X

+

1:20 AM | Generalizing an already tough integral

I did the case $p=1$ here. The generalization to higher $p$ may involve higher-order derivatives as follows: $$\begin{align}K_p &= \int_0^{\pi/2} dx \frac{x^{2 p}}{1+\cos^2{x}} = \frac1{2^{4 p-1}} \int_{-\pi}^{\pi} dy \frac{y^{2 p}}{3+\cos{y}} \end{align}$$ So define, as before, $$J(a) = \int_{-\pi}^{\pi} dy \frac{e^{i a y}}{3+\cos{y}} $$ Then $$K_p = \frac{(-1)^p}{2^{4 p-1}} \left [\frac{d^{2 p}}{da^{2 p}} J(a) \right ]_{a=0}$$ […]

+

cherry-merchant:
Sierpinski transformation

+

mathani:
Get you best paper, cut a circle and fold it so that the circumference falls on a fixed point inside. Repeat, using random folds. Now see the creases. This is how you paper-fold an ellipse.

+

12:30 PM | 4/19/14

4 - 1 = √9 * 1^4
Also:
4 - 1 = √9 = |1 - 4|
Also:
4! = 19 + 1 + 4
Also:
41914 is a palindrome

+

centerofmath:
The Ulam spiral shows a pattern in the (unpredictable) primes

+

Salvador Dalí, Crucifixion (Corpus Hypercubus), 1954

+

centerofmath:
The Ulam spiral shows a pattern in the (unpredictable) primes

+

1:25 PM | The Four-Color Theorem

BIRTHDAY CHALLENGE: Pick a line drawing of the Map of Nigeria (like the one below) with the borders of its states included and try to color each state with one […]

+

12:30 PM | 4/18/14

√4 = 1 * 8 * 1 / 4
Also:
4 = 18 - 14
Also:
|4 + 1 - 8| = |1 - 4|
Also:
41814 is a palindrome

+

12:18 PM | Good Friday

(Click on the cartoon to see the full image.) (C)Copyright 2014, C. Burke.Or possibly labeled "Passion, Death, Resurrection"? Good Friday is part of the Paschal Triduum, which can also be spelled "Pascal", but I wanted to avoid confusion and basically hang a lampshade on it.

+

11:30 AM | Big Helping of Friday Potpourri

ICYM them, quite a mishmash of varied links from the past week (something for everyone perhaps):
1) Martin Gardner's more philosophical writings here:
http://martin-gardner.org/Philosophy.html
2) Who knew there was so much to know about the arrangement of dice? Tanya Khovanova filled us in on what she learned from John Conway:
http://blog.tanyakhovanova.com/?p=489
3) Just in case you've been

+

Επιλεγμένες ασκήσεις (με τις λύσεις τους) για την τελική επανάληψη σας,από τον φίλο και συνάδελφο Γιώργο Μαυρίδη. Οι ασκήσεις προέρχονται από την ανανεωμένη έκδοση του βοηθήματος της Γ Λυκείου που ξεχωρίζει για τα πολλά λυμένα θέματα που περιέχει , την εύληπτη και
[…]

+

mathani:
Get you best paper, cut a circle and fold it, fold it so that the circumference falls on a fixed point inside. Repeat, using random folds. Now see the creases. This is how you paper-fold an ellipse.

+

6:55 PM | "If Tomorrow Starts Without Me"

My cousin Patricia (she'll always be "Patty" to me) read the following poem at her mother's funeral yesterday. (That would be my Aunt Pat.) There wasn't a dry eye in the place. And if that didn't get you, then her sister Debbie, in her eulogy, remembering those who have left us since last August did: my Mom, their sister Barbara, my niece Lisa and most recently, our cousin Mary Ann. To that, I'll add a few more: my Uncle Eddie (on the other side of the family) last week, my colleague Ms.
[…]

+

Just a mathematical meditation for today...:About 10 days ago in a longish post over at MathTango I included some links to Keith Devlin materials -- one of my favorites was a podcast he did for NPR's "On Being," an always-wonderful radio-hour. In that conversation he made the following observation, which I fancy, regarding a transition on the road to becoming a mathematician [bold added]:"...that's when I became a mathematician; that's what I stumbled on at age 15 or 16 when here I was
[…]

+

@pickover: Not only does 12+1=11+2, but the letters “twelve plus one” rearrange to give you “eleven...

+

4:00 PM | The New Math Teacher

Have you met our new colleague? He's a teletubby whose stomach plays Khan Academy videos. Continue reading →

+

1:06 PM | Funny How Things Work Out

It had been my intention, if I could muster the discipline and determination, to have a new comic each day this week because I'm off. Funny how things work out. It seems like every day is booked with something, the first three days being the worst. Sure, I might have something by Thursday, but that's hardly "every day" of the week. That's catch-up time. There will be an update, and if I can spend Friday working on things, I'll plan out next week ahead of time. I hope. And next Monday and
[…]

Sidebar

Filters

+

Here the monotonicity of the EM algorithm is established. $$ f_{o}(Y_{o}|\theta)=f_{o,m}(Y_{o},Y_{m}|\theta)/f_{m|o}(Y_{m}|Y_{o},\theta)$$ $$ \log L_{o}(\theta)=\log L_{o,m}(\theta)-\log f_{m|o}(Y_{m}|Y_{o},\theta) \label{eq:loglikelihood} $$ where \( L_{o}(\theta)\) is the likelihood under the observed data and \(L_{o,m}(\theta)\) is the likelihood under the complete data. Taking the expectation of the second line with respect to the conditional distribution of
[…]

Ruslan R Salakhutdinov, Sam T Roweis & Zoubin Ghahramani (2012). On the Convergence of Bound Optimization Algorithms, arXiv, arXiv: 1212.2490v1

Wu C.F.J. (1983). On the Convergence Properties of the EM Algorithm, The Annals of Statistics, 11 (1) 95-103. DOI: 10.1214/aos/1176346060

McLachlan G. & Peel D. DOI: 10.1002/0471721182

Citation

+

by @ulaulaman about #exoplanets #planetary_transit #kepler_mission #nasa #astronomy The search for extrasolar planets (or exoplanets) had its first success in 1991 with the discovery of some planets orbiting around the pulsar PSR1257+12(1, 2, 3), measuring the variations of the radio pulses coming from the star. The second important milestone in exoplanet research takes place in 1995, with the discovery around the star 51 Pegasi (a star like our Sun) of a Jupiter-like planet, found at a […]

Wolszczan, A. & Frail, D. (1992). A planetary system around the millisecond pulsar PSR1257 + 12, Nature, 355 (6356) 145-147. DOI: 10.1038/355145a0

Wolszczan, A. (1994). Confirmation of Earth-Mass Planets Orbiting the Millisecond Pulsar PSR B1257 + 12, Science, 264 (5158) 538-542. DOI: 10.1126/science.264.5158.538

Mayor, M. & Queloz, D. (1995). A Jupiter-mass companion to a solar-type star, Nature, 378 (6555) 355-359. DOI: 10.1038/378355a0

Borucki, W., Koch, D., Basri, G., Batalha, N., Brown, T., Caldwell, D., Caldwell, J., Christensen-Dalsgaard, J., Cochran, W., DeVore, E. & Dunham, E. (2010). Kepler Planet-Detection Mission: Introduction and First Results, Science, 327 (5968) 977-980. DOI: 10.1126/science.1185402

Lissauer, J., Fabrycky, D., Ford, E., Borucki, W., Fressin, F., Marcy, G., Orosz, J., Rowe, J., Torres, G., Welsh, W. & Batalha, N. (2011). A closely packed system of low-mass, low-density planets transiting Kepler-11, Nature, 470 (7332) 53-58. DOI: 10.1038/nature09760

George, S. (2011). Extrasolar planets in the classroom, Physics Education, 46 (4) 403-406. DOI: 10.1088/0031-9120/46/4/004

LoPresto, M. & McKay, R. (2005). An introductory physics exercise using real extrasolar planet data, Physics Education, 40 (1) 46-50. DOI: 10.1088/0031-9120/40/1/003

Citation

+

Non voglio fare concorrenza alla splendida spiegazione di Amedeo o a quella tecnica di Corrado, ma mia sorella, leggendo il post di pancia scritto nella sera dell'annuncio di BICEP2, ha candidamente confessato di non aver capito cosa era accaduto quel giorno. E allora proviamoci, a raccontarlo. (da The Cartoon History of the Universe #1 di Larry Gonick)C'era una volta un'idea di universo, che era la Terra al centro, quindi il Sole, la Luna e gli altri pianeti e sullo sfondo le stelle fisse,
[…]

Hubble E. (1929). A relation between distance and radial velocity among extra-galactic nebulae, Proceedings of the National Academy of Sciences, 15 (3) 168-173. DOI: 10.1073/pnas.15.3.168

Gamow G. (1948). The Evolution of the Universe, Nature, 162 (4122) 680-682. DOI: 10.1038/162680a0

Alpher R.A. & Herman R. (1948). Evolution of the Universe, Nature, 162 (4124) 774-775. DOI: 10.1038/162774b0

Alpher R., Bethe H. & Gamow G. (1948). The Origin of Chemical Elements, Physical Review, 73 (7) 803-804. DOI: 10.1103/PhysRev.73.803

Peebles P.J.E., Schramm D.N., Turner E.L. & Kron R.G. (1994). The Evolution of the Universe, Scientific American, 271 (4) 52-57. DOI: 10.1038/scientificamerican1094-52

McLeish T.C.B., Bower R.G., Tanner B.K., Smithson H.E., Panti C., Lewis N. & Gasper G.E.M. (2014). History: A medieval multiverse, Nature, 507 (7491) 161-163. DOI: 10.1038/507161a

Citation

+

Much of my undergrad was spent studying physics, and although I still think that a physics background is great for a theorists in any field, there are some downsides. For example, I used to make jokes like: “soft isn’t the opposite of hard sciences, easy is.” Thankfully, over the years I have started to slowly […]

Lazer, D., Kennedy, R., King, G. & Vespignani, A. (2014). Big data. The parable of Google Flu: traps in big data analysis., Science, 343 (6176) 1203-1205. PMID: 24626916

Citation

+

Un triangolo aureo è un triangolo isoscele in cui il rapporto tra uno dei lati uguali con la base è pari alla sezione aurea $\varphi$. Utilizzando un triangolo aureo di lato 1, è possibile dimostrare che \[1 + \frac{1}{\varphi^2} + \frac{1}{\varphi^4} + \cdots = \varphi\] \[\frac{1}{\varphi} + \frac{1}{\varphi^3} + \cdots = 1\] \[\frac{1}{\varphi} + \frac{1}{\varphi^2} + \frac{1}{\varphi^3} + \cdots = \varphi\] Il triangolo aureo qui sopra è lo screenshot della applet realizzata da Irina
[…]

Edwards S. (2014). Proof Without Words: An Infinite Series Using Golden Triangles, The College Mathematics Journal, 45 (2) 120-120. DOI: 10.4169/college.math.j.45.2.120

Citation

rlgordonma posted Generalizing an already tough integral on Math math math math baked beans and math

I did the case $p=1$ here. The generalization to higher $p$ may involve higher-order derivatives as follows: $$\begin{align}K_p &= \int_0^{\pi/2} dx \frac{x^{2 p}}{1+\cos^2{x}} = \frac1{2^{4 p-1}} \int_{-\pi}^{\pi} dy \frac{y^{2 p}}{3+\cos{y}} \end{align}$$ So define, as before, $$J(a) = \int_{-\pi}^{\pi} dy \frac{e^{i a y}}{3+\cos{y}} $$ Then $$K_p = \frac{(-1)^p}{2^{4 p-1}} \left [\frac{d^{2 p}}{da^{2 p}} J(a) \right ]_{a=0}$$ […]

2014-04-20 01:20:32

료칸 미키야에서 체크인을 끝내고 밖으로 나와 봤습니다. 외양 온천 메구리를 즐기기 전에 일단 마을의 이곳저곳을 좀 돌아다녀 보기 위해서… 역시 평상복을 입고 돌아다니는 사람들보다 유카타를 걸치고 있는 사람들이 훨씬 많았습니다. 유카타를 보면 머물고 있는 료칸이 어딘지도 알 수 있죠! 7개의 … Continue reading →Related Posts ?[교토 쿠라마 여행 05] 쿠라마 온천
[…]

2014-04-19 21:42:47

Here the monotonicity of the EM algorithm is established. $$ f_{o}(Y_{o}|\theta)=f_{o,m}(Y_{o},Y_{m}|\theta)/f_{m|o}(Y_{m}|Y_{o},\theta)$$ $$ \log L_{o}(\theta)=\log L_{o,m}(\theta)-\log f_{m|o}(Y_{m}|Y_{o},\theta) \label{eq:loglikelihood} $$ where \( L_{o}(\theta)\) is the likelihood under the observed data and \(L_{o,m}(\theta)\) is the likelihood under the complete data. Taking the expectation of the second line with respect to the conditional distribution of
[…]

2014-04-19 20:25:41

by @ulaulaman about #exoplanets #planetary_transit #kepler_mission #nasa #astronomy The search for extrasolar planets (or exoplanets) had its first success in 1991 with the discovery of some planets orbiting around the pulsar PSR1257+12(1, 2, 3), measuring the variations of the radio pulses coming from the star. The second important milestone in exoplanet research takes place in 1995, with the discovery around the star 51 Pegasi (a star like our Sun) of a Jupiter-like planet, found at a […]

2014-04-19 19:48:00

matthewmaddux posted How I Slayed The Mathematical Beast-Monster by Karla Gabruch on MatthewMaddux Education

Egan…Each semester, on the last day of class, I make prospective elementary school mathematics teachers hand in a Duo-Tang, which contains their weekly math homework and their weekly discussion points (one page responses based on required readings from our text). Ultimately, the Duo-Tang assignment is an ongoing collection of “work” intended to capture prospective elementary school math teachers’ thinking with respect to mathematics and mathematics education over the
[…]

2014-04-19 16:08:00

Here is what I get when I analyze Peirce’s “giver of a horse to a lover of a woman” example along the same lines as the dyadic compositions. We may begin with the mark-up shown in Figure 19. (19) If we … Continue reading →

2014-04-18 22:00:35

[...]

2014-04-18 20:01:53

[...]

2014-04-18 14:13:52

X