Recent Questions - Mathematics Stack Exchange |
- An infinite series with the slow rate of convergence?
- Function satisfying (sort of) differential inequality: $|f'(x) x| \le Cf(x)$
- Is an affine transformation of a multivariate Student-t distribution also a Student-t distribution?
- What is the sum of natural numbers that have $5$ digits or less, and all of the digits are distinct?
- Is the product P = B^T * B positive-definite if B is a square matrix?
- $(G,K)$ is a Gelfand pair iff $C_c(G||K)$ is commutative
- Period of an element in direct product of finite semigroups
- Binary inner-products with XOR operation
- Implicit Euler looks different from Runge-Kutta implicit Euler
- Degree of the maximal $p$-extension inside the ring class field of conductor $p^{\alpha+1}$
- Approximate solutions for a non linear system of equations
- Another improper integral from a Gaussian and a nested exponential.
- Chebyshev polynomial and some perfect squares
- Let $\langle t_1,t_2\rangle = \{f(t_1,t_2) \in \Bbb Q[t_1,t_2] \ \colon f \text{ has constant term } 0 \}$. Show that this is not an PID.
- On the integers $\mathbb{Z}$ acting on a group (not necessarily finitely generated)
- Find isometry on the cylindrical surface
- Find the velocity of the falling object when it touches the water
- Finitely generated torsion-free module which is not torsionless
- What is the negation of $\exists \lim\limits_{x \to a} f(x) \neq f(a)$, in precise FOL terms?
- Using related rates, why can we ignore dimensions and consider rates of change, when in seemingly identical situations, we must consider both?
- Show that, for any sets $A$, $B$ and $C$ , $|A \times (B \oplus C)| = |(A \times B) \oplus (A \times C)|$
- $n$ people round a circular table radius $1$m. Total distance walked by $\frac{n}{2}$ of them to the person $\frac{n}{2}$ spaces away is $\geq n\pi/2$
- How do I determine if variation in data related to decision making process is due to natural variation or noise
- What is the correct method for calculating this trig equation?
- Proof of the Leray-Hirsch theorem in Voisin's book.
- Behavior of solutions of $y'+p(x)y=p(x)\sin(\frac{1}{x+1})$
- Induction on automaton
- How can I show that any two versions of $P(A|X)$ differ in a set of probability zero?
- The value of the zeros of $P'_n(x)$ at $P_n(x)$
- Prove or disprove : $(\mathbb R,\tau)$ is connected and Hausdorff?
| An infinite series with the slow rate of convergence? Posted: 03 Mar 2022 03:53 AM PST I would like to use varying step size in gradient descent. Therefore, I am searching for a series $$\sum_k^\infty \alpha_k = \infty~~\text{ and }~~~\sum_k^\infty \alpha_k^2 < \infty.$$ I use $$\frac{1}{\sqrt{k+1}},$$ but the convergence of this series to zero is very fast. I am wondering if there is better solution or no? | ||||||||||||
| Function satisfying (sort of) differential inequality: $|f'(x) x| \le Cf(x)$ Posted: 03 Mar 2022 03:43 AM PST During my research I stumbled upon some inequality which I tried to tackle using cutoff functions. Ideally, I would need a $C^1$ (at least) function $f:[\delta, 2\delta] \rightarrow [0,1]$, with $\delta>0$, such that $f(\delta)=1, f(2\delta)=0$ and, most importantly, $$ |f'(x) x| \le Cf(x) \, \, \forall x\in(\delta, 2\delta), C>0. $$ It would also be nice if the function was decreasing (since we could use $f'(x)\le 0$ and simplify the inequality) , but this is not required. Nevertheless, I somehow struggle to find an explicit function satisfying those conditions, mostly due to the fact that $|f(x)| \rightarrow 0$ as $x\rightarrow 2\delta$, implying $f'(x) \rightarrow 0$ as well for $x\rightarrow 2\delta$. All examples I had in mind seem to fail because (for them): $$ \lim\limits_{x \rightarrow 2\delta}\frac{|f'(x)x|}{|f(x)|} = \infty. $$ This led me to believe that maybe such functions cannot exist. Then again, I can't see why that would be true. The question is then to either
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| Is an affine transformation of a multivariate Student-t distribution also a Student-t distribution? Posted: 03 Mar 2022 03:42 AM PST I note from Wikipedia that an affine transformation of a Multivariate Normal Distribution is also a Multivariate Normal Distribution, as shown here https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Affine_transformation. Does this also hold for Multivariate Student-t Distributions too? If so, how could I demonstrate/prove this? I am generating a set of correlated random draws by multiplying a vector of independent draws from a univariate Student-t distributions (with zero mean and unit variance for a fixed number of degrees of freedom) by a transformation matrix. I would like to know if I can describe these correlated random draws as being drawn from a Multivariate Student-t Distribution. Is there a general principle for related distributions (Cauchy, LogNormal) that applies? Thanks | ||||||||||||
| What is the sum of natural numbers that have $5$ digits or less, and all of the digits are distinct? Posted: 03 Mar 2022 03:41 AM PST $1+2+3+\dots+7+8+9+10+12+13+\dots+96+97+98+102+103+104+\dots+985+986+987+1023+1024+1025+\dots+9874+9875+9876+10234+10235+10236+\dots+98763+98764+98765=$ The only thing I can do is to evaluate a (bad) upper bound by evaluating the sum of natural numbers from $1$ to $98765$, that is equal to $98765 \times (98765+1)/2=4,877,311,995$. So the desired answer is less than that. Any help would be appreciated. Thanks. | ||||||||||||
| Is the product P = B^T * B positive-definite if B is a square matrix? Posted: 03 Mar 2022 03:35 AM PST I am looking for a matrix parametrization that guarantees positive definiteness. The target matrix P shall be positive-definite. I thought about writing P = B^T * B, where B is some square matrix. Then, for any choice of B, is the result P positive-definite? I am not quite sure. Thanks! Best, JZ | ||||||||||||
| $(G,K)$ is a Gelfand pair iff $C_c(G||K)$ is commutative Posted: 03 Mar 2022 03:36 AM PST In my definition, a pair $(G, K)$, of a group $G$ and a compact subgroup $K$ of $G$, is said to be a Gelfand pair if the subalgebra $L^1(G||K)=\{f\in L^1(G): f^{\#}=f \}$ of $K$-bi invariant $L^1$ functions is commutative. Where $$f^{\#}(x)=\int_K\int_Kf(kxk')dk dk' $$ I tried to prove that $C_c(G||K)$ is commutative then $(G, K)$ is a Gelfand pair as follows: For $f,g\in L(G||K)$ there exist a sequences $(f_n)$ and $(g_n)$ in $C_c(G)$ converges to $f$ and $g$, respectively, in $L^1$ norm. Then, $$f*g(x)=(f*g)^{\#}(x)\\ =\left(\lim_\limits{n\to\infty} f_n*g_n\right)^\#(x)\\=\lim_\limits{n\to\infty} \left(f_n*g_n\right)^\#(x) $$ But from the last step can I take $(f_n*g_n)^\#(x)=f_n^\#*g_n^\#(x)$? I know that it can be done when $f_n,g_n$ are in $C_c(G||K)$. | ||||||||||||
| Period of an element in direct product of finite semigroups Posted: 03 Mar 2022 03:40 AM PST Let $S$ and $T$ be finite semigroups and let $(x,y)\in S\times T$. What is the period of $(x,y)$ ? I know that if $$\mathrm{index}(x)=\mathrm{index}(y)$$, then the period of $(x,y)$ is $$\mathrm{lcm(period}(x),\mathrm{period}(y))$$. What can we say in case $\mathrm{index}(x)\neq \mathrm{index}(y)$? | ||||||||||||
| Binary inner-products with XOR operation Posted: 03 Mar 2022 03:25 AM PST Consider the binary power set $\{\underline{K}\}$ which contains binary strings of length $N$. Example for $N=3$ we have $$\big\{\underline{K}\big\} = \big\{\{000\},\{100\},\{010\},\{001\},\{110\},\{101\},\{011\},\{111\} \big\}.$$ If we now consider two vector states defined by the linear combinations of the tensor product of these two-level basis vector states $$|\psi \rangle := \sum_{\{ \underline{K} \}}a_i|k_1\cdot \cdot \cdot k_N \rangle ~~~\text{and}~~~ |\phi \rangle := \sum_{\{ \underline{K} \}}b_j|(k_1\oplus 1) k_2\cdot \cdot \cdot k_N \rangle,$$ where $a_i$ and $b_j$ are real or complex coefficients and $\oplus$ is the XOR binary operation acting on the first entry of the second vector $| \phi \rangle$ (hence maps $|0\rangle \mapsto |1\rangle$ and $|1\rangle \mapsto |0\rangle$). Is there maybe some formula or pattern that gives the value of the inner-product $$\langle \psi | \phi \rangle$$ if we consider the general $N$-length binary power set. For the $N=3$ case we get $\langle \psi | \phi \rangle = a_1b_2 + a_2b_1 + a_3b_5 + a_4b_6 + a_5b_3 +a_6b_4 + a_7b_8 + a_8b_7$. Thanks for any assistance. | ||||||||||||
| Implicit Euler looks different from Runge-Kutta implicit Euler Posted: 03 Mar 2022 03:41 AM PST In theory Implicit (Backward) Euler should be a Runge-Kutta method with tableu given below, however I find that the standard Backward Euler formula and the Runge-Kutta one differ. $$ {\begin{array}{c|c}1&1\\\hline &1\\\end{array}} $$ Here are the two methods:
Runge Kutta formulasI have used the formulas in Geometric Numerical Integration by Hairer $$ \begin{align} k_i &= f\left(t_0 + c_i h, y_0 + h\sum_{i=1}^s a_{ij} k_j\right) \\ y_1 &= y_0 + h\sum_{i=1}^s b_i k_i \end{align} $$ and Implicit Euler has $c_1 = 1$, $a_{11} = 1$ and $b_1 = 1$. | ||||||||||||
| Degree of the maximal $p$-extension inside the ring class field of conductor $p^{\alpha+1}$ Posted: 03 Mar 2022 03:22 AM PST Fix a prime $p$. Let $K$ be an imaginary quadratic extension of $\mathbb{Q}$ and suppose that the class number of $K$ is prime to $p$. Let $H_{p^{\alpha+1}}$ be the ring class field of conductor $p^{\alpha+1}$, where $\alpha\ge 0$. Call $K_\alpha$ the maximal $p$-extension of $K$ contained in $H_{p^{\alpha+1}}$. It seems that $[K_\alpha:K]=p^\alpha$. Why? | ||||||||||||
| Approximate solutions for a non linear system of equations Posted: 03 Mar 2022 03:20 AM PST As a developer, what is the right approach to solve this system of equations for x, and y, approximately? x and y are real numbers. Thank you, | ||||||||||||
| Another improper integral from a Gaussian and a nested exponential. Posted: 03 Mar 2022 03:17 AM PST In relation to some models in quantitative finance, models based on a geometric Brownian Motion (gBM), I came across a following integral. \begin{equation} {\mathfrak I}^{(\mu\,\sigma)}_\tau(x) := \int\limits_{-\infty}^\infty e^{-\xi^2} \exp\left(- \frac{x}{2} e^{(-\frac{\sigma^2}{2} + \mu) \tau} \cdot e^{\sigma \sqrt{2 \tau} \xi}\right) d\xi \end{equation} where $\mu \in {\mathbb R}$, $\sigma >0 $ are the drift and the volatility of the gBM, $x>0$ is the price at inception and $\tau > 0$ is a time period. This is clearly a integral that converges very fast so I could just evaluate it numerically, but I was wondering whether one could obtain a series representation of that. One could for example expand the second exponential in a series and then integrate term by term-- which I did -- but then the expansion is always divergent-- or converges only asymptotically. Indeed we have: \begin{equation} {\mathfrak I}^{(\mu\,\sigma)}_\tau(x) = \sqrt{\pi} \cdot \sum\limits_{n=0}^\infty \underbrace{\frac{(-\frac{x}{2})^n}{n!} e^{\frac{n^2}{2} \sigma^2 \tau + n (-\frac{\sigma^2}{2} + \mu) \tau}}_{a_n} \end{equation} Indeed, by taking the $n$th root of the modulus of the $n$th coefficient and using the Stirling approximation we have $|a_n|^{1/n} \simeq (x/2) e^{\frac{n}{2} \sigma^2 \tau + (-\frac{\sigma^2}{2} + \mu) \tau}/ ((2\pi n)^{1/(2n)}\cdot n/e) \underbrace{\rightarrow}_{n\rightarrow \infty} \infty$. Yet, on the other hand, using some typical values of drift and volatility-- less then ten percent annually -- and the time period of the order of hundreds truncating the series at the first ten terms gives a pretty good approximation to the integrals as the code below demonstrates. My question would be. Has anyone ever come across those kind of integrals and if yes how would you handle them. Thanks. Another thing would be to expand the first exponential into a series and then integrate term by term but then it is even worse because each term in the series becomes infinity so therefore one needs to take the regular part of those integrals and this is much harder than I thought. | ||||||||||||
| Chebyshev polynomial and some perfect squares Posted: 03 Mar 2022 03:25 AM PST Let $U_{n}(x)$ be the Chebyshev polynomial of the second kind (https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html). I am asking if the following quantity is still a perfect square: $$s= 4(12U_{n-1}(x)-U_{n-2}(x))^2(x^2-1)-4(24x-145)$$ Here $n$ is the degree and $x$ is positive integer variable. | ||||||||||||
| Posted: 03 Mar 2022 03:40 AM PST
To show that something this is not an PID I need to show that $\langle t_1,t_2\rangle$ is not generated by a single element in $\Bbb Q[t_1,t_2]$? So for any $p \in \Bbb Q[t_1,t_2]$ I would need to get $\langle p \rangle \ne \langle t_1,t_2\rangle$? The notation seems off here I don't think that $(t_1,t_2)$ is an element of $\Bbb Q[t_1,t_2]$ so how can they use that to generate the ideal? | ||||||||||||
| On the integers $\mathbb{Z}$ acting on a group (not necessarily finitely generated) Posted: 03 Mar 2022 03:19 AM PST Suppose we have $\mathbb{Z}$ acting on a group $K$ (not necessarily finitely generated) and let $\phi : \Bbb Z \to {\rm Sym}(K)$ be the homomorphism define this action. Suppose that $N$ is a normal subgroup of $K$ with finite index $m$. Is it true that there exists an $n \in \mathbb{Z} \setminus \{0\}$ such that for every $k$ in $N$, $\phi (n) (k) = \phi^n (k) \in N$? I think if we take $n$ to be $m$, it would work, as there are only $m$ distinct cosets, but I couldn't find a way to prove it or find a counterexample. | ||||||||||||
| Find isometry on the cylindrical surface Posted: 03 Mar 2022 03:20 AM PST Assuming that σ is an equidistant transform on the cylindrical surface and π is the covering mapping from Euclid plane to the cylindrical surface which transform (x,y) to (Rcosx,Rsinx,y)，then how to prove that there exists an equidistant transform φ on the Euclid plane such that π ◦ φ = σ ◦ π ? I find it difficult to correctly expand the σ's domain of definition [0,2π)×R to R^2 to get φ and verify that φ is an equidistant transform on the Euclid plane. | ||||||||||||
| Find the velocity of the falling object when it touches the water Posted: 03 Mar 2022 03:25 AM PST The question: An object falls from a hovering surf-lifesaving helicopter over a port at $500$ m above sea level. Find the velocity of the object when it hits the water when the acceleration of the object is $0.2v^2 − g$. (Note that $g \not= 9.8$, because they did not state anywhere.) Since they gave us $a$ in terms of $v$, and a value for $x$, I used:
How do I find the value of $c$, given only one info and where do I go next, after that? | ||||||||||||
| Finitely generated torsion-free module which is not torsionless Posted: 03 Mar 2022 03:27 AM PST
If $R$ is a domain then the answer is no. More generally, if $Q$ is the total ring of fractions of $R$ and $M\otimes Q$ is a free $Q$-module, then the answer is no. But I can't show $M$ is torsionless in the general case nor have I been able to think of an example where it isn't. It may be worth noting that in this case, $M$ is torsionless if and only if it is a submodule of a finite rank free $R$-module. | ||||||||||||
| What is the negation of $\exists \lim\limits_{x \to a} f(x) \neq f(a)$, in precise FOL terms? Posted: 03 Mar 2022 03:18 AM PST Here is a quote from Spivak's Calculus:
It appeared in a problem, not in the main text. It seems to be a definition. I believe it can also be written as a biconditional: $$\exists \lim\limits_{x \to a} f(x) \neq f(a) \iff\ f \text{ has a removable discontinuity at } a\tag{1}$$ In another problem, we are asked to prove that a particular function never has a removable discontinuity. My proof assumed the antecedent $\exists \lim\limits_{x \to a} f(x) \neq f(a)$, and used a proof by cases to show that a contradiction is obtained in all cases. Thus, I concluded that the negation of the antecedent is true, and hence that the negation of the consequent is true. In other words, I concluded that: $$\lnot (\exists \lim\limits_{x \to a} f(x) \neq f(a))\tag{2}$$ Therefore $$\lnot (f \text{ has a removable discontinuity at } a)\tag{3}$$ Negating the entire antecedent is good enough for me to conclude that "for all $a$ f never has a removable discontinuity at $a$". However, my question is about what exactly the negated antecedent is in terms of first-order logic. What is $(2)$, in very precise terms, ie in a way that is correct in a formal first-order logic sense? The antecedent of $(1)$ itself doesn't seem precise. Should it be $$(\exists \lim\limits_{x \to a} f(x)) \land (\lim\limits_{x \to a} f(x) \neq f(a))$$ But then when we negate this we get $$(\nexists \lim\limits_{x \to a} f(x)) \lor (\lim\limits_{x \to a} f(x) = f(a))$$ Seems like the two conjuncts are linked in a way that makes the disjunction feel funny. | ||||||||||||
| Posted: 03 Mar 2022 03:47 AM PST So I was preparing a lesson on related rates for the calc 1 class I am a TA for and I realized that the two problems below in the photo are basically identical: Given a right triangle, x, x', y, y' are known, Find z' (or s').
Problem #1 and $4 are solved identically, but in problem #4, we can use a "cheat" and just consider a right triangle with legs x'=25 and y'=60 and hypotenuse=s'. Solving for $s'... \\s'=\sqrt{x'^2+y'^2}=\sqrt{25^2+60^2}=\sqrt{4225}=65 $ This implies the distance between the cars is changing at constant rate, independent of the location of the cars. But this method does not work for the seemingly identical problem #1. I am conflicted... why is this "cheat" only viable for some instances of these problems and not all? I checked back in my own notes from calc 1 and this "cheat" could be used on other problems too, so it's not something unique with the numbers in #4. | ||||||||||||
| Posted: 03 Mar 2022 03:52 AM PST I want to solve this question by defining a bijection between both sides of the equation, but I do not know how to define that bijection. Let $A \oplus B = \{(0,a): a \in A \} \cup \{(1,b): b \in B\}$; this is the disjoint union of $A$ and $B$ Proposition 1. Suppose that $A$ and $B$ are two sets such that $|A| = m$ and $|B| = n$ for some $m, n \in \omega$. Then
Proposition 2. For any sets $A$, $B$, and $C$,
So, for the multiplicative parts of $(1)$ and $(2)$, we have natural isomorphisms mapping $(a,b)$ to $(b,a)$ and mapping $((a,(b,c))$ to $((a,b),c))$. But for $(3)$ how to define that bijection? My initial thought is like $F(a,(b,c))=((a,b),(a,c))$ | ||||||||||||
| Posted: 03 Mar 2022 03:46 AM PST Let $n$ be an even integer $\geq 4.\ n$ people sit round a circular table with radius $1$ metre. Each person is labelled $x_0,x_1,\ldots,x_{n-1},\ $ in order counter-clockwise round the table. A number $j$ is chosen from $\{0,\ldots,n-1\}.$ They play a game where for each $k\in\{\ j\mod n,\ (j+1)\mod n,\ (j+2)\mod n,\ \ldots,\ (j+\frac{n}{2}-1)\mod n\ \},\ x_k$ walks round the table towards $x_{\left(k+\frac{n}{2}\right)\mod n}.\ $ The total distance walked depends on which $j\in\{0,\ldots,n-1\}\ $ was chosen.
I think this should be easy for $n=4$ by splitting into cases. However I don't see how to use induction to then solve for $n\geq 6.$ I have some ideas, like maybe for $n=4$ there are always at least two such $j'$s, but I'm not sure how to prove this. This also seems like a weird problem to me: like it should be trivial somehow. | ||||||||||||
| Posted: 03 Mar 2022 03:40 AM PST I am analysing some data related to fines issued to people and the contesting of these fines. I looked in to this after reading the book by Kahneman, Sibony and Sunstein called 'Noise', which to give a very poor summary, is about how decision making processes can be affected by noise from factors that shouldn't have a bearing on final decision. This is a fictitious scenario but based on a real world scenario. Imagine library members who receive fines for overdue books can contest the fine if they think they had a valid reason. The contest cases comes through to a team of agents adjudicating these. They can decide to accept the lenders contest or reject it. They should be making these decisions based on a clear set of guidelines i.e. has the contester provided sufficient evidence etc. I have noticed that there is quite a large amount of variability in the percentages of accepted vs rejected contests per agent, which would indicate subjectivity is affecting how these are adjudicated i.e. some agents are very lenient i.e. accept more cases than perhaps they should and others are very stingy. My expectation would be that if the data set is large enough and the agents have clear set of guidelines to follow that determines how they should decide whether a contest is valid or not, then the acceptance rates should be fairly similar among all the agents. We can assume that the contests are assigned to agents at random i.e. no reason to think that certain agents are getting assigned contests that are more likely to be accepted. One other thing to note is that the lender or contester can choose from a list of grounds as to why they are contesting the fine. So the acceptance rates for contests with ground - 'death in the family' might be higher than other grounds like 'Forgot' for example. So I plotted the acceptance rates for all the users by ground as I thought if I could see that if an agent had a high acceptance for all ground codes then it is because they are excessively lenient and likewise if they have a low acceptance rate for all grounds then they are excessively stingy, rather than if I just combined all contests for all grounds. I created a plot (shown below) of acceptance rates for all users for each of three grounds. The plot clearly shows that there is one user who seems to be very lenient with acceptance rates in the 90s for each of the three grounds and one user at the bottom who is very stingy with the lowest or second lowest acceptance rate for each of the grounds. But its a bit hard to tell if this is true for most of the users. Perhaps a different plot might be more revealing but I'm not sure which one I should use. That is my first question. What is clear is that there is a large amount of variation in terms of acceptance rates among agents for all grounds e.g. ground one being between 30% and close to 100%. Note, for each of the grounds I only included agents that had decided on 100 or more contests (i.e. accepted + rejected contests >= 100). So if an agent has an acceptance rate of 33% for a ground its not that they only decided on 3 cases and rejected 2 of them.
I'm fairly sure, that I can assume that I should be seeing a much more closely clustered acceptance rate among agents for the different grounds, I'm not sure what this should be but I don't think it should vary so wildly. But if not please tell me if this is not a good assumption. My main question is what statistical tools or calculations can I use to show that there is too much variation in acceptance rates. Is 100 cases enough of a sample set for each agent/ground combo? Much appreciated. | ||||||||||||
| What is the correct method for calculating this trig equation? Posted: 03 Mar 2022 03:39 AM PST For the question "$f(x)=0.7\cos(2.7x+9.2)+4.4$ find a value of $x$ such that $f(x)=4.5$" I enter in the value of $f(x)$ in as $4.5=0.7\cos(2.7x+9.2)+4.4$ (solving for x) and get the answers:
However, the question wants me to provide the answer as a decimal approximation and I don't see how I am able to do this with these answers. Can someone help me with correctly formatting the equation I need to write in? How do I use the $f(x)$ value of 4.5 in this? | ||||||||||||
| Proof of the Leray-Hirsch theorem in Voisin's book. Posted: 03 Mar 2022 03:33 AM PST I have a problem understanding the proof of the Leray-Hirsch-Theorem from Voisin's Hodge Theory and Complex Algebraic Geometry (Thm 7.33, proof in 8.1.3). Let $\phi: Y \to X$ be a fibration of topological spaces (i.e. in a neighbourhood $U_x$ of $x \in X$, $\phi$ looks like $Y_x \times U_x \to U_x$) where $X$ is contractible. Suppose the cohomology $H^*(Y, \mathbb Z)$ is torsion-free, and we have homogeneous cohomology classes $$\alpha_1, \dotsc, \alpha_N \in H^*(Y, \mathbb Z)$$ such that the graded free subgroup $A^* \subset H^*(Y, \mathbb Z)$ generated by the $\alpha_i$ is isomorphic via pull-back to the cohomology $H^*(Y_x, \mathbb Z)$ of every fiber $Y_x = \phi^{-1}(x), x \in X$. Now choose a flasque resolution $0 \to \mathbb Z_Y \to F^*$ of the constant sheaf $\mathbb Z_Y$ on $Y$. Since $$H^k(Y, \mathbb Z) = H^k(\Gamma(Y, F^*))$$ one can choose closed elements $\beta_i \in \Gamma(Y, F^*)$ representing the $\alpha_i$. As the cohomology of $Y$ is torsion-free, the $\beta_i$ generate a free subgroup of $\Gamma(Y, F^*)$, isomorphic to $A^*$. Now Voisin claims that the graded subsheaf of $\phi_* F^*$, generated by the $\beta_i$, is the constant sheaf $A^*_X$. I'm not sure why this is the case. If we can show that the $\beta_i$ generate a local system, then this would be true because $X$ is contractible, so any local system is trivial. On the other hand, here is an example, where global sections of a flasque sheaf do not give a local system:
Intuitively this example can never happen in our situation, since the $\alpha_i$ generate the cohomology of each fiber, so the $\beta_i$ should not vanish (in a neighbourhood of a fiber?). But I don't know how to make this idea rigorous. How can we relate the sections $\beta_i$ with the pull-back morphisms $H^k(Y, \mathbb Z) \to H^k(Y_x, \mathbb Z), x \in X$? | ||||||||||||
| Behavior of solutions of $y'+p(x)y=p(x)\sin(\frac{1}{x+1})$ Posted: 03 Mar 2022 03:35 AM PST Consider the linear d.e $$(E) \ \ \ \ \ y'+p(x)y=p(x)\sin(\frac{1}{x+1}), x\geq 0$$ which $p(x)=(x+1)(x-1)(x-2), x\geq 0$. I want to study the behavior of solutions of $(E)$. Specifically:
About question 1: I showed that $p(x)$ is finally lower bounded by a positive number (it's easy to show it) and then I calculate $$\lim_{x\to+\infty} p(x)\sin(\frac{1}{x+1}) $$ which is (unfortunately) equals to $+\infty$ and not $0$ as I hoped, before this calculation. About question 2: I have no Idea how exactly can I find (or not) a solution with infinite countable roots. Any hint would be helpful, please. I'm stuck | ||||||||||||
| Posted: 03 Mar 2022 03:29 AM PST Let an automaton defined by the following transition table:
I have this finite automaton, and it recognizes the languages with only an odd number of $1s$ in them. How can we prove this language by induction on the length of its words? $\rightarrow$: Start state $\leftarrow$: Final state | ||||||||||||
| How can I show that any two versions of $P(A|X)$ differ in a set of probability zero? Posted: 03 Mar 2022 03:47 AM PST Given these two definitions(Probility - Breiman): Definition 4.7. The conditional probality $P(A|X=x)$ is defined as any $\mathcal{B}_1 -$measurable function satisfying $$P(A,X\in B)=\int_B P(A|X=x) \hat{P}(dx)$$ all $B\in\mathcal{B}_1$. Definition 4.8. The conditional probality of $A$ given $X(\omega)$ is defined as any random varible $\Omega$, measurable $\mathcal{F}(X)$, and satisfying$$P(A,X\in B)=\int_{\{X\in B\} } P(A|X) P(dx)$$all $B\in\mathcal{B}_1$ How can I show that any two versions of $P(A|X)$ differ in a set of probability zero? | ||||||||||||
| The value of the zeros of $P'_n(x)$ at $P_n(x)$ Posted: 03 Mar 2022 03:16 AM PST I recently come across a problem with respect to Legendre polynomial as follow. For any $n \in \mathbb{N}$, $P_n(x) := \frac{1}{2^n n!}\frac{{\rm d}^n (x^2-1)^n}{ {\rm d} x^n} $ is the classical Legendre polynomial of degree $ n $, orthogonal on $ [-1,1] $ with respect to Lebesgue measure. $\alpha_1,\cdots,\alpha_{n-1}$ are $n-1$ distinct roots of the polynomial $P'_n(x)$, where $P'_n(x)$ is the derivate function of $P_n(x)$. I need to estimate the quantity $\left| \prod_{k=1}^{n-1} P_n(\alpha_k) \right|$, but I don't know how to estimate it. I guess that it may be exponentially decreasing as $n$ increases. The graphs of these polynomials (up to $n = 5$) are shown below (the image is from Wikipedia): According to the graph, it is easy to see that when $n \geq 2$, $\left| \prod_{k=1}^{n-1} P_n(\alpha_k) \right| \leq \left| \prod_{k=1}^{n-1} \frac{1}{2} \right| = \frac{1}{2^{n-1}}$, but I don't know how to prove it. | ||||||||||||
| Prove or disprove : $(\mathbb R,\tau)$ is connected and Hausdorff? Posted: 03 Mar 2022 03:28 AM PST Consider the topology $\tau = \{ G \subseteq \mathbb R: G^c\text{ is compact in }(\mathbb R, \tau_u) \} \cup \{\mathbb R, \varnothing\} \text{ on } \mathbb R$ , where $\tau_u$ is the usual topology on $\mathbb R$. Decide whether $(\mathbb R , \tau)$ is
My attempt for connectedness is: If $G_1 , G_2 \in \tau$ such that $G_1 \cap G_2 = \varnothing$ , then $$G_1^c \cup G_2^c = \mathbb R,$$ so $\mathbb R$ is compact (in the usual topology) because the union of two compact sets is compact, which is a contradiction. So $(\mathbb R, \tau)$ is not Hausdorff. Please tell me about the connectedness. Thank you |
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