How to make animation in mathematica for 3D sphere which randomly its radius increase and decreases? Thanks

So, my story, set in America in 1998, has werewolves. I'm not quite sure how good their hearing is going to be, but bare minimum, even when in human form, they can hear as well as a real-life wolf can. Which is... a lot. Somewhere between 6-10 miles outdoors depending on the level of forestation....

In many analysis courses and books there are two main definitions of connected sets that may not be the same (I know that in both cases a connected set is intended to be defined as a set that has no "holes"). Which one would you say that is a more complete definition? One definition may be: "A m...

I've been trying to replace the windshield wiper of a Toyota Aygo Now, but it seems that I cannot even remove the old one. There are several bits that look like tabs that can be pressed / pushed / moved by a few millimeters, but none of them releases the wiper from the hook. I've searched Toyota ...

I am new to Salesforce Apex coding and I am getting this null pointer exception error when i try to run a schedule batch. Please help me as i am unable to understand the mistake i made. The code is working on insert records, but it fails when i try to run a schedule batch. The issue is related to...

First there was the Roman Kingdom, then the Roman Republic and then the Roman Empire. But what was it that made it an empire the third time, rather than just going back to being a kingdom again? Lucius Tarquinius Superbus was the last rex Romae (=king of Rome), Julius Caesar held the office of di...

In a book in latin, I found the sentence "Adidum vina" — Something (?) the wines. What "adidum" means?

Find six different numbers (positive integers) such that each of them has a common divisor with precisely three of the other numbers. How small can the largest of the six numbers be? What if $2n$, $n>3$, numbers need to be found, each having a common divisor with precisely three of the other numb...

I was reading some papers on delta-hedged option returns and came across an intriguing paper that I found quite interesting. However, I was a bit confused on the authors' methodology of computing option returns for their strategies. For some context, the paper is looking at the returns of a gamma...

John 14:16 "And I will ask the Father, and he will give you another Helper, to be with you forever". ESV. [ton patera/the Father]. John 1:14 "And the Word became flesh and dwelt among us, and we have seen his glory, glory as of the only Son from the Father, full of grace and truth". [patros/t...

I'm struggling with a restoration/color correction task for an old photo that was framed with a mat: As you can see, there's a definite difference in the regions that were covered vs. exposed. I've tried making my own mask, using selection transformations to feather/fade the edges to even out th...

Would likelihood-based frequentist inference amount to the same as Bayesian inference, but where the "frequentist prior" $\pi^F(\theta) = \delta_{\theta^*}(\theta)$ with $\delta_{\theta^*}$ being the Dirac distribution centered at $\theta^*$?

If the PCs plane shift to plane to which they have an attuned tuning fork, do they need an attuned tuning fork to return to the prime material plane?

[] Painting consist of a total of three seals (YING;YANG;& OVAL)with a vertical inscription in kaishu script on my calligraphy ink brush painting.

I was recapping the forced oscillations, and something troubled me. The equation concerning forced oscillation is: $$ x=\frac{F_0}{m(\omega_0^2-\omega^2)}\cos(\omega t) $$ I don't understand why this equation predicts that the amplitude will approach infinity as $\omega$ approaches $\omega_0$. On...

I've run into a situation I've never seen before: I deleted the Screws Kitbash Payware mesh objects from the View Layer display, but they are still visible in the Blender File display. Why? Is this by design? Or is the file corrupted? Thanks. Blender 3.1.2, Windows 11.

During the SENTRI interview, it was not clear to me whether (or not) the officer processed all paperwork and added the vehicle. I did not find a chance to ask while at the office. Below are two screenshots showing what my ttp dashboard looks like... Does it mean that the vehicle is good to go?

I won't go into details, but I am dealing with horrible guilt and shame over cruel acts I have committed in the past. It feels like I am now come to terms with the significance of these actions, and seeing the ripening of this accumulated negative karma into a hellish mental state. I had previo...

When I was learning the Pythagorean theorem(at time A), I was just told to memorize it. I used it often before trying to derive the equation(at time B), and I think actually I have forgotten the derivation now(denoted by time C). A is prior to B, and B is prior to C. I learned from The ABCs of H...

I'm creating a Vim script and a critical part of it is duplicating the selected lines. To duplicate lines, I have this mapping which works as intended: vnoremap <Leader>d :copy '><CR> But, when inside a function/script, the behavior of copy seems different: function Foo() copy '> endfunction...

Could somebody kindly recommend to me some books which deal with the applications of the probabilistic method to problems in real and complex analysis or which consider probabilistic versions of some problems in real and complex analysis? I would also like to know about some books/resources deal...

We can make errors with tex.error() in luatex like \PackageError in .sty or \ClassError in .cls, but how can I make warnings and infos in luatex like \PackageWarning or \PackageInfo?

How can I compute this triple integral? I tried this but it has been running for a long time and not return result yet. Integrate[ Boole[0 <= x^2 + y^2 + z^2 <= 4]*((y - 1)^3/(x^2 + y^2 + z^2 + 1)), {x, 0, 2}, {y, -2, 2}, {z, -2, 2}]

I want to know is there a unite definition of "science"? my guess is "NO"! What do philosophers means by science that they can't define it? what is its difference with knowledge?

JWST’s Primary Mirror has gotten the lion’s share of attention, and rightly so. But recently I’ve been trying to find out how the FSM is moved. I’ve found what appears to be photos of the FSM lab testing setup but cannot see anything of mounting or actuation. I’ve found lots of vendors and infor...

We can use common algorithms such as what is available in firls and firpm in MATLAB for least squares or minimax fitting to target magnitudes at given frequencies. I am familiar with the "Frequency Sampling" method to determine the filter coefficients for an arbitrary magnitude and phase trajecto...

Problem Statement Show that if we strengthen linear programming by also allowing constraints of the form $$ \sum_{i,j = 1}^n a_{ij} x_i x_j = b, $$ for integers $b$ and $a_{ij}$, then the problem becomes NP-complete. This is a practice problem that I am trying to solve. I really do not know how...

In Neverwhere, Richard encounters Lady Serpentine, one of the "Seven Sisters". We later discover that: The Lady Serpentine, who was, but for Olympia, the oldest of the Seven Sisters so we are told the names of two of the seven. Has Gaiman revealed the names of the other five?

How can I use functional code to achieve the same effect as the following procedural code Do[ Do[ f[i,j], {j,1,m} ], {i,1,n} ]

Suppose I have a list of five lists, each with a different number of elements list = {RandomReal[{0, 1}, 7], RandomReal[{0, 1}, 9], RandomReal[{0, 1}, 5], RandomReal[{0, 1}, 11], RandomReal[{0, 1}, 13]} Each of these lists occur in certain unique years (the number of years matching the ...

It is often said that special relativity is more general than Newtonian mechanics. Is there any precise meaning of what is meant by more "general"? I would consider a theory A more general than a theory B if the axioms of B can be considered theorems of A. Is that a proper definition? Consider sp...

I have a line and a point layers. All lines are straight (has two points), parallel to each other and are perpendicular to the direction of points. I want to assign a lane id (1, 2, ...) to the points as in the image. The line names (3021, 3041, ...) increase from left to right. Therefore, the la...

I'm looking at Dijkstra's algorithm for single source shortest paths in a graph $G$ from a vertex $s$ from Introduction to Algorithms by Cormen et al. The $w$ parameter is the weight function such that $w(u,v)$ gives you the weight of the edge from $u$ to $v$. DIJKSTRA(G, w, s) INITIALIZE-SINGL...

I studied the Bohr's model of atom that says electron can jump to higher energy level by absorbing photon, but the quantum model says we have more than that (quantum numbers). Therefore, when the electron of hydrogen atom absorbs a photon, will it move according to the Aufbau principle (1s, 2s, ...

What is the meaning of ばっかっていうのもねえ in the sentence: "せっかくヨットに乗るのに、　野郎ばっかっていうのもねえ"?

From the goo thesaurus, it defines the difference between 道徳 and 倫理 as: 「道徳」は、人間が社会の一員として守るべき行為の基準となるものをいい、「倫理」は、社会的な行動の規範となるものをいう。 While モラル covers both. Unfortunately, I don't really understand the nuance in this description, since they both sound the same to me. There is also a comparison char...

Hashcat list FPGAs as opencl device type in its help. - [ OpenCL Device Types ] - # | Device Type ===+============= 1 | CPU 2 | GPU 3 | FPGA, DSP, Co-Processor So can you really somehow connect and use an FPGA with hashcat?

So my professor gave us this solution to the following integral: $\int \frac{dx}{x\sqrt{x^2-1}}= \int \frac{xdx}{x^2\sqrt{x^2-1}}$ Here he substituted $t=\sqrt{x^2-1}$ from which $t^2=x^2-1$, but what I don't understand is why he then wrote $2xdx=2tdt$ from which $xdx=tdt$ and from here: $\int\fr...

Is it reasonable to have transparent face shields on powered armor or should they be covered with armor? While there are fairly good arguments for why open helmets were fairly popular historically even for soldiers who otherwise wore full armor, this gets more complicated when you're talking abou...

Be it theoretical proposal or anything else, is there even a definition for non one-way (or non measurement-based) quantum computer/computation?

If I do e.g. Flatten[Outer[List, Range[3], Range[3]], 1] I get {{1, 1}, {1, 2}, {1, 3}, {2, 1}, {2, 2}, {2, 3}, {3, 1}, {3, 2}, {3, 3}} but I want to get distinct pairs, i.e. {{1, 1}, {1, 2}, {1, 3}, {2, 2}, {2, 3}, {3, 3}} How can I do this?

Hi i am new to latex and I want to place a gray box to the left side of the document to use the box as a sidebar. Here is what I have tried: \documentclass{article} \usepackage{tikz} \usepackage{xcolor} \definecolor{sidecolor}{HTML}{E7E7E7} \begin{document} \begin{tikzpicture}[remember picture,...

[Control Systems Engineering by Nise] Hey, I have some confusion regarding bode plots and how they can be used to display both open loop and closed loop information. This also partially relates to my understanding of open loop and closed loop poles. In the first image above, the author uses the...

I was thinking about a simple dice game where the goal is to roll all the face values of a six-sided die consecutively in order (1,2,3,4,5,6). If I'm not mistaken, the probability of doing this with a fair die is $\left(\frac{1}{6}\right)^6$. And I think I'm correct in saying that there is no way...

If I were to have a system of 2 masses that rotate around their barycenter such that $m_1$ has a radius of $r_1$ and $m_2$ has a radius of $r_2$, what would the ratio of the periods of the planets be? My thinking: Following Kepler's 3rd law that says $T^2/R^3=4π^2/GM$, and since both would have a...

I'm trying to store a comma-separated list into a key-value property to use it later, but I can't get it to work properly. I don't really understand how latex parses a keyval parameter but my guess is that it's processing the parameter as plain text. This is the code I currently have: \documentcl...

Spacetime interval in one co-ordinate system is given by : $$g_{\mu \nu} dx^{\mu} dx^{\nu} \tag{1}$$ $dx$ is some infinitesimal displacement vector between two events. Spacetime interval after a change of co-ordinate system is give by the algorithm : Change the basis of the matrix $g_{\mu \nu}$, ...

My world is a life bearing moon orbiting a gas giant. There will be plenty of geysers due to gravitational tectonic activity. Is it possible for geysers to happen in the middle of hot and arid biomes? If yes, could they serve as drinking water? What I imagined is that hot and salty water got viol...

Setup: Suppose a coin is tossed 8 times and I'm trying to determine whether the coin is biased in favour of landing on heads. Let $X$ be the number of heads in 8 tosses, so $X \sim B(8,p)$. Conducting a hypothesis test with a significance level of $5\%$, my null hypothesis would be $H_0\!: p = 0....

Get manual with help: help :w_c: :[range]w[rite] [++opt] !{cmd} Execute {cmd} with [range] lines as standard input (note the space in front of the '!'). {cmd} is executed like with ":!{cmd}", any '!' is replaced with the previous command |:!|. What...

Here is a past paper problem which I am struggling to solve currently. Let $\alpha,\beta\in E$, where $E$ an extension of field $F$. We are given $|F(\alpha):F|=6$ and $|F(\beta):F|=15$. What are the possible values of $|F(\alpha, \beta):F|$? I know that $F(\alpha, \beta)$ is a field extension fo...

How else did Job have additional kids? Job’s wife is not explicitly mentioned, or is she? We read: And the Lord restored Job’s losses when he prayed for his friends. Indeed the Lord gave Job twice as much as he had before. Then all his brothers, all his sisters, and all those who had been his a...

I have been reading on orbital mechanics and its strewn everywhere that beyond the escape velocity from any orbit around the earth, the trajectory is a hyperbola that approaches V_infinity and with this velocity we approach other planets (or solar system objects). But in reality, isn't the spacec...

How is it that stems became a part of standard music notation? I was genuinely unable to find an answer to this anywhere on the internet - I couldn't even find an instance of anyone asking the question.

Both 均衡{きんこう} and 平衡{へいこう} express equilibrium or balance. Both can be used with the verb 保{たも}つ. When is one more appropriate than the other?

I mounted a Rogue P4 Pull up bar system on my wall. I used 6 x 1/2" x 4.5" inch lag screws to fix each post to the wall. I did my best to make each post leveled. However, in the end, one of the posts is barely mounted too high, maybe 1/4 of an inch taller than the other post. I can feel the heigh...

I have an exercise to convert a sentence into direct speech: Rajeev said, "But, dad, didn't you promise me a new laptop?" I believe the answer should be Rajeev reminded his dad hadn't he promised him a new laptop. According to my textbook the answer is Rajeev reminded his dad if he hadn't pr...

In undergrad, I did a double major in pure mathematics and linguistics. I ended up loving both fields, and wanting to pursue an academic career. The only problem is that I couldn't decide between them. There is certainly some overlap between the two fields, but not especially much, and this makes...

The verb "antworten" takes the dative as in "ich antworte dem Mann". But when I look it up in my Langenscheidt "Deutsch als Fremdsprache" dictionary it is shown as [Vt]. To make it even more confusing, the first example in the dictionary uses (jemandem) and [Vi] as below. ant-wor-ten; antwortete...

I'm implementing an example in which 2 ESP32's communicate each other using the HTTP Protocol. One acts as Server the other as Client. In the examples provided, I am curious about the declaration on the server, as its implementation is declared in the setup() function (I am using the standard Ard...

My husband’s parents are very abusive now and in the past. Recently they’ve started bringing me into the mix, and so my husband decided to cut them out of our lives. They still text me to curse our marriage and say awful things about both of us. I know that filial piety is very important, but wha...

I have 3 (1D) large datasets, red, green, and blue, each containing about 50,000 continuous values. The datasets are values of pixels from an image, so for example, observation #1 in the red dataset does not correspond to observation #1 in the blue dataset. Put in a non-statistically way, the b...

Planetary invasions are usually depicted using drop-ships which come straight down from the sky. Pretty much armoured Apollo Era capsules. These can be bigger, for example Warhammer40k has company sized drop-ships, or smaller as in Starship Troopers (the books). However they seem very woundeable ...

Who are Saptamatrikas? Where we can find the first reference of Saptamatrikas in our scripture? What is the significance of Saptamatrika worship? Please provide references directly from primary sources.

For visualization purposes I would like to know what part of the valley does the 'mouth' refer to in English. Taken from Sapkowski's "The Lady Of The Lake": it lay right beside the mouth of the enchanted Cwm Pwcca valley

I am trying to calculate the state of this Quirk circuit by hand: where U is a custom gate with 2x2 matrix: From what I've read, I should be able to calculate the resultant amplitudes as: (U⊗I⊗I).(anti_controlled_H⊗I).(anti_Toffoli) which is But the resulting amplitudes in Wolfram Alpha do not...

As written in rigid body physics docs Allows rigid body collisions allocate on different groups (maximum 20). How do I use Multiple collections for Rigid Body simulations ? I see only one collection holder in the rigid body world. It means support for nested collections under the given collection...

I got this rainfall data from the Australian Bureau of Meteorology, but it comes in this .arc format which I can't open in QGIS. After a little bit of googling, I found some relevant instructions for converting it to .tif using 'Raster > Conversion > Translate (Convert Format)'. Whilst this does ...

11 pairs of dancers will participate in a dance competition finale. However, on the day of the competition, 7 dancers were tested Covid positive and had to be eliminated. Given that partners aren’t interchangeable, meaning, if two dancers are left without their pair, they can’t dance together to ...

Denote $S^3_{p/q}(K)$ by performing $p/q$-surgery along a knot $K$ in $S^3$. Let $K$ and $J$ be two arbitrary oriented non-trivial knots in $S^3$. Is there a nice relation between surgery on the connected sum of knots and the connected sum of their surgeries? More explicitly, let say, do we have ...

I believe this is simple, but I can't wrap my head around forming region for RegionPlot3D. I have been reading and trying to understand how to write the region but not success. So, here is the problem. I have this graph: plot1 = Plot[{x^2, -x + 2}, {x, -1, 2}, GridLines -> Automatic]; plot2 = Pl...

I’ve seen a bunch of videos (eg on YouTube) which show a custard and chiffon cake recipe. This custard is essentially: Egg yolks Condensed milk Milk or evaporated milk My preferred custard is quite thick and uses the Delia Smith recipe, which I would usually put on/in a meringue (great with pav...

In (Gottesman and Chuang 1999), when discussing quantum gate teleportation, they mention how it can be used to implement nonlocal gates such as a CNOT, by only using (classically controlled) local operations. Their scheme is shown in Figure 2 in the paper, which I report below: Now, specifics of...

Input: Aliquam ipsum ex, tempus ornare semper ac, varius vitae nibh. Output: A i e, t o s a, v v n. I need a javascript function to solve this. I'm trying something like this: function short_verse(verse) { let result = []; verse.split(' ').map(word => word.charAt(0) != '' ? result.push(w...

\begin{tikzpicture} \node at (1,2) {Β}; \node at (-5,2) {Α}; \draw [thick] (0,0) ellipse [x radius=1cm, y radius=2cm] node [below = 2.3cm] {Σύνολο Άφιξης};; \draw [thick] (-4,0) ellipse [x radius=1cm, y radius=2cm] node [below = 2.3cm] {Πεδίο Ορισμού}; ...

We bought our house about 6 months ago, and there was already a reasonable hole in the enamel (?) of the bathtub. It looks like this: [Edge to edge, the whole thing is about 3cm] I'm afraid I don't know what material the bath is made of, so I'm hoping this picture helps. My guess would be some k...

On a current stay at a mid-level hotel chain in El Paso, Texas, we left the do-not-disturb sign on the door in the morning, until we were done with breakfast, and getting ready for the day. Our room was skipped. The housekeeping staff told us that if they see the do-not-disturb sign on the door, ...

Rules Place some pentominoes into an 8 x 8 grid. They do not touch each other. They can touch only diagonally (with edge). Pentominoes cannot repeat in the grid. Rotations and reflections of a pentomino are considered the same shape. Grid is 8 x 8.

I came across this Short Video on Youtube by MicroType Engineering, link here. His simulation on LTSpice was the below: I tried the simulation on LTSpice myself. However I am curious as to what the voltage means. I entered the following values: Does this mean the voltage source is a DC voltage ...

I am trying to run the following code: f[t_] := {Cos[t], Sin[t], Cos[t]} s = 0.1; s1 = 2; t[u_] := Normalize[f'[u]] n[u_] := Normalize[t'[u]] b[u_] := t[u]\[Cross]n[u] Graphics3D[ { BSplineCurve[Table[f[u], {u, 0, 2 Pi, s}]], Arrow[Table[{f[u], t[u]}, {u, 0, 2 Pi, s1}]], Arrow[Table[{f[u...

The acronym AIR doesn't match with the functional name "Aircraft Certification Service" "AIR (the Aircraft Certification Service) is the department within the FAA that develops and administers safety standards for aircraft and related products that are manufactured in the United States or are use...

As I listen to plans to send asylum seekers to Rwanda, I can’t help think it would be more effective to prevent people smuggling activity by allowing asylum seekers to apply from overseas. It could be a quite simple online process, perhaps supported by small offices in other key countries. Or, in...

Look at the image: Bike model is Rocky Mountain Vertex 990. The frame is broken in a way that it's impossible to install rear disc brake. Possibly there may be a frame deformation. Is it even possible to restore the broken parts and install the brake? How must does it cost and how the process lo...

So there is this class I'm attending on Electronic Instrumentation which involves mostly circuit analysis. For example, RCL circuits, and circuits that also contain opamps, transistors, diodes, etc. Now when all the currents through these circuits are constant I understand we can talk about poten...

I would like to solve the following problem My attempted solution is as follows. Convergence implies that given $\epsilon>0$, the r exists $N$ such that for all $n\geq N$, we have \begin{align*} |a_{n+k}-{a}_{n} |<\epsilon. \end{align*} We are told that this has to hold for all $k$. Consider any...

I have a C-type cable with only three wires. I took apart the cable to get a better look inside and I only have 3 solder points. I'm trying to make an OTG host of the C cable that I have. Where do I attach the 5.1 k resistor if I only have three solder points? I'm trying to make C-type to micro U...

everyone! I have the "big" list of rules {{{Subscript[Abx, 1, 1] -> 182.745, Subscript[Aby, 1, 1] -> -248.194, Subscript[Abz, 1, 1] -> -4.84702}, {Subscript[Abx, 1, 2] -> 150.256, Subscript[Aby, 1, 2] -> -247.769, Subscript[Abz, 1, 2] -> -9.55877}, {Subscript[Abx, 1, 3] -> 130.9...

Is this notationally correct? $$ \begin{align*} 5 - x^2 &< -2, \\[1ex] 5 - x^2 + (-5) &< -2 + (-5), \\[1ex] -x^2 &< -7, \\[1ex] (-x^2)(-1) &> (-7)(-1), \\[1ex] x^2 &> 7, \\[1ex] ...

A teacher has 250 sample test questions and he selects 25 of which to put on the test. Johnny only practices 50 out of the 250 questions. a) What's the expected number of questions that appear on the exam that Johnny have practiced on? My answer: I'm quite confused so I don't think it's the right...

I have a variable (vLUF) in a batch file with a negative number, -5.0, and I need to get the number of characters it has. I'm using a command, posted on above link, but the variable "count" returns with no value. how to count characters in a text file Display input variable: echo input variable %...

We know that in characteristic $0$, all algebraic series are differentiably finite. Is this true in positive characteristic? I look at the proof, indeed we need to the characteristic to be $0$ for the proof to work. If it is not true in positive characteristic, is there a counter-example? Thank y...

For all my documents, I created a template.tex file which is imported in all my files. But it says that I got those issues : /home/rperrod/rp/PERSO/LaTeX/Test/template/t.tex:20: Undefined control sequence. [ \setlength{\footheight}{10mm}] /home/rperrod/rp/PERSO/LaTeX/Test/template/t.tex:20: LaTe...

The context I have a Geopackage file opened in QGIS. It was created by importing a CSV file. However, when importing, something regarding encoding went wrong: I have several special characters like Umlauts (Ä, Ö, Ü) that are not correctly recognized (like Ä for Ä). A possibility would be to fix ...

I'm using an Alliance parallel SRAM IC, and under the Electrical Characteristics table, it says VCC can be min 3.0 V, max 5.5 V. In the Absolute Maximum Characteristics table, it says terminal voltage must be no more than 7 V with respect to VSS (GND in my case). What happens when operating in th...

I have a table with this schema CREATE TABLE `UserIdPhoneNo` ( `userId` int NOT NULL, `phoneNo` bigint NOT NULL, PRIMARY KEY (`userId`,`phoneNo`) ) ENGINE=InnoDB DEFAULT CHARSET=utf8mb3 I am planning to store 1 Billion rows in this table. Queries are mostly reads and rarely writes. To make the...

I’m in the process of having a Mitsubishi mini split system installed. The electrician wired the condenser and didn’t put the cover back on leaving all the electronics exposed to a torrential all night downpour. Will this be ok or should I make them replace the condenser? Thanks

Let $u_0 = 100$, $ \alpha = -200 $ and $ \forall n \in \mathbb{N^*} u_n = u_{0} + n + \alpha \cdot \sum_{i=0}^{n-1} \frac{1}{u_{i}}.$ Is there any way to get the formula of $ u_{n} $ ? or at least get an approximation? Convergence or divergence ? Thank you.

I recently came across this ChessBase India article on Engine Defying problems. Here's a puzzle from the article about Helpmates. We need to get the helpmate in 15 moves. Note: It's black to move. I thought for a long time but couldn't come up with anything useful. Any help/idea on how to solve t...

I have basics polyhedrons like tetrahedrons and I would like to thicken their edges to make them seem like in this picture but with soft rounded edge instead of spheres. Is this possible?

Bob is an expected utility maximizer with utility function $u(x) = −e^{−ax}$, where $a > 0$ is a parameter. Bob has wealth $w$. There are two states of the world, a good state and a bad state. The probability that the state is bad is $π$. If the state is bad, Bob gets sick, and would need to spen...

I am trying to achieve a neat multi line equation. The equation is too big so I had to cut it half way (the first part of the equation). So this is what I ended up doing, the equation break after the equal sign. After this part, I am simplifying the equation. I would like the equal signs to line...

I have a round mesh plane that I'm trying to divide into three equally-sized and separate meshes. I made a cutting tool using three planes I joined together, which I'm trying to use to perform this operation. What workflow can I use to achieve the result I described? Using Blender v2.82a.

I need to draw with tikz a closed curve of the following shape, which is kind of like a trefoil knot suppressed into one plane, but is also rotated. I was wondering if there could be any help or suggestion on that? Thanks in advance.