Recent Questions - Mathematics Stack Exchange

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• A curious identity of the golden ratio, and how to derive it deductively
• Submanifold of Orientable Manifold with Codimension 0 is Orientable
• Functions of finite subsets of a set
• Are mutual information and covariance equivalent if the variables are linearily related?
• cubic residue character of 6
• Eigenvalues of $H=aX+bY+cZ+dI$
• (Concatenated number from 1/(prime number) repeating decimals) mod p = 0, Is this always true?
• Expected value of X*Y^2 or Cov(X,Y^2)
• solve the given initial value problem on the interval (-infinity,0)by Cauchy-Euler Equations
• A Polynomial $ f $ in three variables with real coefficients and any line $l$ in $ \mathbb R^3 $
• How to prove a block matrix is positive definite?
• Is this correct answer for listing sample space elements?
• How to calculate the density of the following using function similar to Totient Function?
• Properties of $\frac{f(x)}{x}$ if $f(x)$ is a convex function [Zorich's book]
• Is my proof of this operator norm correct?
• Finding combinatorial proof of log concavity of certain sequence of binomial coefficients
• A function formula expressed in summation form
• proof the Rank Of Matrix [closed]
• $\operatorname{Length}(\gamma)\neq \int_0^1\|\dot{\gamma}(t)\|\,dt$ possible?
• Prove that the closure of a connected subset is also connected
• $\mathbb Z/n\mathbb Z$ and finding non-zero $y,z$ such that $yz =0$ [duplicate]
• How do I explicitly find a basis that transforms a matrix into a Sylvester form?
• How many fixed polyominoes does it take to force an aperiodic tiling of the plane?
• Smallest non-4-space-filling polytesseract?
• Compute the Laplace transform of $\frac{1}{\sqrt{\pi t}} e^{\frac{-a^2}{4t}}$
• How to show that the hyperbolic space $H^2$ is isometric to $\mathbb{R}^2$ equipped with the following metric?
• What is the smallest possible cardinality of a non-finitely based magma?
• $A(\textbf{r})= \textbf{r}\times \nabla\phi(\textbf{r})$ is orthogonal to $\textbf{r}$ and $\nabla\phi(\textbf{r})$
• At what time $t$ is velocity perpendicular to acceleration?

A curious identity of the golden ratio, and how to derive it deductively Posted: 30 May 2021 07:42 PM PDT I found a nice curious property about the golden ratio. If $$x=\sin\bigg(\frac 12\text{ arcsin } x\bigg)+\cos\bigg(\frac 12\text{ arcsin } x\bigg)$$ then $x=\dfrac{1+\sqrt 5}2$. I was able to find this when I woke up too early in the morning and was bored, so I played around with a basic identity: $2\sin x\cos x=\sin 2x$. This is logically equivalent to $$2\sin\frac x2\cos \frac x2 = \sin x\tag*{$\bigg(x\mapsto \frac x2\bigg)$}$$ or $$\underbrace{\color{red}{\sin^2\frac x2+\cos^2\frac x2}+2\sin\frac x2\cos\frac x2}_{\Large{(\sin\frac x2+\cos\frac x2)^2}}=\color{red}1+\sin x$$ or $$\sin\frac x2+\cos\frac x2=\sqrt{1+\sin x}$$ or $$\sin\bigg(\frac 12\text{ arcsin } x\bigg)+\cos\bigg(\frac 12\text{ arcsin } x\bigg)=\sqrt{1+x}\tag*{$\big(x\mapsto\text{arcsin }x\big)$}$$ and now if you set that equal to $x$, it follows that $x=\frac{1\pm\sqrt 5}2$. Since the minimum value of $\sin$ and $\cos$ is $-1$ when the other is $0$, then $\sin +\cos\geqslant -1$ (no need to look at the interval on which $\text{arcsin}$ is defined), and so $x\geqslant -1$ which means $x$ must be the positive root. My question, if not too broad, is the following: Suppose one is tasked to prove the identity in the sandbox deductively and not inductively like prior (since beginning in such a way by deductive means appears very counter-intuitive)? In other words: How do we deductively solve for $x$ if we are given: $$x=\sin\bigg(\frac 12\text{ arcsin } x\bigg)+\cos\bigg(\frac 12\text{ arcsin } x\bigg)$$

Submanifold of Orientable Manifold with Codimension 0 is Orientable Posted: 30 May 2021 07:37 PM PDT I'm trying to solve Exercise 15.12 from Lee's Introduction to Smooth Manifolds which says "Suppose M is an oriented smooth manifold with or without boundary, and $D\subset M$ is a smooth codimension-$0$ submanifold with or without boundary. Then the orientation of M restricts to an orientation of D. " My initial attempt was just to restrict the orientable charts of M to N, and that seemed to work but I don't see why we need the codimension to be $0$. Is there something I'm missing here?

Functions of finite subsets of a set Posted: 30 May 2021 07:34 PM PDT I found an old exam and this question appeared: Let $X$ be a set and $X^{fin}$ denote the set of all finite subsets of $X$. We say for $n\in\mathbb{N}$ that $X$ is $n$-good if there exists a function $f:X^{fin}\rightarrow X^{fin}$ such that for all subsets $A$ of cardinality $n$ there exists $B\subsetneq A$ such that $A\subseteq f(B)$. Show that $X$ is $n+2$-good iff $|X|\leq \aleph_n$. This question makes no sense to me. Isn't every set $n$-good for all $n$? Just briefly, given an infinite set one has a bijection from the singletons to the sets with $n+1$ elements. I'm assuming this is a mistake. Is there a property similar to this one which would make sense in this problem? I played around with the conditions (i.e. instead of just subsets with cardinality $n$ with cardinality $\leq n$)

Are mutual information and covariance equivalent if the variables are linearily related? Posted: 30 May 2021 07:32 PM PDT I am trying to succinctly summarize the differences between PCA and ICA. Is it fair to say that PCA and ICA give identical results if the components (sources I guess for language used in ICA) are linearly dependent? As I understand it mutual information and covariance are the same if two variables are linearly related, mutual information is a generalization of covariance and ICA minimizes this instead of simply covariance as PCA does. Thus the "sources" aka components or eigenvectors ICA returns are non-linear mixtures of the original data while PCA returns linearly mixed gaussian sources. ICA is thus a generalization of PCA REFS: https://stats.stackexchange.com/questions/454801/difference-between-covariance-and-mutual-information Razak, F. A. (2014, December). The derivation of mutual information and covariance function using centered random variables. In AIP Conference Proceedings (Vol. 1635, No. 1, pp. 883-889). American Institute of Physics. http://web.ist.utl.pt/~andreas.wichert/16_DR.pdf

cubic residue character of 6 Posted: 30 May 2021 07:29 PM PDT can anyone help me with this problem? given p=1(mod 3) prime, prove that the congruence x^3 ≡ 6 (mod p) has integer solution if and only if exists integers A and B such that p=A^2+3B^2 and 9|B or 9|2B+A or 9|2B-A

Eigenvalues of $H=aX+bY+cZ+dI$ Posted: 30 May 2021 07:43 PM PDT Suppose I have the hamiltonian $H=aX+bY+cZ+dI$, where $a,b,c,d$ are some real constants, and $X,Y,Z,I$ are Pauli matrices. I'm trying to figure out the range of possible energy eigenvalues. If I limit the range of $a,b,c,d$ to be $[-1,1]$, then I think the range of eigenvalues should be $[-4,4]$ (linear combination), since the eigenvalues of each Pauli matrix are $-1$ and $1$. is my assumption correct? However, I tried to calculate the eigenvalues using python with such conditions, and it looks like the range of eigenvalues goes from $-\sqrt2-1$ to $\sqrt2+1$. I don't know if my assumption is wrong or the calculation is not working. Thanks!

(Concatenated number from 1/(prime number) repeating decimals) mod p = 0, Is this always true? Posted: 30 May 2021 07:17 PM PDT I'm not mathematician so I can't write it down in mathematical language, and I can only explain rambling words with almost no mathematical language... I'm re-posting this problem, because my former question was closed, and I don't know what to do about that. So, the basic rule is this. choose a random prime number that is not 2, 3, 5 and name it, p ex) p=7 calculate 1/p and find the number of repeating decimal digits, name it r ex) when p=7, r=6 find all the divisors of r, except r itself, and name it rx(x=1,2,3...). ex)when r=6, r1=1, r2=2, r3=3 choose any rx. ex) rx=3 Choose any random rx digit number, and name it n. ex) when rx=3, n=171 Concatenate n for several times to create r digit number (always possible since rx is the divisor of r), and name it N ex) when n=171, N=171171 This final number N you created will be divided by the prime number you chose without remainder. ex)171171/7=24453 Is this always true? for the understanding, I made a video explaining about what I found. https://www.youtube.com/watch?v=g1ZzXkQeN0I Hope this is not against the policy of this page.

Expected value of X*Y^2 or Cov(X,Y^2) Posted: 30 May 2021 07:22 PM PDT I wanted to find the solution to E[X*Y^2] with X and Y random var with given mean and variance. I need it to find Cov(X,Y^2). Thank you. With X and Y independent and normally distributed

solve the given initial value problem on the interval (-infinity,0)by Cauchy-Euler Equations Posted: 30 May 2021 07:16 PM PDT Solve the given initial value problem on the interval $(-\infty,0)$ by Cauchy-Euler Equations $$x^2y''-4xy'+6y=0\\y(-2)=8 , y'(-2)=0$$

A Polynomial $ f $ in three variables with real coefficients and any line $l$ in $ \mathbb R^3 $ Posted: 30 May 2021 07:34 PM PDT Prove that for every polynomial $ f $ in three variables with real coefficients and any line $\ell$ in $ \mathbb R^3 $ either $ \ell \subset Z_f $ or $ |\ell \cap Z_f|\leq \deg f $, where $$ Z_f :=\{(x,y,z)\in \mathbb R^3: f(x,y,z)=0 \}.$$ Here is the case for polynomial $f$ in $2-$variables and line in $\mathbb R^2$ http://diposit.ub.edu/dspace/bitstream/2445/159040/2/159040.pdf (Page 16, Lemma: 2.2.6) which is well understood. But I am stuck on how to solve the case with $3$ variables. Any hint or help is appreciated. Thanks!

How to prove a block matrix is positive definite? Posted: 30 May 2021 07:06 PM PDT Suppose there is a Hermitian matrix $W\in\mathbb{C}^{mn\times mn}$ with size $mn\times mn$. Denote $W_{ij}$ as the submatrix with size $n\times n$ in the following position: \begin{equation} W=\begin{bmatrix} W_{11}& W_{12} &\cdots &W_{1m}\\ W_{21} &W_{22} &\cdots &W_{2m}\\ \vdots &\vdots &\ddots &\vdots \\ W_{m1}& W_{m2}& \cdots &W_{mm}\\ \end{bmatrix} \end{equation} $W_{ij}$ is known to be Hermitian and positive semi-definite. What kind of condition can guarantee that $W$ is positive definite? Do we have some results like if $W_{ii}\succ \sum_{j\neq i}W_{ii} $ for all $i$, then $W$ is positive definite?

Is this correct answer for listing sample space elements? Posted: 30 May 2021 07:40 PM PDT The answer for part (a) is simple. For the part (b), the author states this answer $$ S = \{(x,y) | 1 \leq x, y \leq 6 \} $$ But I feel this is incorrect. My answer is $$ S = \{ (x,y) | 1 \leq x \leq 6, 1 \leq y \leq 6, x \in \mathbb{N}, y \in \mathbb{N} \} $$ What do you think?

How to calculate the density of the following using function similar to Totient Function? Posted: 30 May 2021 07:43 PM PDT You don't need to explain every step: Suppose $\psi\in[0,1]$ and $S=\left\{m/(2n+1):m,n\in\mathbb{Z}\right\}$, calculate $$g(\psi)=\small{\lim\limits_{\left(\epsilon,\omega\right)\to\left(0,\infty\right)}\frac{\sum\limits_{s_{1}=-\omega}^{\omega}\left|S\cap\left\{\frac{\left\lfloor s_{1}\left(\epsilon\pm\psi\right) \right\rfloor}{s_{1}}:s_{1}\in\mathbb{Z},s_{1}\neq 0\right\}\right|+\sum\limits_{s_{2}=-\omega}^{\omega}\left|S\cap\left\{\frac{ s_{2} }{\left\lfloor s_{2}/\left(\epsilon\pm\psi\right)\right\rfloor}:s_{2}\in\mathbb{Z},s_{2}\neq 0\right\}\right|}{\sum\limits_{s_{1}=-\omega}^{\omega}\left|\left\{\frac{\left\lfloor s_{1}\left(\epsilon\pm\psi\right) \right\rfloor}{s_{1}}:s_{1}\in\mathbb{Z},s_{1}\neq 0\right\}\right|+\sum\limits_{s_{2}=-\omega}^{\omega}\left|\left\{\frac{ s_{2} }{\left\lfloor s_{2}/\left(\epsilon\pm\psi\right)\right\rfloor}:s_{2}\in\mathbb{Z},s_{2}\neq 0\right\}\right|}}$$ My Attempt: I believe we may something similar to Euler's Totient function; however, this involves the floor function. I'm not definite whether can be solved. Despite constructing complex functions, especially for research, I don't have the skillset for solving them.

Properties of $\frac{f(x)}{x}$ if $f(x)$ is a convex function [Zorich's book] Posted: 30 May 2021 07:05 PM PDT Show that a) if a convex function $f:\mathbb{R}\to \mathbb{R}$ is bounded, then it is constant; b) if $\lim \limits_{x\to -\infty}\dfrac{f(x)}{x}=\lim \limits_{x\to +\infty}\dfrac{f(x)}{x}=0,$ for a convex function $f:\mathbb{R}\to \mathbb{R}$, then $f$ is constant. c) for any convex function $f$ defined on an open interval $a<x<+\infty$ (or $-\infty<x<a$), the ratio $\dfrac{f(x)}{x}$ tends to a finite limit or to infinity as $x$ tends to infinity in the domain of definition of the function. These problems are from Zorich's book. I have solved parts a) and b) but have some issues with part c). I was trying to solve it by contradiction. WLOG suppose $f(x)$ is defined on $(0,\infty)$ and the $\lim \limits_{x\to +\infty}\dfrac{f(x)}{x}$ does not exist. Then $\forall \delta>0$ we can find $x_0,y_0>\delta$ such that $\left|\frac{f(x_0)}{x_0}-\frac{f(y_0)}{y_0}\right|>c$ for some $c>0$. Then I've tried to apply the definition of convexity to $\delta<x_0<y_0$ to get contradiction but failed. Would be very grateful if you can show how to finish the proof using my approach. Thanks in advance!

Is my proof of this operator norm correct? Posted: 30 May 2021 07:21 PM PDT Let $\alpha \in \ell^\infty$ and $T_\alpha:\ell^p \rightarrow \ell^p$ $(1\leq p \leq \infty)$ given by \begin{equation} T_\alpha(x)=(\alpha_1x_1,\dots,\alpha_nx_n,\dots). \end{equation} Note that $||T_\alpha(x)||\leq ||\alpha|| \ ||x||$, thus $T_\alpha(x) \in \ell^p$. Now, I want to prove that $T_\alpha$ is continuous and $||T_\alpha||=\alpha$. So, let $(x_n)=(x_1^n,x_2^n,\dots)\rightarrow(0,0,\dots)\in \ell^p$. Then $||x_n||\rightarrow 0 $ and by the above inequallity $||T_\alpha(x)||\rightarrow 0$. Then we get that $T_\alpha$ is continuous. Computing the norm, I went for this idea: \begin{equation} \begin{split} ||\alpha||&=\sup_{||x||=1 \\\ \ x\neq 0} ||\alpha|| \ ||x|| \\ &= \sup_{||x||=1 \\\ \ x\neq 0} \sup_{i \in \mathbb{N}} |\alpha_i| \ ||x|| \\ &=\sup_{||x||=1 \\\ \ x\neq 0} \sup_{i \in \mathbb{N}} |\alpha_i| \ \big(\sum_{j=1}^\infty |x_i|^p \big)^{1/p} \\ &=\sup_{i \in \mathbb{N}}\sup_{||x||=1 \\\ \ x\neq 0} \big(\sum_{j=1}^\infty |\alpha_i|^p|x_i|^p \big)^{1/p} = \sup_{i \in \mathbb{N}} ||T_\alpha||=||T_\alpha||. \end{split} \end{equation} Then, in both cases when $p$ is finite or $p=\infty$ we have that $||T_\alpha||=||\alpha||$. Is this proof correct? What I'm missing here? Thanks for sharing!

Finding combinatorial proof of log concavity of certain sequence of binomial coefficients Posted: 30 May 2021 07:29 PM PDT A sequence of numbers $a_{0}, a_{1}, \cdots, a_{n}, \cdots$ is said to be log-concave if for $1 \leq i \leq n-1, a_{i-1} a_{i+1} \leq a_{i}^{2} .$ consider the sequence $\left(\begin{array}{c}k \\ k\end{array}\right),\left(\begin{array}{c}k+1 \\ k\end{array}\right),\left(\begin{array}{c}k+2 \\ k\end{array}\right), \cdots, .$ Show that this sequence is log-concave. Also give a combinatorial proof. I did solve the question using actual computation (That is by expansion of terms). Also I could find a combinatorial proof of a similar sequence, namely sequence $\left(\begin{array}{l}n \\ 0\end{array}\right),\left(\begin{array}{l}n \\ 1\end{array}\right), \cdots,\left(\begin{array}{l}n \\ n\end{array}\right)$ is log-concave, by considering the pairs of subsets. But I couldn't prove that $\left(\begin{array}{c}k \\ k\end{array}\right),\left(\begin{array}{c}k+1 \\ k\end{array}\right),\left(\begin{array}{c}k+2 \\ k\end{array}\right), \cdots, .$ is log concave by any combinatorial argument. (here $k$ is fixed). Any help is highly appreciated, thanks in advance!

A function formula expressed in summation form Posted: 30 May 2021 07:25 PM PDT Can anyone please help me to prove the following, $\displaystyle f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ , if $\displaystyle f$ is an analytical function I tried a lot to search about it in various websites but i was unable to find the proof.

proof the Rank Of Matrix [closed] Posted: 30 May 2021 07:28 PM PDT proof the Rank Of Matrix, The sign R stands for rank, A and b between the multiplication: r(A.B)≤ min (r(A),r(B))

$\operatorname{Length}(\gamma)\neq \int_0^1\|\dot{\gamma}(t)\|\,dt$ possible? Posted: 30 May 2021 07:27 PM PDT Let $\gamma:[0,1]\to\mathbb{R}^d$ be a rectifiable curve and define the length of the curve as $$\operatorname{Length}(\gamma)=\sup\left\{\sum\limits_{k=0}^{n-1}\|\gamma(t_{k+1})-\gamma(t_k)\|,n\ge 1, 0=t_0<t_1<\ldots<t_n=1 \right\}.$$ Is there an example where $\operatorname{Length}(\gamma)\neq \int_0^1\|\dot{\gamma}(t)\|\,dt$ (assuming this quantity is well defined)?

Prove that the closure of a connected subset is also connected Posted: 30 May 2021 07:25 PM PDT So far my proof goes like this. Let $(S, d)$ be a connected subspace of $(X, d)$. By contradiction, suppose $\overline S$ is not connected. Then, we can write $\overline S = A \cup B$, where $A, B$ are non-empty, disjoint, open (in $\overline S$) subsets of $\overline S$. Moreover, since $S \subset \overline S$, then $S = (S \cap A) \cup (S \cap B)$. Take $x \in S$, and assume without loss of generality that $x \in A$. Then it follows that $S\cap A \neq \varnothing$. How can I show that $S \cap B \neq \varnothing$ to get a contradiction?

$\mathbb Z/n\mathbb Z$ and finding non-zero $y,z$ such that $yz =0$ [duplicate] Posted: 30 May 2021 07:19 PM PDT Find non-zero $y$ and $z$ in $\mathbb Z/455\mathbb Z$ Such that $yz=0.$ I really am lost on how to start the question. Can anyone throw me a bone?

How do I explicitly find a basis that transforms a matrix into a Sylvester form? Posted: 30 May 2021 07:29 PM PDT Let $\beta$ be a symmetric bilinear form represented by the transformation matrix with regards to the standard basis: \begin{align*} A:= \begin{pmatrix} 2 & 0 & 2\\ 0 & 6 & 0\\ 2 & 0 & 5 \end{pmatrix} \in \operatorname{Mat}(3, \mathbb{R}). \end{align*} Find a basis $B = \{b_1, b_2, b_3\}$ of $\mathbb{R}^3$ such that $D = (\beta(b_i, b_j))$ of $\beta$ with regards to $B$ is given through: \begin{align*} D = \begin{pmatrix} \epsilon_1 & 0 & 0 \\ 0 & \epsilon_2 & 0 \\ 0 & 0 & \epsilon_3 \end{pmatrix}, \end{align*} where $\epsilon_1, \epsilon_2, \epsilon_3 \in \{-1, 0, 1\}$. I started with the eigenvalues and eigenvectors of $A$ and found the orthonormal basis.$$a_1=\begin{pmatrix} 0\\1\\0 \end{pmatrix}, a_2=\frac{1}{\sqrt{5}}\begin{pmatrix} 1\\0\\2 \end{pmatrix}, a_3=\frac{1}{\sqrt{5}}\begin{pmatrix} -2\\0\\1 \end{pmatrix}$$ With this basis, I would get $\operatorname{diag}(6,6,1)$. How can I find the form mentioned above?

How many fixed polyominoes does it take to force an aperiodic tiling of the plane? Posted: 30 May 2021 06:59 PM PDT A longstanding open problem asks whether there is a single connected tile that only tiles the plane aperiodically. As far as I know, this is still open even in the case of polyominoes: we do not know if there is a single polyomino that forces an aperiodic tiling of the plane. I am curious about the case where we treat the polyominoes as fixed, i.e., when we consider rotated/reflected copies of a tile as distinct, and only permit translations. In this situation, we do know that more than one tile is needed; as shown in this paper, a single fixed polyomino tiles the plane periodically if it forms any tessellation at all. (In fact, we can ensure the tiling is isohedral!) Conversely, there exist finite sets of fixed polyominoes which tile the plane, but not in a periodic manner. If we use Wang tiles to generate polyominoes, we can obtain a solution with $11$ tiles (which is minimal), but there are smaller solutions. For instance, consider Matthew Cook's set of three polyominoes which force an aperiodic tiling (with rotations and reflections allowed): We observe that the tiling only makes use of $4$ orientations each of two of the tiles, and a single orientation of the third tile, so only $9$ fixed polyominoes are needed in total. The answer is therefore somewhere between $2$ and $9$ inclusive; I am interested in learning of any improvements to either the upper or lower bounds, or pointers to discussion of this problem in the literature. One potential avenue for a smaller upper bound is the aperiodic set of polyominoes described at this Wolfram MathWorld page as being announced by Roger Penrose in 1994, but it gives no further details, and I have been unable to track down a reference.

Smallest non-4-space-filling polytesseract? Posted: 30 May 2021 07:14 PM PDT As a follow-up to Smallest non-space-filling polycube?, what's the smallest polychoron produced by fusing tesseracts 3-face to 3-face which does not fill 4-space?

Compute the Laplace transform of $\frac{1}{\sqrt{\pi t}} e^{\frac{-a^2}{4t}}$ Posted: 30 May 2021 06:56 PM PDT I'm struggling to find this transform. I tried to complete squares but the expression I had left was too messy. Could you please provide any suggestions/general hints/solution?

How to show that the hyperbolic space $H^2$ is isometric to $\mathbb{R}^2$ equipped with the following metric? Posted: 30 May 2021 07:24 PM PDT $H^2=\{(x,y,z)\in\mathbb{R}^3:x>0,-x^2+y^2+z^2=-1\}$ is the hyperbolic space, $\mathbb{R}^2$ endowed with the metric $$g=(\mathrm{d}r)^2+\sinh^2r(\mathrm{d}\theta)^2$$ in polar coordinates, how to show that these two spaces are isometric？I tried the natural projection $\pi: H^2\to\mathbb{R}^2$ but not succeeded. Well, I had some new tries, since $H^2$ is isometric to the Poincaré Disk $\left(\mathbb{D}^2, \frac{4(\mathrm{d}x^2+\mathrm{d}y^2)}{(1-x^2-y^2)^2}\right)$, we can consider the map $\pi: \mathbb{D}^2\longrightarrow(\mathbb{R}^2,g)$, via (in local coordinates of the Poincaré disk) $$\pi(x,y)=\left(\log\frac{x^2+y^2}{1-(x^2+y^2)},\arctan\frac{y}{x}\right):=(r,\theta)$$ Here I take the $\log$ to cancellate with $\sinh$, and I had checked that its Jacobian matrix DOES always non-degenerate, but hadn't checked whether an isometric yet.

What is the smallest possible cardinality of a non-finitely based magma? Posted: 30 May 2021 06:55 PM PDT I was told that every magma $(S,*)$ whose base set $S$ has 2 elements has a finite basis of identities. The natural question is, what is the smallest possible cardinality of a non-finitely based magma? I would be very interested to see a 3-element set and a binary operation on it which is not finitely based.

$A(\textbf{r})= \textbf{r}\times \nabla\phi(\textbf{r})$ is orthogonal to $\textbf{r}$ and $\nabla\phi(\textbf{r})$ Posted: 30 May 2021 07:43 PM PDT Let ${\textbf{r}} = (x,y,z)$ be the position vector. Show that the vector field $A(\textbf{r})= \textbf{r}\times \nabla\phi(\textbf{r})$ is orthogonal to $\textbf{r}$ and $\nabla\phi(\textbf{r})$, that is, the following expressions are true : $A(\textbf{r})\cdot \textbf{r} = 0$ and $A(\textbf{r})\cdot \nabla\phi(\textbf{r}) = 0$. I managed to prove this by direct application of the definitions in cartesian coordinates. Now I am trying to do it using index notation since I started to learn it recently : $A(\textbf{r})\cdot \textbf{r} = (\textbf{r}\times \nabla\phi(\textbf{r})) \cdot \textbf{r} = \epsilon_{ijk}\textbf{r}_j\partial_k\phi\textbf{r}_i$ and $A(\textbf{r})\cdot \nabla\phi(\textbf{r}) =(\textbf{r}\times \nabla\phi(\textbf{r}))\cdot \nabla\phi(\textbf{r}) = \epsilon_{ijk}\textbf{r}_j\partial_k\phi\partial_i\phi $. But I am not sure how to proceed from here, I can't see how these two expressions are going to be zero. Any help will be appreciated . Thank you

At what time $t$ is velocity perpendicular to acceleration? Posted: 30 May 2021 07:04 PM PDT So, for $\textbf{r}=\textbf{r}(t)$, $\textbf{r}=(t+3)\textbf{i}+(2t-t^{2})\textbf{j}+(3t-t^{2})\textbf{k}$ = displacement Therefore, $\textbf{v}=\textbf{r}'=\textbf{i}+2\textbf{j}-2t\textbf{j}+3\textbf{k}-2t\textbf{k}$ = velocity And, $\textbf{a}=\textbf{r}''=-2\textbf{j}-2\textbf{k}$ = acceleration To show orthogonality, I know $\textbf{a} \cdot \textbf{b}=0$, so for velocity and acceleration to be orthogonal (or perpendicular), $\textbf{v} \cdot \textbf{a}=0$. $\textbf{v} \cdot \textbf{a}= (\textbf{i}+2\textbf{j}-2t\textbf{j}+3\textbf{k}-2t\textbf{k}) \cdot (-2\textbf{j}-2\textbf{k})=[\textbf{i}+(2+2t)\textbf{j}+(3-2t)\textbf{k}] \cdot (-2\textbf{j}-2\textbf{k})=(1)(0)\textbf{i}+(2+2t)(-2)\textbf{j}+(3-2t)(-2)\textbf{k}=-4\textbf{j}-4t\textbf{j}-6\textbf{k}+4t\textbf{k}=0$ $\implies 4t\textbf{j}-4t\textbf{k}=-4\textbf{j}-6\textbf{k}$ Obviously, there is no value for $t$ which satisfies the above, so I think I may have made a mistake! Thanks.

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