Cozyno Casino Review - Unbiased opinion on its game offer, customer support and withdrawal policy. See AnswerSee AnswerSee Answer done. Trigonometry. We will use the unit circle definitions for sine and cosine, the Pythagorean identity. It assigns the value of i=1 and the loop continues till the condition of the for loop is true. 6293… x 30. Cos (x) = cosx + (pow (x, i) / fact) * sign. Give values for the Fourier cosine series C(x)=A02+∑n=1∞Ancos(nπ3x)C(x)=A02+∑n=1∞Ancos(nπ3x). Recall the Fourier series is a serie of the form ansin(2πnx T) + bncos(2πnx T) where T is the period of the function. Nutzen Sie einfach unseren Link, um zum Cozyno Casino zu gelangen. Their reciprocals, though used, are less common in modern mathematics. When you join the casino we give you. Fast Barrier Option Pricing by the COS BEM Method in Heston Model. Cozyno Casino Review -. Join me on Coursera: notes at example, Chau et al. . Problem 4. Math. 2 Problem 3. This function describes the shape of a hanging cable, known as the catenary. This function describes the shape of a hanging cable, known as the catenary. After a description of its use in the Black and Scholes (BS) model. Previous question Next question. I know that sounds very confusing but it’s basically a way to measure how unique a certain word is relative to. In this case we compute [Math Processing Error] a 0 = 2 3 and [Math Processing Error] a n = 4 ( − 1) n n 2 π 2. We have to find C1's value. 0 out of 5 on our portal. Author: Swokowski. Meldungen über einen Cozyno Betrug sind uns nicht bekannt. I Integration and LT tables provided. An illustration of a periodic function with period . Miscellaneous. For n a positive integer, expressions of the form sin(nx), cos(nx), and tan(nx) can be expressed in terms of sinx and cosx only using the Euler formula and binomial. Y{array-like, sparse matrix} of shape (n_samples_Y, n_features), default=None. Consider the familiar trigonometric identity: cos3(x) = 34cos(x) + 14cos(3x) cos 3 ( x) = 3 4 cos ( x) + 1 4 cos ( 3 x) Show that the identity above can be interpreted as Fourier series expansion. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this siteFinal answer. Play List of Trigon. At π 4, we can see that the x -value is √2 2. The computer has as many memory units as needed. the infinite product = = is defined to be the limit of the partial products a 1 a 2. Share. This video works on the cosine terms. I Heat Eq. 2. cosz. 78539. Betrug konnten wir keinen feststellen. With boundary value problems we will have a differential equation and we will specify the function and/or derivatives at different points, which we’ll call boundary values. 4. STEP 4: Read the number of terms in the Series to n. Geometrically, these are identities involving certain functions of one or more angles. It is described first in Cooley and Tukey’s classic paper in 1965, but the idea actually can be traced back to Gauss’s unpublished work in 1805. Simulation examples reveal that our. 1) (9. . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 /* cos example */ #include <stdio. Kein Wunder, denn die Betreiber werden von der Regierung auf Curacao stets überprüft. All reviews were written by real casino players. In fact, cos z = 1 2(eiz +e−iz), sin z = 1 2i(eiz −e−iz) cos z = 1 2 ( e i z + e − i z), sin z = 1 2 i ( e i z − e − i z) so. After constructing an even extension, and evaluating over − π ≤ x ≤ π. 465), also denoted arccosz (Abramowitz and Stegun 1972, p. Let f(x), f 1 (x), and f 2 (x) be as defined above. Pricing requires resorting only to odd moments in. 3. Retrieved from "Casino is an online casino where members have access to video slots and casino games from a multitude of software providers. The inverse hyperbolic cosine cosh^(-1)z (Beyer 1987, p. Well, from the above output, you can see that the simple recommender did a great job!. W. ISBN: 9781133382119. Cos is the cosine function, which is one of the basic functions encountered in trigonometry. 6. See for example Wikipedia for a MATLAB implementation which takes radians as input. Moreover we propose a kind of Fourier cosine expansions with polynomials factors such that the. You can't beat that with a Newton-Raphson iteration starting. Let φ (x) = 0 for 0 < x < 1 and φ (x) = 1 for 1 < x < 3. Both one-dimensional and two-dimensional Fourier-cosine series expansions are used to approximate the density of the time to ruin, and the approximation errors are also obtained. The Fourier cosine expansion (COS) method is used for pricing European options numerically very efficiently. 4. For a cosine series. Often you will also be able to get spins that you can use a slot machine for free. Sine (series) For , the series. TRANTER AN. 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It assigns the value of i=1 and the loop continues till the condition of the for loop is true. xxix). 1 FOURIER SERIES FOR PERIODIC FUNCTIONS This section explains three Fourier series: sines, cosines, and exponentials eikx. You can play classic, golden. I plot a cosine sampled at 400 points in the interval -5 . Special values include cosh0 = 1 (2) cosh (lnphi) =. In mathematics, sine and cosine are trigonometric functions of an angle. I'm aware for Fourier Cosine Series you have an even extension of f(x) and the Sine Series has an odd extension, the former requiring a_o, a_n, and cosine as the periodic function, with the latter Even and Odd Extensions. Add a comment. + 60 ilmaiskierrosta. To find the Fourier sine series and Fourier cosine series of the function f ( x) = 2 x on the interval [ 0, 1]] we need. Cozyno online casino is a legit casino and we recommend this casino to our visitors. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Take in the number of terms to find the sum of the series for. I have a discontinuous function: F(x) ={0, πx, −π < x < 0 0 < x < π F ( x) = { 0, − π < x < 0 π x, 0 < x < π. Cozyno Casino Review. An arbitrary point on the plane is a linear combination of the points (1;0) and (0;1), and one can see that multiplication by ei will act as rotation by on any such linear combination, knowing that it does so for the cases of (1;0) In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. A novel family of Cosine series Quantum Sampling (QCoSamp) operators appropriate for quantum computing is described. For Complete Video Series visit More Learning Resources and Full videos are only available at ww. Player rating 6 reviews. B) / (||A||. First of all, am i right in thinking this function, because discontinuous, is neither odd or even. We will also define the even extension for a function and work several examples finding the Fourier Cosine Series for a function. com Discover cozyno. UPDATE. STEP 3: Declare the variables x, sign, cosx, fact as type float. Viewed 5k times. It is defined in <math. Clearly, the Series vanishes if k = 1. 481), sometimes called the area hyperbolic cosine (Harris and Stocker 1998, p. cos (−θ) = cos θ. Explore math with our beautiful, free online graphing calculator. Because the selection of design points where data are collected greatly affects the modeling process, we study the choice of fractional factorial designs for fitting Fourier-cosine models. x2y′′ +3xy′ +λy = 0 y(1) = 0 y(2) = 0 x 2 y ″ + 3 x y ′ + λ y = 0 y ( 1) = 0 y ( 2) = 0. An arbitrary point on the plane is a linear combination of the points (1;0) and (0;1), and one can see that multiplication by ei will act as rotation by on any such linear combination, knowing that it does so for the cases of (1;0)In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. In most cases, the convergence rate of the COS method is. 1. This is the general formula for Fourier Series, which includes both cosine and sine terms. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this siteThis paper studies the statistical estimation of the Gerber-Shiu discounted penalty functions in a general spectrally negative Lévy risk model. Ortiz-Gracia, S. Part (i) #a_n = 1/L int_0^(2L) f(x) cos ((n pi x)/L) dx# # implies a_n = 5/pi int_0^(2 pi) sin (theta/2) cos (n theta) d theta#Here is the list of formulas for Class 11 students as per the NCERT curriculum. Some simulation examples are illustrated to show that. We show that the estimator is easily computed and has a fast convergence rate. Check out our casino games, bonus codes, latest complaints, new. or inverse1 Fourier integrals. This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this siteIt consists of a series of three rectangular pulses symmetric to the origin with the outer two pulses having the height —1 and the width w. Cos 30° =Length of the adjacent/length of the hypotenuse = PR/PQ. f(x)={−xfor 0<x<2,0for 2<x<3. For example, if you have two vectors X1 and X2, and your Pearson correlation function is called pearson (), pearson (X1, X2) == pearson (X1,. Remark: Notice that the approximation is INCORRECT since the first three coefficients are zero by default in the simulation. The product is said to converge when the limit exists and is not zero. com Cozino Online Casino Review 2023 - Discover what's waiting for you at this online casino saloon. (infinitely) That is the idea of a Fourier series. The Fourier coefficients of F are defined byVIDEO ANSWER: We are going to find the Fourier series a0, an and bn for the given function. is called the cosine series, and the series. In this C program, library function defined in <math. R. Sin squared x means sin x whole squared. 3. I would be very happy to resolve all yo. 1. Similarity = (A. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. In complex analysis, Euler's formula provides a fundamental bridge between the exponential function and the trigonometric functions. First we have the following asymptotic expansion cosn √n + cosn = cosn √n 1 1 + cosn √n = cosn √n (1 − cosn √n + cos2n n + O( 1 n3 / 2)) = cosn √n + cos(2n) 2n − 1 2n + O( 1 n3 / 2) Now we are going to prove that the series ∑ cosn √n and ∑ cos ( 2n) 2n are convergent using the Abel transform. Check our article about Taylor series if you need more details. representing a function with a series in the form Sum ( A_n cos (n pi x / L) ) from n=0 to n=infinity. (b) For each x (0 ≤ x ≤ 3), what is the sum of this series? (c). Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. To start consider only the constant terms. In this C program, library function defined in <math. Whether to return dense output even when the input is sparse. In the following discussion it is assumed that xT is real so an and bn are real. e, $$cos ext{A}cdotcos ext{B. Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. 5. Because a full rotation equals 2*pi radians, one degree is equivalent to pi/180 radians. The key insight is in the close relation of the characteristic. Dirichlet’s condition for Fourier expansion of a given function f are: The function is well-defined and single-valued except at a finite number of points in (–L, L). Oosterlee 2. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. Im Gegenteil: Nahezu alle Online Buchmacher begrüßen ihre neuen Kunden mit einem attraktiven Willkommensgeschenk. Use a calculator to find sin 39°: d/30 = 0. Question: Determine if the function shown below is an even or odd function, and what is the coresponding sine or cosine series representation of the function? f(x)={x+1x−1−130 F. Fourier Cosine Series Examples January 7, 2011 It is an remarkable fact that (almost) any function can be expressed as an infinite sum of cosines, the Fourier cosine series. Obviously, the minimization is simply given as the k-th de-gree expansion: µ$ i = #k l=0 (8) ζ i,ψ l ψ l. First of all, am i right in thinking this function, because discontinuous, is neither odd or even. Use a for loop ranging from 1 to the number and find the sum of the series. equivalently if one of $Sigma a_i$,$Sigma b_i$ have unbounded partial sums, then that implies that the partial sums of the Fourier series itself. Play 1200+ top games and get started with 100 free spins!250%. It is easy to check that these two functions are defined and integrable on and are equal to f(x) on . The program is compiled using Dev-C++ compiler, but you can use any standard C compiler to compile and run the program. That is, all the energy of a sinusoidal function of frequency A is entirely localized at the frequencies given by |f|=A. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Bonus code ASK150. De nition. In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The computer can write the content of a memory unit into another memory unit. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). Initialize the sum variable to 0. Special values include cosh0 = 1 (2) cosh (lnphi) = 1/2sqrt (5), (3. Casino Site Global Rank. The dot() function. (6)The program to compute cosine series is based on the mathematical concept of sequences and series – particularly power series. x = x * 3. STEP 2: Declare the integer variables n, x1, i, j. 1 Space, the Final Frontier 337 To be neater, we again use the notation f−(ξ) = f(−ξ), and with this definition the duality results read exactly as in the one-dimensional case:Free Law of Cosines calculator - Calculate sides and angles for triangles using law of cosines step-by-step. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. W. Plotting cosine with Python. comWe discuss the Fourier sine and cosine series, and talk about how to sketch the relevant graphsHalf range cosine series is given by $a_{ heta} = frac{1}{l}intlimits_{0}^{l}f(x)dx$ $a_{n} = frac{2}{l}intlimits_{0}^{l} f(x)cos frac{n pi x}{l}dx , b_{n} = 0$Here we develop an option pricing method for European options based on the Fourier-cosine series, and call it the COS method. As you can see from the plot you included sin(nπ) = 0 for any integer n. Sine, cosine, and tangent are the most widely used trigonometric functions. W. Both one-dimensional and two-dimensional Fourier-cosine series expansions are used to approximate the density of the time to ruin, and the approximation errors are also obtained. e^ {ix} = cos {x} + i sin {x}. Some numerical examples are also presented to. ] A very important fact is that the. Input data. In this case we compute [Math Processing Error] a 0 = 2 3 and [Math Processing Error] a n = 4 ( − 1) n n 2 π 2. Pearson correlation and cosine similarity are invariant to scaling, i. Why Fourier series is so important? A periodic signal is just aFor More Free Videos Download Vibrant Phoenix App :. 1) (9. Let f f be expressed by a half-range Fourier sine series : f(x) ∼ ∑n= 1∞ bn sin nπx λ f ( x) ∼ ∑ n =. The hyperbolic cosine is defined as coshz=1/2 (e^z+e^ (-z)). For any complex number z ∈ C : cosz = exp(iz) + exp( − iz) 2. (10pts) problem 3. Ist das Cozyno Casino. Get the list of similarity scores of the movies concerning all the movies. f(x) = a0 2 +∑n=1∞ an cos nπx L, f(x) even. Pearson correlation is also invariant to adding any constant to all elements. Find the Exact Value cos (pi/2) cos ( π 2) cos ( π 2) The exact value of cos(π 2) cos ( π 2) is 0 0. The hyperbolic trigonometric functions extend the notion of the parametric equations for a unit circle ((x = cos t) and (y = sin t)) to the parametric equations for a hyperbola, which yield the following two fundamental hyperbolic equations: [x = cosh a = dfrac{e^a + e^{-a}}{2},quad y = sinh a = dfrac{e^a - e^{-a}}{2}. From my understding, if the given function is even, there will be only cosine c o s i n e term in the series, thus bn = 0 b n = 0. 5 cm. Cozyno Casino offers you games powered by regulated game suppliers and the casino has reliable licenses. Unlock. Advanced Math questions and answers. Advanced Math questions and answers. φ = arg z = atan2 (y, x). denotes the exponential function. Das deine Anmeldung auf jeden Fall gut abgesichert ist, das haben wir auch in unserem Cozyno. 3. . [1] The function, with length and amplitude is given by: [a] For digital signal processing, the function is sampled symmetrically (with spacing and amplitude ): which is a sequence of samples, and can be even or. The function series_cosine_similarity_fl() is a user-defined function (UDF) that takes an expression containing two dynamic numerical arrays as input and calculates their cosine similarity. 71471090, 71671166) and the Natural Science Foundation of Jiangsu Province, PR China (No. So we have values of derivatives in the following sequence : katex is not defined and so on. Fourier transform has many applications in Engineering and Physics, such as signal processing, RADAR, and so on. 263) are sometimes used to refer to. Find the values of the Fourier cosine series of f (x) for x in [0,L], and justify your answer using the convergence properties presented. It can be verified that only a single term ( the first harmonic) can be evaluated here. f ( x) = C 0 2 + ∑ n = 1 ∞ C n cos n π x L. Multiply both sides by 30: d = 0. W. Minimum Deposit €20. The Fourier series for an even function with period 2L 2 L is thus. 1. v. Cosine similarity is a measure of similarity, often used to measure document similarity in text analysis. φ(ω) = Z R eixωf(x)dx, (2) f(x) = 1 2π Z R e−iωxφ(ω)dω. 1) a n = 2 L ∫ 0 L f ( x) cos n π x L d x, b n = 0. Most websites like Amazon, YouTube, and Netflix use collaborative filtering as a part of their sophisticated recommendation systems. Recalling the right-triangle definitions of sine and cosine, it follows that. It's named after Joseph Fourier, a French mathematician and physicist who introduced the series in the early 19th century. 17) (seeIn this section we go through the complete separation of variables process, including solving the two ordinary differential equations the process generates. Write the series in terms of consecutive integers n. Minimum Deposit €20. We made a special deal with the casino and we have a great Cozyno bonus for you when you join the casino. Thus, both series are absolutely convergent for all . Sin, cos, and tan are trigonometric ratios that relate the angles and sides of right triangles. ISOTROPIC RECTANGULAR PLATES SUBJECTED TO NON-UNIFORM HEATING. From the first integration by parts we got I = − cos x cos nx − nJ. Look at the main equation for f (t) at the beginning of the video. We are reading the number of the terms and the degree value of the series using ‘n’ and ‘x’ variables. In this paper, we investigate the pricing problem of barrier options under the Heston model. Ao = 9/4 . There are six trigonometric functions: sine, cosine, tangent and their reciprocals cosecant, secant, and cotangent, respectively. This video works on the cosine terms. 6. 33x (d+b) Cashable Yes. The general solution consists of (4. . The formula for finding a0 is 1 by pi integral of f of x dx that is a0 1 by pi and the integral varies from minus 3 to minus 2 and 2 to 2. Let us evaluate cos (90º - 30º) to understand this better. Comparison of Fourier Sine and Cosine Series Expansions for Beams with Arbitrary Boundary ConditionsJust by offering phone support to its player, Cozyno Casino is better than roughly 60% of all other online casinos. – Klaramun. Look at the main equation for f (t) at the beginning of the video. User must enter the upper range limit and the lower range limit. 4. Recall, the Fourier cosine series is formed from the eigenfunctions of the eigenvalue problem on [0,L] with boundary conditions u'(0) = u'(L)=0. (2018), where the method is used to denoise EEG. The material properties of. . Let $\displaystyle\text{C}=\prod_{r=1}^{7}\cos{\left(\dfrac{r\pi}{15}\right)}$ . 0 2 Let f (x) for 0 < x <3, for 3 < 3 < 4. Although I think my solution is correct (please correct me if I'm wrong) I still would like to see other solutions and in particular understand the author's hint since I can. 1. In the function factorial you are doing an int multiply before assigned to the double return value of the function. My goal is to find the Fourier cosine series for f(x) = cos(πx) f ( x) = cos ( π x) on the interval (0, 1). h> header file is used to compute mathematical functions. , Beyer 1987, p. The Hann function is named after the Austrian meteorologist Julius von Hann. . My understanding is this: If instead of a symmetric interval [−L, L] [ − L, L] you are provided by an interval [0, L] [ 0, L] then you will have to extend the graph to 2L 2 L to come up with a function that is either even or. The function of the half-range expansion is necessary: O For a sine series. 1. The next video works on the sine terms. Expires in. Here we develop an option pricing method for European options based on the Fourier-cosine series, and call it the COS method. In the case of Neumann boundary conditions, one has u(t) = a 0 = f. f ( x) = cos x is even and all A n vanish when you consider interval between 0 and L = π. If we integrate weFinding the coefficients, F’ m, in a Fourier Sine Series Fourier Sine Series: To find F m, multiply each side by sin(m’t), where m’ is another integer, and integrate: But: So: Åonly the m’ = m term contributes Dropping the ‘ from the m: Åyields the coefficients for any f(t)! f (t) = 1 π F m′ sin(mt) m=0 ∑∞ 0For the function f (x)=1−x,0≤x≤1 find (a) the | Chegg. Let I = − cos x cos nx − nJ where. 4. This can be done in two ways: We can construct the even extension of f (x) : or the odd extension of f (x) : For the even function, the Fourier. But if the function is odd, there will only be sine s i n e term in. 4: import matplotlib. Calculate the Fourier series. By adding infinite sine (and or cosine) waves we can make other functions, even if they are a bit weird. Write a VBA function, MacCos_n (x,nf), that finds the sum of the. Express each of the trigonometric Fourier series found in Problem. Publisher: Cengage. 5. We consider all Blacklisted operators to be. Square waves (1 or 0 or −1) are great examples, with delta functions in the derivative. In most cases, the convergence rate of the COS method is exponential. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Guardasoni, L. With this motivation in mind, we have the following k-thIn this paper, the density of the time to ruin is studied in the context of the classical compound Poisson risk model. The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. Therefore, we have [Math Processing Error] f ( x) ∼ 1 3 + 4 π 2 ∑ n = 1. Enumerate them (create tuples) with the first element being the index and the second element is the cosine similarity score. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse ), and the cosine is. The method of computing Fourier series is quite different from the methods of computing power series. Proof. 4. Of course, this is a pretty large tensor — at 512x768 — and we want a vector to apply our. Again the forth term is negative. We write the Fourier series for [Math Processing Error] f ( t). Step 2: We know, cos (a - b) = cos a cos b + sin a sin b. Use an orthogonality argument to derive the formula (2. Sketch the even 2 - f ( x) given by: There are 2 steps to solve this one. Here, a = 90º and b = 30º. View the full answer Step 2. The graph of y = cos (x). The density and its characteristic function, f(x) and φ(ω), form an example of a Fourier pair. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. (it is not necessary to generate them). Na nossa avaliação, nós consideramos as reclamações dos jogadores sobre o casino, receitas estimadas, licença, genuinidade dos jogos, qualidade do apoio ao. The Tukey window is a rectangular window with the first and last r/2 percent of the samples equal to parts of a cosine. The value returned by cos () is always in the range: -1 to 1. while Loop. Type Welcome Bonus. Product of Cosine Series, Cosine Product Series, Trigonometric Ratio and Identities Class-11. We use the below formula to compute the cosine similarity. e. Note: VBA does not have a factorial function, but you can borrow Excel’s factorial function using WorksheetFunction. 500539546249695 cos (x) by default function is -0. With boundary value problems we will have a differential equation and we will specify the function and/or derivatives at different points, which we’ll call boundary values. Geometrically, these identities involve certain trigonometric functions (such as sine, cosine, tangent) of one or more angles. CasinoRank. In the geometrical proof of cos(a+b) formula, let us initially assume that 'a', 'b', and (a+b) are positive acute angles, such that (a+b) < 90. Compute the Fourier cosine coefficients for f (x). To find the sum of cos (x) series, the following formula is used. There’s no need in reinventing the bicycle each time, when you face one of common functions. The Fourier cosine series of f(x) on [0;L] is a 0 2 + X1 n=1 a ncos nˇx L; (4) where a n= 2 L ZSome doubts regarding half range cosine series. So the first you need to fins is the period of such a function, and then express this as a sum of these sines and cosines. 4. Include lengths: sin 39° = d/30. Compute the Fourier cosine coefficients for f (x). For second order differential equations, which will be looking at pretty much exclusively here, any of. Complex Form of Fourier SeriesFinal answer. If we calculate the limit at k = 1 k = 1, we get 1 1. Included is an example solving the heat equation on a bar of length L but instead on a thin. 141; Bronshtein and Semendyayev, 1997, p. Avaliamos Cozyno Casino completamente e atribuímos um Índice de Segurança muito baixo ,o que significa que é dos piores casinos para apostar. . 3Blue1Brown, by Grant Sanderson, is some combination of math and entertainment, depending on your disposition.