Recent Questions - Mathematics Stack Exchange |
- θ and φ are complementary angles. tan θ = 0.5 What is tan φ?
- How do I solve the following combined operations, can someone explain to me?
- Does anyone recognize this formula? $\sin\varphi_2=\frac{–A_1 A_3\pm A_2\sqrt{ A_1^2 – A_3^2 + A_2 ^2}}{A_1^2 + A_2^2}$
- Explicit smooth function $f:[a_0,a]\to\mathbb{R}$ which equals the identity near $a$.
- Are Lebesgue integral definitions equivalent for nonnegative bounded functions on finite support?
- Is it decidable whether the value of a given definite integral has a closed-form expression?
- Change of variables in Logistic Map
- why does the slope of a vertical tangent equal $ \infty $ or $ - \infty $?
- Tangent bundle on a smooth scheme
- Calculate the translation required to zoom in on a pivot point
- Finding the order of the inverse image of the multiplicative group of units. Is it possible without brute-force?
- why does $0.888888888889 \times 9 = 8$?
- About Construction a one-to-one function from $(a,b)$ onto $[a,b]$
- Prove the function f(x) = {1 if x is rational, -1 if x is irrational} is not Riemann integrable on the interval [0,2].
- Complex vector space, but with a real inner product (Hilbert sub-spaces...?)
- Prove $f:[a,b]\to [a,b]$ is a homeomorphism, then $a$ and $b$ are fixed points or $f(a)=b$ & $f(b)=a$
- Proximal operator of $L_1$ norm with indicator of a subspace
- Integral of exponential times $\frac{x^2}{x^2+a}$
- Deducing irreducible characters from non-irreducible characters
- A variant coin toss game with weighted value
- Question on summation of binomial coefficient
- A harmonic function is non-decreasing
- Solve the non-square matrix to get the closest unitary matrix
- Conjugate prior Gamma distribution on Poisson intensity
- Checking and proving unicity of solution of a system of equations
- Fokker Planck with d'Alembert operator
- Proof Continuity of Power Functions
- Maximum and minimum of a fractional function
- Which mathematicians have influenced you the most? [closed]
| θ and φ are complementary angles. tan θ = 0.5 What is tan φ? Posted: 06 Jun 2021 08:36 PM PDT θ and φ are complementary angles. tan θ = 0.5 What is tan φ? Hi, this is trigonometry and I need help with this. I don't understand, thanks. I know how to solve the question using trial and error by plausible values of tan θ, and then solving for tan φ, but I don't get how to logically solve it.  |
| How do I solve the following combined operations, can someone explain to me? Posted: 06 Jun 2021 08:38 PM PDT How do I solve the following combined operations, can someone explain to me? $\sqrt0.3\hat6 . \frac{15}{22}$ $\sqrt(\frac{3}{5}- 1) . {\frac{5}{16}}$ I have tried to reduce the periodic number to fraction, but it does not work. The first result has to be = 1/2 The second result has to be = -1/2 But I can't.  |
| Posted: 06 Jun 2021 08:38 PM PDT Does anyone recognize this formula? $$\sin\varphi_2=\frac{–A_1 A_3\pm A_2\sqrt{ A_1^2 – A_3^2 + A_2 ^2}}{A_1^2 + A_2^2}$$ The kinematic sketch view:
The formulas:
I have looked at my textbook and searched through online websites but still do not know where this formula comes from. I hope someone can help me.  |
| Explicit smooth function $f:[a_0,a]\to\mathbb{R}$ which equals the identity near $a$. Posted: 06 Jun 2021 08:29 PM PDT Given $c<a$, I'd like to find a smooth map $f:[a_0,a]\to\mathbb{R}$ such that $f(a_0)=c$, $f'>0$ and $f(t)=t$ for all $t$ sufficiently close to $a$. (To be clear, it's ok to have $f'(a_0)=0$, but I'd prefer not to). So far this is what I've got: if $c=a_0$, then it's obvious. If $c<a_0$, then take any $0<\delta<\frac{a-a_0}{2}$, so that $$f(t) = c + (t-c)\gamma(t)$$ does the trick, where $\gamma:\mathbb{R}\to\mathbb{R}$ is the classic smooth map such that $\gamma|_{(-\infty,a_0]} = 0$, $\gamma_{[a-\delta,+\infty)} = 1$, $0<\gamma(t)<1$ for all $t\in (a_0,a-\delta)$ and $\gamma$ is strictly increasing in $(a_0,a-\delta)$. This works because $f(a_0)=c$, $f(t)=t$ for every $t\in [a-\delta,a]$ and the equality $$f'(t) = \gamma(t) + (t-c)\gamma'(t)$$ implies that $f'(t)>0$ for all $t>a_0>c$. However, the case where $c> a_0$ doesn't seem to work the same way, since the derivative can be negative somewhere in $(a_0,c)$. Does anyone know how to construct this? I'd like to do this without having to explicitly compute $\gamma'$. Remark: Let me say who $\gamma$ is. First, we have the map $\beta:\mathbb{R}\to\mathbb{R}$, $$\beta(t) = \begin{cases} e^{-1/x}, & x>0\\ 0, & x\leq 0 \end{cases}$$ which is smooth. Then, $\gamma$ is defined as follows: $$\gamma(t) = \frac{\beta(t-a_0)}{\beta(t-a_0)+\beta(a-\delta-t)}.$$ Remark 2: Also, if anyone knows how to get $f'(a_0)>0$ in the case $c<a_0$, then I can solve the rest by taking the inverse. Thanks in advance.  |
| Are Lebesgue integral definitions equivalent for nonnegative bounded functions on finite support? Posted: 06 Jun 2021 08:16 PM PDT According to Royden's Real Analysis, if $f$ is a bounded measurable function, with $m(E)<\infty$, then the integral is defined as $$\int_E f = \sup \{\int_E\phi:\phi \text{ simple}, \phi \le f\}$$ On the next chapter, if $f$ is a nonnegative measurable function, the integral is defined as $$\int_E f = \sup \{\int_Eh: 0 \le h \le f \text{ is bounded, measurable, finite support}\}$$ Then if $f$ is a nonnegative, bounded function on a set of $m(E)<\infty$, it only makes sense for the two definitions to be equivalent. However, this is not obvious to me, so I want to show it. This is what I have so far. \begin{equation} \label{eq1} \begin{split} \int f & = \sup \{\int h: 0 \le h \le f, h \text{ is measurable}\} \\ & = \sup \{\sup \{ \int \phi: \phi \text{ simple}, \phi \le h\}: 0 \le h \le f, h \text{ is measurable}\} \end{split} \end{equation} I feel like I'm almost done here. In fact, it looks equal to the first definition because intuitively, it's like the squeeze theorem. But I can't quite express it formally, so help would be appreciated.  |
| Is it decidable whether the value of a given definite integral has a closed-form expression? Posted: 06 Jun 2021 08:00 PM PDT There are many elementary functions such as $e^{-x^2}$ which don't have an elementary antiderivative, but a definite integral of the same integrand has a closed-form value, e.g. $$\int_{-\infty}^\infty e^{-x^2} \mathrm{d}x = \sqrt{\pi}$$ I'm curious whether it is decidable whether or not such a definite integral has a closed-form value. In other words:
where those "standard operations" are elementary functions, and the only allowed inputs are integers. (Note that rational numbers can be made using the function $f(x)=x^{-1}$ and certain well-known constants can be made such as $\pi = \arccos(-1)$ and $e = \exp(1)$) I know that for indefinite integrals, the Risch algorithm does this, but I couldn't find anything about definite integrals. Now for my follow-up question, define $S$ to be an arbitrary set of integrable functions and $R$ to be an arbitrary set of constants.
I would guess that the answer is no, but I couldn't find anything about it online.  |
| Change of variables in Logistic Map Posted: 06 Jun 2021 07:59 PM PDT How do I transform the difference equation x_{n+1}=rx_n(1-x_n) to x_{n+1}=1-a*[x_n]^2?  |
| why does the slope of a vertical tangent equal $ \infty $ or $ - \infty $? Posted: 06 Jun 2021 08:37 PM PDT we know that for any point on a curve, the slope of its tangent is m: $$ m = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} $$ But i can't understand that for a point which has a vertical tangent, why does the slope of its tangent be $ \infty $ or $ - \infty $? They both are this case: as the denominator $h$ becomes close to $0,$ the numerator $f(x+h)-f(x)$ also becomes close to $0.$ But why does m for the point which has a non-vertical tangent equals a number, but for the point which has a non-vertical tangent equals $ \infty $ or $ - \infty $ ?  |
| Tangent bundle on a smooth scheme Posted: 06 Jun 2021 07:52 PM PDT My question is, how might I construct the tangent bundle on a smooth scheme? It is clear how to define the tangent space at a point: the Zariski tangent space. It is also clear what we should do in the setting of manifolds: we assemble the tangent spaces at all the stalks and put a topology on it. This, along with the evident projection map, give a vector bundle. But the situation for schemes is perhaps more difficult. The other question I have is about why this construction is not more well known or used in algebraic geometry.  |
| Calculate the translation required to zoom in on a pivot point Posted: 06 Jun 2021 07:54 PM PDT I cannot think of a better way to explain this. I hope it makes sense. Mathematical question: Let the origin be $O=\left[0,0\right]$. How does one calculate $T$ in order to simulate zooming in to or away from the pivot point? Graphical example: Zooming in on the top-left white square. Before zoom:
After zoom:
 |
| Posted: 06 Jun 2021 08:35 PM PDT Let's suppose you know the prime factorization of $n$ and you want to compute the order of the surjective inverse image of a mutliplicative group of units modulo $m$. That is we wish to compute $|\pi^{-1}(\Bbb{Z}_m^{\times})|$ where $\pi:\Bbb{Z}_n\twoheadrightarrow \Bbb{Z}_m$ is the natural surjective ring homomorphism, and where $m = \text{rad}(n)$. Is there a way to do it? Does it help if I know the factorization of $n$, so say if $n = p_1^{k_1} p_2^{k_2}$, then $\Bbb{Z}_n \simeq \Bbb{Z}_{p_1^{k_1}} \times \Bbb{Z}_{p_2^{k_2}}$ by CRT. Because then isn't the inverse image under $\pi$ of $$\Bbb{Z}_m^{\times} \simeq (\Bbb{Z}_{{p_1}^{k_1}})^{\times} \times (\Bbb{Z}_{{p_2}^{k_2}})^{\times}$$, which would be the intersection: $$ (\text{pr}_1 \circ \pi)^{-1}(\Bbb{Z}_{p_1^{k_1}}^{\times}) \cap (\text{pr}_2 \circ \pi)^{-1}(\Bbb{Z}_{{p_2}^{k_2}}^{\times}) $$ Is there a formula for the order or is it still an open problem? Does it help if $\text{rad}(m) = m$ so that $m$ is a product of distinct primes?  |
| why does $0.888888888889 \times 9 = 8$? Posted: 06 Jun 2021 07:57 PM PDT So im teaching myself maths, watching alot of youtube videos about topics way beyond my head. Im trying to unlearn the rigid way school taught maths, such as the rules and procedures to solve specific problem, which I hated. Feynman said maths is 'too abstract' and some book somewhere said 'math is about logical reasoning and pattern recognition', which I enjoy. I like to think logically and abstractly but dont have a way to show it or express it (except with coding and basic social skills). Anyway heres my question: Given 0.888888888889 * 9 = 8 0.88888888889 * 9 = 8.00000000001 0.8888888889 * 9 = 8.0000000001 ... 0.8889 * 9 = 8.0001 0.889 * 9 = 8.001 0.89 * 9 = 8.01 First question why are 11 8's the magic number for it to be a whole number again? What is the relationship between 8's and the 0's on the output when there are less than 11 8's? Why are they the same when less than 11? Then why is more the 11 8's still equal 8? Any sources would be great, like what field of mathematics deals with these kinds of questions, its because the calculator blah blah cant calculate that high, the history behind it showing how these things work etc. Also if this is too dumb of a question, how? (do you see things in your head that just makes sense of these symbols?) What are good math questions? Why did you get into maths? Again I just want to learn maths, working as business rule translator (software developer) is kinda boring and I feel like im just coasting, if i can understand maths better then hopefully I atleast have a chance to learn/work on more interesting topics like theoritcal computer science (algorithms specifically), theoretical physics (how the universe works), dynamical systems (how unpredictable phenomas work), bioinformatics etc.  |
| About Construction a one-to-one function from $(a,b)$ onto $[a,b]$ Posted: 06 Jun 2021 08:06 PM PDT This question asked to construct one-to-one function from $(a,b)$ onto $[a,b]$. I know there is a function but it seems the question to define this function explicitly. How this can be done? Edit I did first by transfinite induction But I was wrong. Thanks to user @ Arture Magidin. He told me this wrong and he answered the question.  |
| Posted: 06 Jun 2021 08:17 PM PDT I am currently arriving at the fact that U(P,f) >= 2 and that L(P,f) >= -2, which does not imply inability to integrate, as L is not bound from above.  |
| Complex vector space, but with a real inner product (Hilbert sub-spaces...?) Posted: 06 Jun 2021 08:37 PM PDT Say I have a complex-valued vector space $\mathbb{C}^n$ with the following inner product: $$ \langle u,v\rangle=u^Tv $$ If one picks only vectors that have real entries, and only allow linear transformations from $\mathbb{R}^n\to \mathbb{R}^n$. Then, can we say that this complex vector space embeds a real Hilbert sub-space, in some sense?  |
| Posted: 06 Jun 2021 08:21 PM PDT
Hello. I've been struggling with this question. I found something related:
I know a homeomorphism is equivalent to (1) Bijection (2) $f$ continuous (3) $f$ inverse continuous. I am not sure how these conditions are related to prove the function has 2 fixed points or $f(a)=b$ and $f(b)=a$. This is a question from my introduction to real analysis course. The only theorems proved are: (1) Brouwer fixed-point theorem (case $n$=1). (2) Banach theorem. I really appreciate some help. Thanks in advance.  |
| Proximal operator of $L_1$ norm with indicator of a subspace Posted: 06 Jun 2021 08:02 PM PDT How could one solve the problem $$ \operatorname{Prox}(y) = \operatorname*{argmin}_x \frac{1}{2}\|x-y\|^2_2+\lambda\|x\|_1 + I_C(x) $$ where $C$ is the set $\{x:Bx=0\}$, with $B$ being a projection matrix, and $$I_C(x)=\cases{0,&if $x\in C$\cr \infty,&otherwise \cr}$$  |
| Integral of exponential times $\frac{x^2}{x^2+a}$ Posted: 06 Jun 2021 08:03 PM PDT Is it possible to find the indefinite integral of: $$ y = \frac{x^2}{x^2 + a} e^{-(x+b)^2} $$ I have tried integration by parts but haven't been successful. Maybe its possible considering a special function?  |
| Deducing irreducible characters from non-irreducible characters Posted: 06 Jun 2021 08:02 PM PDT Say I have a complex character $\alpha$ of a finite group $G$ with the inner product $(\alpha, \alpha) = 2$. Since the only decomposition of $2$ as a sum of squares is $2 = 1^2 + 1^2$, the representation belonging two $\alpha$ must be a direct sum of two irreducible representations. In the particular example I am working with, it is easy to calculate that for $\chi_1$ the trivial representation, the class function $\gamma(g) = \alpha(g) - \chi_1(g)$ has inner product $(\gamma, \gamma) = 1$. Is this enough to show that $\alpha$ is the direct sum of $\chi_1$ and $\gamma$? Since it's not even guaranteed that the representation for $\alpha$ has the subrepresentation for $\chi_1$, this does not seem like something I could deduce. What about the case that $(\gamma, \chi_1) = 0$? Since this means that $\gamma$ is orthogonal to $\chi_1$ in the vector space of class functions on $G$, and all the irreducible characters are orthogonal to each other, does this imply $\gamma$ is a character (and thus irreducible)? I would be interested in knowing this in the most generality possible, but I can supply concrete numbers if necessary, I just omitted them as the calculations are somewhat tedious.  |
| A variant coin toss game with weighted value Posted: 06 Jun 2021 08:15 PM PDT This is a coin toss game with weighted values for the outcome. If you get Head, you get 100 dollars; if you get Tail, you get 80 dollars. You have 50/50 chance getting H/T.
Here is an example for one gameplay with 500 Gems: Round 1: Round 2: Round 3: Round 4: For a limited amount of Gems，what is the best strategy for maximizing the earning? (eg: How many rounds should the player play in each game? And in what condition should the player end/stay in the game?) The tricky part of the question is due to the entrance fee. Otherwise, it would be best to keep restarting the game and purchase only for the coins with Head.  |
| Question on summation of binomial coefficient Posted: 06 Jun 2021 08:07 PM PDT Find the value of $$\sum\sum_{1\leq i\lt j\leq{n-1}}(ij)(^nC_i.^nC_j)$$ My answer : $n^2(\frac{2^{2(n-1)}-2^n+2-^{2(n-1)}C_{n-1}}{2})$ Answer in book: $n^2(\frac{2^{2(n-1)}-^{2(n-1)}C_{n-1}}{2})$ My approach: It was quite difficult to write so I have attached the pic. IS THERE A MISPRINT IN THE BOOK OR I HAVE DONE SOME MISTAKE ?  |
| A harmonic function is non-decreasing Posted: 06 Jun 2021 07:59 PM PDT We consider $u:U\to\mathbb{R}$ be a harmonic function. Now, I need to prove that for every $x_{0}\in U$, the function $\left( {0,{\rm{dist}}\left( {{x_0},\partial U} \right)} \right)\ni\rho \mapsto \dfrac{1}{{{\rho ^{n - 2}}}}\int_{{B_\rho }\left( {{x_0}} \right)} {{{\left| {Du} \right|}^2}dx} $ is non-decreasing. We know that if $u$ is a harmonic function then $u$ be also a subharmonic function. But I am referring Example 2 in Section 8.6.2 in Partial Differential Equations Lawrence C.Evan with Monotonicity formula for harmonic functions . But I do not have an idea approaching the problem to solve because I confuse with them. Thank you advance for your supports  |
| Solve the non-square matrix to get the closest unitary matrix Posted: 06 Jun 2021 07:54 PM PDT First, I would like to say that I have asked similar question to this before, but I noticed that many details are missing so I decided to rewrite the question with clear details. I have a real matrix $D$ with size $m$ x $n$ , and $m < n$ such that $D'D = I_n$, where $D'$ denoted the transpose of the matrix $D$. I need to get the real matrix $V$ with size $n$ x $m$ such that $DV = Y$ where the matrix $Y$ has $Y'Y = I_m$ (It means get the matrix $V$ which make $(DV)'*(DV) = I_m)$. Does it exist? In other words it's to get the closest unitary matrix $Y$ by non-square matrix $V$. In case if the matrix $V$ is square, I think the solution can be straightforward, but what's about it's not square? let's take an example that $m$=6 and $n$=4.  |
| Conjugate prior Gamma distribution on Poisson intensity Posted: 06 Jun 2021 08:05 PM PDT I don't know what I am missing in following in my understanding. Whether it is my mathematica code that is incorrect or my mathematical skill is short. Gamma distribution is the conjugate prior when the likelihood function is Poisson distribution. By this, I understood the following: $$\overbrace{f(x|y)}^{\rm Gamma(\alpha+x, \beta + 1)}=\frac{\overbrace{f(y|x)}^{\rm Poisson(\lambda)} \cdot \overbrace{f(x)}^{\rm Gamma(\alpha, \beta)}}{\underbrace{f(y)}_{NB(\alpha, \frac{1}{1+\beta}\:)}} $$ If my understanding above is correct then why the LHS is not equal to RHS in following Mathematica code ? Correction on Wikipedia page conjugate prior $$\overbrace{f(\lambda|x)}^{\text{Gamma}(k+x,\frac{\theta}{\theta + 1})}=\frac{\overbrace{f(x|\lambda)}^{\text{Poisson}(\lambda)} \cdot \overbrace{f(\lambda)}^{\text{Gamma}(k, \theta)}}{\underbrace{f(y)}_{\text{NB}(k, \frac{1}{1+\theta}\:)}} $$ and $$\overbrace{f(\lambda|x)}^{\text{Gamma}(\alpha+x,\beta+1)}=\frac{\overbrace{f(x|\lambda)}^{\text{Poisson}(\lambda)} \cdot \overbrace{f(\lambda)}^{\text{Gamma}(\alpha, \beta)}}{\underbrace{f(y)}_{\text{NB}(\alpha, \frac{\beta}{\beta +1}\:)}} $$ i.e. the two Negative Binomial distributions are mistakenly swapped on wiki page.  |
| Checking and proving unicity of solution of a system of equations Posted: 06 Jun 2021 08:14 PM PDT Consider the following system of equations: $$\prod_{j=1}^K\alpha_j^{R_j} p_i+\prod_{j=1}^K(1-\alpha_j)^{R_j}(1-p_i)=y_{i,(R_1,\cdots,R_K)}$$ for each $i\in\{1,\cdots,I\}$ and each $(R_1,R_2,\cdots,R_K)\in\{0,1,\cdots,T\}^K$ satisfying $\sum\limits_{k=1}^K R_k=T$. Here:
Question: does the system has a unique solution? If yes, can we show it? Further discussion: The system above is a generalised version of a system that I have studied in a simplified setting where unicity holds. For example, for $I=2$, $K=2$, $T=2$, the system is: $$ \begin{cases} \alpha^2_1 p_1+(1-\alpha_1)^2(1-p_1)]=y_{1,(2,0)}\\ \alpha^2_2 p_1+(1-\alpha_2)^2(1-p_1)]=y_{1,(0,2)}\\ \alpha_1 \alpha_2 p_1+(1-\alpha_1)(1-\alpha_2)(1-p_1)]=y_{1,(1,1)}\\ {}^{\underline{\hphantom{\Huge------------}}}\\ \alpha^2_1 p_2+(1-\alpha_1)^2(1-p_2)]=y_{2,(2,0)}\\ \alpha^2_2 p_2+(1-\alpha_2)^2(1-p_2)] =y_{2,(0,2)}\\ \alpha_1 \alpha_2 p_2+(1-\alpha_1)(1-\alpha_2)(1-p_2)]=y_{2,(1,1)} \end{cases} $$ which can be shown to have a unique solution with respect to $\alpha_1, \alpha_2, p_1, p_2$ if $\alpha_1>1/2$. Just recalling that $\alpha_1=1-\alpha_2$, derivations are easy. I am unable to generalise such derivations though. Can you see some patterns/properties?  |
| Fokker Planck with d'Alembert operator Posted: 06 Jun 2021 08:34 PM PDT Question:Consider the d'Alembert operator with only one space-variable: $\Box:=\frac{1}{c^2} \partial_t^2 -\partial_x^2$ and then the PDE $$\frac{\partial p}{\partial t}=\frac 12 \Box p, \,\,\,\,\,\, \forall (t, x) \in [0, \infty) \times \mathbb{R}$$ with IC $p(0, x)=\delta_{x_0}(x)$ and $p_t(0,x)=0$ (for limiting BC, see below). Does this PDE have a (semi)-closed-form solution? Some comments:If we discretize this using a basic implicit scheme, we see solutions that evolve like this:
So, analogous to the classic Fokker-Planck evolution $p_t = \frac 12 p_{xx},$ except that instead of one Gaussian that disperses over time around it's initial point $x_0$, we see two traveling Gaussian-like curves dispersing, where the peaks occur roughly at $\pm ct$ and move away from the initial point $x_0=0$, in this case. This, I think, intuitively captures wave-like properties and diffusion-properties. Additionally, judging by the numerical solutions, we can further impose some regularity conditions/limiting behavior: $\forall t >0$, $\lim_{x\to \pm \infty} p(t, x)=0$ and $\forall x$, $\lim_{t\to \infty} p(t, x)=0$. (The label price should be space, by the way, the time-span was $T=3/365$, $c\approx 0.307$, and $100$ time subintervals and $500$, space sub-intervals, please comment if you want me to include the R-code that implements the solver; it is not too long). However, if we guess $p(t, x)$ is a $1/2$-mixture of two Gaussians with opposite means $\pm ct$, which we write as $u^+$ and $u^-$ with same variance $\sigma^2 t$, then $p$, it can be seen, satisfies the standard Fokker-Planck, $$\frac{\partial p}{\partial t} = -\frac{\partial}{\partial_x} \left[\frac{c}{2}(u^+(t, x)-u^-(t,x))\right]+\frac12 \sigma^2 \frac{\partial^2 p}{\partial x^2}.$$ Now we might set the RHS side equal to $\frac 12 \Box p$ and see what this implies on $p=\frac12 u^+ +\frac 12 u^-$, but I have not been able to carry out anything fruitful with moderate effort. I have tried separation of variables and transform techniques (Laplace followed by Fourier) and I do not quite have the fortitude (yet) to brute-force check whether this Gaussian mixture guess satisfies the PDE by computing the derivatives $p_t, p_{tt}, p_{xx}$. Naturally, this mixture guess (or something like it) seems to match characteristics of the basic heat-diffusion and the d'Alembert formula for wave-equations simultaneously (since the Gaussians depend on $(x-ct)$ and $(x+ct)$), at least informally. To be honest, judging from many numerical computations, the densities look slightly skewed, with the density moving house-left of $x_0$ left-skewed, and the density moving house-right of $x_0$, right-skewed, so perhaps then, a mixture of Gaussians is not entirely accurate, but is an okay first-guess. Another appealing idea, but for different purposes, is to try to discretize the PDE into a difference equation that we could build a random walk $S_n$ whose PMF satisfies, and then do a scaling limit of some sort. This I am working on today. Of course this won't give us the density form but it'll at least give us a stochastic object (the RW) that is associated with it and this is useful/interesting to me. Please comment for clarifications of the question, context, or corrections on typos/mistakes I missed, thank you.  |
| Proof Continuity of Power Functions Posted: 06 Jun 2021 08:04 PM PDT I am trying to prove: $$\lim_{x\to c}Ax^k=Ac^k.$$ What I have: For $k > 0$ For all $\varepsilon >0$ there exists $\delta>0$ such that $0<|x−c|<\delta$ implies $|Ax^k−Ac^k|<\varepsilon$ $|Ax^k−Ac^k|=|A||x^k−c^k|<|A||(\delta-c)^k−c^k|=k.root((((|A||\varepsilon^k−c^k|)/|A|)+c^k))=\varepsilon$ I have doubts that I did this correctly.  |
| Maximum and minimum of a fractional function Posted: 06 Jun 2021 08:10 PM PDT Let $x, y \in \mathbb{R}$, $a, b, c$ are three real parameters with $c\neq 0$. Find the maximum and minimum of $\dfrac{ax+by+c}{\sqrt{x^2+y^2+1}}$ This is quite complicated if I calculate the derivative. Is there any other ways? Please help me. Thanks. I know that some people have voted my question down, I know how to use Cauchy-Schwarz inequality, but this only gives me the maximum, not the minimum. I'm not good at this kind of math, so, instead of voting down, please explain for me.  |
| Which mathematicians have influenced you the most? [closed] Posted: 06 Jun 2021 08:05 PM PDT This question is lifted from Mathoverflow.. I feel it belongs here too. There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, which can inspire the desire, even the need for doing mathematics, or can make one to confront some kind of problems, dedicate his life to a branch of math, or choose an specific research topic. I think that this kind of force must not be underestimated; on the contrary, we have the duty to take advantage of it in order to improve the mathematical education of those who may come after us, using the work of those gifted mathematicians (and even their own words) to inspire them as they inspired ourselves. So, I'm interested on knowing who (the mathematician), when (in which moment of your career), where (which specific work) and why this person had an impact on your way of looking at math. Like this, we will have an statistic about which mathematicians are more akin to appeal to our students at any moment of their development. Please, keep one mathematician for post, so that votes are really representative.  |
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