Recent Questions - Mathematics Stack Exchange |
- converting a differential algebraic system of equation into a lagrange equation
- Conceptual question about the dot product
- Let $A$ and $B$ be nonempty disjoint closed sets such that $A$ is compact. There exists a neighborhood $V$ of $0$ such that $(A+V) \cap B=\emptyset$
- Changing the Frequency of a Complex Sinusoid
- Areas of polygons in the the language of Riemannian geometry
- Solving $Q {(1-E)}^A + E^{AQ} < 1$
- Proof for one of the theorem mentioned in the "Advance Calculus" book
- $O_{K_1}$(Ring of integers of $K_1$) is also written as direct limit of $O_{K_i}$, where $K_i$ is finite extension of $K$
- The Converse of the Radon-Nikolydym Theorem for Signed Measures
- Let $f$ be a real-valued function on $[0,1]$ with a continuous second derivative. Assume that $f(0)=0, f'(0)= 1, f''(0)\neq 0$, and $0<f'(x)< $ ....
- Eigenfunction of a matrix that only acts on second variable cannot depend on first var.
- How to solve the equation $x + \sin x = A$?
- Orthogonality of Sine and Cos [closed]
- Need pre-requisites for Machine Learning by Kevin P Murphy
- How to compute $\lim\limits_{\theta \to \frac{\pi}{2}^+} \frac{\cos{\theta}-2\sin{\theta}\cos{\theta}}{\sin^2{\theta}-\cos^2{\theta}-\sin{\theta}}$?
- Show that the following function is continuous on R^2
- Frobenius norm derivative for a quadratic form
- Integrating $\frac{1}{x}$
- About using fix-points to define partial recursive functions
- Question about a particular property of sheaves
- Cumulative distribution of mixed random variable function
- Finding the weak form of a PDE with a tensor argument
- Fokker-Planck: uniqueness and attractiveness of stationary distribution (gradient systems)
- Rank nullity theorm proof
- Fundamental Theorem of Calculus when the integrand is logarithmic derivative
- $2$ Urns, $2$ colors, pick 1 ball from each
- Least Prime in Arithmetic Progression Conjecture
- Are these equations "properly" defined differential equations? (finite duration solutions to diff. eqs.)
- Evaluate $\int_{y=-\infty}^1\int_{x=0}^\infty 4xy\sqrt{x^2+y^2}\mathrm dx\mathrm dy$
- Doubt in Definition of Inverse of $y=x^n$
| converting a differential algebraic system of equation into a lagrange equation Posted: 06 Jun 2022 11:41 AM PDT I have searched in quite a few different books now, but I'm still confused about the following: Consider a differential algebraic system of equation of Hessenberg Index-2 form: \begin{align} \dot{x} &= f(t,x,z), \\ st. 0 &= g(t,x). \end{align} I wish to convert this system into a differential algebraic system of equations using a Lagrange multiplier. I believe it should end up looking like this: \begin{align} M(t,x,\dot{x}) \ddot{x} &= f(t,x,\dot{x}) + C^\top \lambda, \\ 0 &= g(t,x), \end{align} where $M$ is some unspecified mass matrix that is potentially just the identity in my case?, and $C=\frac{\partial g}{\partial x}$. Apparently deriving this result is a fairly standard thing to do, and I can find variations of it in a lot of books, but I have yet to find anything that actually gives me all the steps I need in order to understand it. From what I understand you treat the problem as a constrained optimization problem and define: \begin{equation} \mathcal{F}({x}) = \int_{t_0}^{t_1} L(t,x, \dot{x}) dt \end{equation} subject to the constraints \begin{equation} \phi(x) = \int_{t_0}^{t_1} h(t,x) dt = 0 \end{equation} Now I'm not sure exactly how $\mathcal{F}/L$ is related to $f(t,x,z)$ or how $\phi / h$ is related to $g(t,x)$. But I understand that the trick is then to use the Euler-Lagrange equation: \begin{equation} \frac{\partial L}{\partial x} - \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}} \right) = \frac{\partial h(x)}{\partial x} \lambda \label{eq:1} \end{equation} |
| Conceptual question about the dot product Posted: 06 Jun 2022 11:37 AM PDT If $u$ and $v$ are vectors orthogonal to $q$ and $c$ is a scalar, then $q ·(u − cv) = 0$ I created the diagram below to understand what's going on: Is drawing a picture the best way to look at this or is there a better way to deal with these sorts of conceptual questions? |
| Posted: 06 Jun 2022 11:32 AM PDT I'm reading this theorem and its proof.
Proof: For each $a \in A$, fix $U_{a} \in \mathcal{U}(0)$ such that $\left(a+U_{a}\right) \cap B=\emptyset$, and an open $V_{a} \in \mathcal{U}(0)$ with $V_{a}+V_{a} \subset U_{a}$. Since the sets $a+V_{a}(a \in A)$ form an open covering of the compact set $A$, there exists $a_{1}, \ldots, a_{n} \in A$ such that $A \subset \bigcup_{i=1}^{n}\left(a_{n}+V_{a_{i}}\right)$. The set $$ V=\bigcap_{i=1}^{n} V_{a_{i}} $$ is a neighborhood of 0 , and $A+V \subset \bigcup_{i=1}^{n}\left(a_{i}+V_{a_{i}}+V\right) \subset \bigcup_{i=1}^{n}\left(a_{i}+V_{a_{i}}+V_{a_{i}}\right) \subset$ $\bigcup_{i=1}^{n}\left(a_{i}+U_{a_{i}}\right)$. Then $(A+V) \cap B \subset \bigcup_{i=1}^{n}\left(a_{i}+U_{a_{i}}\right) \cap B=\emptyset$. Below my shorter proof which I found more intuitive. Could you have a check on it? My attempt: Assume the contrary that for each neighborhood $V$ of $0$, there is $x_V \in (A+V) \cap B$. Let $a_V \in A$ and $y_V \in V$ such that $x_V = a_V + y_V$. Because $A$ is compact, there is a subnet $(a_{\psi(d)})_{d\in D}$ and $\overline a\in A$ such that $a_{\psi(d)} \to \overline a$. By construction, $y_{\psi(d)} \to 0$, so $x_{\psi(d)} \to \overline a$. Because $A, B$ are closed, we get $\overline a \in A \cap B$, which is a contradiction. |
| Changing the Frequency of a Complex Sinusoid Posted: 06 Jun 2022 11:29 AM PDT Given a complex sinusoidal wave $f(t) = Amplitude *e^{i*2*pi*Frequency*t}*e^{i*Phase}$. How do I have to manipulate $e^{i*2*pi*Frequency*t}$, to effectively half the frequency without touching the term itself. For example I know that ${(e^{i*2*pi*Frequency*t})}^2 = e^{i*2*pi*2*Frequency*t}$ essentially doubles the Frequency. I was trying $\sqrt[2]{e^{i*2*pi*Frequency*t}}$, but apparently roots do weird things with complex numbers |
| Areas of polygons in the the language of Riemannian geometry Posted: 06 Jun 2022 11:28 AM PDT I have a pedantic and somewhat vague question that has been bothering me a bit and that I was hoping for some clarification on. I'm sure it is a standard issue that there is a standard way of dealing with and I was hoping for some pointers. I've been thinking a bit about differential forms and volume of Riemannian manifolds which is what prompts the following question. Suppose $X^2$ is for simplicity a compact Riemannian manifold of dimension 2. I know how to define the area of codimension 0 submanifold $M$ of $X$, I take the volume form on $X$ and I integrate it over $M$. What about when $M$ is not a submanifold but is still "reasonably nice"? For me, reasonably nice means, maybe $M$ is a compact subset whose boundary is a union of a bunch of smooth arcs. Or maybe the boundary is a union of geodesics - so we just have a polygon. Well, then I imagine I can define the area of $M$ as the supremum of the areas of all subsets of $M$ that are submanifolds. Or maybe I should just develop all the necessary material for manifolds with corners so that it covers this case... I was wondering what the proper way to formalize all of this might be and possibly if there was a reference that does this? I also don't know exactly what "reasonably nice" should mean exactly - I guess in dimension 2 I'd like it to cover poygons and in dimension 3 for it to cover polytopes.... |
| Solving $Q {(1-E)}^A + E^{AQ} < 1$ Posted: 06 Jun 2022 11:32 AM PDT Let $\lambda_1 = Q {(1-E)}^A$ and $\lambda_2 = 1 - E^{AQ}$. I'd like to find the conditions where $\lambda_1 < \lambda_2$, i.e. solve: $Q {(1-E)}^A + E^{AQ} < 1$, subject to: $0 \le E \le 1$. Any help is appreciated... |
| Proof for one of the theorem mentioned in the "Advance Calculus" book Posted: 06 Jun 2022 11:35 AM PDT I'm unable to find a proof for one of the theorem menionted in an advanced Calculus book. Please help with a link to the text containing the proof or explain the proof. Image of the page containing the theorem Link to the text - https://people.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf Page 75 Theorem: If P1(t) and P2(t) are relatively prime polynomials, then there exist polynomials a1(t) and a2(t) such that a1(t)P1(t) + a2(t)P2(t) = 1 Thanks in advance. |
| Posted: 06 Jun 2022 11:30 AM PDT Let $K$ be a field. Let $K_1$ be algebraic extension of $K$. Then, $K_1$ can be written as direct limit of finite extensions of $K$. Then, I heard
How can I prove this ? My thought(try) : My intuition is that, when $K_1⊆K_2$, $O_{K1K2}＝O_{K_2}$, in this way, union(though we cannot always take union), is $O_{\cup{K_i}}$. Maybe I need to use universality of direct limit. |
| The Converse of the Radon-Nikolydym Theorem for Signed Measures Posted: 06 Jun 2022 11:32 AM PDT I noticed that on Wikipedia there is a statement of Radon Nikoldym Theorem for signed measures: https://en.wikipedia.org/wiki/Radon–Nikodym_theorem#For_signed_and_complex_measures. I would like to see if we can show the converse statement as I haven't seen it being shown anywhere else: Statement in Interest: Let $(X, \mathcal{F}, \mu)$ be a measure space and $f: X \to \mathbf{R}$ be a $\mu$-absolutely-integrable function. Then there exists a finite signed measure $\lambda: \mathcal{F} \to \overline{\mathbf{R}}$ such that for all $E \in \mathcal{F}$, we have $$ \lambda(E) = \int_E f \,d\mu. $$ Moreover, we have $\lambda: \mathcal{F} \to \overline{\mathbf{R}}$ satisfies the property that if $A \in \mathcal{F}$ is such that $\mu(A) = 0$, then $\lambda(A) = 0$. Furthermore, if $g: X \to \mathbf{R}$ is a $\lambda$-semi-integrable function, then $gf: X \to \mathbf{R}$ is a $\mu$-semi-integrable function and for any $E \in \mathcal{F}$ we have $$ \int_E g \,d\lambda = \int_E g f \,d\mu. $$ For the question to make sense, we need to define integration with respect to signed measures: We shall define $g \in \mathcal{M}(X, \mathcal{F})$ be absolutely integrable as functions such that $\int_X |g| \,d\lambda^+ < +\infty$ and $\int_X |g| \,d\lambda^- < +\infty$, where $\lambda^+ - \lambda^- = \lambda$ is the Hahn Decomposition of the signed measure $\lambda$. Moreover, we define the absolutely integrable integration as \begin{align*} \int_X g \,d\lambda &:= \int_X g \,d\lambda^+ - \int_X g \,d\lambda^-. \end{align*} I was able to show that $\lambda$ is a finite signed measure and it is absolutely continuous with respect to $\mu$. However, I could not prove $\int_E g \,d\lambda = \int_E gf \,d\mu$, not even for simple functions. Here is my attempt so far: To show this ($\int_E g \,d\lambda = \int_E gf \,d\mu$), we start with the case where $g$ is a simple function. That is, suppose we have $g(X) = \{ t_1, \cdots, t_n \}$ where $n \in \mathbf{N}$. Then we have that $$ g = \sum_{i = 1} ^n t_i \chi_{g^{-1}(\{ t_i \})}. $$ Therefore, we have by the sequence of sets $\{ g^{-1}(\{ t_i \}) \}_{i = 1} ^n$ are pairwise disjoint that \begin{align*} \int_E g \,d\lambda &= \sum_{i = 1} ^n t_i \lambda(g^{-1}(\{ t_i \}) \cap E) \\ &= \sum_{i = 1} ^n t_i \int_{g^{-1}(\{ t_i \}) \cap E} f \,d\mu \\ &= \sum_{i = 1} ^n \int_E t_i f \chi_{g^{-1}(\{ t_i \})} \,d\mu \\ &= \int_E f \sum_{i = 1} ^n t_i \chi_{g^{-1}(\{ t_i \})} \,d\mu \\ &= \int_E f g \,d\mu. \end{align*} I am stuck at this point, not being able to justify the second line of equality I wrote above. I need to claim some kind of unique representation here, but I am not able to do so. Any advice is appreciated. |
| Posted: 06 Jun 2022 11:20 AM PDT Let $f$ be a real-valued function on $[0,1]$ with a continuous second derivative. Assume that $f(0)=0, f'(0)= 1, f''(0)\neq 0$, and $0<f'(x)<1 $ for all $x\in(0, 1]$. Let $x_1, x_2,\ldots$ be a sequence with $0< x_1 \leq 1$ and with $$x_n = f\left(\dfrac{x_1+x_2+\cdots+x_{n-1}}{n-1}\right)$$ for $n \geq 2$. Prove that $\displaystyle\lim_{n\to\infty}x_n\ln n = −2/f'' (0)$. I tried to use fixed point theorem by converting it to form $x_n=\phi(x_{n+1})$ but it is not coming. Please help regarding this. |
| Eigenfunction of a matrix that only acts on second variable cannot depend on first var. Posted: 06 Jun 2022 11:11 AM PDT I'd like to prove that for a matrix that only acts on $y$ the eigenvectors of this matrix cannot depend on $x$, but instead are functions of $y$ only. Note: the involved matrix is a hermitian matrix |
| How to solve the equation $x + \sin x = A$? Posted: 06 Jun 2022 11:11 AM PDT I have tried Wolfram Alpha, with no success. I would be satisfied with an answer in special functions. |
| Orthogonality of Sine and Cos [closed] Posted: 06 Jun 2022 11:07 AM PDT |
| Need pre-requisites for Machine Learning by Kevin P Murphy Posted: 06 Jun 2022 11:06 AM PDT Hi and thanks for taking the time to read my question. I have a goal to read Kevin P Murphy's Machine Learning, A probabilistic Perspective. I have only read half way through an introductory real analysis textbook (Steven L Ray's Analysis with an introduction to proof). I also own Zorich's Mathematical Analysis I and II, as well as Paul L Halmos' Finite Dimensional Vector Spaces. My current plan is to go through the vector algebra and the real analysis books, then read through the chapters on higher dimensional derivatives/integration from Zorich's Analysis I. For anyone who has read Kevin's Machine Learning textbook, will this be a good enough background in the linear algebra/calculus required? If not, what would you recommend and why? Also, are there any other prerequisites for reading Kevin's book aside from linear algebra and calculus? I am also considering reading the chapters on measure theory in Zorich's Analysis II, but unsure if this would be really necessary (as I am learning the measure theoretic approach to probability at University). I would like to be as rigorous as possible (which is why I am learning calculus from an analysis book rather than a "calculus" textbook), but I am happy to leave the derivations of the probability distributions until later. It is quite difficult to offer a 100% accurate description of what exactly I am after, so I hope I have given enough detail to enable you to come up with some tips/advice. Finally, in case it were not clear, I would prefer to learn the theory of machine learning techniques, as I can always read a more "applied" text thereafter. Thus the focus is on the theory of machine learning techniques. Please do correct me if I have made any errors. Thanks! |
| Posted: 06 Jun 2022 11:39 AM PDT How can we compute the limit $$\lim\limits_{\theta \to \frac{\pi}{2}^+} \frac{\cos{\theta}-2\sin{\theta}\cos{\theta}}{\sin^2{\theta}-\cos^2{\theta}-\sin{\theta}}\tag{1}$$ Here is context on why I want to solve this limit. Consider the function $f(\theta)=1-\sin{\theta}$. If $(r,\theta)$ are polar coordinates, then $$r=f(\theta)\tag{2}$$ represents the following curve in polar coordinates (it's called a cardioid)
Now suppose we want to verify that the tangent at $x=0$ is vertical. Note that $r=0 \implies 1-\sin{\theta}=0 \implies \theta=\frac{\pi}{2}$. That is, The point $(0,0)$ on the graph corresponds to $(0,\pi/2)$ in polar coordinates. The parametric equations for this curve are $$(x(\theta),y(\theta))= ((1-\sin{\theta})\cos{\theta},(1-\sin{\theta})\sin{\theta})$$ If $x'(\theta) \neq 0$ we have $$y'(x)=\frac{y'(\theta)}{x'(\theta)}=\frac{\cos{\theta}-2\sin{\theta}\cos{\theta}}{\sin^2{\theta}-\cos^2{\theta}-\sin{\theta}}\tag{3}$$ However, $x'(\frac{\pi}{2})=0$. I'd like to compute the limit of $(3)$ as $\theta \to \left ( \frac{\pi}{2} \right ) ^+$ and show that it is $\infty$. |
| Show that the following function is continuous on R^2 Posted: 06 Jun 2022 11:22 AM PDT Consider the function f : $\displaystyle\mathbb{R^2}→ \mathbb{R}$ with $$f(x_1,x_2)=\begin{cases} \frac{sin(|x_2|)cot(|x_1|)+cos(|x_2|)}{sin(|x_2|)-cot(|x_1|)cos(|x_2|)}(|x_1|+|x_2|)^{-1}, & \text{if $(x_1,x_2)\neq(0,0)$} \\ 0, & \text{if $(x_1,x_2)=(0,0)$} \end{cases}$$ Is the function continuous on $\displaystyle\mathbb{R^2}$? $$$$ I tried to rewrite the function $f(x_1,x_2)$ as a function only depending on a norm of $(x_1,x_2)$, and I also tried to show if it is continuous by using a sequence $x_k=(\frac{1}{k},\frac{1}{k})$, which converges to 0, and replace it to $f(x_1,x_2)$ but still I can't get anywhere with that and to be honest I don't know how to continue anymore. Any solution is appreciated. |
| Frobenius norm derivative for a quadratic form Posted: 06 Jun 2022 11:41 AM PDT I have the following loss function: $\vert\vert Y - XBX^T \vert\vert_F^2$ where $Y, X$ and $B$ are square and symmetric matrices and $\vert \vert \cdot \vert \vert$ denotes the Frobenius norm. Which is the derivative of the loss function with respect to $X$? |
| Posted: 06 Jun 2022 11:22 AM PDT There is a general (mis)conception that $$\int \frac{1}{x} \, \mathrm{d}x = \ln|x| + C \label{1} \tag{1}$$ However, taking the indefinite integral as the set of all functions whose derivative is $\frac{1}{x}$, the technically correct answer should be $$\int \frac{1}{x} \, \mathrm{d}x = \begin{cases} \ln(x) + C & \text{if $x > 0$} \\ \ln(-x) + D & \text{if $x < 0$} \end{cases} \label{2} \tag{2}$$ See Your calculus prof lied to you (probably) for more details.
I suppose $\eqref{2}$ is more correct but it perhaps doesn't have any additional value compared to $\eqref{1}$ as the definite integral will give the same result for both (as the integral diverges if we move from the negative to positive $x$-axis).
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| About using fix-points to define partial recursive functions Posted: 06 Jun 2022 11:37 AM PDT In untyped lambda calculus, the fix-point operator $\mathcal{Y}$ can be used "to create" recursion. This is ok to me. But when we think about fix-points in the context of recursive functions, although we have a similar theorem that presents fix-points, this fix-points operates only on indices of recursion functions. The theorem says that, given a total function $f$, there is a fix point $n$ such that $\phi_n = \phi_{f(n)}$. In Odifreedi book, Classical Recursion Theory, page 157, it says we can take only this components to define all the partial recursive functions:
I didn't get what are this fixed-point definitions. How we can understand this? It seems to me if we take only the fix-point theorem as a way to define other functions, we would get stuck in the indices. For example, if $f$ is the function $f(x) = xf(x-1)$, if $x \not = 0$, and $f(x) = 1$, if $x= 0$, we would get: $$\phi_n \simeq \phi_{f(n)} \simeq \phi_{f\circ f(n)} \simeq ... \simeq \phi_{n!}$$ But it seems to me we would have to use the S-m-n theorem to get the $n!$ out of the indices. In wikipedia (link here) it says we have to use the second recursion theorem to get this (and the second recursion theorem needs the S-m-n plus fix-point theorems). The second theorem says that, for every partial recursive functions $\psi$, there is a index $e$ such that: $$\phi_e(x) \simeq \psi(e,x).$$ This theorem uses the S-m-n theorem in the proof. And it seems it can be used to get a function $\psi$ out of the index. But I didn't understand the proof used in wikipedia to show that we can eliminate primitive recursion in this context. So my questions are:
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| Question about a particular property of sheaves Posted: 06 Jun 2022 11:27 AM PDT Let $F$ be a Hausdorff topological field. Let us require all algebras to be associative commutative unital, and algebra morphisms to be unital. Let us say a sheaf $\mathcal{O}$ of $F$-algebras on a topological space $X$ has property $\mathcal{P}$ if: for every open $U\subset X$ and $x \in U$ and $f \in \mathcal{O}(U)$, there exists a unique $\lambda \in F$ satisfying the property that $f-\lambda$ is non-invertible when restricted to any open neighborhood of $x$ (contained in $U$). Could anyone help me verify if the following proof is correct? Claim: the sheaf of continuous $F$-valued functions on $X$ has property $\mathcal{P}$. Proof: Given $U,f,x$ as above; Existence: let $\lambda = f(x)$; then $f-\lambda$, restricted to any open neighborhood of $x$, has a zero at $x$ and therefore cannot be invertible. Uniqueness: suppose $\alpha \in F$ is such that $f-\alpha$ is non-invertible in any neighborhood of $x$; it follows that $f-\alpha$ has a zero in any neighborhood of $x$; thus $x$ is a limit point of $f^{-1}(\{\alpha\})$, but the latter is closed since $f$ is continuous and $F$ is Hausdorff, hence $x \in f^{-1}(\{\alpha\})$, hence $\alpha = f(x)$. Edit: generalized from $\mathbb{R}$ to any Hausdorff topological field $F$. |
| Cumulative distribution of mixed random variable function Posted: 06 Jun 2022 11:08 AM PDT Random variable $X$ has probability density function: $$ f_X(x) = \begin{cases} 2e^{−2x}, & \text{for } x\geq 0,\\ 0, & \text{otherwise.}\\ \end{cases} $$ Furthermore, the random variable Y is defined as $$ Y = \begin{cases} 2X, & \text{if } X < 1/2,\\ 0, & \text{if } X\geq 1/2.\\ \end{cases} $$ Determine the cumulative distribution $F_Y(y)$ of the random variable $Y$. How do I solve this question? If $Y$ was simply defined as $Y=2X$ then it would simply be finding $P[Y\leq y]=P[2X\leq y]=P[X\leq y/2]$ and thus solving the integral $$ \int^{y/2}_0{2e^{-2x}}dx $$ |
| Finding the weak form of a PDE with a tensor argument Posted: 06 Jun 2022 11:15 AM PDT Let $A$ be a $3×3$ matrix, a (possibly) complex function of $x, z \in \mathbb{R}$ representing the Order Parameter in the Ginzburg Landau equations, $$ A = \begin{pmatrix} A_{uu} & A_{uv} & A_{uw} \\ A_{vu} & A_{vv} & A_{vw} \\ A_{wu} & A_{wv} & A_{ww} \end{pmatrix}. $$ $K_i$,$\beta_i$, and $\alpha (T)$ are known parameters. I need to solve the following equation, (with the implied sum over $j$, for each component identified by $\mu$, $i$), $$ K_1 \partial^2_j A_{\mu i} + K_{23} \partial_i (\partial_j A_{\mu j}) = 2 \beta_1 Tr(AA^T) A_{\mu i} + 2 \beta_2 Tr(AA^\dagger)A_{\mu i} + 2 \beta_3[AA^TA]_{\mu i} + 2 \beta_4[AA^\dagger A]_{\mu i} + 2 \beta_5[AA^TA]_{\mu i} + \alpha(T) Tr(AA^\dagger) = (\text{rhs})_{\mu i} $$ I have a code in C++ that implements the FDM with relaxation, but we have found that our mixed derivative approximations have large enough error that the code doesn't always converge to a solution. We are looking at the option of using others' FEM solvers (like FreeFEM or MOOSE), but I'm having a hard time getting the weak form of our set of equations. I am following the description here, and obtained $$ 0 = K_1 \oint_{\Gamma} {\psi \vec{\nabla}A_{\mu i} \cdot \hat{n}} - K_1 \int_{\Omega} {\vec{\nabla} \psi \cdot \vec{\nabla}A_{\mu i}} - \int_{\Omega} {\psi (\text{rhs})_{\mu i}} + K_{23}\int_{\Omega} {\psi \partial_i (\partial_j A_{\mu j})}, $$ but I don't know how to get the last term. I saw this post and thought that maybe I'd have to explicitly write out all 9 equations and use 9 test functions? Edit: I played around with the equations a little more and figured I could write them generally as, $$ (\text{rhs})_{\mu i} = K_1 \vec{\nabla}^2 A_{\mu i} + K_{23} \partial_i \Big[ \vec{\nabla} \cdot \vec{A}_{\mu;r} \Big], $$ where $\vec{A}_{\mu;r}$ is the vector formed from the $\mu$-th row of $A$ (is there a better way to represent that?). Thus, $$ 0 = K_1 \oint_{\Gamma} {\psi \vec{\nabla}A_{\mu i} \cdot \hat{n}} - K_1 \int_{\Omega} {\vec{\nabla} \psi \cdot \vec{\nabla}A_{\mu i}} - \int_{\Omega} {\psi (\text{rhs})_{\mu i}} + K_{23}\int_{\Omega} {\psi \partial_i \Big[ \vec{\nabla} \cdot \vec{A}_{\mu;r} \Big]}, $$ which might be easier to integrate by parts? So, would I still have to use 9 test functions? We expect the solution function for each element to be different, so I might still need to have all 9 separately... |
| Fokker-Planck: uniqueness and attractiveness of stationary distribution (gradient systems) Posted: 06 Jun 2022 11:23 AM PDT consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of equations: $\frac{\partial x_n}{\partial t} = \frac{\partial}{\partial x_n} U(\vec{x}) + \sqrt{2c} \eta_n$ the problem can be reformulated in terms of the probability distribution $P(\vec{x},t)$, through the following fokker-planck equation: $\frac{\partial P(\vec{x},t)}{\partial t} = \bigg( - \sum_{i=1}^N \frac{\partial}{\partial x_i} \big( \frac{\partial}{\partial x_i}U(\vec{x})\big) + c \sum_{i,j=1}^N \frac{\partial^2}{\partial x_i \partial x_j} \bigg) P(\vec{x},t)$ The equation above admits the following stationary solution: $P^s(\vec{x}) = \mathcal{N} e^{\frac{-U(\vec{x})}{c}}$ Is there a simple way to convince yourself that, in this case, given any initial distribution I always converge only to above $P^s(\vec{x})$? |
| Posted: 06 Jun 2022 11:30 AM PDT So I'm watching a proof about the rank nullity theorem and it says this:
He then says this:
Now I'm honestly really confused because, well, the way he said it, that all the rows after the $x_{k}$th row are just linear combinations of the previous rows. But then I have a question, is this just a general solution to the equation $Ax = 0$, right? Why do you need linear combinations here? Like there must be some thing I'm not understanding, because all these rows are just going to be numbers when I solve for the vector, right? So would I not be able to represent it with just a scalar multiple of x1 for every single row? Like I really do not understand this. Could anyone help? |
| Fundamental Theorem of Calculus when the integrand is logarithmic derivative Posted: 06 Jun 2022 11:18 AM PDT Let $f(z)$ be an entire function and let $C$ be a broken line segment from $2$ to $2+2i$ to $1+2i$, and given that $f(z)$ is non zero on $C$. Fundamental theorem of Calculus(FToC): If a continuous function $f$ has a primitive $F$ in $\Omega$, and $\gamma$ is a curve in $\Omega$ that begins at $w_{1}$ and ends at $w_{2}$, then $\int_{\gamma}f(z)dz = F(w_{2}) - F(w_{1})$ Question: How by the fundamental theorem of Calculus we have $$\Im\left(\int_{C} \frac{f'(z)}{f(z)}dz\right)=\Im\left(\log\left(f\left(1+2i\right)\right)\right)$$ where $\Im$ denotes the imaginary part and $f(z)$ is non zero on $C$. Since $f(z)$ is entire and non zero on $C$ but then $f(z)$ might be $0$ in some domain containing curve $C$. Then how FToC applies here? Do we require parametrizing the segment? Edit If we consider $f(z)=\left(z-(\frac{1}{4}+i)\right)\left(z-(\frac{3}{4}+i)\right)\left(z-(\frac{1}{4}-i)\right)\left(z-(\frac{3}{4}-i)\right)$. Consider a rectangle $R=\{x+ iy\in\mathbb{C}\mid 0\leq x\leq 1, 0\leq y\leq 2\}$ and $\#Z$ be the no. of zeros of $f$ inside $R$. By Argument principle on $R$, $$\#Z= \frac{1}{2\pi i}\int_{\partial R} \frac{f'(z)}{f(z)}dz=2\tag{1}$$ Since $f(z)=f(1-z)$ so $$\#Z= \frac{1}{\pi}\Im\left(\int_{L} \frac{f'(z)}{f(z)}dz\right)\tag{2}$$ where $L: 1\to 1+2i\to \frac {1}{2}+2i$. By FToC, $(2)$ becomes $$\#Z= \frac{1}{\pi}\Im\left(\int_{L} \frac{f'(z)}{f(z)}dz\right)=\frac{1}{\pi}\Im\left(\log\left(f\left(\frac{1}{2}+2i\right)\right)\right)\tag{3}=0$$ Why $(1)$ and $(3)$ contradict each other? |
| $2$ Urns, $2$ colors, pick 1 ball from each Posted: 06 Jun 2022 11:22 AM PDT There are $2$ urns, urn $1$ contains $20$ white and $30$ red balls, urn $2$ contains $40$ white and $45$ red balls. We pick one ball from each urn, then discard $1$ of the $2$ picked balls. What is the probability, that the leftover ball is white? All balls have equal chance to be picked and the picked ball is discarded randomly. So far, I have only managed to calculate that chance of picking white is $2/3$ for urn $1$ and $8/9$ for urn $2$. However I'm not sure how to get probability after discarding $1$ ball. I'm not sure how to solve this problem, could someone help me? Please explain the solution. |
| Least Prime in Arithmetic Progression Conjecture Posted: 06 Jun 2022 11:11 AM PDT Let $q>2$ be a prime, and let $p(a,q)$ denote the least prime in arithmetic progression $$a+nq,$$ where $n$ runs through the positive integers. It is conjectured that $p(1,q)\le q(q+1)+1$. Research suggests that the above statement is related to Linnik's theorem, albeit a special case. How would one go about determining if this conjecture is valid/interesting or not and where would one start to prove this true or false and would proving something like this in any way help to decrease the bound of $L$ in Linnik's Theorem? Background: The Pocklington Criterion can be used to prove the primality of an integer $N$, given a partial factorization of $N-1$, specifically given that $N-1$ has a prime factor $q>\sqrt{N}-1$. For a given odd prime $q$, the Pocklington Criterion can be used to test exactly $\frac{q+1}{2}$ integers for primality, $1+nq$, where $n=2, 4, ..., q+1$. For the special case $q=2$, $n=1,2,3$ can be tested for primality, which is $3$, $5$ and $7$. As an example, take $q=7$, the following $4$ integers can be tested:
From this list it is easy to see that $29$ and $43$ are primes and the Pocklington Criterion will confirm this fact. Odd $n$ terms are omitted as they are all divisible by $2$. What we can now do is use these two primes and generate other primes. What I am interested in, and where the question comes from, is whether this process continues ad infinitum or not. Can we generate an infinite number of primes given a starting prime $q$, or stated differently, can the Pocklington Criterion be used to generate at least one prime given a starting prime $q$. The other part of the question is whether this is in any way useful. We already know, according to Dirichlet's Theorem on Arithmetic Progressions, that there are an infinite number of primes of the form $1+nq$. Linnik's Theorem has to do with what the smallest prime in an Arithmetic Progression is. Research Update: There are a couple of theorems and conjectures about what the smallest prime in an Arithmetic Progression is, and these are mostly based on Linnik's Theorem. It is conjectured that $p(a,d)<d^2$. Heath-Brown, Roger (1992). "Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression". This seems to be the smallest conjectured bound. If the above conjecture is true, then the conjecture that $p(1,q)\le q(q+1)+1$ is also true, and not necessarily useful. So, the conjecture in the question is not unique. What seems to be unique is the very specific bound $p(1,q)\le q(q+1)+1$. The other part that seems to be unique is the thinking that led to the conjecture. |
| Posted: 06 Jun 2022 11:32 AM PDT Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it. I am trying to find out if there exists any exact/accurate/non-approximated mathematical framework to work with physics under the assumption that phenomena have finite ending-times, instead of solutions that at best vanishes at infinity. Intro Definition 1 - Solutions of finite-duration: the solution $y(t)$ becomes exactly zero at a finite time $T<\infty$ by its own dynamics and stays there forever after $(t\geq T\Rightarrow y(t)=0)$. So, they are different of just a piecewise section made by any arbitrarily function multiplied by rectangular function: it must solve the differential equation in the whole domain. (Here I just pick the shorter name its look more natural to me for several I found: "finite-time", "finite-time-convergence", "finite-duration", "time-limited", "compact-supported time", "finite ending time", "singular solutions", among others). Recently, in this answer I figure out that the following autonomous scalar ODE: $$\dot{y}=-\sqrt[n]{y} \tag{Eq. 1}\label{Eq. 1}$$ Could stand the finite-duration solutions, for any real-valued $n>1$ and positive ending time $T>0$: $$ y(t)=\left[\frac{(n-1)}{n}\left(T-t\right)\right]^{\frac{n}{(n-1)}}\,\theta(T-t)\equiv y(0)\left[\frac{1}{2}\left(1-\frac{t}{T}+\left|1-\frac{t}{T}\right|\right)\right]^{\frac{n}{(n-1)}} \tag{Eq. 2}\label{Eq. 2}$$ with $\theta(t)$ the standard Heaviside step function. But later, working with the solution of \eqref{Eq. 2} I figure out that if I take its derivative I will end with: $$\begin{array}{r c l} \dot{y} & = & -\left[\frac{(n-1)}{n}\left(T-t\right)\right]^{\frac{1}{(n-1)}}\,\theta(T-t)+\underbrace{\left[\frac{(n-1)}{n}\left(T-t\right)\right]^{\frac{n}{(n-1)}}\,\delta(T-t)}_{=\,0\,\text{due}\,(x-a)^n\delta(x-a)=0} \\ & = & -\left[\frac{(n-1)}{n}\left(T-t\right)\right]^{\frac{1}{(n-1)}}\,\theta(T-t) \\ & = & -\frac{n}{(n-1)}\frac{1}{(T-t)}y(t) \end{array}$$ Which implies that the solutions $y(t)$ are also solving the non-autononous scalar ODE (is time-variant since the time $t$ is explicitely involved): $$\dot{y}+\frac{n}{(n-1)}\cdot \frac{y}{(T-t)} = 0 \tag{Eq. 3}\label{Eq. 3}$$ This last equation of \eqref{Eq. 3} resembles very much this another question were I found that the finite duration solution: $$z(t)=\exp\left(\frac{t}{(t-1)}\right)\cdot\theta(1-t) \tag{Eq. 4}\label{Eq. 4}$$ can solve the equation: $$\dot{z}+\frac{z}{(1-t)^2} = 0,\quad z(0)=1\tag{Eq. 5}\label{Eq. 5}$$ So I started to try another finite-duration solutions to see if their differential equations follows the same "trend" of having terms $(T-t)$ on it: which means that an standard integration constant is becoming part of the differential equation as its finite ending time $T$. Main analysis On this question is found that similarly for equations \eqref{Eq. 3}: $$x(t)=\frac{1}{4}(2-t)^2\theta(2-t) \tag{Eq. 13}\label{Eq. 13}$$ is a finite duration solution to: $$\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1\tag{Eq. 14}\label{Eq. 14}$$ so I will modify this solution to see what happens: $$q(t) = \frac{1}{4}(2-t)^2\cos(20\,t)\,\theta(2-t)\tag{Eq. 6}\label{Eq. 6}$$ which can be verified is solves the equation: $$\ddot{q}+\frac{4}{(2-t)}\cdot \dot{q}+\left(400+\frac{6}{(2-t)^2}\right)\cdot q = 0,\quad q(0)=1,\,\,\dot{q}(0)=-1\tag{Eq. 7}\label{Eq. 7}$$ Again, on \eqref{Eq. 7} there terms $(2-t) \cong (T-t)$ indicating that there exists solutions that are behaving like finite duration solutions (see plot of \eqref{Eq. 6}). But these "construction" of making finite duration solutions where these two conditions are met:
both mentioned on this answer, becomes less obvious where more than one variable is considered. Let think and a slight modification to the function of this answer: $$U(t,x) = \exp\left(1-\frac{1}{(1-(x-t)^2)}\right)\theta(1-(x-t)^2) \tag{Eq. 8}\label{Eq. 8}$$ it will show to be compact supported in space as it moves in the time variable, fulfilling the standard wave equation with a velocity $c=1$: $$\left(\frac{1}{c}\frac{\partial^2}{\partial t^2}-\nabla^2\right)U(t,x)=0 \overset{c=1}{\Rightarrow} U_{tt}-U_{xx}=0 \tag{Eq. 9}\label{Eq. 9}$$ Here the partial equation doesn't show the term $(T-t)$, but it does make sense if the countour plot of \eqref{Eq. 6} is viewed:
It is only compacted-supported in space and it is still never-ending in time. But if I test this modified version of \eqref{Eq. 8} mixed with the terms of \eqref{Eq. 2}: $$E(t,x) = U(t,x)\cdot \frac{1}{4}(2-t)^2\,\theta(2-t) = \exp\left(1-\frac{1}{(1-(x-t)^2)}\right)\theta(1-(x-t)^2)\cdot\frac{1}{4}(2-t)^2\,\theta(2-t) \tag{Eq. 10}\label{Eq. 10}$$ where this time it can be seen in its plot that is indeed behaving as also a finite duration solution in time:
It could be verified that is solves the equation: $$E_{tt}-E_{xx}+\frac{4}{(2-t)}E_{t}+\frac{6}{(2-t)^2}E = 0 \tag{Eq. 11}\label{Eq. 11}$$ within the domain \eqref{Eq. 10} is different from zero (see here). So again the terms $(T-t)$ are required for having finite-duration solutions. I beforehand know that some of these functions should be defined piecewise because has a undefined value, as example: $$ f(t) = \begin{cases} \exp\left(\frac{t}{(t-1)}\right),\quad t < 1 \\ 0, \qquad t\geq 1 \end{cases} \tag{Eq. 12}\label{Eq. 12}$$ instead of the one-line version of \eqref{Eq. 4}, so I ask you now to forgive this shortcut for now: the functions still continuous on these points and for me are easy to work with them finding their differential equations on this way (I have a extended discussion of this here), but if this annoying you, please just assume it is defined piecewise such as the function is zero in the undefined point. This comment is important, since I want to focus the discussion in the other singularities that appears on these equations (and sometimes the point could be coincident), and are the discontinuities on the differential equations due the terms $(T-t)$. The questions 1) Are all these differential equations with terms $(T-t)$ "well-defined" as differential equations? 2) Which are the name of this kind of differential equations? (noting here there are ODEs and PDEs with these terms - hope there are any references) 3) Does this $(T-t)$ terms on the denominator make these differential equation Non-Lipschitz? Here looking for parallels with the papers by V. T. Haimo: Finite Time Differential Equations and Finite Time Controllers. 4) Does this $(T-t)$ terms on the denominator make possible the existence of solutions of finite duration? Remembering here that solutions of finite duration don't fulfill uniqueness (see the recently cited papers for better explanation), a singularity in my opinion makes sense since there are multiple values rising from a single point, which is similar with traditional mistakes made when dividing by zero - but formally speaking, I have no idea of what it is happening. Motivation I am trying to find a mathematical framework to work with dynamical systems under the assumption: the system achieves an end in finite time, which seems really logical to me through daily life experience, but I have found this is really messy when it is translated to math (I still didn't find a related framework). Current physics solutions at best "vanishes at infinity", which is not exact but fairly good for almost every possible application, so good, that I have had discussions with many people that affirm as true that movement last forever, which at least under the assumption that thermal noise is Gaussian, you can say if it still moving (neither if it has stopped), since already all the information is lost: Gaussian distribution is the Maximum entropy probability distribution for a phenomena with finite mean and finite power. Don't meaning the assumption of never ending movement is wrong, but affirming is true is indeed false (see the comments of this question to a more detailed discussion applied to the pendulum movement). This subtle distinction is seen for example, in the Galton board toy, where many people affirm that somehow order have risen from chaos, due the appearance of the Normal Distribution, when the right interpretation is that all the information of the path each ball have take when falling is already lost, and the Gaussian distribution is the maximum possible disordered way of been for this system (to have an intuition of why there is a lobe in the middle, take a look to Concentration inequality). But nowadays there is a lot of theoretical physics squeezing their models to the very last drop and many counter-intuitive things are reaching mainstream: many Youtube videos talking about the "nonexistence" of time (is an illusion), and also many videos about the impossibility of current physics laws to determine "the direction of the arrow of time", and I would like to know if any of this things is due current models are not aware of the effects of keeping the assumption that phenomena have finite ending times. As example, in all the differential equations of this question there indeed a clue of which way the time is considered to be flowing (due the term $(T-t)$), and for the solutions, going backwards in time will lead to spontaneous rise from zero (being zero from a non-zero meassure interval in time). I know that current models are on the way they are because the models were designed to keep holding uniqueness of solutions, which if not could lead to a lot of struggles (see here some of this issues with the Norton's Dome example). But I also know now that for the existence of solutions of finite duration uniqueness of solutions must be abandoned, and I would like to know if this were known by physicists, if the current models would still been having their current formulations (to keep this on perspective, main models have about a hundred years old, and the first papers about finite duration solutions I have found are from 1985). And maybe because they are "relativately new", I only found these finite duration solutions on control theory papers (where they started assuming a high-knowledge background from the reader), and every book I find on differential equation keep their analysis tied to uniqueness, so they don't touch this topic. With this, any related book reference will be useful. Answering two firsts comments by @Desura Some user named @Desura mentioned the existence of the Frobenius method and the Fuchs' theorem for solving differential equations of the form $u''+\frac{p(t)}{t}u'+\frac{q(t)}{t^2}u=0$, and also what is a Regular singular point. On all these pages you can see differential equations that really resemble the ones I have displayed above, so very much for share it. I would like to comment why I think these kind of methods aren't able to find solutions of finite duration by "themselves", and also how I think I am already using them through Wolfram-Alpha. On the mentioned papers by Vardia T. Haimo (1985) Finite Time Controllers and Finite Time Differential Equations, where only scalar autonomous ODEs of 1st and 2nd order are studied ($x'=G(x)$ or $x''=G(x',x)$), then by assuming without lost of generality that $T=0$, and also assuming that the right-hand-side of the ODE is at least class $C^1(\mathbb{R}\setminus\{0\})$, the author mentioned that: "One notices immediately that finite time differential equations cannot be Lipschitz at the origin. As all solutions reach zero in finite time, there is non-uniqueness of solutions through zero in backwards time. This, of course, violates the uniqueness condition for solutions of Lipschitz differential equations." So, if the author is right, NO Lipschitz 1st or 2nd order scalar autonomous ODEs could stand solutions of finite duration, so classic linear ODEs are discarded. But reaching zero in finite time have also another consequence, and is that also are discarded classical solutions through Power Series which are analytical in the whole domain, since there exist a non-zero measure compact set of points identical to zero, the only analytical function that could support this matching is the zero function, due the Identity Theorem. Because of this last paragraph, is why I believe that a method that find the coefficients for a function of the form $\sum_k a_k(T-t)^k$ aren't able to describe a finite duration solution "alone", which is a traditional method for finding solution to ODEs (as the method shown by @Desura). But fortunately, being analytic in a piecewise part of the time domain and zero on the part "looks like" working if the two restrictions are met (now labeled after \eqref{Eq. 7}), as they are used for finding the solution of the now labeled \eqref{Eq. 14}, which is demonstrated on this answer. Here is where I am using the traditional ways of finding solutions to differential equations, to find the "non-zero section". But How this "construction" by "stitching solutions" is formally working, or if they are being or not formal solutions to the differential equations is something I don´t know (I am "highly abusing" of an answer without any formal prove - but because it "make sense"), neither when we have to "choose" one solution or another (since there are multiple options now - at least with this "construction": the full solutions, the finite duration alternative, or the other piece that rise spontaneously from zero). This is why I am looking for a framework, but because as requirement uniqueness must be left aside, traditional books in differential equations don't touch this kind of solutions. Even so, on the mentioned papers of V. T. Haimo, this kind of solutions aren't studied at all, and more interestingly, there are given a 2nd order ODE example where is shown that depending on its parameters, the differential equation could allow having finite duration solutions, but outside "this restriction" the solutions are not of finite duration (unfortunately no exact solution is given). |
| Evaluate $\int_{y=-\infty}^1\int_{x=0}^\infty 4xy\sqrt{x^2+y^2}\mathrm dx\mathrm dy$ Posted: 06 Jun 2022 11:22 AM PDT Evaluate $$\int_{y=-\infty}^1\int_{x=0}^\infty 4xy\sqrt{x^2+y^2}\mathrm dx\mathrm dy$$ I tried to solve this for hours without any success. I tried substitution method of $u = x^2+y^2$ but doesn't seem to work. Here is my attempt:
Attaching image for reference
EDITS Original Post asked |
| Doubt in Definition of Inverse of $y=x^n$ Posted: 06 Jun 2022 11:34 AM PDT Attaching Snapshot
I can't figure out whether it is a typo or I don't understand something. The very last sentence, why it says "inverse function at any $x > 0$", shouldn't it be "at any $x \ge 0$", because the domain includes $0$? |
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