Recent Questions - Mathematics Stack Exchange |
- Finding Minimum value of following integral
- How to get the rational equation given this graph
- Will there be a proper subset B(x,d) of B(x,D) in a metric space?
- Learning Cohomology theory in homological algebra for cohomology theory in algebraic topology
- Average Length of Longest Segment: What am I Doing Wrong?
- closed subscheme of $\mathbb{P}_Z^1$ which is affine over $Z$
- Is the following a Bijective Function?
- How to do this limit by expressing it into definite integral?
- Evaluate double integration over positive quadrant $x^2+y^2=1$
- Simplifying Summation Sigma when there is a log function
- Besicovitch via Baire: questions
- A standard graded algebra with krull dimension zer
- Hypothesis testing - When to subtract one during type I and type II testing?
- Finding the area of my lake
- logarithm with trigonometric problem
- Is it possible to identify a unique square within a grid given its vertices?
- What is the correct answer to this infamously ambiguous arithemetic problem 6÷2(1+2)?
- Accessible explanation of the link between the distribution of primes and the Riemann zeta function.
- Proof of the upper bound
- Proving a formula for $\sum_{j=1}^n\frac{x_j^k}{f'(x_j)}$ for $f$ an $n$-th degree polynomial with $n$ distinct real roots $x_j$
- Why is the graph of the reciprocal function $f(x)=\frac{1}{x}$ not a one-to-one function?
- Intuition behind Group Action
- Signature/index of a finite branched covering space
- What are the equivalence relation in the set {a, b, c, d} that define the pairs (a, b) and (c,d)? [closed]
- Struggling in solving a logic theorem with quantifiers
- Probability of going to school $A_1$ when there are 9 more other possibilities
- Combinatorics proof counting
- Proving $\sum \limits _{n=1}^{\infty}\frac{\sin (n)}{n}$ convergent
- An easy way to count sum of squares of first x prime numbers?
- Differentiation question find the normal to the curve
| Finding Minimum value of following integral Posted: 23 Mar 2021 09:06 PM PDT Let f:[0,1]- R be differentiable function with continuous derivative f'(x) on [0,1] and f(0)= 0 and f(1)=1 then minimum value of definite integration of (1+x^2)^0.5 {f'(X)}^2 from 0 to 1.  |
| How to get the rational equation given this graph Posted: 23 Mar 2021 09:05 PM PDT So basically I was looking for the equation of the graph in the picture and I ended up with (x^2+2)(x-2)/(x-1)(x-2)(x+2) the problem is that the horizontal asymptote of the given graph I believed is 0.5 but my equation is 1. I tried to change the coefficient of the terms but the hole of the graph changes.  |
| Will there be a proper subset B(x,d) of B(x,D) in a metric space? Posted: 23 Mar 2021 09:02 PM PDT I am working on this: If $(X,\rho)$ is a non-discrete metric space, $x\in X$ is not an isolated point. Assume $U$ is an open set which contains $x$. It's obvious that $\exists B(x,D) \subset U.$ I suppose that there must be a proper subset $B(x,d) \subsetneqq B(x,D)$ Here is my proof: First of all, U can't be finite. If so, assume $U=\{x,x_1,x_2,...,x_n\}$, then take $0<\varepsilon<min \{\rho(x,x_1),...,\rho(x,x_n)\}$. Then take $B(x,\varepsilon)$. By defination of $\varepsilon$, $B(x,\varepsilon)=\{x\}$. However, $x$ is not an isolated point. So U is not finite. Then,take $y\ne x \in B(x,D)$, and $\frac{\rho(x,y)}{2}$ as d, then $B(x,d) \subsetneqq B(x,D)$. My tutor said there is something wrong with my proof. But I can't see where's the problem. If there really something wrong? If so ,where is the mistake?  |
| Learning Cohomology theory in homological algebra for cohomology theory in algebraic topology Posted: 23 Mar 2021 09:02 PM PDT I wonder if it's helpful to learn cohomology theory in homological algebra for learning cohomology theory in algebraic topology. I found it's helpful to learn homology theory in algebra for algebraic topology.  |
| Average Length of Longest Segment: What am I Doing Wrong? Posted: 23 Mar 2021 08:55 PM PDT I'm following this problem Average length of the longest segment, but I have a mistake in how I compute part of it. The problem is that a rope of 1m is divided into three pieces by two random points. Find the average length of the largest segment. They start by noting (their text copied) Here we have two independent random variables $X,Y$, both uniform on $[0,1]$. Let $A=\min (X,Y), B=\max (X,Y)$ and $C=\max (A, 1-B, B-A)$. First we want to find the probability density function $f_C(a)$ of $C$. Let $F_C(a)$ be the cumulative distribution function. Then $$ F_C(a) = P(C\le a)=P(A\le a, 1-B\le a, B-A\le a).$$ By rewriting this probability as area in the unit square, I get $$F_C(a)=\left\{\begin{array}{ll} (3a-1)^2 & \frac{1}{3}\le a\le \frac{1}{2}\\ 1-3(1-a)^2 & \frac{1}{2}\le a\le 1\end{array}\right.$$ (my text) However, I can't follow how they get those probabilities. I noted that the inequalities can be written as \begin{align} A&\leq a\\ B&\geq 1-a\\ B&\leq A+a \end{align} and we also have that $A\in [0,1],B\in [0,1]$. For $\frac{1}{2}\leq a\leq 1$, I 'think' that the first two inequalities give us a square of area $a^2$, and the third forces us to remove a triangle with two sides of length $1-a$ and thus area $\frac{(1-a)^2}{2}$. This is shown below for $a=2/3$ in the shaded area with the x-axis being $A$ and the y-axis being $B$ using wolfram alpha. However, this doesn't match what they did or anyone else's solution: what am I doing wrong?  |
| closed subscheme of $\mathbb{P}_Z^1$ which is affine over $Z$ Posted: 23 Mar 2021 08:54 PM PDT Given an algebraically closed field $k$ and a scheme $Z$ over $k$. Consider the projective scheme $X=\mathbb{P}^1_Z=\mathbb{P}_k^1\times_kZ$ and $Y\subset X$ a closed subscheme. Assume that $Y$ does not contain any closed fibre of the projection $p:X\rightarrow Z$, try to show that $\pi:=p|_Y:Y\rightarrow Z$ is an affine morphism, i.e. one can find an affine covering $\{U_i\}$ of $Z$ such that each $\pi^{-1}(U_i)$ is affine. My attempt: In the extremely special case $Z=\text{Spec}(k)$, one has $X=\mathbb{P}_k^1$ and $X_z\cong\mathbb{P}_k^1$. Here any closed subscheme $Y$ of $X$ satisfies the condition that $X_z\not\subseteq Y$. So we want the canonical map $Y\hookrightarrow\mathbb{P}_k^1$ to be affine, which is clearly true. Now let $Z=\text{Spec}(A)$ over $k$, we also have $X_z\cong\mathbb{P}_k^1$ and now $X=\mathbb{P}_A^1$. We know that any closed subscheme of $\mathbb{P}_A^1=\text{Proj}(A[x_0,x_1])$ is determined by a homogeneous ideal $I\subset A[x_0,x_1]$, i.e. $$Y=V_+(I)=\text{Proj}(A[x_0,x_1]/(I))$$ Then I want to define a scheme $Y_k$ such that $Y=Y_k\times_kZ$. If I can do it, them I am done. However, I have no ideal how to apply the setting $X_z\not\subseteq Y$ here.  |
| Is the following a Bijective Function? Posted: 23 Mar 2021 09:06 PM PDT In a bijective function is it necessary that all the elements of the domain correspond to a value in the range? Like for example can the following be a bijective function - If not then what type of function is it?  |
| How to do this limit by expressing it into definite integral? Posted: 23 Mar 2021 08:50 PM PDT I have been asked to find the limit of the following series by expressing it as definite integral: If na=1 always and n tends to infinity, find the limiting value of $$\prod_{r=1}^{n}[1+(ra)^2]^{1÷r}$$. I took log on both sides and set up my integral as $$\int_{0}^{1}log(1+x^2)dx$$ But my answer came out wrong. The answer given is e^((pi^2/24)). Thanks in advance  |
| Evaluate double integration over positive quadrant $x^2+y^2=1$ Posted: 23 Mar 2021 08:55 PM PDT Evaluate $\displaystyle\int \int \sqrt\frac{1-x^2-y^2}{1+x^2+y^2} \,dx \,dy$ over the positive quadrant $x^2+y^2=1$  |
| Simplifying Summation Sigma when there is a log function Posted: 23 Mar 2021 08:55 PM PDT Can i write use exponetial to simplify $ \sum_k \log S_p^k $ such that taking the exponential it becomes $\sum_k S_p^k$  |
| Besicovitch via Baire: questions Posted: 23 Mar 2021 08:42 PM PDT I have some problems understanding the proof in the LEMMA 2.4 in TW Körner's paper Besicovitch via Baire (2003) (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1090.5109&rep=rep1&type=pdf).
In the proof, he constructed a translated set from the set $P$ and claims that $P'\in \mathcal{P}$. However, it is not true if a line segment $l$ through $(x,v)$ which joins $(x_0,0)$ and $(1,1)$, where $x<0$. Right translation makes $l$ not in $[-1,1] \times [0,1]$ anymore. Could anyone who had read the paper offer some help?  |
| A standard graded algebra with krull dimension zer Posted: 23 Mar 2021 08:42 PM PDT Let $R$ be a graded standard $k$-algebra. Show that, the Krull dimension of $R$, $K \text{dim}(R)=0$ iff there is $N \in \mathbb{Z}^+$ such that $R_i=\{0\}$, where $R=\bigoplus_{i \geq 0}R_i$. $\mathbf{Definition}$: Let $R= \bigoplus_{i \geq 0}R_i$ be graded algebra. Then, $R$ is standard if it is generated by finite number of homogenous elements of $R$ of degree 1. Suppose $K \text{dim}(R)=0$, and let's assume for the contrary that for any positive integer $N$ there is $i>N$ such that $R_i \neq \{0\}$. Let $R_0=K$ as its from the decomposition of $R$, and for each $1 \leq n \in \mathbb{Z^+}$, let $R_{n}=R_{n_i}$, where $R_{n_i} \neq \{0\}$, i.e $n_i$ represent the index by assumption. Here I am getting stuck with having those elements in the right order in the decomposition. Also, I wanted to use the Nother Normalization lemma but I got confused with finding the finite number satisfies the related conditions. For the another direction, if Kdim$(R)=n>0$, then the Hilber function $$F(R;t)=\sum_{n \geq 0}H(R;n)t^n$$ is not a polynomial, where $H(R;n)=dim_K (R_n).$ This is what I have done so far. I am looking for any comments on that.  |
| Hypothesis testing - When to subtract one during type I and type II testing? Posted: 23 Mar 2021 09:04 PM PDT I'm currently studying statistics and I'm reviewing notes I took during a class last week. However, there's something I'm confused about. My professor subtracted 1 from the "successes" and I can't seem to figure out why. Essentially, the scenario was like this: Someone flipped a coin 16 times. $$H_0:p=0.5\text{ v.s. }H_a:p=0.55$$ Test 1: Reject $p = 0.50$ if 10 or more heads are observed out of 16. $\text{Pr}(X \ge 10 \text{ when } p= 0.5)$ where X is a binomial with $n = 16$ and $p = 0.50$. This is the part I don't understand. When using R, he did the following: Where did the 9 come from? Why did he subtract 1 from the initial 10 tosses? If I was in a scenario where I had: $$\text{Pr}(X \le 15 \text{ when } p = 0.50)$$ Would I still subtract 1? Would the R solution be ( Sorry if my formatting is bad! I've never posted math equations like this to these forums and I can't seem to figure it out)  |
| Posted: 23 Mar 2021 08:59 PM PDT Last year I took AP Calculus BC and learned some basic information on areas of irregular objects (Riemann sums and thats about it). I thought it was very interesting and this year we were assigned a math project on the topic of our choice. I have a lake behind my house that I decided to find the area of for the project. I went into google maps and clicked on some points, but I'm not sure what to do from here. I've been reading online and saw some interesting methods like Green's theorem and Planimetry, but I'm not sure how to utilize those in the context of my lake since I don't have an equation. I assumed I would have to plot these points on a graph to start off with, but I'm not sure where or how to do that using the information from my Google Map. Can anyone help me mathematically go about this? For reference, an image of the lake is attached below: (I know it shows the area, I just want to learn - and show - how to do that) Heres a link to the lake for anyone that wants to experiment with the 'points' on a google map. You just right click and press 'measure distance'. https://goo.gl/maps/M8WH6wPYGyxMP97z9  |
| logarithm with trigonometric problem Posted: 23 Mar 2021 08:54 PM PDT It is not a homework. I have been trying to solve $\log_{\sin x}2\log_{\cos x}2 + \log_{\sin x} 2 + \log_{\cos x}2 = 0$ but I cannot figure out how to do it. Please help with hints or solution.  |
| Is it possible to identify a unique square within a grid given its vertices? Posted: 23 Mar 2021 08:39 PM PDT
The y-axis goes from 3.0 to -3.0 with an interval of 0.3 as well. How can I identify a unique square within this grid just by its vertices? I tried to sum up the vertices which make up the square, but the sum of the vertices is not unique and can apply to a few other squares within the grid as well. EDIT FOR CLARIFICATION: Is there a way to use vertices like these to help me identify the square in the graph? I am designing a game where I want to be able to check if the box is currently occupied. I only have access to the vertices surrounding the box.  |
| What is the correct answer to this infamously ambiguous arithemetic problem 6÷2(1+2)? Posted: 23 Mar 2021 08:55 PM PDT I am going to make the case that $6÷2(1+2) = 1$. Many people often quote the PEMDAS rule and get the answer $9$ but look at this way: $$ 6 \div 2(1+2)= \frac{6}{2(1+2)}$$ Algebraically, we can treat the division operator(obelus or solidus) as a fraction. Everything to the left of the operator is in the numerator and everything to the right is in the denominator. So then you apply the usual PEMDAS rule to simplify the denominator: $$\frac{6}{2(1+2)}=\frac{6}{2(3)}=\frac{6}{2\ast 3}=\frac{6}{6}=1$$ A lot of engineering types just work left to right and end up with $9$ because they don't appreciate how division lacks 2 properties of multiplication: commutativity and distributivity. But if there is a good mathematical reason why I'm wrong, I'd like to hear it!  |
| Accessible explanation of the link between the distribution of primes and the Riemann zeta function. Posted: 23 Mar 2021 08:55 PM PDT I am looking for an accessible explanation of the link between the distribution of the primes and the Riemann zeta function. I have read the related questions and answers here (eg this), and also via internet search, and also key recommended books (popular maths, like Prime Obsession, texts like Apostol, etc) - and they either gloss over the key points, or are aimed at readers with university level maths. I would appreciate answers here, or pointers to explanations elsewhere. My students understand the Euler product formula, complex functions, calculus. Note - this is a repeat of a previous question (now deleted) as it was marked negative without explanation.  |
| Posted: 23 Mar 2021 08:57 PM PDT Suppose $B \in M_{n \times n} (\mathbb C)$ is Hermitian. Prove that $$\lambda_{\min}\leq b_{kk} \leq \lambda_{\max}, \text{ for } 1 \leq k \leq n.$$ With the equality in both sides for some $k$ only if $b_{kj}=b_{jk}=0,$ for all $k\neq j.$ Here $\lambda_{\min} \;\& \; \lambda_{\max}$ are the minimum and maximum eigenvalues of $B$ and $b_{kj}$ is the $(k,j)$- entry of $B$. I already cracked a proof for the lower bound using Rayleigh's theorem but got confused on how to show that it is bounded above by $\lambda_{\max}$. Do we use same approach as to the lower bound? Please someone give me a heads up. Thanks.  |
| Posted: 23 Mar 2021 09:01 PM PDT If polynomial $f(x)=a_0 x^n+a_1 x^{n-1}+\cdots+a_{n-1} x+a_n$ has $n$ different real roots $x_1,x_2,\cdots,x_n$, prove the following formula: $$ \sum_{j=1}^n \frac{x_j^k}{f^{\prime}(x_j)}=\left\{\begin{array}{ll} 0, & 0 \leq k \leq n-2 \\ a_0^{-1}, & k=n-1 \end{array}\right. $$  |
| Why is the graph of the reciprocal function $f(x)=\frac{1}{x}$ not a one-to-one function? Posted: 23 Mar 2021 08:50 PM PDT The horizontal line test seems to show that no horizontal line intersects the graph of $f$ at more than one point. However, the textbook tells me it's not one-to-one. It is related to the asymptotes, I believe, but I'm not sure how.  |
| Posted: 23 Mar 2021 08:48 PM PDT I am going over the group action and unfortunately I am not sure if I understand it very well. So for this reason, I have a few questions that possibly could help me to understand this concept better. A group action is defined as: Group $G$ acts on a set $X$ if there is a homomorphism $\sigma:G \rightarrow S_X$. I am aware group action has other definitions but for me this one is easier to understand. Now $S_X$ is the group of permutations of $X$. It is not crystal clear what is meant by this to me. Is $S_X$ the same thing as $S_{|X|}$? When we talk about $S_n$, I have a clear mental picture; $n$ is an integer and I think of all the permutations with $n$ elements. But in $S_X$, $X$ is not an integer but a set. So suppose $X=\{1,2,3\}$. Is $S_X$ the same thing as $S_3$ in this case? The other part of my question is about what the definition implies: it says "... if there is a homomorphism". Is it possible that homomorphism does not exist? Is the homomorphism unique? Or is it possible to have more than one homomorphism? Once we specify a group and a set, how do we find such homomorphism(s)? Is there a requirement for the size of the group and the set to make this definition work, for instance $|G| \geq |X|$? Please let me know if you would like me to clarify my question. Any help with making a better intuition is appreciated!  |
| Signature/index of a finite branched covering space Posted: 23 Mar 2021 08:59 PM PDT Suppose $\tilde{M}\rightarrow M$ is a finite ($n$-fold) covering of the smooth, oriented, compact 4-manifold $M$ with branched locus $\tilde S$. Then is $$\operatorname{Sign}(\tilde M)=n\operatorname{Sign}(M)-\operatorname{Sign}(\tilde S\cdot\tilde S)?$$ I know when $n=2$, Atiyah says $\operatorname{Sign}(\tilde M)=2\operatorname{Sign}(M)-\operatorname{Sign}(\tilde S\cdot\tilde S)$ in p.78 of here. I wonder if it's true for any $n$? In Atiyah's statement, he claimed this formula is from the 6.15 of this paper, which says that Let $X$ be a compact oriented manifold of dimension $4k$, and let $g$ be an orientation preserving involution with fixed point set $X^g$. Let $(X^g)^2$ denote the oriented cobordism class of the self-intersection of $X^g$ in $X$. Then $\operatorname{Sign}(g, X)=\operatorname{Sign}((X^g)^2)$. I cant' see how is the formula related to the 6.15, in particular, how is that related to the involution?  |
| Posted: 23 Mar 2021 08:45 PM PDT What are the equivalence relation in the set {a, b, c, d} that define the pairs (a, b) and (c,d)?  |
| Struggling in solving a logic theorem with quantifiers Posted: 23 Mar 2021 09:00 PM PDT Prove that $(∀ x • f x) ∨ (∃ y ❙ g y • (∀ x • x = y ⇒ f x)) ⇒ (∃ x ❙ h x ∨ f x • g x) ∨ (∀ z • f z)$ Proof: $$(∀ x • f x) ∨ (∃ y ❙ g y • (∀ x • x = y ⇒ f x))$$ $$= ⟨ \text{Renaming for } ∀ ⟩$$ $$(∀ z • f z) ∨ (∃ y ❙ g y • (∀ x • x = y ⇒ f x))$$ $$= ⟨ \text{Commutativity of } ∨ ⟩$$ $$(∃ y ❙ g y • (∀ x • x = y ⇒ f x)) ∨ (∀ z • f z)$$ Not sure how to continue this. Can you help? Be aware that $f x = f(x)$ and $h x ∨ f x = h(x) ∨ f(x)$  |
| Probability of going to school $A_1$ when there are 9 more other possibilities Posted: 23 Mar 2021 08:39 PM PDT I am curious to an extension of the previous question Probability of going to school A and want to extend the schools from 3 to 10. Now the story begins : I commute to teach at one of $A_i$ (i from 1 to 10) schools everyday by train. I live at the terminal station where every 5 x i mins (i.e.,5,10,15,20,... mins)there is a train leaving for school $Ai_i$ punctually, but the actual departure times are unknown to me. I will jump on the earliest train available at the station and teach at that school(if they arrive the same time, I have equal probability for each one of them). My question is: What is the probability I teach at school A$_1$? I can manually do it for cases of 2,3,4 schools following the procedure provided from the answer. Is there a formula I can use to directly calculate when the school options grow to 10? EDIT:
 |
| Posted: 23 Mar 2021 08:59 PM PDT The question is counting the number $b_{p,q}$ of binary strings with no consecutive $1$'s, with a $0$ at each end. With q 1's and p 0's.
Any help would be appreciated!  |
| Proving $\sum \limits _{n=1}^{\infty}\frac{\sin (n)}{n}$ convergent Posted: 23 Mar 2021 08:58 PM PDT I need to prove that $\sum \limits _{n=1}^{\infty}\frac{\sin (n)}{n}$ convergent only by Cauchy's test. Cauchy's test for sequences convergence : The $\sum \limits _{n=1}^{\infty}a_n$ convergent $\iff$ $\forall \varepsilon >0$ $ \exists$ $ n_0\in \mathbb{N}$ such that $\forall $ $n_0 \le n$ , $\forall$ $p\in \mathbb{N}$ the following holds $\mid S_{n+p}-S_n\mid = \mid a_{n+1}+\dots+a_{n+p}\mid<\varepsilon$ My Attempt: Let $\varepsilon>0$,$\ p\in\mathbb{N}$. We define $n_0=\frac{p}{\varepsilon}$, and since $-1\le \sin(x)\le 1$ therefore, for each $n\le n_0$: $$\mid \frac{\sin (n+1)}{n+1}+\frac{\sin (n+2)}{n+2}+...+\frac{\sin (n+p)}{n+p} \mid \ \le \ \mid \frac{\sin (n+1)}{n+1}\mid + \mid \frac{\sin (n+2)}{n+2}\mid+...+\mid \frac{\sin (n+p)}{n+p} \mid \le \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+p}\le \frac{1}{n}+\frac{1}{n}+...+\frac{1}{n}=\frac{p}{n}<\varepsilon$$ Is my proof correct? if not so how can I prove it only by Cauchy's test as I said earlier, without any further test - such as Dirichlet, Abel...  |
| An easy way to count sum of squares of first x prime numbers? Posted: 23 Mar 2021 08:52 PM PDT
 |
| Differentiation question find the normal to the curve Posted: 23 Mar 2021 09:02 PM PDT Hi I was struggling to this question, can anyone please help me :P The curve $C$ has equation $2x^2+y^2=18$. Determine the coordinates of the four points on $C$ at which the normal passes through the point $(1,0)$. I got the gradient of the normal to be: (I called the co-ordinates $a,b$) $y = (b/2a)x+b/2$, then I don't know how to continue, note i found this by differentiating, thanks  |
| You are subscribed to email updates from Recent Questions - Mathematics Stack Exchange. To stop receiving these emails, you may unsubscribe now. | Email delivery powered by Google |
| Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, United States | |
No comments:
Post a Comment