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Recent Questions - Mathematics Stack Exchange |
| Laplace transform of 3, simple but what happens to $e^{st}$ term? Posted: 31 Jan 2022 09:08 AM PST I want to find the Laplace transform of f(t)=3, simply it is calculated by the integration procedure: \begin{equation} \int_{-\infty}^{\infty}f(t)e^{-st}dt=3\int_{-\infty}^{\infty}e^{-st}dt=\bigg[\frac{3}{s}e^{-st}\bigg]_{\infty}^\infty=\bigg(\frac{3}{s}e^{-s\infty}+\frac{3}{s}e^{s\infty}\bigg)=\frac{3}{s}\bigg(e^{-s\infty}+e^{s\infty}\bigg) \end{equation} But here the last term in the brackets is certainly not 1, in fact it evolves towards infinity, as given here. But wolfram suggests on $f(s)=\frac{3}{s}$. What happened with $e^{-st}$ terms then? Thanks |
| Very basic trig question: how to calculate an angle given a slope (an angle from a ratio of sides) Posted: 31 Jan 2022 09:07 AM PST I am trying to write a very basic trig primer, from scratch. I asked a basic question on Mathematics Educators, but they suggest to ask here. Here it is. Say I discuss slopes (say, the slope of a line through the origin), and wish to give the 'right' hints on how the slope of a line is related to the angle of the line with the x-axis. It would be nice to show how to calculate the angle, given the slope. I am looking for easy examples, say for lines with slopes 1,1/2,1/3,1/4,...; or any suitable family of lines. I am not completely sure, but my question seems to be something like: how do I design an algorithm (and/or a geometric construction) to determine the angle from the slope (i.e, from the ratio of sides). Sorry to be quite vague. |
| Convergence in the operator norm Posted: 31 Jan 2022 08:58 AM PST Given the following operator: $T:\ell^2\to\ell^2$, which acts in the following way on the standard basis vectors: $$Te_{2k-1}=\frac{1}{k}(e_{2k-1}-ie_{2k})$$ $$Te_{2k}=\frac{1}{k}(ie_{2k-1}+e_{2k})$$ I need to prove that this operator is compact by showing that it is the limit of finite rank operators. My attempt: Define $(T_nx)_j=(Tx)_j$, for $j\le 2n$, and for $j\gt 2n: (T_nx)_j = 0$. Intuitively this should converge to $T$ In the operator norm and all these operators $(T_n)_{n\ge1}$ are of finite rank thus $T$ is compact! Let us try and show that $\|T-T_n\|\to0$, for $n\to \infty$. $$\|T-T_n\|=\sup_{\|x\|=1}\|(T-T_n)x\|$$, but from here I am lost I don't see how this expression goes to $0$. Can anyone show me how it works? |
| Boolean Algebra; comprehending a given solution Posted: 31 Jan 2022 08:52 AM PST I am reading a text on Boolean Algebra. Right after the definition of the disjunctive normal form and the conjunctive normal form the author gives an example. Unfortunately I am not able to understand a transformation, that is needed to find the CNF of the given Boolean expression using the four Axioms. (The DNF was easy to find, so I know that there are probably no mistakes in the text.) Let $z(x'y'+xy)'+x'y$ be a Boolean expression. Using De Morgan's laws we have $$z(x+y)(x'+y')+xy'.$$ That was easy. The author then goes on stating:
I do not understand what is happening in these two lines, so I would be grateful if someone could tell me in detail what is going on. If you are interested, the "final" CNF supposedly is $$(x+y'+z)(x+y+z)(x'+y'+z)(x+y+z')(x'+y'+z')$$ |
| Area of Triangle and Diophantine Equations Posted: 31 Jan 2022 08:56 AM PST We are given some arbitrary solution $(x_0, y_0)$ to $ax-by=1$ and another point $(b,a)$. We are asked to prove that the area of the triangle with vertices at $(0,0)$, $(b,a)$, and $(x_0, y_0)$ is $\frac{1}{2}$. I figured that I should use the equation for a right triangle, and apply some substitution as follows: $\frac{1}{2}\sqrt{x_0^2+y_0^2}\sqrt{(b-x_0)^2+(a-y_0)^2}$ plugging in $x_0$ and $y_0$ in terms of the diophantine equation doesn't seem to cancel out any terms and the algebra is messy. |
| Derivative of a matrix w.r.t to another? Posted: 31 Jan 2022 08:46 AM PST What's the general formula for taking the derivative of a matrix w.r.t to another matrix? For example, what's $\frac{\partial A}{X}$, $\frac{\partial A}{X^\top}$, $\frac{\partial A}{\tilde{X}}$, $\frac{\partial A}{\tilde{X}^\top}$ for $A = X\tilde{X}^\top$, $A = X\tilde{X}$, $A = X^\top\tilde{X}$, and $A = X^\top\tilde{X}^\top$. Or how about $\frac{\partial A}{X}$, $\frac{\partial A}{X^\top}$ for $A = XX^\top$, $A = X^2$, $A = X^\top X$ I struggle to derive these generally, are there known formulas for these? How do I in general go about deriving these? There are also more complicated cases of $\frac{\partial X\tilde{X}}{X^\top \tilde{X}}$ and so on. |
| Confusion about the necessity of the axiom of choice for infinite sets [duplicate] Posted: 31 Jan 2022 08:46 AM PST Given a family $(X_i)_{i \in I}$ of non-empty sets, why cannot one introduce a function $f : I \to \bigcup_{i \in I} X_i$ as follows : for all $i \in I$, there exists $x \in X_i$ — let $f(i) := x$, thus producing a "choice function" ? Why isn't the above construction valid in $\sf {ZF}$ ? |
| Solve the complex number equation $|z|-\bar{z}=i$ Posted: 31 Jan 2022 08:39 AM PST Solve the complex number equation $|z|-\bar{z}=i$. The following was my thought process: $$|z|=i+\bar{z}$$ Given that $Im(|z|)=0$, $Im(\bar{z})=-i$. Hence, $\bar{z}=a-i$ and $z=a+i$. $$|z|=i+(a-i)$$ $$|z|=a$$ But since $z=a+i$, $|z|=\sqrt{a^2+1}$ and hence I've reached a contradiction, as $\sqrt{a^2+1}\neq a$. Hence, there is no solution for $|z|-\bar{z}=i$, for $z\in\mathbb{C}$. |
| How do we calculate the inverse Laplace transform of $F(s)=\frac{s^2+1}{(s+1)(s-1)}$? Posted: 31 Jan 2022 08:43 AM PST We have \begin{equation} F(s)=\frac{s^2+1}{(s+1)(s-1)} \end{equation} which I want to use Heavisde method to find the fractions. We start \begin{equation} F(s)=\frac{s^2+1}{(s+1)(s-1)}=\frac{A}{(s+1)}+\frac{B}{(s-1)} \end{equation} By Heaviside method, we multiply both sides by the denumerator of A, ($s+1$) \begin{equation} (s+1)\frac{s^2+1}{(s-1)}=A+\frac{B(s+1)}{(s-1)} \end{equation} We insert for the pole of the A-fraction, for s, that is $s=-1$ and we obtain \begin{equation} (-1+1)\frac{s^2+1}{(s-1)}=A+\frac{B(0)}{(-2)} \end{equation} which gives the incorrect results, that $A=0$ What should I do here to prepare the fractions correctly for the Heaviside method to be applied? Thanks Update, I devise the method suggested below in the accepted solution, and try the Heaviside method on that. \begin{equation} F(s)=\frac{s^2+1}{(s+1)(s-1)}=\frac{(s^2-1)+2}{(s+1)(s-1)} \end{equation} \begin{equation} F(s)=\frac{(s^2-1)+2}{(s+1)(s-1)}=\frac{(s+1)(s-1)+2}{(s+1)(s-1)} \end{equation} \begin{equation} F(s)=\frac{(s+1)(s-1)+2}{(s+1)(s-1)}=\frac{(s+1)(s-1)}{(s+1)(s-1)}+\frac{2}{{(s+1)(s-1)}} \end{equation} \begin{equation} F(s)=1+\frac{2}{{(s+1)(s-1)}} \end{equation} So from here we treat the last fraction with the Heaviside method: \begin{equation} F(s)=\frac{2}{{(s+1)(s-1)}}=\frac{A}{(s+1)}+\frac{B}{(s-1)} \end{equation} Find A by multiplying with $(s+1)$ on both sides and obtain \begin{equation} (s-1)=A+\frac{B(s+1)}{(s-1)} \end{equation} Insert for the pole of the A-fraction, $s=-1$ \begin{equation} (-1-1)=A+\frac{B(-1+1)}{(s-1)} \end{equation} This yields that $A=-2$. We do the same for B, \begin{equation} (s+1)=\frac{A(s-1)}{(s+1)}+B \end{equation} Insert for $s=1$, and get that $B=2$, hence: \begin{equation} F(s)=1+\frac{2}{(s-1)}-\frac{2}{(s+1)} \end{equation} Inverse transform of F(s): \begin{equation} \mathscr{L}^{-1}\{1+\frac{2}{(s-1)}-\frac{2}{(s+1)}\}=\delta(t)+2e^{t}-2e^{-t} \end{equation} |
| Hard Integral without Partial Fraction Decomposition Posted: 31 Jan 2022 08:48 AM PST So I was trying to use partial fraction decomposition on this problem, and I realized that it didn't work, as it is already in partial fraction decomposition form. $\int{\frac{3x+4}{(x^2+5)^2}dx}=$ Well, $\frac{3x+4}{(x^2+5)^2}=\frac{Ax+B}{x^2+5}+\frac{Cx+D}{(x^2+5)^2}$ $(x^2+5)^2[\frac{3x+4}{(x^2+5)^2}]=(x^2+5)^2[\frac{Ax+B}{x^2+5}+\frac{Cx+D}{(x^2+5)^2}]$ $3x+4=(Ax+B)(x^2+5)+(Cx+D)$ $3x+4=Ax^3+5Ax+Bx^2+5B+Cx+D$ $3x+4=(A)x^3+(B)x^2+(5A+C)x+(5B+D)$ So, $A=0$, $B=0$, $5A+C=3$, and $5B+D=4$ So, $A=0$, $B=0$, $C=3$, and $D=4$ $\int{\frac{3x+4}{(x^2+5)^2}dx}=\int{\frac{(0)x+(0)}{x^2+5}+\frac{(3)x+(4)}{(x^2+5)^2}}dx$ $\int{\frac{3x+4}{(x^2+5)^2}dx}=\int{\frac{3x+4}{(x^2+5)^2}}dx.$ As this clearly doesn't work, I am wondering what the integral would be. WolframAlpha tells me that the integral is as follows: $\int{\frac{3x+4}{(x^2+5)^2}dx}=\frac{1}{50}(\frac{5(4x-15)}{x^2+5}+4\sqrt{5}\tan^{-1}(\frac{x}{\sqrt{5}}))+c$ I am not sure how to get here. Any help would be greatly appreciated. |
| Posted: 31 Jan 2022 08:54 AM PST Given the following time varying model: $Z(t) = \alpha*t*z(t-1)$, how do I go about proving linearity? The issue I keep running into with a number of approaches is that because Z(t) is dependent on Z(t-1), I always end up with some inscrutable function within the function. For transparency this is homework, some I'm happy for general guidance over a full solution. Edit: The question explicitly says "Prove linearity" not "prove or disprove linearity." And the professor has used these homework's for years. So... take that for what it's worth. This is in the context of time series. |
| Posted: 31 Jan 2022 08:47 AM PST Exercise If $x\in X$ and $r > 0$, then for every $y\in Y\cap B^{X}_{r}(x)$ there is $s > 0$ such that $y\in B^{Y}_{s}(y)\subset Y\cap B^{X}_{r}(x)$. My attempt So far it has not been discussed the characterization of open sets as sets that each point is contained in an open ball contained in the set. Having said, could anyone give me some hint to solve such exercise? It is not homework. I am really interested in understanding the theory properly. Notation The notation $B^{X}_{r}(x)$ represents the open ball centered in $x$ with radius $r > 0$ within the set $X$. |
| Verifying continuity by topology Posted: 31 Jan 2022 08:46 AM PST I want to show that sum of two continuous functions is continuous using the topological definition only . I know the question could be answered by composing two continuous functions $(.,.):X\to\mathbb{K}^2$ and $+:\mathbb{K}^2\to\mathbb{K}$ over some infinite field $\mathbb{K}$ , and that has been answered several times in this site as well . But if I would want to use the basic open set definition , what would be the correct approach ? Here's what I tried . Consider two continuous maps $f:X\to\mathbb{K}$ and $g:X\to\mathbb{K}$ where $X$ is some topological space . Then for all open sets $U$ in $\mathbb{K}$ , we have $f^{-1}(U)=\{x\in X:f(x)\in U\}$ and $g^{-1}(U)=\{x\in X:g(x)\in U\}$ are open in $X$ . We need to show $(f+g)^{-1}(U)=\{x\in X:(f+g)(x)\in U\}=\{x\in X:f(x)+g(x)\in U\}$ is open in $X$ . But how to relate the $f^{-1}(U),g^{-1}(U)$ with $(f+g)^{-1}(U)$ I don't understand . I have a feeling that the fact union of two open sets is open might be useful here . Any help is appreciated . |
| Posted: 31 Jan 2022 09:02 AM PST I'm having trouble solving this one and would really appreciate any help. thank you in advance! so, the problem is: given a flow network with integer capacities on the edges and a max flow f on that network, I need to write an algorithm that determine whether there are at least 4 more different max flows on that given network. thanks again! |
| Solving second order differential equations for forced oscillation Posted: 31 Jan 2022 09:05 AM PST
Please refer to this image I have several doubts regarding the stapes involved in solving this Differential equations
2.How we reached $A(\omega)=\frac{1}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+\gamma^{2} \omega^{2}}}$ (Please give some detailed Solution) Please someone help |
| Show that two random variables have equal distribution Posted: 31 Jan 2022 08:49 AM PST Let $(X, \{\mathbb{P}_\vartheta; \vartheta \in (0,1)\})$ be a statistical model whith the density of $X$ given as $$ f_\vartheta(x) = \frac{\vartheta^{-1}}{2}\mathbb{1}_{[0,\vartheta]}(x) + \frac{(1 - \vartheta)^{-1}}{2}\mathbb{1}_{(\vartheta,1]}(x). $$
I tried this exercise for a very long time and didn't get any further so some help would be nice. |
| Posted: 31 Jan 2022 09:06 AM PST It is fairly well known that for any $k$-coloring of $\mathbb{Z}^2$, it is impossible to color the vertices such that you never get a rectangle with axes parallel to the x and y-axis with vertices of the same color. In fact, it is known that you cannot color an $(k+1) \times (\frac{k^2(k+1)}{2}+1)$ lattice of points that meets this criterion. For example, you cannot 2-color the $3\times 7$ lattice in this manner. You can, however, color the $(k+1) \times (\frac{k^2(k+1)}{2})$ lattice in this way. One such successful 2-coloring of the $3\times 6$ lattice is below.
So, if the $3\times 6$ lattice is 2-colorable in this fashion, it is a natural question to ask, "what about the $6\times 6$ lattice?". As it turns out, the answer to this is "no". We can prove this by first showing you cannot 2-color the $5\times 6$ lattice. After some experimentation, one sees that the only successful 2-coloring of the $4\times 6$ lattice is some permutation of the rows and columns of the below.
If one adds a column to the right, it becomes apparent that the top-right vertex can neither be red nor blue: in the case of the former, we eventually see that the fourth-row vertex cannot be red or blue, and in the case of the latter, we see the same for the third-row vertex. Therefore, the $6\times 6$ lattice cannot be colored like this - QED. However, what about the $5\times 5$ lattice? It becomes clear that we essentially have six unique ways to color in the first three rows, seen below (again, barring permutations of the rows and columns). I know that out of these six, you cannot 2-color the top-left one successfully. I have yet to try the others, but would not be surprised to find you cannot achieve this for any configuration.
For what it is worth, it can be shown that yes, the $4\times 4$ lattice can be successfully colored like this, which will be shown at the end. However, this is only for 2-coloring. What about 3-coloring, or 4-coloring? What about $n$-coloring in general? If I had to make a conjecture, the most I would be willing to say is that you can never make a $(\frac{k^2(k+1)}{2})\times (\frac{k^2(k+1)}{2})$ lattice that can be colored like this, but even for that I don't know where to start proving it, and I don't know how to move that upper bound any lower. Any thoughts?
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| Posted: 31 Jan 2022 09:04 AM PST Consider $W_t$ is a standard Wiener process. Assume $\tau = \inf\{t : W_t = a\}$. Consider $X_t = W_t \mathbb{1}(\tau > t) + (2W_{\tau} - W_t) \mathbb{1}(\tau \le t)$. We want to show that $(X_t, \mathcal{F}_t)$ is a martingale, where $\{\mathcal{F}_t\}$ is natural filtration of the Wiener process. I know that it's possible to show that the process is also Wiener, but it's much more complicated (since we use strong Markov property). Here I guess the situation is much easier. It's not hard to show that $X_t$ is $\mathcal{F}_t$ measurable and $L_1$. The major problem is to show $\mathbb{E}(X_t | F_s) = X_s$ precisely. I tried so: consider different cases
But I guess it's not so fair to write so, since $\tau$ is random variable and for some $\omega \in \Omega$ it might be true and for others not. Any hints? |
| Equation of a cubic Bezier curve Posted: 31 Jan 2022 09:08 AM PST
For a quadratic Bezier Curve defined by points $A, B, C$, with point $M$ on the curve interpolated by $i$, When points $A, M, C$ and angle $\alpha$ are given, $i$ and $B$ are: $$i^2 = \frac{(y_A-y_M)\sin\alpha-(x_A-x_M)\cos\alpha}{(y_A-y_C)\sin\alpha-(x_A-x_C)\cos\alpha}$$ $$x_B=\frac{x_A(1-i)^2+x_Ci^2-x_M}{2i(i-1)}$$ $$y_B=\frac{y_A(1-i)^2+y_Ci^2-y_M}{2i(i-1)}$$ for a full working out: $$x_D = x_A+i(x_B-x_A)$$ $$x_E = x_B+i(x_C-x_B)$$ $$x_M = x_D+i(x_E-x_D)$$ $$x_M = x_A(1-i)^2+2ix_B(1-i)+i^2x_C$$ $$x_B=\frac{x_A(1-i)^2+i^2x_C-x_M}{-2i(1-i)} = AB\sin\alpha+x_A$$ $$y_B=\frac{y_A(1-i)^2+i^2y_C-y_M}{-2i(1-i)} = AB\cos\alpha+y_A$$ $$AB = \frac{i^2(x_A-x_C)+x_A-x_M}{2i\sin\alpha(1-i)}= \frac{i^2(y_A-y_C)+y_A-y_M}{2i\cos\alpha(1-i)}$$ hence resulting the above 3 equations.
Now consider a cubic Bezier curve defined by points $A, B, C, D$ $1)$With given points $A, D$, $M$ interpolated by $i$,angles $\alpha, \beta$, Can $B, C$ and $i$ be derived? $2)$ if not, With given points $A, D$, $M$ (interpolated by $i$), $N$ (interpolated by $j$), angles $\alpha, \beta$, What are the equations for $B, C$, $i$ and $j$? |
| Posted: 31 Jan 2022 08:45 AM PST So, we are trying to find the distance of all of the data points about the mean. But, isn't it the other way around? When we use things like variance and MAD, aren't we actually finding the average distance of mean to all of the data points and not the distance of all of the data points from the mean? Isn't a more convincing way of finding a good number for a spread would be like this: Sum of "average distance" of each data point to the other divided by n(sample pop.). Now, think of this thing for a second. Because it has been over a week that I am trying to make sense of why do are they finding just the average distance of one number, i.e. mean, in variance? Formula is: Sum of the "average distances" of each point divided by n. By the sum of average distances I meant is: Let's say you have 4 data points: 2,4,8,12
I agree that variance and SD are the conventions. But, they give mean a special position. They say let's find the distance of all of the data points from the mean, but the way I see it they actually measure how far is mean to all of the data points. The analysis is incomplete. The way I proposed the formula, I know it's a little bit lengthy, but I take into account each number in the data set, find how far is each of them from each other, and then take an average to finally get to the point. I didn't give any special position to any number. Note: I understand that it will take time to find the measure for spread relative to what we are used to, i.e., variance and standard deviation. But, I think the important thing to realize is that we will, at last, use all of the data and measure it through programs. And, I can make a code in R that can easily calculate the spread with the formula I just proposed. |
| Existence elimination in Lean 3 Posted: 31 Jan 2022 08:58 AM PST Lean 3 is a theorem prover that implements the calculus of inductive constructions. Differently than Coq, Lean 3s kernel works proof irrelevant. This means that in the kernel of Lean all proofs of the same theorem are judgmentally (definitionally) equal and can be substituted for each other. Does that mean that if I have I thought being able to extract witnesses from existence proofs is one of the main arguments for Martin-Löf style type theories. |
| Posted: 31 Jan 2022 08:51 AM PST
I found the answers for small values of $n$. For $n=1$, the answer is $2$. For $n=2$, the answer is $6$. For $n=3$, the answer is $12$. But how to solve this problem for any $n$? |
| Equivalent condition to boundedness of a linear operator Posted: 31 Jan 2022 08:44 AM PST I have to prove the following statement:
I have been stuck on this problem for days, yet I feel I'm really close to solving it. Any hint, idea or advice would be appreciated. I haven't been able to prove any of the implications, but this is what I have right now: For the $(\implies)$ direction, assume that $T$ is bounded, and let $\{x_n\}$ be a sequence such that $\sum_{n=1}^\infty x_n$ converges absolutely. I'm trying to prove that $\sum_{n=1}^\infty T(x_n)$ converges on two steps. First I prove that the sequence $\{S_k\}_{k=1}^\infty$ where $$ S_k=\sum_{n=1}^kT(x_n) $$ is Cauchy. This is easy to show using the boundedness and linearity of $T$, and the fact that $\{\sum_{n=1}^k ||x_n||\}_{k=1}^\infty$ is Cauchy, because it converges. Then I would like to prove that the sequence $\{S_k\}$ has a convergent subsequence, as that would imply the result. However, I don't know how to show this. The only idea I had was to prove and use this statement:
as the continuity and linearity of $T$ would then imply what I want. However, I don't think that statement is true, as it would then imply that every normed space is Banach, which is obviously not true. So I have no idea on how to proceed now. In general I am having trouble to prove the convergence of almost anything, any advice for that would be really helpful. For the ($\impliedby$) direction, I am trying to use the sequential criterion for continuity to prove that $T$ is continuous, and since $T$ is linear that would prove that $T$ is bounded. Let $a\in X$ and $\{x_n\}$ a sequence in $X$ that converges to $a$. I want to prove that $T(x_n)\to T(a)$. I trying to use the same steps as in the other implication, i.e. I need to show that $\{T(x_n)\}$ is Cauchy and has a subsequence that converges to $T(a)$. Here however, I managed to show the existence of a convergent subsequence, but I don't see how to show that the sequence is Cauchy. To see that it has a subsequence convergent to $T(a)$, notice the following: As $x_n\to a$, for each $k\in\mathbb{Z}^+$ there exists an $n_k\in\mathbb{Z}^+$ such that $$ ||x_{n_k}-a||<\dfrac{1}{2^k} $$ And since $\sum_{k=1}^\infty 2^{-k}$ converges, we have that $\sum_{k=1}^\infty||x_{n_k}-a||$ converges. By hypothesis, this implies that $\sum_{k=1}^\infty T(x_{n_k}-a)$ converges, and from here we get that $||T(x_{n_k})-T(a)||=||T(x_{n_k}-a)||\to 0$ as $k\to\infty$, and as such $T(x_{n_k})\to T(a)$. If now I manage to prove that $\{T(x_n)\}$ is Cauchy, I would get the result. However I have no idea on how to do this. Thanks in advance for any help you could give me! |
| Posted: 31 Jan 2022 08:48 AM PST Consider $$f(x) = \begin{cases} 5 x^2 (2 - \sin \tfrac{1}{x}) & \text{ if } x \neq 0 \\ 0 & \text{ if } x = 0. \end{cases} $$ For any given $\delta > 0$ it is very clear from the graph of $f$ that there is at least one (in fact infinitely many) point $x_\delta$ in $(-\delta, \delta)$ at which $f'(x_\delta) = 0$. Somehow, I am not getting how to mathematically prove this. Can anyone help me out?
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| Is this derivation of $\int{\frac{1}{x^2-a^2}}dx$ correct? Posted: 31 Jan 2022 08:41 AM PST Let $a>0.$ Using substitution, determine $$\int{\frac{1}{x^2-a^2}}\,\mathbb{d}x.$$ My book's attempt: $$\int{\frac{1}{x^2-a^2}}\mathbb{d}x\\ [\text{let}\ x=a\sec\theta,\ \therefore dx=a\sec\theta\tan\theta d\theta]\\ =\int{\frac{1}{a^2\sec^2\theta-a^2}}a\sec^2\theta d\theta\\ =\int{\frac{1}{a^2(\sec^2\theta-1)}}a\sec\theta\tan\theta d\theta\\ =\int{\frac{1}{a^2\tan^2\theta}}a\sec\theta\tan\theta d\theta\\ =\frac{1}{a}\int{\frac{\sec\theta}{\tan\theta}}d\theta\\ =\frac{1}{a}\int{\frac{\frac{1}{\cos\theta}}{\frac{\sin\theta}{\cos\theta}}}d\theta\\ =\frac{1}{a}\int{\csc\theta d\theta}\\ =\frac{1}{a}\int{\frac{\csc\theta(\csc\theta+\cot\theta)}{(\csc\theta+\cot\theta)} d\theta}\\ =-\frac{1}{a}\int{-\frac{\csc^2\theta+\csc\theta\cot\theta}{(\csc\theta+\cot\theta)} d\theta}\\ =-\frac{1}{a}\ln|\csc\theta+\cot\theta|+C$$
$$=-\frac{1}{a}\ln\left|\frac{x}{\sqrt{x^2-a^2}}+\frac{a}{\sqrt{x^2-a^2}}\right|+C\\ =-\frac{1}{a}\ln\left|\frac{x+a}{\sqrt{x-a}\sqrt{x+a}}\right|+C\\ =-\frac{1}{a}\ln\left|\frac{\sqrt{x+a}}{\sqrt{x-a}}\right|+C\\ =-\frac{1}{a}{\ln\left|\frac{x+a}{x-a}\right|}^{\frac{1}{2}}+C\\ =-\frac{1}{2}.\frac{1}{a}\ln\left|\frac{x+a}{x-a}\right|+C\\ =\frac{1}{2a}{\ln(\left|\frac{x+a}{x-a}\right|)}^{-1}+C\\ =\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C.$$ Someone told me that this work contains many errors. The second line clearly has a typo ($\sec^2\theta$ should be replaced with $\sec\theta\tan\theta$ there); what are the other errors? |
| finding derivatives of variables in multivariable taylor polynomial Posted: 31 Jan 2022 08:51 AM PST Given: $$F(x,y) = -6 -4(x-4) +6(y-6) +8(x-4)^2 +9(x-4)(y-6) -4(y-6)^2 + R_2$$ and that $F(4,6)=-6$ |
| An 8th grade contest-math puzzle Posted: 31 Jan 2022 08:56 AM PST In a competitive sort of exam, the following question was aimed at 8th graders and above-
On top of this, the question may have multiple answers (and the student is expected to report all of them) as well. I don't have any idea about how to attack this problem. Of course, brute forcing is not a solution since in that case you have to check through $\sim 22^{14}\approx 6.2\times 10^{18}$ cases which is an impossibility even for a computer programme. After doing some Google search recently, I found out something similar called Hexagonal Tortoise Problem and a paper about it. But, these couldn't take me anywhere. One random observation is that if we indeed need to brute force the solution with some educated guesses, then maybe the disc connecting $4,7,12$ is the one to start with (since that's the one which is most restricted). But, even in that case, there are too many choices to deal with. All ideas are welcome. |
| Prove |Re z| less than or equal to |z| and |Im z| less than or |z| Posted: 31 Jan 2022 09:07 AM PST |Re z| less than or equal to |z| ; z = x + yi |Re z|= |x| --> sqrt(x^2) |z| = sqrt(x^2 + y^2) sqrt(x^2) less than or equal to sqrt(x^2 + y^2) (therefore y^2 greater than or equal to 0) then the answer is sqrt(x^2) less than or equal to sqrt(x^2 + y^2) is this correct? and do the same thing for Im z? is this correct? Thanks! |
| Good approximation to $\ln(x)$ for $x$ in $1 < x < e$ Posted: 31 Jan 2022 09:05 AM PST I'm looking for a simple function that gives a good approximation to $\ln(x) $ within $1 < x < e$. Do you have anything in mind? I'm not looking for an infinite function, but a short and finite version of it might be good, if it's also a simple solution as well. |
| RYB and RGB Color Space Conversion Posted: 31 Jan 2022 09:06 AM PST I am working on a project where I need to convert colors defined in RGB (Red, Green, Blue) color space to RYB (Red Yellow Blue). I managed to solve converting a color from RYB to RGB space based on the article - Paint Inspired Color Mixing and Compositing for Visualization. I convert a color from RYB to RGB with this "algorithm": So the values of here is how I calculate the value of for My problem is that now I want to convert from RGB to RYB back. In other words, knowing the values of Any help is appreciated. |
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Recent Questions - Stack Overflow |
| Creat a list with repeated terms by adding a multiplier index in Pandas Dataframe Posted: 31 Jan 2022 08:00 AM PST Given a dataframe like this: which leads to:
I would like to generate the following dataframe:
where the final column List contain a multiplier index that indicates how many times the term appears consecutively in the list. Could you suggest a pandas instruction that solves this task? | ||||||||||||
| Integrating DynamoDB Table with Lambda to fetch data for Lex Chatbot - No response Posted: 31 Jan 2022 07:59 AM PST I am building a AWS Lex bot integrated with Lambda which can fetch data from table in DynamoDB. I have granted Full Access to DynamoDB for Lambda by adding new role. But every time it is showing- "Invalid Lambda responses". Below I am sharing the Lambda code in python. | ||||||||||||
| How to request refresh token from Etsy using OAuth-2.0 from C# Posted: 31 Jan 2022 07:59 AM PST Having failed miserably using C# code to get an Etsy OAuth 2 token, I resorted to using Postman for the initial request figuring that it would then be a simple matter of requesting a refresh token as and when needed. So, from Postman, all good, I can sign in and grant access, which then causes Etsy to respond with a shortlived access token (and a refresh token to use to request another access token without having to go through the whole grant process again). However, I cannot figure out how to get another access token using this refresh token as Etsy keeps on responding with grant_type is required. OK, the initial response from Postman following the success grant access is: And following their docs to request a refresh OAuth token, it implies that all I need to do is to make a POST request to https://openapi.etsy.com/v3/public/oauth/token, adding the following parameters to the JSON body with content type: application/x-www-form-urlencoded: Thus, setting all this up in Postman and looking at the generated code for the request gives: *NOTE - I had to add in the header the parameter x-api-key otherwise it would fail with "error": "Missing x-api-key header or client_id" Alas, the response from Etsy is always This post guided me in making the initial request. Why is it moaning about grant_type when it's in the body as requested!?!?!? | ||||||||||||
| Posted: 31 Jan 2022 07:59 AM PST With this code I'd got data from oracle, postgres or sql server many times, how ever this time receiving error message in the subject Comlete log message is below: py4j.protocol.Py4JJavaError: An error occurred while calling o33.showString. : org.apache.spark.SparkException: Job aborted due to stage failure: Task 0 in stage 0.0 failed 4 times, most recent failure: Lost task 0.3 in stage 0.0 (TID 3) (10.42.8.125 executor 1): UnknownReason Will be appreciated with any help. Regards, Mert. | ||||||||||||
| SFTP How to List Large # of Files Posted: 31 Jan 2022 07:59 AM PST I am currently working on SFTP load to GCS bucket. However, I am able to do it for a limited number of files in any given SFTP directory. However, if the directory has too many files (or files within another folder), I am not able to do a simple ls & get the list of files to download from SFTP. Following is the working code to get the list of files in any given directory recursively from sftp: I get the following exception after some time of execution: | ||||||||||||
| Gstreamer how to get element information from STATE_CHANGED message from the bus Posted: 31 Jan 2022 07:59 AM PST I am working with Gstreamer and its Python bindings. Consider the following When the message type is | ||||||||||||
| app update in google developer console for payment subcribtion Posted: 31 Jan 2022 07:59 AM PST I had completed my first app using android studio and firebase and I am now trying to upload to google play store through google play console. I have two sets of users in my app , 1. Buyers and 2. Sellers. Currently my app is free for both buyers and sellers, so in data safety section in developer account, in Financial info section I didn't select any thing like credit card or debit card , but in next update I am planning to make seller subscription charges or 1 year or 2 year like that , so I will be collecting their credit or debit card information... I like to know whether financial information can be changed in every update or I need to plan prior and select payment information in first upload itself. | ||||||||||||
| spark trying to convert .Mf4 to save as .txt files, getting issue Posted: 31 Jan 2022 07:59 AM PST I have a spark code which reads .mf4 files and write as a .txt file. Also running this code in databricks. Could you please suggest? Even after increasing the executor memory in data bricks in cluster, still having the issue. I am running this code in databricks , I have 5Gb mf4 file as a input. When I try to run small file there is no issue. But when I use this 5GB mf4 file getting | ||||||||||||
| unit testing in springboot - best approach Posted: 31 Jan 2022 07:59 AM PST I'm trying to figure out what is the best approch to unittesting in SpringBoot app. Before last friday i thought that best approach is to test:
But then i watched this -Ian Cooper - TDD, Where Did It All Go Wrong and I've realized that my testswas implementation tests, and I should only mock db, filesystem, other systems, not my application. And I'am wondering how should I be doing this? Because when I will be creating all objects myself it will still be "testing implementation", so I can use Can someone clever than me could share his experience or thougths on this? | ||||||||||||
| node-polyfill-webpack-plugin not recognizing in webpack.config.js Posted: 31 Jan 2022 07:59 AM PST I am trying to use knex for sqlite local storage in an electron application and have gotten the "Breaking Change" message due to knex uses some of the polyfll components that seize to exist on webpack v5. I heard that I can use node-polyfill-webpack-plugin to put the polyfills back in. For this project, I created a "webpack.config.js" at root and put the following contents inside: Here is my package.json: However, when I run "npm start", I am still getting all of the webpack breaking change messages. Any help would be much appreciated. I have never seen this problem until I am trying to use any type of local storage (such as sqlite or NEDB) and have done an extensive search on Stack Overflow and tried several of those suggested methods (in fact the node-polyfill-webpack-plugin was from reading this site) but I still cannot get it to work. | ||||||||||||
| Xamarin: Modal from not popping Posted: 31 Jan 2022 07:59 AM PST since a couple of weeks I have the problem that a modal form which provides a spinner and a label is not popping anymore. It was running in older versions but not with all NuGet Packages updated. Any idea what this can be? Funnywise it is working on some forms and not on others. | ||||||||||||
| Posted: 31 Jan 2022 07:59 AM PST Can I use JSONPath expressions to get a string out of an array of string? Example, here is an array of strings: and the expresion will result into however I want to have something like or better is this possible? | ||||||||||||
| Posted: 31 Jan 2022 07:58 AM PST I am exploring AWS sage maker for deploying the machine learning models with single container/docker image and wants to achieve scalability too. For your info, we have written the training and prediction code in such a way that so, same code is being used for different customers. Only the ENV variables,VERSION_NUMBER are different according to customer. VERSION_NUMBER : Refers to S3 folder name as current timestamp which contains latest model artifact. for e.g. If you read it carefully, you will see same docker image is being refereed everywhere So,my doubts are as follow.
Please let me know your inputs on the same. | ||||||||||||
| Replace strings inside of a file with the same name file for multiple files Posted: 31 Jan 2022 07:59 AM PST Im working with coco datasets, so I have thank you. | ||||||||||||
| best approche to asynchronous programming Posted: 31 Jan 2022 07:59 AM PST I'm building an app with NodeJs that get data from a websocket server and then store or update these data on an array of objects, after that the data on the array is analysed (heavy calculations) none stop. I need help to choose the best way to execute these two tasks independently from each other. I tried parallel tasking but it's didn't work, the calculation part takes 250 ms and the websocket takes only 1ms, the app waits untill calculation is done and miss all data from websocket server.. What's the best approach to achieve this? PS: i'm a self taught please don't down vote :) | ||||||||||||
| How do you POST html form data with javascript Posted: 31 Jan 2022 08:00 AM PST Im trying to POST data from a html form using javascript however on submission I believe no data is actually being sent because my alert is showing no data. when i added an alert box the output came up as "{}" here is my full code below if anyone can spot my errora | ||||||||||||
| how to return the pointer of a specific value in a tree in C? Posted: 31 Jan 2022 07:59 AM PST using this struct : I'm trying to write a code that return the pointer of the the node if node ->content== x I have tried this so far but I don't think its right : can u help me correct my code? | ||||||||||||
| Common Lisp Package Definition with Dependencies for Exploloration at the REPL? Posted: 31 Jan 2022 07:59 AM PST This is another take at the question of packages and systems especially for usage locally in personal projects (maybe similiar to HOWTO definition and usage of Common Lisp packages (libraries)? from 2014). What are recommendations for how to approach this and can links be provided for sources that describe the recommended approach in 2022? CL's long half-life means lots of old blogs and books are still useful, but unfortunately not all of them, and it is hard for the novice to know which are the still viable guides. Here is the start of a file that I might write for a simple personal exploration project. My basic working mode would be to start slime, load and compile this in emacs, and then change to the slime repl and use The leading If I am thinking of using my package elsewhere someday is it recommended that I make it a system, perhaps with quickproject? Or is it the more traditional approach to just In summary, the questions are about the writing of code that is for personal use, might be a small file, or might be a few files that depend on each other and a few external systems. What are some of the reasonable approaches and resources that describe those approaches? And which are the ones that would be easiest to scale up if the project grew into something that one wanted to share? | ||||||||||||
| GravityForms Dynamic Confirmation Redirects Posted: 31 Jan 2022 08:00 AM PST I have set up a gravity form with a single line text field. I am using it to check for zip codes and cities. If someone types in the correct zip code it will redirect to a specific page. Also if someone types in a zip code or city and there is no matching value it will also redirect to a certain page. I have tried this code. But I can not seem to get it to work correctly. It works if I have one number but if I add multiple numbers separated by a comma it does not work. Please help if possible. | ||||||||||||
| Firebird reading database issue on iis asp.net core 5 Posted: 31 Jan 2022 07:58 AM PST I want to develop a demo project in ASP.NET Core MVC with embedded Firebird v2.5. I read Firebird connection string information from a text file. Because I will change database in project presentation thus I must read from a text file. Or what is the your idea? In this project I create a When I work locally, everything is fine. I can do all CRUD operations. But when I upload my project, I see this screen:
I added all privileges is IIS but I can't make this work - how can I solve this? | ||||||||||||
| Wrong number of item passed 13, placement implies 1 while pivot using pandas Posted: 31 Jan 2022 07:59 AM PST I am getting the above error if I add 'column' in the pivot table function of pandas. Without 'Column' it works fine. It would be great help, I am struck mid of a project. Thanks You | ||||||||||||
| Posted: 31 Jan 2022 07:59 AM PST This component renders window width when mounted for first time for which useEffect() runs, Ok... then when the browser window is resized the and then it triggers for which the component re-renders with the state change of | ||||||||||||
| how to add and remove specific attribute values Posted: 31 Jan 2022 08:00 AM PST In angular it is possible to add and remove class attribute values from an element by specifying a boolean that is bound to a specific variable in the controller: Is it possible to do the same for other attributes? Similar to below (which does not work) | ||||||||||||
| nested for loop issue for subfolders and files makefile Posted: 31 Jan 2022 07:59 AM PST I have a logical issue, trying to do a nested for loop for a fpöder which holds two subfolders with code files in it. But I receive the following error Anyone knows what Iam doing wrong? | ||||||||||||
| Posted: 31 Jan 2022 07:59 AM PST I want to make my first Wordpress plugin. I made perfect work js, css, html and php widget. Now I want to insert it into plugin. I watched some Udemy.com curses but still do not know exactly how to make. Please give me support. First, I want to add scripts and here style and html i have a second script there is no error, what's making me proud, but first it look ugly because look different then in localhost. Second, javascript do not work at all. Update: Finally I know what is require to change the look. My plugin is container into div which have width: 600px, I tried change its possition to another one using script which had 1000px but not work; also change params of the first one. Another problem is words are callopse one by one like a s d f I tried it remove by Update2: Yuap, it was a jQuery conflict. I change all $ to jQuery and start running. But still have issue with appearance of plugin. The main | ||||||||||||
| Emscripten: How to catch JS exception? Posted: 31 Jan 2022 08:00 AM PST Emscripten 'val.h' API allows calling methods of JS objects, however, C++ try-catch won't catch JS exception. Consider this example: The JS exception makes the 'test' function stop and no There's 1 issue here but it's still open: https://github.com/emscripten-core/emscripten/issues/11496 | ||||||||||||
| TVCardView rounded corners strange behaviour Posted: 31 Jan 2022 07:59 AM PST I'm setting up a new tvOS app with a Has anyone experience with | ||||||||||||
| Posted: 31 Jan 2022 07:58 AM PST OK, this is driving me crazy and the solution I am sure will be pretty simple and I will probably bang my head into a wall. I have a query where I have client data. On some queries I ONLY want clients between a certain date that have had a client visit at some point within that date range, which correspond to Call_Report_ID's 3, 8 or 12. For these clients, I want it to bring back ALL data within the date range, not just the rows with this client visit code. For the other clients who DON'T have a client visit, I want to exclude them from the data. Here is what I have been playing around with and cannot get it working at all. So here is an example of What I want: In the above when I run this query, I want it to return both rows from Customer 5 since they have a call report ID of 3,8, or 12 in ANY row within the date range, and exclude both rows from Customer 6 since they do not have a call_report_Id with those types in a date range However, every time I run it, it returns all rows for this user, not just the ones specified in the WHERE EXISTS SubQuery. What am I dong wrong here? | ||||||||||||
| How do I notify VueJS/Vuex of an value change in an array? Posted: 31 Jan 2022 07:59 AM PST when I update a value in an array of the store, the interface doesn't reflect the change. Goal: I'm trying to display a simple minesweeper grid. Once I click on a cell, the object attach to that cell should update isRevealed to true to say is was revealed and Vuejs should add the class dug to that cell. What's working: isRevealed is switched to true. What was working: using only props everything was working, but trying to learn VueJS and including Vuex store the UI doesn't update anymore with the array. Structure: App shows a grid and grid shows multiple cell. store.js for Vuex GameGrid contain the following: 10 arrays of 10 objects: grid.vue, nothing special, display every cell cell.vue: Edit accoridng to pirs answer:Grid.vue file Note that I get My functions to set isRevealed are on the cell.vue though. I'm still seeing no UI updates (except when I'm saving?). Cells with Here is how I'm populating GameGridThis function is calling when the user press a button on the hmm, home page? App.vue file. | ||||||||||||
| Simpler way of sorting three numbers Posted: 31 Jan 2022 07:58 AM PST Is there a simpler and better way to solve this problem because
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