Wikipedia:Reference desk/Archives/Mathematics/2016 January 20

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January 20

Convexity

Let $f:[0,1]^{2}\to \mathbb {R} ,f(x,y)=x^{2}\cdot y^{2}$. Is it a convex function? 213.8.204.77 (talk) 11:57, 20 January 2016 (UTC)

Consider the values at (1,0) and (0,1) and at half way in between at (0.5, 0.5). Is the value in between less than or equal to the average of the end values? Dmcq (talk) 12:12, 20 January 2016 (UTC)

Or, using calculus, the determinant of the Hessian matrix is negative almost everywhere. Loraof (talk) 15:49, 20 January 2016 (UTC)

Oh, great! Thanx! 213.8.204.77 (talk) 17:45, 20 January 2016 (UTC)

Matrix Determinant

Let $a_{0},\cdots ,a_{n}\in \mathbb {R} ,$

$M={\begin{pmatrix}a_{n}&a_{n-1}&\cdots &a_{1}&a_{0}&0\\0&a_{n}&a_{n-1}&\cdots &a_{1}&a_{0}\\1&-1&0&\cdots &0&0\\0&1&-1&0&\cdots &0\\&&\ddots &\ddots &&\end{pmatrix}}$

That is, M is a $(n+2)\times (n+2)$ matrix, whose first two rows are the a's, and the a's shifted by one column, and the next rows are $1,-1,0,\cdots ,0$ and then $0,1,-1,0,\cdots ,0$ and then $0,0,1,-1,0,\cdots ,0$ etc. , so the last row is $0,\cdots ,0,1,-1,0$

What is $det(M)$? 213.8.204.77 (talk) 17:43, 20 January 2016 (UTC)

Are you sure about that last row? That gives a matrix with one more row than it has columns. Dmcq (talk) 19:02, 20 January 2016 (UTC)

You're right. I fixed the question. 213.8.204.77 (talk) 19:24, 20 January 2016 (UTC)

Notice that this is a Sylvester matrix for the polynomials f(x) = a_nxⁿ+...+a₁x+a₀ and g(x)=x²-x. Since g has roots 0 and 1, the determinant is ±f(0)f(1) = ±a₀(a_n+...+a₁+a₀). --RDBury (talk) 23:23, 20 January 2016 (UTC)

Wow, this was genius! Thank you! 213.8.204.77 (talk) 08:43, 21 January 2016 (UTC)

Accurate date sought

Counting from Christmas Eve day 2015, what would be the exact date three and a half years ago? -- Mr. Zoot Cig Bunner (talk) 20:18, 20 January 2016 (UTC)

Step one. Find the date for Christmas Eve. Answer 24 Dec 2015.

Step two. Find the Julian Day for that date. Answer 2457381.

Step three. Define three and a half year ago. Answer 3.5 * 365.25 = 1278.375.

Step four. Round up or down to the nearest day. Answer 1278 or 1279.

Step five. Subtract that number from 2457381. Answer 2457381 - {1278,1279} = {2456103,2456102}.

Step six. Convert back to Gregorian date. Answer: {24 Jun 2012,23 Jun 2012} 175.45.116.66 (talk) 23:37, 20 January 2016 (UTC)

You can use the Julian Date Converter here. BEWARE THAT Julian Day starts at NOON!!! http://aa.usno.navy.mil/data/docs/JulianDate.php 175.45.116.66 (talk) 23:42, 20 January 2016 (UTC)

For a more exact and accurate calculation, please choose your poison, I mean choose your definition of a year.

Summary

Days	Year type
346.62	Draconic, also called eclipse.
354.37	Lunar.
365	Vague, and a common year in many solar calendars.
365.24219	Tropical, also called solar, averaged and then rounded for epoch J2000.0.
365.2425	Gregorian, on average.
365.25	Julian.
365.25636	Sidereal, for epoch J2000.0.
365.259636	Anomalistic, averaged and then rounded for epoch J2011.0.
366	Leap in many solar calendars.

An average Gregorian year is 365.2425 days (52.1775 weeks, 8765.82 hours, 525949.2 minutes or 31556952 seconds). For this calendar, a common year is 365 days (8760 hours, 525600 minutes or 31536000 seconds), and a leap year is 366 days (8784 hours, 527040 minutes or 31622400 seconds). The 400-year cycle of the Gregorian calendar has 146097 days and hence exactly 20871 weeks. 175.45.116.66 (talk) 00:14, 21 January 2016 (UTC)

I think I posted it in the wrong place; I shoul've posted it in the Miscellaneous Desk.

Do you mean 25 June 2012? -- Mr. Zoot Cig Bunner (talk) 20:30, 21 January 2016 (UTC)

The calculations are correct. If you believe that there is a mistake in the calculations, kindly point out where the mistake resides.. 175.45.116.66 (talk) 23:18, 21 January 2016 (UTC)

The mistake resides in my understanding. I salute you mate. Thank you for your time. The way you explained it... Regards. -- Mr. Zoot Cig Bunner (talk) 07:57, 22 January 2016 (UTC)