Recent Questions - Mathematics Stack Exchange |
- How can the alpha level be increased or decreased even though it is dependent on the reality that the null hypothesis is true?
- Let $f:[0,\pi]\to R$ be a continuous function and $\int_0^{\pi}f(x)\sin x.dx=\int_0^{\pi}\cos x.dx=0$. Find min no. solution of $f(x)=0$ in$(0,\pi)$
- Is there any reason to restrict the Hermitian adjoint to linear operators?
- How to find the first non-zero number after the decimal point?
- Largest positive domain of a linear ODE system
- Coin Flip Strategy Problem
- differentiating a Summation for the mean energy - Statistical Physics
- Markov Chain: How to Calculate the Probability that two states are equal?
- Stochastic process-tandem queue
- Mean squared error and Chebyshev's inequality
- If $k + 1$ is prime and $(k + 1) \mid (q - 1)$, then $\sigma(q^k)$ is divisible by $k + 1$, but not by $(k + 1)^2$ (unless $k+1=2$).
- Isomorphism of schemes restricts down to local isomorphism
- Determine if they are the same plane.
- How to get value of variable B?
- Showing that $\frac{x^{2x}}{(x+1)^{x+1}}\rightarrow +\infty$ as $x\rightarrow +\infty$
- Why isn't the surface area of a shell in a sphere the difference between two spherical caps?
- Show $x \mapsto \frac{x}{\Vert x\Vert^2}$ is orientation reversing.
- Term for number only divisible by 1, itself, and its square root
- Using chain rule for derivative of multivariable function
- Let the pair $(X, Y)$ have the bivariate normal distribution. Show that $aX + bY$ has a univariate normal distribution
- Suppose B is diagonalizable, then $e^{B}$ is diagonalizable and $e^{\lambda}$ is an eigenvalue of $e^{B}$ if and only if $\lambda$ is an eigenval of B
- Conditional Independence and probability
- Show that $P(W_t \in [-1,1] \; \forall t \geq 0) = 0$
- Does $\partial_{\varepsilon_k} f(x^k)$ approximate $ \partial f (\bar{x})$ for $ x^k \to \bar{x} $ and $\varepsilon_{k} \to 0 $?
- Find an angle to accelerate at to most quickly go from one movement vector to another
- Arranging numbers in an array (Swedish Math Olympiad 1986)
- Manageable project to learn some arithmetic geometry
- Prove $7^{71}>75^{32}$
- Finding orthogonal functions
| Posted: 25 Mar 2021 09:02 PM PDT How does it make sense that the alpha level be increased/decreased even though it is dependent on the reality that the null hypothesis is true? By a similar token, how does it make sense to alter the beta level?  |
| Posted: 25 Mar 2021 09:01 PM PDT I don't know how to solve this at all. I have no idea how the final line should be written or how the minimum number of solutions can even be obtained. Can I get a hint? I have never done a question like this before. Thanks!  |
| Is there any reason to restrict the Hermitian adjoint to linear operators? Posted: 25 Mar 2021 08:51 PM PDT The Wikipedia page on the Hermitian adjoint is inconsistent about whether that operation is only defined for linear operators on a single Hilbert space, or more generally for arbitrary linear maps between (not necessarily identical) Hilbert spaces. It seems to me that the concept works equally well for general linear maps, but there may be some subtle technicality that requires restricting to linear operators. Is there any reason why it's mathematically necessary to restrict to linear operators? If not, then is there any reason why doing so is more natural, convenient, or useful?  |
| How to find the first non-zero number after the decimal point? Posted: 25 Mar 2021 09:01 PM PDT If (35/6)^40 is written as a decimal 0.……, which number below is the first non-zero number after the decimal point? (log2≒0.3010，log3≒0.4771，log7≒0.8451)  |
| Largest positive domain of a linear ODE system Posted: 25 Mar 2021 08:49 PM PDT Given a linear system of ODEs of $n$ dimensions ($\textbf{x}'=\textbf{A}\textbf{x}+\textbf{b}$), with $\textbf{x}(t):\mathbb{R}^+\cup\{0\}\rightarrow\mathbb{R}^n$, I want to know how to find the largest set $C$ of $\textbf{x}(0)$, such that $C \subseteq \mathbb{R}^{n+}$, and for every $t$, $\textbf{x}(t):\mathbb{R}^+\cup\{0\}\rightarrow \mathbb{R}^{n+}$. That is, I want to know the conditions for initial values, such that the solutions of a given system of ODEs are strictly positive.  |
| Posted: 25 Mar 2021 08:57 PM PDT My friend designed a strategy for a coin flip game, and wanted to see where the flaw in the logic is, or if the math is sound. Visual depiction of the strategy. The rules of the game is you place an initial bet. You flip a fair coin. If you win, you double your bet. If you lose, you lose your bet. You can continue doing this until you run out of money. To explain the strategy, essentially you do the following Given an initial balance, you initially bet \$100. If you win, you restart. If you lose, you then you bet \$50. If you win, you restart. If you lose, then you bet \$200, and then restart. The idea is in the first bet will gain you money. If you lose, the \$50 will break even. If you lose that, the \$200 will net you \$50 (\$200-\$150). if you lose, you're out $350 and eat the loss. This has (according to our math), a positive expected value, as given by $(\$100\cdot\frac{1}{2})+(\$0\cdot\frac{1}{4})+(\$50\cdot\frac{1}{8})+(-\$350\cdot\frac{1}{8}) = \$12.5$ The first term describes the percentage of winning the first roll, the second the second, the third the third, and the final term of losing all three. Is there a flaw in the math? Is there a more optimized version of this if there is not? Here is some non-rigorous evidence. The figure was created using the code above and running it 10000 times on an initial balance of \$10k. Overall, the strategy seems positive. So is the math sound? And if so, is there a way to optimize this? Edit: Here is a larger simulation of the strategy (starting bal \$100k)  |
| differentiating a Summation for the mean energy - Statistical Physics Posted: 25 Mar 2021 08:43 PM PDT I am currently taking a Physics course Statistical Physics, and this pop up regularly: this is from Fundamental of statistical and thermal physics My question is: I understand Z is the partition theorem equal to the summation from zero to infinity of exponential to the - beta times the energy. How is the derivative related to the summation? why can I go from the left side of the equation meaning the side where both summations are present to the right side where theres a partial derivative of z with respect to beta. what is the math process to go from summations to derivatives?  |
| Markov Chain: How to Calculate the Probability that two states are equal? Posted: 25 Mar 2021 08:40 PM PDT Given a Markov Chain (discrete time) with transitional probabilities: \begin{bmatrix} a&b\\ c&d\\ \end{bmatrix} and state space $\mathcal{S} = $ {0,1}. Initial Probability distribution $P(X_{o} = 0) = \alpha = 1-P(X_{0} = 1)$. How can we calculate $P(X_{1} = X_{2})$?  |
| Stochastic process-tandem queue Posted: 25 Mar 2021 08:48 PM PDT The question New customers arrive at Server 1 according to a Poisson process with rate λ. After processing by Server 1, customers go on to Server 2. After processing by Server 2, customers go on to Server 3. After processing by Server 3, each customer independently with probability 1/2 leaves the system, and with probability 1/2 goes back to Server 1. Each server can handle one customer at a time; any more customers must wait in line. The time it takes for a server to handle a customer is exponentially distributed with mean 1/μ. In equilibrium, solve for the probability that there are i customers in line at Server 1, j customers at Server 2, and k customers at Server 3. (assume 2λ<μ) The formula said the expected number of waiting customers is ρ/(1-ρ), but for the queue of server 2 and 3, ρ would be 1. How does this work?  |
| Mean squared error and Chebyshev's inequality Posted: 25 Mar 2021 08:49 PM PDT How do I prove the following using Chebyshev's inequality? Let $X$ and $Y$ be two random variables satisfying that $\mathbb{E}\left[(Y-X)^2\right]=0$, then $P(X\neq Y)=0$.  |
| Posted: 25 Mar 2021 08:48 PM PDT I tried Googling for the keywords "the theory of odd perfect numbers" and one of the search results that came up was this document, titled ON THE DIVISORS OF THE SUM OF A GEOMETRICAL SERIES WHOSE FIRST TERM IS UNITY AND COMMON RATIO ANY POSITIVE OR NEGATIVE INTEGER. (The author turns out to be J. J. Sylvester.) Let me copy the first two paragraphs from that document:
Now, let me translate the second paragraph into modern language, fitted to the case of the divisor-sum function of the special prime power component of an odd perfect number $N = q^k n^2$, where $q$ is the special prime and satisfies $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. That is, we will be considering $$\sigma(q^k) = \frac{q^{k+1} - 1}{q - 1},$$ where $\sigma(x)=\sigma_1(x)$ is the classical divisor sum function of $x$. (Descartes, Frenicle, and subsequently Sorli conjectured that $k=1$ always holds.) Let us now see what we have got. Translating to modern language (in the context of $\sigma(q^k)$):
Here is my:
MY ATTEMPT Suppose that the Descartes-Frenicle-Sorli Conjecture holds. Then $k=1$. Then $k+1=2$ is prime. Also, since $q \equiv 1 \pmod 4$, then $q - 1 \equiv 0 \pmod 4$, so that $2 = (k + 1) \mid (q - 1)$. Then $\sigma(q^k)$ is divisible by $(k+1)=2$ but not by $(k+1)^2=4$, which is true. So what's with the UNLESS? Here is a proof for $\sigma(q^k) \equiv 2 \pmod 4$:
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| Isomorphism of schemes restricts down to local isomorphism Posted: 25 Mar 2021 08:46 PM PDT I just want to make sure I am understanding the definitions I'm learning correctly. Suppose we have an isomorphism of schemes $\pi: X \to Y$, and suppose we have some affine cover of $X$, $\{U_i\}$. Is it true that $\pi|_{U_i}$ is an isomorphism for each $i$?  |
| Determine if they are the same plane. Posted: 25 Mar 2021 08:53 PM PDT How do I solve this problem?
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| How to get value of variable B? Posted: 25 Mar 2021 08:58 PM PDT I am not a regular math user. I need to solve an equation to apply on timeseries variables. I have value of $A$ but don't have value of $B$ variable. Here time series variables (mentioned as variable A) with known values are canopy cover values (range 40 to 100). whereas variable B ( unknown values) are the leaf area index (LAI).The output LAI values (variable B) might be in 0.5 to 10. The equation is given below as written in MS Excel:
Standard format of the equation- $$ A = 94(1 - e^{-0.43 B})^{0.52} $$ What would be the equation that can be applied to get value of This might be quite simple but for me its not. please help.  |
| Showing that $\frac{x^{2x}}{(x+1)^{x+1}}\rightarrow +\infty$ as $x\rightarrow +\infty$ Posted: 25 Mar 2021 08:56 PM PDT I am trying to show that $$\frac{x^{2x}}{(x+1)^{x+1}}\rightarrow +\infty \ \ \text{as} \ \ x\rightarrow +\infty.$$ My attempt is as follows: \begin{align} \frac{x^{2x}}{(x+1)^{x+1}}&=\frac{x^{x}}{x+1}\left(\frac{x^x}{(x+1)^x}\right) \\ &=\frac{x^{x}}{x+1}\left(\frac{x}{x+1}\right)^x \\ &=\frac{x^{x}}{x+1}\left(\frac{1}{(1+1/x)^x}\right). \end{align} I can see that the second fraction will converge to $1/e$, but I am unsure of how to approach the first fraction.  |
| Why isn't the surface area of a shell in a sphere the difference between two spherical caps? Posted: 25 Mar 2021 09:03 PM PDT In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. [(Wikipedia)][1] [In a proof of the Shell Theorem, it is stated that the area of a thin ring of shell of a sphere is 2πR2sin(θ)dθ][2] The thing is, I can actually understand the derivation of this formula but even so, I'm stuck on why it can not be the difference of caps, as: $2πRh_1 - 2πRh_2$ Is this something similar to the volume of a cone not being $πr2h$? [1]: https://en.wikipedia.org/wiki/Spherical_cap [2]: https://i.stack.imgur.com/h9LMs.png  |
| Show $x \mapsto \frac{x}{\Vert x\Vert^2}$ is orientation reversing. Posted: 25 Mar 2021 08:59 PM PDT I wanted to show that the map $\mathbb{R}^n- \{0\} → \mathbb{R}^n- \{0\},\ x\mapsto \frac{x}{\Vert x\Vert^2}$ is orientation reversing. But i don't know how to tackle it appropriately. The solution proposed the following:
Then they justify it by connectedness of $\mathbb{R}^{n}$ for $n$ greater than $1$. I am not sure whether I fully understand the solution. For simplicity let's assume $n = 2$, then at $(1,0)$ the differential of the map is $(-1,1)$, similarly $(-1,1,1...,1)$ for arbitrary $n$. How is it the identity on $0 \oplus \mathbb{R}^{n-1} \subset T_p\mathbb{R}^n$? Could someone elaborate? I tried to look up similar problems in my books but couldn't find any. I would really like to understand this. Thanks for any help!  |
| Term for number only divisible by 1, itself, and its square root Posted: 25 Mar 2021 09:00 PM PDT Do numbers only divisible by 1, themselves, and their square roots have a specific term? Seems like they're all squares of primes: 1, 9, 25, 49, 121, 169... I feel like I'm missing something really obvious, I apologize if that's the case. Searching using the obvious keywords didn't bring anything directly referencing this class of numbers to light.  |
| Using chain rule for derivative of multivariable function Posted: 25 Mar 2021 08:50 PM PDT I've been trying to learn more about the chain rule of a multivariable function, but I'm a bit confused on a special case I've encountered. I'm given a single valued function $\displaystyle f(x_1,x_2,x_3)$ evaluated at $x_3 = g(x_1,x_2)$, which is also a single valved function. Note that $x_n$ are all scalars. I think I would summarize the functions as $f:\mathbb{R}^3\rightarrow \mathbb{R}$ and $g:\mathbb{R}^2\rightarrow \mathbb{R}$. I'm going to represent the function composition as $\displaystyle h(x_1,x_2) = f(x_1,x_2,g(x_1,x_2)) \equiv (f \circ g)(x_1,x_2)$ in which $h:\mathbb{R}^2\rightarrow \mathbb{R}$. I'm told to find the derivative of $h$ with respect to $x_2$. However, I want to write down the general expression for the derivative. Following the link above (and material within), I can use the so called derivative operator $\textbf{D}$ and write $\displaystyle \textbf{D}h = \textbf{D}f|_{x_3=g}\textbf{D}g$ where I've introduced the notation $|_{x_3=g}$ to denote "is evaluated at $x_3 = g(x_1,x_2)$" and dropped the other function variables for clarity. What I don't follow is how to write this in terms of vector/matrix products. In some of the linked material, it says the derivative of a scalar valued function is expressed as a $1 \times n$ row vector. Therefore, $\displaystyle \textbf{D}h = \left[ \frac{\partial h}{\partial x_1} \quad \frac{\partial h}{\partial x_2}\right]$ $\displaystyle \textbf{D}f|_{x_3=g} = \left[ \frac{\partial f}{\partial x_1}|_{x_3=g} \quad \frac{\partial f}{\partial x_2}|_{x_3=g} \quad \frac{\partial f}{\partial x_3}|_{x_3=g} \right]$ $\displaystyle \textbf{D}g = \left[ \frac{\partial g}{\partial x_1} \quad \frac{\partial g}{\partial x_2}\right]$ I know the answer, specific to the question: "find the derivative of $h$ with respect to $x_2$" is $\displaystyle \frac{\partial h}{\partial x_2} = \frac{\partial f}{\partial x_2}|_{x_3=g} + \frac{\partial f}{\partial x_3}|_{x_3=g} \frac{\partial g}{\partial x_2}$ so, my question is not about the answer. I'm trying to see how I would arrive at that using vector/matrix products? Specifically, the form "$\displaystyle \textbf{D}h = \textbf{D}f|_{x_3=g}\textbf{D}g$" is easy to remember (from single variable days), and if I can understand how to use it for all special cases, that would be helpful (using the case presented here to showcase).  |
| Posted: 25 Mar 2021 08:52 PM PDT Let the pair $(X, Y)$ have the bivariate normal distribution of (6.76), and let $a, b \in \mathbb{R}$. Show that $aX + bY$ has a univariate normal distribution, possibly with zero variance. $$g(x,y)=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}e^{-\frac{1}{2}Q(x,y)} \text{ for } x,y \in \mathbb{R}, \tag{6.76} $$ where $$Q(x,y)=\frac{1}{1-\rho^2}\left[\left(\frac{x-\mu_1}{\sigma_1} \right)^2-2\rho \left(\frac{x-\mu_1}{\sigma_1}\right) \left(\frac{y-\mu_2}{\sigma_2}\right) + \left(\frac{y-\mu_2}{\sigma_2}\right)^2 \right]$$ for $\mu_1,\mu_2 \in \mathbb{R}, \sigma_1,\sigma_2 >0, -1<\rho<1$. I have no idea how to do this problem. I was thinking maybe trying to split $g(x,y)$ into separate marginal density functions but that math seems way too complicated considering what $g$ is.  |
| Posted: 25 Mar 2021 09:01 PM PDT As stated above, Suppose B is diagonalizable, then $e^{B}$ is diagonalizable and $e^{\lambda}$ is an eigenvalue of $e^{B}$ if and only if $\lambda$ is an eigenvalue of B. I'm struggling to determine if this is true or not, is there proof or theorem to justify this one way or the other?  |
| Conditional Independence and probability Posted: 25 Mar 2021 08:57 PM PDT There are two coins each having probability of heads as $0.9$ and $0.1$ respectively. And probability of choosing a coin for experiment is $0.5$ ( a coin once picked at beginning is used for every toss ). P(Heads on $11$th toss) = $0.5$ P(Heads on $11$th toss | Heads in first $10$ tosses ) = P(Heads on $11$th toss ∩ Heads in first $10$ tosses )/ P(Heads in first $10$ tosses) P(Heads on $11$th toss ∩ Heads in first $10$ tosses ) = $0.5 * ( (0.9)^{11}) + 0.5 * ( (0.1)^{11}) $ P(Heads in first $10$ tosses) = $0.5 * ( (0.9)^{10}) + 0.5 * ( (0.1)^{10})$ P(Heads on $11$th toss | Heads in first $10$ tosses ) = $( 0.9^{11} + 0.1^{11})/(0.9^{10}+0.1^{10}) = 0.8999$ But there is another way to write P(Heads on $11$th toss ∩ Heads in first $10$ tosses ) as P(Heads in first $10$ tosses|Heads on $11$th toss ) * P(Heads on $11$th toss) I am not able to figure out what would be the value of P(Heads in first $10$ tosses|Heads on $11$th toss ) and how to find it? If there were no biases in the coins would that make the above events independent?  |
| Show that $P(W_t \in [-1,1] \; \forall t \geq 0) = 0$ Posted: 25 Mar 2021 09:00 PM PDT
My attempt: It holds that $W_t = W_t - 0 = W_t - W_0 \sim N(0,t) \Longrightarrow \dfrac{W_t}{\sqrt{t}} \sim N(0,1)$. So we have \begin{aligned} P(W_t \in [-1,1]) &= P(-1 \leq W_t \leq 1) = P\left(-\frac{1}{\sqrt{t}} \leq \frac{W_t}{\sqrt{t}} \leq \frac{1}{\sqrt{t}}\right) \\ &= \Phi\left(\frac{1}{\sqrt{t}}\right) - \Phi\left(-\frac{1}{\sqrt{t}}\right) = \Phi\left(\frac{1}{\sqrt{t}}\right) - \left(1 - \Phi\left(\frac{1}{\sqrt{t}}\right)\right)\\ &= 2\Phi\left(\frac{1}{\sqrt{t}}\right) - 1 \end{aligned} but this obviously does not equal $0$ for all $t\geq 0$. Anyone can explain what I'm doing wrong? I recently started to study Stochastic processes.  |
| Posted: 25 Mar 2021 08:43 PM PDT Suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is convex and globally Lipschitz continuous. Let $(x^k) \subset \mathbb{R}^n$ be a sequence such that $ x^k \to \bar{x} $ for some $\bar{x} \in \mathbb{R}^n$. Assume $(\varepsilon_k) \subset \mathbb{R}_{+}$ a sequence of positive scalars such that $\varepsilon_{k} \to 0 \, .$ Does the following hold: $$ \operatorname{dist} \left( \, \partial_{\varepsilon_k} f(x^k) \, , \, \partial f (\bar{x}) \, \right):= \sup_{g \in \partial_{\varepsilon_k} f(x^k)} \, \, \inf_{h \in \partial f (\bar{x})} \, \| g - h \|_2 \to 0 $$ where
The following is all from Convex Analysis and Minimization Algorithms II and might be of use: Definition 1.1.1 Given $x \in \mathbb{R}^n$ the vector $s \in \mathbb{R}^{n}$ is called an $\varepsilon$ -subgradient of $f$ at $x$ when the following property holds: $$ f(y) \geqslant f(x)+\langle s, y-x\rangle-\varepsilon \quad \text { for all } y \in \mathbb{R}^{n} $$ It follows immediately from the definition that
Theorem 1.1.4 For $\varepsilon \geqslant 0, \partial_{\varepsilon} f(x)$ is a closed convex set, which is nonempty and bounded. Proposition 4.1.1 Let $\left\{\left(\varepsilon_{k}, x_{k}, s_{k}\right)\right\}$ be a sequence converging to $(\varepsilon, x, s),$ with $s_{k} \in$ $\partial_{\varepsilon_{k}} f\left(x_{k}\right)$ for all $k .$ Then $s \in \partial_{\varepsilon} f(x)$. Proposition 4.1.2 Let $\delta>0$ and $L$ be such that $f$ is Lipschitzian with constant $L$ on some ball $B(x, \delta),$ where $x \in \mathbb{R}^n$. Then, for all $\delta^{\prime}<\delta$ $$ \|s\| \leqslant L+\frac{\varepsilon}{\delta-\delta^{\prime}} $$ whenever $s \in \partial_{\varepsilon} f(y),$ with $y \in B\left(x, \delta^{\prime}\right)$. As a result, the multifunction $\partial_{\varepsilon} f$ is outer semi-continuous, just as is the exact subdifferential. Theorem 4.1.3 Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be a convex Lipschitzian function on $\mathbb{R}^{n} .$ Then there exists $K>0$ such that, for all $x, x^{\prime}$ in $\mathbb{R}^{n}$ and $\varepsilon, \varepsilon^{\prime}$ positive: $$ \Delta_{H}\left(\partial_{\varepsilon} f(x), \partial_{\varepsilon^{\prime}} f\left(x^{\prime}\right)\right) \leqslant \frac{K}{\min \left\{\varepsilon, \varepsilon^{\prime}\right\}}\left(\left\|x-x^{\prime}\right\|+\left|\varepsilon-\varepsilon^{\prime}\right|\right) $$ This result implies the inner semi-continuity of $(x, \varepsilon) \longmapsto \partial_{\varepsilon} f(x)$ for a Lipschitz-continuous $f$. In particular, for fixed $\varepsilon>0$ $$\partial_{\varepsilon} f(y) \subset \partial_{\varepsilon} f(x)+\|y-x\| B(0, \frac{K}{\varepsilon}) \quad \text{for all } x \text{ and } y \, .$$ Corollary 4.1.5 Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be convex. For any $\delta \geqslant 0,$ there is $K_{\delta}>0$ such that $$\Delta_{H}\left(\partial_{\varepsilon} f(x), \partial_{\varepsilon} f\left(x^{\prime}\right)\right) \leqslant \frac{K_{\delta}}{\varepsilon}\left\|x-x^{\prime}\right\| \quad \text{ for all } x \text{ and } x^{\prime} \in B(0, \delta) \, .$$ Theorem 4.2.1 Let be given $f: \mathbb{R}^n \to \mathbb{R}$, $x \in \mathbb{R}^n$ and $\varepsilon \geqslant 0 .$ For any $\eta>0$ and $s \in \partial_{\varepsilon} f(x),$ there exist $x_{\eta} \in B(x, \eta)$ and $s_{\eta} \in \partial f\left(x_{\eta}\right)$ such that $\left\|s_{\eta}-s\right\| \leqslant \varepsilon / \eta$. This result can be written in a set formulation: $$ \partial_{\varepsilon} f(x) \subset \bigcap_{\eta>0} \bigcup_{\|y-x\| \leqslant \eta}\left\{\partial f(y)+B(0, \frac{\varepsilon}{\eta})\right\} . $$ The following one is not from the above mentioned source but easy to prove: Theorem Assume $f: \mathbb{R}^n \to \mathbb{R}$ is convex and let $x \in \mathbb{R}^n$. Then for every $\varepsilon > 0$ there is a $\delta > 0$ such that $$ \bigcup_{y \in B_{\delta}(x)} \partial f(y) \subset \partial_{\varepsilon} f(x) \,. $$  |
| Find an angle to accelerate at to most quickly go from one movement vector to another Posted: 25 Mar 2021 09:03 PM PDT Okay, so this is with respect to game design, so that's where I'm coming from (please try to use smol words, I am no mathematician) I have a 2D space ship. Its velocity is defined by vector A, let's say (1i,0j), and the ship's current position is (0,0). I also have a point P, let's define it as (0,3). I'm trying to figure out how to select an angle at which the ship can be constantly accelerated at a rate of 1 unit/sec^2 in order to go from traveling along vector A to having a velocity that is directed exactly at point P. Ideally, this angle would result in the ship reaching the correct vector before it crosses point P in order for it to have time to turn around and negatively accelerate. I'm assuming I could find a way to plot a parabola using the calculus and stuff but it's a little beyond me at the moment. Any help is appreciated :)  |
| Arranging numbers in an array (Swedish Math Olympiad 1986) Posted: 25 Mar 2021 09:00 PM PDT Consider an $m\times n$ array of real numbers. Let $d>0$. Suppose that the difference between the maximum number and the minimum number in each row is at most $d$. We then sort the numbers in each column so that the maximum element is in the first row and the minimum element is in the last row (so each column is arranged in decreasing order). Show that the difference between the maximum number and the minimum number in each row (after the change) is still $\le d$. Source : Swedish Math Olympiad 1986, also Principles and Techniques in Combinatorics Here is my attempt. Let $M_k$ and $m_k$ be the original largest and smallest elements of the $k$th row. Let $M'_k$ and $m'_k$ be the new largest and smallest elements of the $k$th row. We have $M'_1\ge M'_2 \ge ... \ge M'_m$ and $m_1'\ge m_2' \ge ... \ge m'_m$. Let's try induction on $k$. For $k=1$, suppose that $M'_1$ was originally in row $i$. Then $M'_1\le M_i$. We know that $m'_1$ is in the same column as some element originally from row $i$. Let that element be $a_i$. Therefore $m'_1\ge a_i\ge m_i$ so $M'_1-m'_1\le M_i-m_i\le d$. Let $1<k\le m$. Suppose that $M'_k$ was originally in row $i$ so $M'_k\le M_i$. We know that $m'_k$ is in the same column as some element $a_i$ originally from row $i$. Suppose that $a_i$ is in row $j$ after the change. If $j\ge k$, then $m'_k\ge a_i\ge m_i$ so $M'_k-m'_k\le M_i-m_i\le d$. We now assume $j<k$. What to do here? BTW in the book "Principles and Techniques in Combinatorics" this problem is in chapter 3 which is about pigeonhole principle. So if you have a solution with pigeonhole that'd be great.  |
| Manageable project to learn some arithmetic geometry Posted: 25 Mar 2021 08:41 PM PDT OK, this may be an unusual question, since I'm asking you for advice, the way I would ask a supervisor were I a grad student. I am a thirty-something maths teacher with an unresolved attraction to arithmetic geometry, and would like to know more about themes I find fascinating when I read texts about arithmetic geometry, such as:
One of the reasons this attraction is unresolved is that I never managed to seriously learn enough algebraic number theory, algebraic geometry, etc. in a "bottom-to-top" way: I have often started to read for instance R. Vakil's The rising sea, D. Eisenbud & J. Harris The geometry of schemes or Q. Liu's Algebraic geometry and arithmetic curves and find them to be very good texts, but I don't have the stamina to ingest hundreds of pages without a more specific goal. Because of this there are a number of subjects I'm vaguely familiar with, but don't know in any depth. For instance, I know what a scheme is, I even think I understand in part their raison d'être, but am probably unable to pass an scheme theory exam (and the same goes for elliptic curves, derived categories, class field theory...). To change this, I would like to try and understand a specific result and working my way from the top down to acquire the needed skills. The result has to be interesting enough to preserve my motivation in the long run, but "small" enough so that I could realistically understand it in a few months. I would love it to really mix number theory and geometry. So the question is:
Personal facts that might be relevant: my PhD is in low-dimensional topology, so I'm quite comfortable with things like cohomology and Riemann surfaces, and I like it when geometric intuition is useful. Large doses of homological algebra or category theory don't frighten me.  |
| Posted: 25 Mar 2021 09:01 PM PDT My math teacher left two questions last week, prove (1) $6^9>10^7$ and (2) $7^{71}>75^{32}.$ I did the first question: \begin{align}\frac{6^9}{10^7}&=\frac{4}{5}\times\frac{27^3}{25^3}\\&=0.8\times1.08^3\\&>0.8\times(1+3\times0.08+3\times0.08^2)\\&>0.8\times(1+3\times0.086)\\&>0.8\times1.25=1.\end{align} But I can't work out the second, I calculated it out on my computer, $\frac{7^{71}}{75^{32}}=1.000000949\cdots$  |
| Posted: 25 Mar 2021 09:01 PM PDT Let an inner product be defined on $\mathcal{C}[-1,1]$ in the following manner: $\langle f,g \rangle=\int_{-1}^{1} g(x)f(x) dx$. It is easy to check that this is an inner product and that $x \perp x^2$. My question is, is there a way of finding a function $h$ that is $(h \perp x) \wedge (h \perp x^2).$ Normalizing $h$ is easy since I can divide by its norm as a scalar which carries over under the $\int$. sign. $\textbf{Question: can I find it without using Gram Schmidt}$? I already have a solution $h(x)=\frac{-15}{8}(x^2-3/5)$, but since we haven't discussed that process in class there must be a simpler way $\int_{-1}^{1} xh(x) dx=\int_{-1}^{1} x^2h(x) dx=0$  |
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