in mathbb {P}, ; p_ {m + 1} notin mathbb {P} end {align}

My strategy is simple: try $$(1) Rightarrow (2)$$ false or test $$(2) Rightarrow (1)$$ false.

Through some computer work, I found that $$m ge 9$$ for $$p = 2397347207$$:

``````2397347207, 2397347209, 2397347213, 2397347221, 2397347237, 2397347269, 2397347333, 2397347461, 2397347717, 2397348229
``````

If someone can resolve this and / or direct me to useful material, please post it.

## co.combinatorics – "Injective routes" for the function \$ f: mathbb {Z} times mathbb {Z} to mathbb {Z} \$ with finite fibers

We make $$mathbb {Z} times mathbb {Z}$$ on a graph saying $$(a, b), (x, y) in mathbb {Z} times mathbb {Z}$$ form an edge if and only if $$| (a, b) – (x, y) | = 1$$, where $$| cdot |$$ Denotes the Euclidean metric.

Suppose $$f: mathbb {Z} times mathbb {Z} to mathbb {Z}$$ it's a function that $$f ^ {- 1} ( {z })$$ it is finite for each $$z in mathbb {Z}$$. Dice $$n en mathbb {N}$$, there is a path $$P subseteq mathbb {Z} times mathbb {Z}$$ of length $$n$$ such that the restriction $$f | _P$$ Is he injective?

## Theory of numbers: Is each cousin contained in the set \$ , {n in mathbb {N}: n | 10 ^ k-1 } cup {2,5 } , \$ for \$ k = 1, 2,3, … \$?

It is each cousin contained in the set
$$, {n in mathbb {N}: n | 10 ^ k-1 } cup {2,5 } ,$$ for $$k = 1,2,3, …$$?

Leave $$p$$ be a prime number Then $$frac {1} {p}$$ has a maximum decimal period of length $$p-1$$. Denote $$frac {1} {p}$$ by $$frac {1} {p} = 0. overline {d_1d_2 … d_r}$$ where $$1 leq r lt p$$ Y $$d_i in {0,1,2, …, 9 }$$.

Leave $$d = 0.d_1d_2 … d_r cdot10 ^ r ;$$ (For example, yes $$p = 7$$, so $$frac {1} {7} = 0. overline {142857}$$, $$r = 6$$Y $$d = 142857$$).

Let's consider $$frac {1} {10 ^ m-1}$$ for some $$m en mathbb {N}$$:
$$frac {1} {10 ^ m-1} = frac {1} { underbrace {99 … 9} _ { text { m times}}} = 0. overline { underbrace {00 … 0} _ {m-1 \ text {times}} 1} = sum_ {i = 1} ^ infty frac {1} {10 ^ {i cdot m}}$$

So
begin {align} frac {1} {p} & = 0. overline {d_1d_2 … d_r} = d cdot left ( frac {1} {10 ^ r} right) + d cdot left ( frac) {1} {10 ^ {2r}} right) + d cdot left ( frac {1} {10 ^ {3r}} right) + cdots \ & = d cdot left ( frac) {1} {10 ^ r} + frac {1} {10 ^ {2r}} + frac {1} {10 ^ {3r}} + cdots right) \ & = d cdot sum_ { i = 1} ^ infty frac {1} {10 ^ {i cdot r}} \ & = frac {d} {10 ^ r-1} end {align}

Reorganizing the terms, we have $$, 10 ^ r-1 = p cdot d , Longrightarrow , p | 10 ^ r-1$$.

Does this prove that the set of prime numbers is contained in the set? $${n in mathbb {N}: n | 10 ^ k-1 } cup {2,5 }$$?

Clearly, $$2$$ Y $$5$$ missing in $${n in mathbb {N}: n | 10 ^ k-1 }$$. Are there other cousins ​​not included in this set?

## Domain in the form of a star in \$ mathbb {C} P ^ 2 \$

To consider $$( mathbb {C} P ^ 2, omega_ {FS})$$ where $$omega_ {FS}$$ It's Fubini's standard study form. Leave $$L$$ denotes a sphere in $$mathbb {C} P ^ 2$$ in the class $$mathbb {C} P ^ 1$$. Also let $$int_ {L} omega_ {FS} = pi$$.

Then the map

begin {align} i: (B (1), omega_0) & a ( mathbb {C} P ^ 2 setminus L, omega_ {FS}) \ (z, w) and mapsto [sqrt{1 – |z|^2 -|w|^2} : z : w] end {align}

It is a simplectomorphism. Where $$(B (1), omega_0)$$ denotes the standard ball of radius 1 in $$mathbb {R} ^ 4$$ with the restriction of the symplectic standard form in $$mathbb {R} ^ 4$$.

Leave $$B _ { mathbb {C} P ^ 2} (x_0, r)$$ denotes a metric ball in $$mathbb {C} P ^ 2$$ (for the standard Kahler metric) of radius r and centered on $$x_0 in L$$.

So the paper I'm reading says that $$i ^ {- 1} ( mathbb {C} P ^ 2 setminus (L cup B _ { mathbb {C} P ^ 2} (x_0, r))$$ it's a star-shaped neighborhood in $$B (1)$$.

Could someone help me try the previous statement?

## Actual analysis: each interval \$ I subset mathbb {r} \$ is connected. [Proof clarification]

I struggled to understand a part of the next test.

Topological test that each Interval \$ I subset mathbb {R} \$ is connected

Definition: A topological space is connected if, and only if, it can not be divided into two
subsets that are not empty, open and separate, or, similarly, if the empty set and the complete set are the
Only subsets that are open and closed at the same time.

Test. Suppose $$I = A cup B$$ Y $$A cap B = emptyset$$, $$A$$ Y $$B$$ They are not empty and they are open in the subspace-topology of $$I subset mathbb {R}$$. Choose $$a in A$$ Y $$b in B$$ and suppose $$a . Leave $$s: = mathrm {inf} {x in B ~ | ~ a . Then in each neighborhood of $$s$$ there are points of $$B$$ (due to the definition of infimum), but also of $$A$$then if not $$s = a$$, so $$a and the open intervall $$(a, s)$$ It is entirely in $$A$$. Y so $$s$$ it can not be an internal point of $$A$$ neither $$B$$, but this is a contradiction to the property that both $$A$$ Y $$B$$ be open and $$s in A cup B$$.

My problems
I'm struggling with bold parts. Probably because I am more familiar with metric spaces than with topological spaces. I know that $$forall epsilon> 0, exist and in B: s + epsilon> y$$ And this implies that each ball open. $$beta (s, epsilon)$$ contains a point of $$B$$. Furthermore, if $$a = s in A$$ clearly $$beta (s, epsilon)$$ contains a point of $$A$$otherwise, if $$a so $$(a, s) subset A$$ which also means that $$beta (s, epsilon)$$ contain a point $$A$$. If this argument is correct, then my problem is the translation of these ideas of open balls and distances to the topological framework. I have the intuition that he is trying to show that $$s$$ It is a limit point of $$B$$ in $$I$$. As $$A$$ Y $$B$$ coordinated $$I$$, $$B ^ c = A$$. So, $$s$$ can never be (an inner point) either in $$A$$ or in $$B$$.

I am trying to provide a test in the context of a topological space. Can someone clarify my doubts?

## Algebraic geometry – flat family of general points in mathbb {P} ^ {3} \$

Let's fix a plane $$mathbb {P} ^ {2} subset mathbb {P} ^ {3}$$ Y $$gamma in mathbb {P} ^ {2}$$

Leave, $$t in mathbb K$$ ( where, $$mathbb K$$ It is the underlying closed field of the characteristic. $$0$$), then, how can we choose a flat family of general points? $$delta_t in mathbb {P} ^ {3}$$ and a family of aircraft $$H_t$$ such that the following is fulfilled:

$$(1)$$ $$delta_t in H_t$$ For any $$t$$,

$$(2)$$ $$delta_t notin mathbb {P} ^ {2}$$ For any $$t neq 0$$

$$(3)$$ $$H_0 = mathbb {P} ^ {2}$$ Y $$delta_0 = gamma in mathbb {P} ^ {2}$$ ?

I know by any point in $$mathbb {P} ^ {3}$$, this content in at least $$3$$ The hyperplanes, but how to build a flat family of such points?

Can anyone give any reference? Any help from anyone is welcome.