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https://mathematica.stackexchange.com/questions/210318/dsolve-gives-the-wrong-solution-for-an-inhomogeneous-differential-equation | # DSolve gives the wrong solution for an inhomogeneous differential equation
I have the following inhomogeneous differential equation (sorry, I cannot come up with a simpler example)
Hom = -(-1 +
z) z^4 (8 (-5 + 6 z) f0[z] + (20 - 70 z + 52 z^2) f0'[
z] + (-1 + z) z (2 (-5 + 7 z) f0''[z] + (-1 + z) z f0'''[z]))
InHom = -((24 z (24 + z (-48 + z (23 + 2 z))))/(-1 + z)) + (
48 Log[1 - z] (z (18 + z (-24 + 5 z)) + 3 (-2 + z) (-1 + z) Log[1 - z]))/z
myeq = Hom + InHom
DSolve gives me a solution but it doesn't work, meaning that
DSolve[myeq == 0, f0[z], z][[1, 1, 2]];
myeq /. f0 -> ((% /. z -> #) &) // FullSimplify
% === 0
gives False. Any idea why this is happening?
• Could be a bug... but the issue unfortunately seems to depend sensitively on the form of the rather complicated equation you are solving. – Will.Mo Nov 26 '19 at 18:15
• @Bill, the range of $z$ i'm interested is $0<z<1$, I tried to use Assuming but it doesn't change. The solution I'm interested in should be $0$ at $z=0$ and for small $z$ behave linearly in $z$. – bnado Nov 26 '19 at 19:57
• Yeah, it says that the solution diverges as $z^{-5}$ but I don't trust that. For example, I can easily solve the differential equation as a power series. For example Nmax = 15; powser = ((1/z^3 myeq) /. f0 -> ((Sum[\[Alpha][n] #^n, {n, 1, Nmax}]) &)); li = CoefficientList[Series[powser, {z, 0, Nmax}], z] // Normal; lis = Solve[li == 0 li]; fsol = (Sum[\[Alpha][n] z^n, {n, 1, Nmax}]) /. lis[[1]]; Series[myeq /. f0 -> (fsol /. z -> # &), {z, 0, Nmax}] gives the first Nmax terms of the solution I'm interested in. It does not diverge at $z=0$ – bnado Nov 26 '19 at 21:00
• Your checking is wrong. Guess what aaaa === 0 will evaluate to? – xzczd Nov 27 '19 at 4:06
• In fact, a simpler example is Hom + InHom[[2]]. – bbgodfrey Nov 27 '19 at 5:27
DSolve gives incorrect answer
With Hom, InHom, and myeq as defined in the question, DSolve indeed gives an incorrect answer,
sol = DSolveValue[myeq == 0, f0, z];
FullSimplify[(myeq == 0) /. f0 -> %]
(* ((-1 + z) (z^2 - 5 (-2 + z) z Log[1 - z] + (14 + z (-14 + 3 z)) Log[1 - z]^2))/z == 0 *)
which is not, in general, True, as can be seen by evaluating the expression for a few values of z. Note that none of the constants of integration appear in this expression, indicating that the error is not associated with the homogeneous equation. This also can be demonstrated by solving the homogeneous equation directly.
DSolveValue[Hom == 0, f0, z]
FullSimplify[(Hom == 0) /. f0 -> %]
(* Function[{z}, C[1]/z^4 - (C[2] Log[1 - z])/z^4 +
(C[3] (1/(1 - z) - z - 1/4 (-1 - 2 Log[-1 + z])^2))/z^4] *)
(* True *)
Somewhat surprisingly (to me, at least), the correct answer can be obtained as follows. Restructure InHom as
InHomsim = Collect[InHom, _Log, Simplify]
(* -((24 z (24 + z (-48 + z (23 + 2 z))))/(-1 + z)) +
48 (18 + z (-24 + 5 z)) Log[1 - z] + (144 (-2 + z) (-1 + z) Log[1 - z]^2)/z *)
Next, solve
sol1 = DSolveValue[Hom + InHomsim[[1]] == 0, f0, z];
FullSimplify[(Hom + InHomsim[[1]] == 0) /. f0 -> %]
(* True *)
sol2 = DSolveValue[Hom + InHomsim[[2]] == 0, f0, z];
FullSimplify[(Hom + InHomsim[[2]] == 0) /. f0 -> %]
(* True *)
sol3 = DSolveValue[Hom + InHomsim[[3]] == 0, f0, z];
FullSimplify[(Hom + InHomsim[[3]] == 0) /. f0 -> %]
(* True *)
Thus, correct solutions for each of the three additive components of InHomsim can be obtained without difficulty. Simply add them to obtain the correct answer to the original equation.
Collect[(sol1[[2]] + sol2[[2]] + sol3[[2]]) /. C[i_] -> C[i]/3, C[_], FullSimplify];
soltr = Function[{z}, Evaluate@%]
FullSimplify[(myeq == 0) /. f0 -> soltr]
(* Function[{z}, C[1]/z^4 - (C[2] Log[1 - z])/z^4 +
(C[3] (1/(1 - z) - z - 1/4 (1 + 2 Log[-1 + z])^2))/z^4 +
(1/((-1 + z) z^4)) 4 (-6 (7 + 6 \[Pi]^2 (-1 + z) - z (7 + 4 z)) -
2 (-1 + z) Log[1 - z]^3 - 6 Log[1 - z] (-16 + z (15 + 2 z) +
(-1 + z) (-23 + Log[-1 + z]) Log[-1 + z]) + (-1 + z) Log[-1 + z]
(12 + Log[-1 + z] (-69 + 4 Log[-1 + z])) - 6 (-1 + z) Log[1 - z]^2 (5 + Log[z]) +
12 (-1 + z) Log[1 - z] PolyLog[2, z] - 12 (-1 + z) PolyLog[3, 1 - z])] *)
(* True *)
Evidently, DSolve can handle each of the three components of InHomsim but not the whole expression at once.
Why the error?
DSolve also can solve the general equation, Hom + g[z] == 0, correctly.
solgen = DSolveValue[Hom + g[z] == 0, f0, z];
FullSimplify[(Hom + g[z] == 0) /. f0 -> %]
(* True *)
Not surprisingly solgen contains three integrals over g[z] , two of them fairly complicated. Because the g[z] in the question (namely InHom) and (as it happens) the kernels of the integrals both contain expressions with branch cuts, DSolve needs to select the proper contours along which to perform the integrals. Probably, DSolve chose incorrect contours.
• thanks, it does work now. I thought that Assuming would take care of the branch cuts but it's not the case. Only one thing, you have a sol1 written instead of a sol3 in your answer. Thanks again! – bnado Nov 27 '19 at 21:55 | 2021-04-18 08:52:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49783220887184143, "perplexity": 3627.5515934038594}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038469494.59/warc/CC-MAIN-20210418073623-20210418103623-00243.warc.gz"} |
https://mathoverflow.net/questions/191752/could-quadratic-variation-determine-distribution | # Could quadratic variation determine distribution?
Let $M=\{M_t,\mathcal{F}_t;0\le t<+\infty\}$, $N=\{N_t,\mathcal{F}_t;0\le t<+\infty\}$ be two continuous local martingales with $M_0=N_0=0\text{ a.s.}$. If $\langle M\rangle=\langle N\rangle$, then could we say that $M$ and $N$ have the same distribution?
No, consider Brownian motion $W_t$ and $$M_t=\frac{W_t^2-t}{2},$$ $$N_t = -M_t.$$ Source: slides by David Heath page 5. | 2019-10-20 20:21:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9864569902420044, "perplexity": 254.28400116083776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986718918.77/warc/CC-MAIN-20191020183709-20191020211209-00213.warc.gz"} |
https://web2.0calc.com/questions/help_77412 | +0
# Help
0
40
1
You have linear functions p(x) and q(x). You know p(2)=3, and p(q(x))=4x+7 for all x. Find q(-1).
Feb 26, 2020
#1
+24366
+1
You have linear functions $$p(x)$$ and $$q(x)$$. You know $$p(2)=3$$, and $$p(q(x))=4x+7$$ for all $$x$$.
Find $$q(-1)$$.
$$\begin{array}{|l|} \hline p(q(x)) = 4x+7,\\ \text{if q(x) = 2, then 4x+7 = 3, so 4x= 3-7 or 4x = -4 hence \mathbf{x=-1} } \\ \mathbf{q(-1)=2} \\ \hline \end{array}$$
Feb 26, 2020 | 2020-03-31 23:14:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9977450370788574, "perplexity": 6315.5101967308365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370504930.16/warc/CC-MAIN-20200331212647-20200401002647-00272.warc.gz"} |
https://ask.sagemath.org/question/31087/piecewise-function-of-several-variables-and-how-to-display-it/ | Piecewise function of several variables and how to display it
Hi, I need to define a piecewise function of several variables. For a function of one variable I would use probably Piecewise. I would like to work with a function G of two variables t and s that is of this form:
def G(t,s):
if (t < s):
return 1
else:
return 0
Although this works, I would like to have a nice displaying of this function, like a function generated by Piecewise does. Is there another way how to define this function?
edit retag close merge delete
I don't think so, although you could use floor(min(1, s/t))... | 2018-02-20 07:30:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3105950951576233, "perplexity": 537.6102210895807}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812913.37/warc/CC-MAIN-20180220070423-20180220090423-00164.warc.gz"} |
https://math.stackexchange.com/questions/3253476/find-integrable-combination-to-solve-system-of-differential-equations | # Find integrable combination to solve system of differential equations
There is the system:
$$\left\{ \begin{array}{c} \dot{x} = -xy^2+x+y \\ \dot{y} = -x-y+x^2y \end{array} \right.$$
the way it should be solved is to find an integrable combination there is the description (page 349 of pdf document).
I've already tried multiplying the first equation by $$x$$, the second one by $$y$$ then adding first to second so I got: $$xdx+ydy = x^2 - y^2$$, which I have no idea how to integrate.
Also I tried multiplying the first by $$y$$, the second by $$x$$, also adding first to second so I got: $$ydx+xdy=(xy - 1)(x^2 - y^2)$$, which I also don't know how to integrate.
Could you plese provide any integrating combinations or ideas how to deal with equations I got.
That's a good start, except that you should write $$\dot x$$ and $$\dot y$$ instead of $$dx$$ and $$dy$$.
Anyway, you have $$(\tfrac12 (x^2+y^2))\dot{} = x \dot x + y \dot y = x^2-y^2$$ and $$(xy)\dot{} = \dot x y + x \dot y = (xy-1)(x^2-y^2) .$$ These can be combined to give $$(xy)\dot{} = (xy-1) (\tfrac12 (x^2+y^2))\dot{}$$ so that either $$xy-1=0$$ identically or $$\frac{(xy)\dot{}}{xy-1} = (\tfrac12 (x^2+y^2))\dot{}$$ where both sides can be integrated to find a constant of motion.
• Thank you for the answer. I used your steps and got $ce^{x^2+y^2}=(xy-1)^2$. As I know after that we should express $x$ or $y$ from there and substitute the result to one of the equations so we get $x(t)$ or $y(t)$. Am I right or there is some another way get the answer (cause mine seems to be too complicated :)) – FoRRestDp Jun 6 '19 at 20:57
• There also was Cauchy problem in the task so I've tried to use to simplify my task. It is: $x(2)=1$, $y(2)=1$. I substituted it to the result above and got that $c=0$ hence $1=xy$ hence $x=\frac{1}{y}$. Then I substituted this to the second equation and got $\frac{dy}{y} = -dt$. Am I right doing this? – FoRRestDp Jun 6 '19 at 21:25
• @ЕгорПономарёв: Looks fine! I doubt it's possible to find expressions for $x(t)$ and $y(t)$ in the general case (in practice, that is; it's always possible in principle once you have a constant of motion). – Hans Lundmark Jun 7 '19 at 5:02
• Also, $\dot{x} + \dot{y} \equiv 0$, hence $\frac{d}{dt}(x(t) + y(t)) \equiv 0$ and $x(t) + y(t) \equiv x(0) + y(0)$. – Evgeny Jun 7 '19 at 20:01
• @Evgeny: No, that would have been too easy! Note that it's $\dot x + \dot y = -xy^2 + x^2 y = xy (x-y) \neq 0$. – Hans Lundmark Jun 7 '19 at 20:07 | 2021-04-18 14:50:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 16, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8477517366409302, "perplexity": 212.61123047937792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038492417.61/warc/CC-MAIN-20210418133614-20210418163614-00239.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=594576 | MathSciNet bibliographic data MR594576 (81k:82033) 82A42 (81E99) McKane, A. J. Reformulation of $n\rightarrow 0$$n\rightarrow 0$ models using anticommuting scalar fields. Phys. Lett. A 76 (1980), no. 1, 22–24. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews. | 2015-07-06 14:20:46 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9962298274040222, "perplexity": 10755.769225727689}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098464.55/warc/CC-MAIN-20150627031818-00147-ip-10-179-60-89.ec2.internal.warc.gz"} |
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I'm trying to obtain the Causal Green Function for the D'Alembertian in D=2, I'm finding $$G^{(2)}_C (t; \vec x) = \frac{1}{4\pi} \frac{\Theta(t^2-\vec x^2)}{\sqrt{t^2-\vec x^2}} - \frac{i}{4\pi^2} ... 1answer 17 views ### Work done from force-extension graph The answer is C, but I got B... F at 3cm = 10, F at 6cm = 20 W=F\cdot s (20-10)\times 3=30 I must have made a mistake or stupid assumption then, how am I meant to get 45 N cm? 0answers 20 views ### Gregory-Laflamme Instability Certain solutions to the low energy effective action of string theory, namely black strings and branes, were shown to exhibit an instability via perturbation theory. Specifically, the metric is ... 0answers 17 views ### Arrow of time and virtual reality [on hold] As we know that in classical world we have a clear arrow of time which points towards future..We can see glass falling from table and break but not the opposite in our world and whole universe... If ... 0answers 16 views ### Wannier functions on a ring Let's say I have a single particle hamiltonian in a periodic potential. For example a 1D lattice such that:$$H = -\frac{\partial_x^2}{2m} + V(x) with $V(x+a) = V(x)$ where $a$ is the lattice ...
A topological space is defined as a non-empty set $X$ together with a given collection of subsets $T$ (topology) of $X$, such that, (i) any union of these subsets is one of the subsets. (ii) any ... | 2014-04-21 06:03:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9169428944587708, "perplexity": 611.4745698288316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00421-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/tags/terminology/hot?filter=year | # Tag Info
30
Which year? The sidereal year? The tropical year? The anomalistic year? The calendar year (and whose calendar)? The sidereal year is the average amount of time it takes the Earth to make one complete orbit about the Sun with respect to the fixed stars. The tropical year is the amount of average amount of time between successive spring equinoxes. The ...
17
The definition of a wave is not that it is the oscillation of a medium. Waves are called waves because they are solutions to a wave equation, which is, for a generic "excitation" $A(t,x)$ depending on the time $t$ and some spatial coordinate $x\in\mathbb{R}^n$, of the general form $$\frac{\partial^2 A}{\partial t^2} = c^2\Delta A$$ where $\Delta$ is the ...
15
A fermion is any particle, elementary or composite, that obeys Fermi-Dirac (as opposed to Bose-Einstein) statistics relating to how identical particles behave when you swap two of them. Due to an important but complicated result, this is taken to amount to having half-integer spin. A lepton is one type of elementary particle with spin 1/2. The only leptons ...
13
If I saw the word "amp" written as such in a paper in my field (astrophysics) it would strike me as a bit informal. I would expect to see the full "ampere" written. That said, it is rare to actually write out the full name of a unit; usually it follows a number and is given its standard abbreviation. When abbreviated to e.g. "$5\ \mathrm{A}$", I would ...
13
What is “special” and what is “general” in Relativity? The "special" in special relativity refers to the fact that it is not a universal theory. Predictions made by special relativity only apply under certain special circumstances. Those special circumstances are where gravitation is not present or can essentially be ignored. Initially I thought in ...
11
SR: Flat Space-time (Minkowski metric), no gravity, Lorentz coordinates transformations (usually $\Lambda \in SO^+(3,1)$, the proper orthochronous Lorentz group). Acceleration is allowed, but you usually want to work with inertial frames. GR: Curved Space-time (non trivial and dynamic metric tensor), theory of gravitation, generic coordinates ...
11
Technically, apparently, your teacher is correct. BIPM and NIST In the official brochure from the Bureau international des poids et mesures (BIPM, the keepers of SI units) in §5.1 Unit symbols we find: It is not permissible to use abbreviations for unit symbols or unit names, such as sec (for either s or second), sq. mm (for either mm2 or ...
11
First, you have system with some energy, named $U$ by physicists. You think you have all the information you need to characterize the system but then some guy comes near and says: "Whoa, that's bad, the volume of your system can change." You say: "No problem, we just add here $pV$. Our new energy is $H=U+pV$." "But hey," they say, "your temperature can ...
10
Short answer: Gibbs free energy $G = U + PV - TS$ combines internal energy $U$, pressure $P$, volume $V$, temperature $T$, and entropy $S$ into a single quantity that measures spontaneity. With that, I mean that processes that lower the Gibbs free energy of your system will spontaneously occur, and equilibrium is reached when the Gibbs free energy reaches ...
10
In addition to the other answers, back in the olden days they were thought of as oscillations in the ether. As a result of the Michelson-Morley experiment back in 1887, physicists began to think that there was no ether. But the term didn't change.
9
A fermion is any particle characterized by Fermi–Dirac statistics and obeying the Pauli exclusion principle. So for example quarks are fermions, as are Helium-3 atoms. A fermion does not have to be an elementary particle. I'm not even sure that it has to be spin $\tfrac{1}{2}$, though I can't think of any fermions that aren't. A lepton is a spin ...
9
Below follows a handful of excerpts from the book Introduction to the Classical Theory of Particles and Fields (2007) by B. Kosyakov. Controversial/misleading/wrong statements are marked in $\color{Red}{\rm red}$. We agree with OP that the statements marked in $\color{Red}{\rm red}$ are opposite standard terminology/conventions. Some (not all) correct ...
9
If one of the rules to be a planet is that it needs to clear ALL objects from their orbit, does this also make Neptune a non-planet? This is a somewhat common misconception of the meaning of the term "clearing the neighborhood". None of the planets could be called "planets" if clearing ALL objects from the vicinity of the orbit was what that term meant. ...
9
The "shift in the meaning" refers to some attempts to reinterpret the terminology that were made by a metrological document, ISO 5725, in 2008. That may be described as a bureaucratic effort by a few officials – really bureaucrats of a sort – and as far as I know, the "shift in the meaning" hasn't penetrated to the community of professionals. The people ...
8
The Big Bang was originally defined as the zero time limit of the FLRW metric, so it's a mathematical construct and not primarily something physical. We have chosen to apply it to the zero time limit of the universe because we thought the FLRW metric was a good description of the universe, but then inflation gatecrashed the party and spoiled the fun. So if ...
8
Special relativity is physics in a $3+1$ dimensional Lorentzian spacetime, with the additional requirement that the spacetime is flat, which determines spacetime completely. General relativity is physics in a $3+1$ dimensional Lorentzian spacetime, with no additional geometric requirement. An equation for the metric is required to determine the spacetime, ...
8
OK I don't understand anything.when I placed my mobile phone on the ground, its accelerometer shows nine point something m/s^2. So is that the value of its acceleration? That is the value of the phone's proper acceleration. From the Wikipedia article "Proper acceleration": proper acceleration is the physical acceleration (i.e., measurable ...
7
what does memorylessness mean? Essentially, it means that the length of a rod and the rate of a clock depend on their current state only. The alternative would require that, e.g., two otherwise identical clocks at rest with respect to each other may run at different rates if their histories differed.
7
The first bullet would be read "$A$ dot $B$" or "The dot product of $A$ and $B$" The second bullet would be read "$A$ cross $B$" or "The cross product of $A$ and $B$"
7
Neptune actually is the dominant gravitational force in the region of the Kuiper belt in which Pluto resides. In fact, if you look at the image below, the belt is being cleared out by Neptune: In fact, there is a class of objects, suitably named the plutinos, that have been captured by Neptune. Solar system models have actually shown that Neptune was ...
7
Sanaris's answer is a great, succinct list of what each term in the free energy expression stands for: I'm going to concentrate on the $T\,S$ term (which you likely find the most mysterious) and hopefully give a little more physical intuition. Let's also think of a chemical or other reaction, so that we can concretely talk about a system changing and thus ...
6
In the sense of "Copenhagen Interpretation", what exactly is an interpretation? What purpose does an interpretation serve? I would describe interpretations of quantum mechanics as part of the philosophy of physics. Here is a well-known quote by Bertrand Russell: "As soon as definite knowledge concerning any subject becomes possible, this subject ceases ...
6
In normal usage, a gauge is a particular choice, or specification, of vector and scalar potentials $\mathbf A$ and $\phi$ which will generate a given set of physical force fields $\mathbf E$ and $\mathbf B$. More specifically, a physical situation is specified by the electric and magnetic fields, $\mathbf E$ and $\mathbf B$. A set of potentials $\mathbf A$ ...
6
The sort of trick involved in removing the $|P\rangle$ on both sides to get the conjugate imaginary equation $$\langle P|\xi|P\rangle = \langle P|a|P\rangle \tag1$$ is quite common but it is indeed nontrivial to grasp the first time. In essence, you leverage the fact that in an equation of the form $$⟨\psi|\hat A|\phi⟩=⟨\psi|\hat B|\phi⟩\tag2 ... 6 We use the term mass, when we mean the mass of a weight, and we use the term weight, when we mean the weight of a mass. :-) The important thing to remember is, that the mass is the same everywhere, while the weight varies with the local gravity. So if you are referring to the constant mass of an object, you use mass expressed in kg. If, however, you mean ... 6 The bound state is defined such that the probability density average will be finite at some particular space region when time passes. While for unbounded states, as time passes, the probability density will tends to zero. See Landau Quantum Mechanics section 10. This can be understand as this, if the state is bounded, i.e. it is exist only within some ... 6 I took a quick look at pages 59 and 60 of "Gravitation", section 2.6 "Gradients and Directional Derivatives", to see if there's anything there we can use to clarify this issue. In this section, the gradient of f is \mathbf df, the directional derivative along the vector \mathbf v is \partial_{\mathbf v}f and the following relationship holds: ... 6 "To clear an orbit" has a specific meaning which may not entirely intuitive. "Clearing an orbit" specifically does not mean emptying an orbit of all other bodies. It means the planet gravitationally dominates other bodies at approximately the same distance to the sun. Now you can wonder perhaps whether Neptune dominates Pluto or Pluto dominates Neptune. ... 6 The more common names for what you are talking about are the abbreviated action$$S_0[q] := \int p \mathrm{d}q$$versus the action$$ S[q] := \int_{t_1}^{t_2}L(q,\dot q,t)\mathrm{d}t Both are used in different formulations of classical mechanics, and deliver a different "flavor" of solutions. On both one can do variations calculus and obtains the ...
6
The term 'equation of motion' is somewhat subjective as it depends on the context, but for any given context there is usually one single equation, or set of equations, which can be described as an equation of motion. These are typically differential equations in time, usually of second order, and for simple objects in Newtonian mechanics they do not involve ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2015-05-24 15:48:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8275399208068848, "perplexity": 528.7175551091689}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928019.82/warc/CC-MAIN-20150521113208-00001-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/visual-interpretation-of-fundamental-theorem-of-calculus.898415/ | # I Visual interpretation of Fundamental Theorem of Calculus
1. Dec 27, 2016
Hi, this is a newbee question. Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))? That is, the two-dimensional area under a curve in [a,b] for f(x) is always equals to the one-dimensional distance F(b)-F(a)? If you graph x^2 and 2x, they look nothing alike, and there’s no clue as to how they are related, but the area from 1 to 2 under the curve y=2x is always equal to (2)^2 – (1)^2. The units work out also.
2. Dec 27, 2016
### dkotschessaa
3. Dec 28, 2016
### Ssnow
The link is the ''area function''. If we permit $x$ to varies in an intervall $[a,b]$ then the area under $f(x)$ depends by $x$ and is a function in one variable $\mathcal{A}(x)$ given by:
$\mathcal{A}(x)=\int_{a}^{x}f(s)ds= F(x)-F(a)=\text{Area under} \ \ f \ \ \text{between} \ \ a \ \ \text{and} \ \ x$
so the link is the Area that you can write in integral from $\int_{a}^{x}f(s)ds$ or as the difference $F(x)-F(a)$ (where $F'(x)=f(x)$ and we assume $f$ continuous on $[a,b]$). As @dkotschessaa said I suggest the same link where this can be visualize very well...
Ssnow
4. Jan 2, 2017
Thank you!
5. Jan 5, 2017
### Stephen Tashi
My comment is that there are instances where algebra is a better way of understanding theorems than pictures. The Calculus of Finite Differences makes the fundamental theorem of calculus seem very natural.
6. Jan 9, 2017
### newjerseyrunner
The derivative is simply the rate of change. The first graphic here is a good example, where the wavy line is f(x) and the red bars represent the derivative at each point. http://m.sparknotes.com/math/calcab/applicationsofthederivative/section5.rhtml
The easiest way to think about how derivatives work is by thinking of the sine wave and costume wave. Why are they derivatives of each other? Visually, it becomes quite obvious when you put them on top of each other. When the sine wave crosses the y axis, it's going up with a slope of exactly 1, so where sine crosses the y axis from beneath, its derivative is 1, which is the cosines of the same x. When the sine wave is at a value of 1, what's it doing? It's at the top of its period and headed back down, so it's not going up or down at all, giving it a derivative of zero.
Oh, and if you look carefully, you can tell why 2x is the derivative of x^2. Look at how the graph changes on x^2. What is the slope of the line at any given x alone that line? It's a curve so you know it has to be changing. How's it changing? 2x.
A better example with something concrete: your bank account. Your bank account value is f(x). So today u have 50, tomorrow you have 75... so f(1) = 50, f(2) = 75... So from your real values, what was the rate of change? 25. That's the first derivative of your bank account. So next week, you have 100 in your account for f(3), the rate of change f'(x) was again 25. If you take it one step further, you'll notice that the account went up 25 each time. So what was the rate at which the rate itself changed? Well the rate didn't change at all, it was 25 both times, so 0. That's the second derivative. That's essentially the same as position, velocity, acceleration. Derivatives tell you how much a function above it changes.
Last edited: Jan 9, 2017 | 2017-08-21 16:48:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7721794843673706, "perplexity": 519.7803768171824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109157.57/warc/CC-MAIN-20170821152953-20170821172953-00409.warc.gz"} |
https://phvu.net/tag/hmm/ | # Kalman filters (and how they relate to HMMs)
Kalman filters are insanely popular in many engineering fields, especially those involve sensors and motion tracking. Consider how to design a radar system to track military aircrafts (or warships, submarines, … for that matter), how to track people or vehicles in a video stream, how to predict location of a vehicle carrying a GPS sensor… In all these cases, some (advanced) variation of Kalman filter is probably what you would need.
Learning and teaching Kalman filters is therefore quite challenging, not only because of the mere complexity of the algorithms, but also because there are many variations of them.
With a Computer Science background, I encountered Kalman filters several years ago, but never managed to put them into the global context of the field. I had chances to look at them again recently, and rediscovered yet another way to present and explain Kalman filters. It made a lot of sense to me, and hopefully it does to you too.
Note that there are a lot of details missing from this post (if you are building a radar system to track military aircrafts, look somewhere else!). I was just unhappy to see many introductory material on Kalman filters are either too engineering or too simplified. I want something more Machine Learning-friendly, so this is my attempt.
Let’s say you want to track an enemy’s aircraft. All you have is a lame radar (bought from Russia probably) which, when oriented properly, will give you a 3-tuple of range, angle and angular velocity $[r \;\phi\;\dot{\phi}]^{T}$ of the aircraft. This vector is called the observation $\mathbf{z}_k$ (subscript $k$ because it depends on time). The actual position of the aircraft, though, is a vector in cartesian coordinates $\mathbf{x}_k = [x_1\;x_2\;x_3]^{T}$. Since it is an enemy’s aircraft, you can only observe $\mathbf{z}_k$, and you want to track the state vector $\mathbf{x}_k$ over time, every time you receive a measurement $\mathbf{z}_k$ from the radar.
Visualised as a Bayesian network, it looks like this:
With all the Markov properties hold, i.e. $\mathbf{x}_k$ only depends on $\mathbf{x}_{k-1}$ and $\mathbf{z}_k$ only depends on $\mathbf{x}_k$, does this look familiar?
# Sweet implementation of Viterbi in Python
An implementation of the Viterbi algorithm for HMM in Python can be as short as 10 lines of code like this:
def Decode(self, obs):
trellis = np.zeros((self.N, len(obs)))
backpt = np.ones((self.N, len(obs)), 'int32') * -1
trellis[:, 0] = np.squeeze(self.initialProb * self.Obs(obs[0]))
for t in xrange(1, len(obs)):
trellis[:, t] = (trellis[:, t-1, None].dot(self.Obs(obs[t]).T) * self.transProb).max(0)
backpt[:, t] = (np.tile(trellis[:, t-1, None], [1, self.N]) * self.transProb).argmax(0)
tokens = [trellis[:, -1].argmax()]
for i in xrange(len(obs)-1, 0, -1):
tokens.append(backpt[tokens[-1], i])
This is the implementation of Viterbi I’ve completed recently. Holy python, how sweet it is! The code is even shorter than the pseudo-code, which is taken from this book.
It takes no more than 30 lines for the complete class:
import numpy as np
class Decoder(object):
def __init__(self, initialProb, transProb, obsProb):
self.N = initialProb.shape[0]
self.initialProb = initialProb
self.transProb = transProb
self.obsProb = obsProb
assert self.initialProb.shape == (self.N, 1)
assert self.transProb.shape == (self.N, self.N)
assert self.obsProb.shape[0] == self.N
def Obs(self, obs):
return self.obsProb[:, obs, None]
def Decode(self, obs):
trellis = np.zeros((self.N, len(obs)))
backpt = np.ones((self.N, len(obs)), 'int32') * -1
# initialization
trellis[:, 0] = np.squeeze(self.initialProb * self.Obs(obs[0]))
for t in xrange(1, len(obs)):
trellis[:, t] = (trellis[:, t-1, None].dot(self.Obs(obs[t]).T) * self.transProb).max(0)
backpt[:, t] = (np.tile(trellis[:, t-1, None], [1, self.N]) * self.transProb).argmax(0)
# termination
tokens = [trellis[:, -1].argmax()]
for i in xrange(len(obs)-1, 0, -1):
tokens.append(backpt[tokens[-1], i])
The full source code and a simple test case, as always, can be found on github.
# Markov models
Fortunately, I have another opportunity to complete my presentation on Markov models which I have partly published before. This time I have made some significant changes:
– Added a section discussing the motivation of Markov chains.
– Included three algorithms of HMM with detailed ideas and equations.
– Finally, to complete a talk on Markov models, I have also included a short review on Markov Random Field and its application in image segmentation. However I don’t have enough time so this section is pretty short and lack of some details. This is a fundamental subject so you can read about it in various textbooks.
Well, honestly this is the longest presentation I have ever made (with 107 pages). Actually I will have to present it this afternoon. I’m not sure how much time it will take to present this slide completely, but you can be sure that it takes me a lot of time (and many stay-up-late nights) to be completed. So if you are gonna adopt it for your own purpose, please take it with care 😉
Due to this slide, the tutorial on QR algorithm is temporarily corrupted. I will post the final section on QR algorithm soon (hopefully before the weekend).
# Hidden Markov Models
This hasn’t been finished yet, and I am still working on this. Maybe I will upload a completed version on tomorrow (Dec. 6th).
Updated March 30, 2011: You can view and download the completed version here.
Updated December 8, 2010: I have updated the latest version, though it is still uncompleted. We will need about 10-20 slides for the three famous HMM tasks. But I’m too tired to finish this right now … | 2017-11-19 05:11:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 11, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3651597797870636, "perplexity": 2181.737610557435}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805362.48/warc/CC-MAIN-20171119042717-20171119062717-00277.warc.gz"} |
http://mitpress.mit.edu/books/living-complexity | Hardcover | $26.95 Trade | £18.95 | ISBN: 9780262014861 | 312 pp. | 5.375 x 8 in | 88 figures| October 2010 ebook |$18.95 Trade | ISBN: 9780262296854 | 312 pp. | 5.375 x 8 in | 88 figures| October 2010
# Living with Complexity
## Overview
If only today’s technology were simpler! It’s the universal lament, but it’s wrong. We don't want simplicity. Simple tools are not up to the task. The world is complex; our tools need to match that complexity. Simplicity turns out to be more complex than we thought. In this provocative and informative book, Don Norman writes that the complexity of our technology must mirror the complexity and richness of our lives. It’s not complexity that’s the problem, it’s bad design. Bad design complicates things unnecessarily and confuses us. Good design can tame complexity.Norman gives us a crash course in the virtues of complexity. But even such simple things as salt and pepper shakers, doors, and light switches become complicated when we have to deal with many of them, each somewhat different. Managing complexity, says Norman, is a partnership. Designers have to produce things that tame complexity. But we too have to do our part: we have to take the time to learn the structure and practice the skills. This is how we mastered reading and writing, driving a car, and playing sports, and this is how we can master our complex tools. Complexity is good. Simplicity is misleading. The good life is complex, rich, and rewarding—but only if it is understandable, sensible, and meaningful. | 2015-03-05 15:23:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20029228925704956, "perplexity": 2114.8555554524123}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936464193.61/warc/CC-MAIN-20150226074104-00113-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://dash.harvard.edu/handle/1/8808827 | # Modeling Spatial Patterns of Traffic-Related Air Pollutants in Complex Urban Terrain
Title: Modeling Spatial Patterns of Traffic-Related Air Pollutants in Complex Urban Terrain Author: Zwack, Leonard M.; Paciorek, Christopher Joseph; Spengler, John D.; Levy, Jonathan Ian Note: Order does not necessarily reflect citation order of authors. Citation: Zwack, Leonard M., Christopher Joseph Paciorek, John D. Spengler, and Jonathan Ian Levy. 2011. Modeling spatial patterns of traffic-related air pollutants in complex urban terrain. Environmental Health Perspectives 119(6): 852-859. Full Text & Related Files: 3114822.pdf (759.9Kb; PDF) Abstract: Background: The relationship between traffic emissions and mobile-source air pollutant concentrations is highly variable over space and time and therefore difficult to model accurately, especially in urban settings with complex terrain. Regression-based approaches using continuous real-time mobile measurements may be able to characterize spatiotemporal variability in traffic-related pollutant concentrations but require methods to incorporate temporally varying meteorology and source strength in a physically interpretable fashion. Objective: We developed a statistical model to assess the joint impact of both meteorology and traffic on measured concentrations of mobile-source air pollutants over space and time. Methods: In this study, traffic-related air pollutants were continuously measured in the Williamsburg neighborhood of Brooklyn, New York (USA), which is affected by traffic on a large bridge and major highway. One-minute average concentrations of ultrafine particulate matter (UFP), fine particulate matter [$$\leq 2.5 \mu m$$ in aerodynamic diameter $$(PM_{2.5})$$], and particle-bound polycyclic aromatic hydrocarbons were measured using a mobile-monitoring protocol. Regression modeling approaches to quantify the influence of meteorology, traffic volume, and proximity to major roadways on pollutant concentrations were used. These models incorporated techniques to capture spatial variability, long- and short-term temporal trends, and multiple sources. Results: We observed spatial heterogeneity of both UFP and $$PM_{2.5}$$ concentrations. A variety of statistical methods consistently found a 15–20% decrease in UFP concentrations within the first 100 m from each of the two major roadways. For $$PM_{2.5}$$, temporal variability dominated spatial variability, but we observed a consistent linear decrease in concentrations from the roadways. Conclusions: The combination of mobile monitoring and regression analysis was able to quantify local source contributions relative to background while accounting for physically interpretable parameters. Our results provide insight into urban exposure gradients. Published Version: doi:10.1289/ehp.1002519 Other Sources: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3114822 Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:8808827 Downloads of this work:
Advanced Search | 2018-08-19 21:15:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4475398361682892, "perplexity": 8077.212024038504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221215393.63/warc/CC-MAIN-20180819204348-20180819224348-00189.warc.gz"} |
http://www.lmfdb.org/knowledge/show/lfunction.gamma_factor | show · lfunction.gamma_factor all knowls · up · search:
The complex functions $\Gamma_\R(s) := \pi^{-s/2}\Gamma(s/2)\qquad\text{and}\qquad \Gamma_\C(s):= 2(2\pi)^{-s}\Gamma(s)$ that appear in the functional equation of an L-function are known as gamma factors. Here $\Gamma(s):=\int_0^\infty e^{-t}t^{s-1}dt$ is Euler's gamma function.
The gamma factors can also be viewed as “missing” factors of the Euler product of an L-function corresponding to (real or complex) archimedean places.
Authors:
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• Last edited by Andrew Sutherland on 2019-06-05 16:37:58
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Differences | 2019-08-21 09:57:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7901589274406433, "perplexity": 2365.4137492786867}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315865.44/warc/CC-MAIN-20190821085942-20190821111942-00148.warc.gz"} |
https://www.tutorvista.com/content/math/ellipse/ | Top
# Ellipse
Conics are the curves formed by the intersection of a plane and a double napped cone. Ellipse is one of the four basic conics which are formed when the Plane does not pass through the vertex of the cone.
A point is called the degenerating conic corresponding to an Ellipse, which is formed when the intersecting plane passes through the vertex of the cone. We find the orbits of planets are in the form of Ellipses.
The equations of ellipses are used as models to solve many real life problems.
Basic Conic The Ellipse Degenerating Conic The Point.
A conic section is defined as the locus of a point (x, y) which moves such that the ratio of its distances from a fixed point and a fixed line is a constant. The constant is called the Eccentricity of the Ellipse and is represented by 'e'.The fixed point is called the Focus and the fixed line is called the Directrix of the Conic.
The conic formed is an Ellipse when 0 < e < 1.
An Ellipse has two Foci corresponding to two directirces.
An Ellipse is also generally defined as the locus of a point (x, y) which moves such that the sum of its distances from two fixed points (Foci) is always a constant.
Related Calculators Ellipse Calculator Perimeter of an Ellipse Area of a Ellipse Calculator
## Ellipse Equations
The equation of an ellipse with center at the origin can be written in two forms based on its orientation.
The parent Ellipse is a closed curve symmetric about both x and y axis.
The lengths intercepted on the two axes are called the major and the minor axes of the Ellipse, the longer being the major and the shorter the minor.
Ellipse with Horizontal Major Axis
The equation of the ellipse with center at (0, 0) and the major axis along the x axis is,
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$=1 where a > b
The length of the major axis = AA' = 2a
The length of the minor axis = BB' = 2b
The Vertices are A (a, 0) and A' (-a, 0). The covertices are B (0, b) and B' (0, -b).
The two Foci have the coordinates C (c, 0) and C' (-c, 0)
where c2 = a2 - b2.
For example, if the equation of the Ellipse is
$\frac{x^{2}}{25}+\frac{y^{2}}{16}$=1
a = 5 and b = 4
Hence the length of the major axis = 10 and the length of the minor axis = 8.
The focal length c = $\sqrt{a^{2}-b^{2}}$ = $\sqrt{25-16}$ = $\sqrt{9}$ = 3.
The vertices are (5, 0) and (-5, 0). The covertices are (0, 4) and (0, -4).
The foci are (3, 0) and (-3, 0).
Ellipse with Vertical Major Axis
The equation of the ellipse with center (0, 0) and the major axis along the y axis is,
$\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}$=1 where a > b
The length of the major axis = BB' = 2a
The length of the minor axis AA' = 2b
The vertices are B (0, a) and B' (0, -a). The covertices are A (b, 0) and A' (-b, 0).
The two Foci are C (0, c) and C' (0, -c) and c2 = a2 - b2.
For example, consider the equation $\frac{x^{2}}{36}+\frac{y^{2}}{64}$=1
a = 8 and b = 6
The length of the major axis = 16 and the length of the minor axis = 12
The Focal length c = $\sqrt{a^{2}-b^{2}}$ = $\sqrt{64-36}$ = $\sqrt{28}$ = $2\sqrt{7}$
The vertices are (0, 8) and (0, -8). The covertices are (6, 0) and (-6, 0).
The foci are ( (0, $2\sqrt{7}$) and (0, -$2\sqrt{7}$).
The eccentricity e of an ellipse describes its shape and e = $\frac{c}{a}$
The descriptions of Ellipses with center translated to (h, k) are as follows:
Horizontal Orientation
$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}$=1 where a > b and c2 = a2 - b2.
Center (h, k).
Foci ( h ± c, k)
Major axis: y = k
Vertices (h ± a, k)
Minor axis: x = h
Covertices (h, k ± b).
Vertical Orientation
$\frac{(x-h)^{2}}{b^{2}}+\frac{(y-k)^{2}}{a^{2}}$=1 where a > b and c2 = a2 - b2.
Center (h, k)
Foci (h, k ± c)
Major axis; x = h
Verrices ( h, k ± a)
Foci (h, k ± c)
Minor axis: y = k
Covertices ( h ± b, k).
More topics in Ellipse Area of an Ellipse
*AP and SAT are registered trademarks of the College Board. | 2019-09-23 05:47:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.616750955581665, "perplexity": 876.2753526267761}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514576047.85/warc/CC-MAIN-20190923043830-20190923065830-00420.warc.gz"} |
http://math.stackexchange.com/questions/237709/breadth-first-search-and-bipartiteness | # Breadth first search and bipartiteness
I was just wondering what the correlation is between a breadth-first search tree of a graph and that graph being bipartite?
-
A bipartite graph is a graph that you can $2$-color. If you fix the color of one vertex (the source), then the rest of the coloring (provided that it exists) is totally defined by the distance to the source vertex. Hence, your graph is bipartite if and only if there is no "horizontal" edge in your BFS tree. | 2014-07-11 02:19:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6540043950080872, "perplexity": 157.4869844170573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776424634.96/warc/CC-MAIN-20140707234024-00047-ip-10-180-212-248.ec2.internal.warc.gz"} |
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## User: ramy-e-ali
### Title: Consistency Analysis of Replication-Based Probabilistic Key-Value Stores
Partial quorum systems are widely used in distributed key-value stores due to
their latency benefits at the expense of providing weaker consistency
guarantees. The probabilistically bounded staleness framework (PBS) studied the
latency-consistency trade-off of Dynamo-style partial quorum systems through
Monte Carlo event-based simulations. In this paper, we study the
latency-consistency trade-off for such systems analytically and derive a
closed-form expression for the inconsistency probability. Our approach allows
fine-tuning of latency and consistency guarantees in key-value stores, which is
intractable using Monte Carlo event-based simulations.
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https://it.mathworks.com/help/econ/msvar.summarize.html | # summarize
Summarize Markov-switching dynamic regression model estimation results
## Syntax
``summarize(Mdl)``
``summarize(Mdl,state)``
``results = summarize(___)``
## Description
example
````summarize(Mdl)` displays a summary of the Markov-switching dynamic regression model `Mdl`. If `Mdl` is an estimated model returned by `estimate`, then `summarize` displays estimation results to the MATLAB® Command Window. The display includes: A model descriptionEstimated transition probabilitiesFit statistics, which include the effective sample size, number of estimated submodel parameters and constraints, loglikelihood, and information criteria (AIC and BIC)A table of submodel estimates and inferences, which includes coefficient estimates with standard errors, t-statistics, and p-values. If `Mdl` is an unestimated Markov-switching model returned by `msVAR`, `summarize` prints the standard object display (the same display that `msVAR` prints during model creation). ```
example
````summarize(Mdl,state)` displays only summary information for the submodel with name `state`.```
example
````results = summarize(___)` returns one of the following variables and does not print to the Command Window. If `Mdl` is an estimated Markov-switching model, `results` is a table containing the submodel estimates and inferences.If `Mdl` is an unestimated model, `results` is an `msVAR` object that is equal to `Mdl`. ```
## Examples
collapse all
Consider a two-state Markov-switching dynamic regression model of the postwar US real GDP growth rate, as estimated in [1].
Create Partially Specified Model for Estimation
Create a Markov-switching dynamic regression model for the naive estimator by specifying a two-state discrete-time Markov chain with an unknown transition matrix and AR(0) (constant only) submodels for both regimes. Label the regimes.
```P = NaN(2); mc = dtmc(P,'StateNames',["Expansion" "Recession"]); mdl = arima(0,0,0); Mdl = msVAR(mc,[mdl; mdl]);```
`Mdl` is a partially specified `msVAR` object. `NaN`-valued elements of the `Switch` and `SubModels` properties indicate estimable parameters.
Create Fully Specified Model Containing Initial Values
The estimation procedure requires initial values for all estimable parameters. Create a fully specified Markov-switching dynamic regression model that has the same structure as `Mdl`, but set all estimable parameters to initial values. This example uses arbitrary initial values.
```P0 = 0.5*ones(2); mc0 = dtmc(P0,'StateNames',Mdl.StateNames); mdl01 = arima('Constant',1,'Variance',1); mdl02 = arima('Constant',-1,'Variance',1); Mdl0 = msVAR(mc0,[mdl01; mdl02]);```
`Mdl0` is a fully specified `msVAR` object.
Load the US GDP data set.
`load Data_GDP`
`Data` contains quarterly measurements of the US real GDP in the period 1947:Q1–2005:Q2. The estimation period in [1] is 1947:Q2–2004:Q2. For more details on the data set, enter `Description` at the command line.
Transform the data to an annualized rate series:
1. Convert the data to a quarterly rate within the estimation period.
2. Annualize the quarterly rates.
```qrate = diff(Data(2:230))./Data(2:229); % Quarterly rate arate = 100*((1 + qrate).^4 - 1); % Annualized rate```
Estimate Model
Fit the model `Mdl` to the annualized rate series `arate`. Specify `Mdl0` as the model containing the initial estimable parameter values.
`EstMdl = estimate(Mdl,Mdl0,arate);`
`EstMdl` is an estimated (fully specified) Markov-switching dynamic regression model. `EstMdl.Switch` is an estimated discrete-time Markov chain model (`dtmc` object), and `EstMdl.Submodels` is a vector of estimated univariate VAR(0) models (`varm` objects).
Display the estimated state-specific dynamic models.
`EstMdlExp = EstMdl.Submodels(1)`
```EstMdlExp = varm with properties: Description: "1-Dimensional VAR(0) Model" SeriesNames: "Y1" NumSeries: 1 P: 0 Constant: 4.90146 AR: {} Trend: 0 Beta: [1×0 matrix] Covariance: 12.087 ```
`EstMdlRec = EstMdl.Submodels(2)`
```EstMdlRec = varm with properties: Description: "1-Dimensional VAR(0) Model" SeriesNames: "Y1" NumSeries: 1 P: 0 Constant: 0.0084884 AR: {} Trend: 0 Beta: [1×0 matrix] Covariance: 12.6876 ```
Display the estimated state transition matrix.
`EstP = EstMdl.Switch.P`
```EstP = 2×2 0.9088 0.0912 0.2303 0.7697 ```
Display an estimation summary containing parameter estimates and inferences.
`summarize(EstMdl)`
```Description 1-Dimensional msVAR Model with 2 Submodels Switch Estimated Transition Matrix: 0.909 0.091 0.230 0.770 Fit Effective Sample Size: 228 Number of Estimated Parameters: 2 Number of Constrained Parameters: 0 LogLikelihood: -639.496 AIC: 1282.992 BIC: 1289.851 Submodels Estimate StandardError TStatistic PValue _________ _____________ __________ ___________ State 1 Constant(1) 4.9015 0.23023 21.289 1.4301e-100 State 2 Constant(1) 0.0084884 0.2359 0.035983 0.9713 ```
Create the following fully specified Markov-switching model the DGP.
• State transition matrix: $P=\left[\begin{array}{ccc}0.5& 0.2& 0.3\\ 0.2& 0.6& 0.2\\ 0.2& 0.1& 0.7\end{array}\right]$.
• State 1: $\left[\begin{array}{c}{y}_{1,t}\\ {y}_{2,t}\end{array}\right]=\left[\begin{array}{c}-1\\ -1\end{array}\right]+\left[\begin{array}{cc}-0.5& 0.1\\ 0.2& -0.75\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ {y}_{2,t-1}\end{array}\right]+{\epsilon }_{1,t}$, where ${\epsilon }_{1,\mathit{t}}\sim {\mathit{N}}_{2}\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}0.5& 0\\ 0& 1\end{array}\right]\right)$.
• State 2: $\left[\begin{array}{c}{y}_{1,t}\\ {y}_{2,t}\end{array}\right]=\left[\begin{array}{c}-1\\ 2\end{array}\right]+{\epsilon }_{2,t}$, where ${\epsilon }_{2,\mathit{t}}\sim {\mathit{N}}_{2}\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right)$.
• State 3:$\left[\begin{array}{c}{y}_{1,t}\\ {y}_{2,t}\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]+\left[\begin{array}{cc}0.5& 0.1\\ 0.2& 0.75\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ {y}_{2,t-1}\end{array}\right]+{\epsilon }_{3,t}$, where ${\epsilon }_{3,\mathit{t}}\sim {\mathit{N}}_{2}\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}1& -0.1\\ -0.1& 2\end{array}\right]\right)$.
```PDGP = [0.5 0.2 0.3; 0.2 0.6 0.2; 0.2 0.1 0.7]; mcDGP = dtmc(PDGP); constant1 = [-1; -1]; constant2 = [-1; 2]; constant3 = [1; 2]; AR1 = [-0.5 0.1; 0.2 -0.75]; AR3 = [0.5 0.1; 0.2 0.75]; Sigma1 = [0.5 0; 0 1]; Sigma2 = eye(2); Sigma3 = [1 -0.1; -0.1 2]; mdl1DGP = varm(Constant=constant1,AR={AR1},Covariance=Sigma1); mdl2DGP = varm(Constant=constant2,Covariance=Sigma2); mdl3DGP = varm(Constant=constant3,AR={AR3},Covariance=Sigma3); mdlDGP = [mdl1DGP; mdl2DGP; mdl3DGP]; MdlDGP = msVAR(mcDGP,mdlDGP);```
Generate a random response path of length 1000 from the DGP.
```rng(1) % For reproducibiliy Y = simulate(MdlDGP,1000);```
Create a partially specified Markov-switching model that has the same structure as the DGP, but the transition matrix, and all submodel coefficients and innovations covariance matrices are unknown and estimable.
```mc = dtmc(nan(3)); mdlar = varm(2,1); mdlc = varm(2,0); Mdl = msVAR(mc,[mdlar; mdlc; mdlar]);```
Initialize the estimation procedure by fully specifying a Markov-switching model that has the same structure as `Mdl`, but has the following parameter values:
• A randomly drawn transition matrix
• Randomly drawn contant vectors for each model
• AR self lags of 0.1 and cross lags of 0
• The identify matrix for the innovations covariance
```P0 = randi(10,3,3); mc0 = dtmc(P0); constant01 = randn(2,1); constant02 = randn(2,1); constant03 = randn(2,1); AR0 = 0.1*eye(2); Sigma0 = eye(2); mdl01 = varm(Constant=constant01,AR={AR0},Covariance=Sigma0); mdl02 = varm(Constant=constant02,Covariance=Sigma0); mdl03 = varm(Constant=constant03,AR={AR0},Covariance=Sigma0); submdl0 = [mdl01; mdl02; mdl03]; Mdl0 = msVAR(mc0,submdl0);```
Fit the Markov-switching model to the simulated series. Plot the loglikelihood after each iteration of the EM algorithm.
`EstMdl = estimate(Mdl,Mdl0,Y,IterationPlot=true);`
The plot displays the evolution of the loglikelihood with increasing iterations of the EM algorithm. The procedure terminates when one of the stopping criteria is satisfied.
Display an estimation summary of the model.
`summarize(EstMdl)`
```Description 2-Dimensional msVAR Model with 3 Submodels Switch Estimated Transition Matrix: 0.501 0.245 0.254 0.204 0.549 0.247 0.188 0.102 0.710 Fit Effective Sample Size: 999 Number of Estimated Parameters: 14 Number of Constrained Parameters: 0 LogLikelihood: -3634.005 AIC: 7296.010 BIC: 7364.704 Submodels Estimate StandardError TStatistic PValue ________ _____________ __________ ___________ State 1 Constant(1) -0.98929 0.023779 -41.603 0 State 1 Constant(2) -1.0884 0.030164 -36.083 4.1957e-285 State 1 AR{1}(1,1) -0.48446 0.01547 -31.316 2.8121e-215 State 1 AR{1}(2,1) 0.1835 0.019624 9.3509 8.6868e-21 State 1 AR{1}(1,2) 0.083953 0.0070162 11.966 5.3839e-33 State 1 AR{1}(2,2) -0.72972 0.0089002 -81.989 0 State 2 Constant(1) -0.9082 0.030103 -30.17 5.9064e-200 State 2 Constant(2) 1.9514 0.030483 64.016 0 State 3 Constant(1) 1.1212 0.044427 25.237 1.5818e-140 State 3 Constant(2) 1.9561 0.0593 32.986 1.2831e-238 State 3 AR{1}(1,1) 0.48965 0.023149 21.152 2.6484e-99 State 3 AR{1}(2,1) 0.22688 0.030899 7.3427 2.0936e-13 State 3 AR{1}(1,2) 0.095847 0.012005 7.9838 1.4188e-15 State 3 AR{1}(2,2) 0.72766 0.016024 45.41 0 ```
Display an estimation summary separately for each state.
`summarize(EstMdl,1)`
```Description 2-Dimensional VAR Submodel, State 1 Submodel Estimate StandardError TStatistic PValue ________ _____________ __________ ___________ State 1 Constant(1) -0.98929 0.023779 -41.603 0 State 1 Constant(2) -1.0884 0.030164 -36.083 4.1957e-285 State 1 AR{1}(1,1) -0.48446 0.01547 -31.316 2.8121e-215 State 1 AR{1}(2,1) 0.1835 0.019624 9.3509 8.6868e-21 State 1 AR{1}(1,2) 0.083953 0.0070162 11.966 5.3839e-33 State 1 AR{1}(2,2) -0.72972 0.0089002 -81.989 0 ```
`summarize(EstMdl,2)`
```Description 2-Dimensional VAR Submodel, State 2 Submodel Estimate StandardError TStatistic PValue ________ _____________ __________ ___________ State 2 Constant(1) -0.9082 0.030103 -30.17 5.9064e-200 State 2 Constant(2) 1.9514 0.030483 64.016 0 ```
`summarize(EstMdl,3)`
```Description 2-Dimensional VAR Submodel, State 3 Submodel Estimate StandardError TStatistic PValue ________ _____________ __________ ___________ State 3 Constant(1) 1.1212 0.044427 25.237 1.5818e-140 State 3 Constant(2) 1.9561 0.0593 32.986 1.2831e-238 State 3 AR{1}(1,1) 0.48965 0.023149 21.152 2.6484e-99 State 3 AR{1}(2,1) 0.22688 0.030899 7.3427 2.0936e-13 State 3 AR{1}(1,2) 0.095847 0.012005 7.9838 1.4188e-15 State 3 AR{1}(2,2) 0.72766 0.016024 45.41 0 ```
Consider the model for the US GDP growth rate in Estimate Markov-Switching Dynamic Regression Model.
Create a Markov-switching dynamic regression model for the naive estimator.
```P = NaN(2); mc = dtmc(P,'StateNames',["Expansion" "Recession"]); mdl = arima(0,0,0); Mdl = msVAR(mc,[mdl; mdl]);```
Create a fully specified Markov-switching dynamic regression model that has the same structure as `Mdl`, but set all estimable parameters to initial values.
```P0 = 0.5*ones(2); mc0 = dtmc(P0,'StateNames',Mdl.StateNames); mdl01 = arima('Constant',1,'Variance',1); mdl02 = arima('Constant',-1,'Variance',1); Mdl0 = msVAR(mc0,[mdl01; mdl02]);```
Load the US GDP data set. Preprocess the data.
```load Data_GDP qrate = diff(Data(2:230))./Data(2:229); % Quarterly rate arate = 100*((1 + qrate).^4 - 1); % Annualized rate```
Fit the model `Mdl` to the annualized rate series `arate`. Specify `Mdl0` as the model containing the initial estimable parameter values.
`EstMdl = estimate(Mdl,Mdl0,arate);`
Return an estimation summary table.
`results = summarize(EstMdl)`
```results=2×4 table Estimate StandardError TStatistic PValue _________ _____________ __________ ___________ State 1 Constant(1) 4.9015 0.23023 21.289 1.4301e-100 State 2 Constant(1) 0.0084884 0.2359 0.035983 0.9713 ```
`results` is a table containing estimates and inferences for all submodel coefficients.
Identify significant coefficient estimates.
`results.Properties.RowNames(results.PValue < 0.05)`
```ans = 1x1 cell array {'State 1 Constant(1)'} ```
## Input Arguments
collapse all
Markov-switching dynamic regression model, specified as an `msVAR` object returned by `estimate` or `msVAR`.
State to summarize, specified as an integer in `1:Mdl.NumStates` or a state name in `Mdl.StateNames`.
The default summarizes all states.
Example: `summarize(Mdl,3)` summarizes the third state in `Mdl`.
Example: `summarize(Mdl,"Recession")` summarizes the state labeled `"Recession"` in `Mdl`.
Data Types: `double` | `char` | `string`
## Output Arguments
collapse all
Model summary, returned as a table or an `msVAR` object.
• If `Mdl` is an estimated Markov-switching model returned by `estimate`, `results` is a table of summary information for the submodel parameter estimates. Each row corresponds to a submodel coefficient. Columns correspond to the estimate (`Estimate`), standard error (`StandardError`), t-statistic (`TStatistic`), and the p-value (`PValue`).
When the summary includes all states (the default), `results.Properties` stores the following fit statistics:
FieldDescription
`Description`Model summary description (character vector)
`EffectiveSampleSize`Effective sample size (numeric scalar)
`NumEstimatedParameters`Number of estimated parameters (numeric scalar)
`NumConstraints`Number of equality constraints (numeric scalar)
`LogLikelihood`Optimized loglikelihood value (numeric scalar)
`AIC`Akaike information criterion (numeric scalar)
`BIC`Bayesian information criterion (numeric scalar)
• If `Mdl` is an unestimated model, `results` is an `msVAR` object that is equal to `Mdl`.
Note
When `results` is a table, it contains only submodel parameter estimates:
• `Mdl.Switch` contains estimated transition probabilities.
• `Mdl.Submodels(j).Covariance` contains the estimated residual covariance matrix of state `j`. For details, see `msVAR`.
## Algorithms
`estimate` implements a version of Hamilton's Expectation-Maximization (EM) algorithm, as described in [3]. The standard errors, loglikelihood, and information criteria are conditional on optimal parameter values in the estimated transition matrix `Mdl.Switch`. In particular, standard errors do not account for variation in estimated transition probabilities.
## References
[1] Chauvet, M., and J. D. Hamilton. "Dating Business Cycle Turning Points." In Nonlinear Analysis of Business Cycles (Contributions to Economic Analysis, Volume 276). (C. Milas, P. Rothman, and D. van Dijk, eds.). Amsterdam: Emerald Group Publishing Limited, 2006.
[2] Hamilton, J. D. "Analysis of Time Series Subject to Changes in Regime." Journal of Econometrics. Vol. 45, 1990, pp. 39–70.
[3] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[4] Hamilton, J. D. "Macroeconomic Regimes and Regime Shifts." In Handbook of Macroeconomics. (H. Uhlig and J. Taylor, eds.). Amsterdam: Elsevier, 2016.
## Version History
Introduced in R2021b | 2022-09-27 21:36:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.786408007144928, "perplexity": 5330.408090934872}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335058.80/warc/CC-MAIN-20220927194248-20220927224248-00638.warc.gz"} |
http://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-12th-edition/chapter-2-section-2-5-introduction-to-relations-and-functions-2-5-exercises-page-195/47 | ## Intermediate Algebra (12th Edition)
$y \text{ is a function of }x \\\text{Domain: } (-\infty,\infty)$
$\bf{\text{Solution Outline:}}$ To determine if the given equation, $y=2x-6 ,$ is a function, check if every value of $x$ will produce a different value of $y.$ To find the domain, find the set of all possible values of $x.$ $\bf{\text{Solution Details:}}$ For each value of $x,$ multiplying it by $2$ and then subtracting $6$ will produce a different value of $y.$ Hence, $y$ is a function of $x.$ The variable $x$ can be replaced with any number. Hence, the domain is the set of all real numbers. Hence, the given equation has the following characteristics: \begin{array}{l}\require{cancel} y \text{ is a function of }x \\\text{Domain: } (-\infty,\infty) .\end{array} | 2018-04-21 10:42:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.952608048915863, "perplexity": 117.15986171676774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945111.79/warc/CC-MAIN-20180421090739-20180421110739-00373.warc.gz"} |
https://math.stackexchange.com/questions/1503187/i-have-a-free-summer-before-university-what-should-i-learn | # I have a free summer before university. What should I learn? [closed]
Note: This is a soft question.
It may be a bit early to be thinking about this, but I figured I'd ask now and see what responses I get.
I'm currently a high school senior, and I quite like pure math. At the moment, I'm taking Graduate Analysis at a highly ranked university out of Big Rudin. I've already taken linear algebra, differential equations, linear algebra and differential equations, real analysis out of chapters 1-8 of Baby Rudin, and topology. I'm also a course grader for Baby Rudin at the uni.
I have a few questions, and a few concerns.
Concerns:
(i) Though I've only been writing formal proofs now for less than a year, I find some of the exercises in Big Rudin to be hard to the point that I feel as though I could never prove such a thing. Some exercises seem near impossible from "down here''.
(ii) I feel as though there may be "gaps" in my mathematical education, though I'm not sure how important these are. For example, I've never written one of those super messy $\epsilon-\delta$ proofs, nor have I had to work too much with weird algebraic estimates as in some basic sequences and series. Though these "gaps" haven't seemed to pose any issues yet.
(iii) Just from studying a lot of analysis, I see truly how much information there is to be learnt. I'm concerned that when I try to learn other subjects, such as alg top, general algebra, diff top, etc, I'll slowly forget what I've learned in analysis. Like a lot of "oh yeah I forgot you could do that" when doing something simple in analysis.
Questions:
(i) Realistically, in less than a year of writing formal proofs, how able should I be in constructing more complicated proofs? How can I set goals for the future? Is there a way to quantify this?
(ii) If I maintain this sort of pace, (beginning the first year in university with algebra, functional analysis, measure theory, etc) what track does this put me on for potential graduate schools? What are students that are accepted into to 20 programs doing early on in college?
(iii) Before entering uni next fall, I have a free summer. What should I learn? What should I touch? What should I master? Naturally, algebra would be a good place to start (though I know much of it informally), but beyond this, what will give a solid foundation? Perhaps an area whose proofs are quite unique and thus give a nice broad proof-writing basis.
(iv) How should one who is just entering into the world of "real" mathematics (graduate and beyond) cope with the desire to learn all necessary graduate subjects at once? For example, I find myself wanting to learn alg top, diff geo, diff top, adv funct analysis, and operator algebras all at the same time. So is it better to get a taste of each? Or to pick them off one at a time in depth?
Finally, thank you for reading this post. Any responses are welcome and entirely appreciated. Answers can surely address my specific questions, but I've included questions that may be answered in general as to not seek individual advice.
## closed as primarily opinion-based by Najib Idrissi, JonMark Perry, Claude Leibovici, Grigory M, draks ...Dec 14 '15 at 10:46
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
• I'd say if you maintain this sort of pace you're on a really good track for potential graduate schools. And I think it's not a problem at all that you struggle with some problems in Big Rudin, I think plenty of grad students at top universities struggle with some problems in Big Rudin. Do you have an option to attend a math summer program like the Ross program this summer? Might be worth considering. – littleO Oct 29 '15 at 9:15
• @littleO I've definitely looked into Ross, but I'm not sure any of the topics (mod arithmetic, continued fractions, etc) appeal too much to me at the moment solely because I've never really had a taste (or a knack) for competition-math-type problems. My plan for the summer is hopefully to read some texts and work some problems with the help of math se – Anthony Peter Oct 29 '15 at 9:17
• In my experience, just interacting with the other students and counselors at a place like Ross is an extremely valuable experience, even if you're not that interested in learning number theory right now. But your plan sounds good too. – littleO Oct 29 '15 at 9:22
• @littleO Ahh. I will admit this has been one of my least favourite things about taking grad uni classes in high school: I end up doing most problems sets entirely on my own. I can't meet and do the problems with students or profs because it's a significant drive. So, I understand completely the value of this interaction – Anthony Peter Oct 29 '15 at 9:23
• amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071 maybe something to look at. – Willie Wong Nov 11 '15 at 5:12 | 2019-07-21 15:13:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4693504571914673, "perplexity": 786.1799168991494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195527048.80/warc/CC-MAIN-20190721144008-20190721170008-00451.warc.gz"} |
https://tex.stackexchange.com/questions/419845/adding-line-break-in-displayed-math-only-one-equation?noredirect=1 | Adding line break in displayed math, only one equation [duplicate]
This question may seem like a duplicate, but I haven't seen any satisfactory answer in similar questions.
Is there a way to add a line break in displayed math mode without using multiple-line environments such as gather or eqnarray? I want to display an equation that spans multiple lines inside an equation environment (so that the entire thing only gets one number).
My current code:
$$A_i=A_0-\sum_{j=0}^{i-1}[(2^i\cdot3)\cdot\frac{s_i^2}{18}\cdot sin(\alpha)]=% \sum_{j=0}^{i-1}[(2^i\cdot3)\cdot\frac{(\frac{s_0}{3^i})^2}{18}\cdot sin(180º-\frac{120º}{2^i})]=% \sum_{j=0}^{i-1}[(2^i\cdot3)\cdot\ddfrac{s_0^2}{18\cdot3^{2i}}\cdot sin(180º-\frac{120º}{2^i})]$$
This "overflows" the line:
I want to add a line break in between (after one of the = signs), but still have the whole thing labelled as (4.7). | 2019-10-19 05:07:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.821938693523407, "perplexity": 1036.8299219001567}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986688826.38/warc/CC-MAIN-20191019040458-20191019063958-00336.warc.gz"} |
http://thesassway.com/intermediate/using-source-maps-with-sass | # Using source maps with Sass 3.3
One of the exciting new features in Sass 3.3 that every developer should take advantage of is source maps.
As CSS pre-processors, minifiers, and JavaScript transpilers have become mainstream it is increasingly difficult to debug the code running in the browser because of differences with the original source code.
For example, if you use CoffeeScript (which compiles to JavaScript) you won't see CoffeeScript while debugging in the browser. Instead, you'll see compiled JavaScript. The same problem exists for Sass which compiles down to CSS.
Source maps seek to bridge the gap between higher-level languages like CoffeeScript and Sass and the lower-level languages they compile down to (JavaScript and CSS). Source maps allow you to see the original source (the CoffeeScript or Sass) instead of the compiled JavaScript or CSS while debugging.
In practice, for Sass users, this means that when you inspect an element with developer tools, rather than seeing the CSS styles associated with that element, you can see the code we really care about: the pre-compiled Sass.
## Generating source maps for Sass
To get access to this feature in the browser, you need to generate a source map file for each source file.
For Sass, this is as easy as adding a flag to Sass's command line tool:
\$ sass sass/screen.scss:stylesheets/screen.css --sourcemap
If you look in your output folder after running that command, you'll notice that a comment has been added to the end of the generated CSS file:
/*# sourceMappingURL=screen.css.map */
This points to an additional file that was created during compilation: screen.css.map, which - as the name implies - is what maps all of the compiled CSS back to the source Sass declarations. If you're interested in the details of this file and how source maps actually work, check out Ryan Seddon's Introduction to JavaScript Source Maps over at HTML5Rocks. (Even though the article implies that it's only about JavaScript, all source maps work the same.)
## Enabling source maps in the browser
The second thing we need to do to take advantage of source maps is to make sure that our browser knows look for them. Chrome, Firefox and Safari all have support for source maps.
### Chrome
If you're using Chrome, source maps are now part of the core feature set, so you don't have to monkey around in chrome://flags any more. Simply open up the DevTools settings and toggle the Enable CSS Source Maps option:
### Firefox
For Firefox users, source maps are in version 29. You can enable them in the Toolbox Options menu (the gear icon) or by right-clicking anywhere in the Style Inspector's rule view and selecting the Show original sources option. (More info is available at the Mozilla blog.)
### Safari
Safari is a bit ahead of the curve in terms of source map support. If a map file is detected, references are automatically changed to the source-mapped files, no configuration necessary.
## Another tool in our belt
Once source maps are enabled in your browser of choice, the source reference for every style will change from the generated CSS to the source Sass, right down to the line number. Amazing!
Just as Firebug and its successors drastically improved our ability to debug in the browser, source maps increase the depth of our diagnostic capabilities. By allowing us to directly access our pre-compiled code, we can find and fix problems faster than ever. Now that I've been using source maps for a few months, I can't imagine working without them.
Now that you have source maps enabled for Sass, you may want to learn how to use source maps with JavaScript.
June 7, 2014 ~ Intermediate, Guides and Tutorials, Tim Hettler | 2018-08-19 15:14:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23960410058498383, "perplexity": 2865.0309414187573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221215222.74/warc/CC-MAIN-20180819145405-20180819165405-00397.warc.gz"} |
https://openreview.net/forum?id=lXMlDL78Alx | ## Causal Attention to Exploit Transient Emergence of Causal Effect
Abstract: We propose a causal reasoning mechanism called $\textit{causal attention}$ that can improve performance of machine learning models on a class of causal inference tasks by revealing the generation process behind the observed data. We consider the problem of reconstructing causal networks (e.g., biological neural networks) connecting large numbers of variables (e.g., nerve cells), of which evolution is governed by nonlinear dynamics consisting of weak coupling-drive (i.e., causal effect) and strong self-drive (dominants the evolution). The core difficulty is sparseness of causal effect that emerges (the coupling force is significant) only momentarily and otherwise remains dormant in the neural activity sequence. $\textit{Causal attention}$ is designed to guide the model to make inference focusing on the critical regions of time series data where causality may manifest. Specifically, attention coefficients are assigned autonomously by a neural network trained to maximise the Attention-extended Transfer Entropy, which is a novel generalization of the iconic transfer entropy metric. Our results show that, without any prior knowledge of dynamics, $\textit{causal attention}$ explicitly identifies areas where the strength of coupling-drive is distinctly greater than zero. This innovation substantially improves reconstruction performance for both synthetic and real causal networks using data generated by neuronal models widely used in neuroscience. | 2023-03-23 23:29:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3312482237815857, "perplexity": 1056.2488868008093}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945218.30/warc/CC-MAIN-20230323225049-20230324015049-00291.warc.gz"} |
http://eprint.iacr.org/2008/144 | ## Cryptology ePrint Archive: Report 2008/144
A Note on Differential Privacy: Defining Resistance to Arbitrary Side Information
Abstract: In this note we give a precise formulation of "resistance to arbitrary side information" and show that several relaxations of differential privacy imply it. The formulation follows the ideas originally due to Dwork and McSherry, stated implicitly in [Dwork06]. This is, to our knowledge, the first place such a formulation appears explicitly. The proof that relaxed definitions satisfy the Bayesian formulation is new.
Category / Keywords: foundations / Differential Privacy, Composition | 2016-05-03 05:09:34 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8716897368431091, "perplexity": 2255.3170875254364}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860118790.25/warc/CC-MAIN-20160428161518-00169-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.maa.org/press/maa-reviews/fundamentals-of-technical-mathematics | # Fundamentals of Technical Mathematics
###### Sarhan M. Musa
Publisher:
Publication Date:
2016
Number of Pages:
394
Format:
Paperback
Price:
100.00
ISBN:
9780128019870
Category:
Textbook
[Reviewed by
Tom Schulte
, on
12/21/2016
]
Fundamentals of Technical Mathematics introduces basic mathematics from fractions, decimals, percentages, order of operations, and more to fundamentals of algebra, equations, inequalities, vectors, modelling, graphs and functions, geometry, and trigonometry. The theme of applied mathematics for engineering technologists and technicians comes out through strategically placed story problems drawn from such areas as circuit design, astrophysics, measurement systems, and more. This is the minority of the largely general content. About once per chapter there are samples in MATLAB and Maple.
The material is appropriate for secondary education or even the first year of higher education. The text includes plenty of examples and chapter exercises backed up by, for all exercises, detailed solutions. Also at the back of the book are fairly complete primers for MATLAB and Maple, which I both appreciate and find unusual considering comparable texts.
Like many textbooks at this level, the chapters feature highlighted definitions and key points. In my experience, even students unwilling to invest in reading an entire chapter will pay attention to these, so when they are present care should be taken with them. Consider the definition here of a quadratic equation: “A quadratic equation is a second order equation written as $ax^2+bx+c=0$…” This definition is too restrictive, really; it should have ended at word eight. On the other extreme, some definitions are incomplete to the point of seeming ill-conceived or hastily assembled: “A determinant of a matrix is a scalar number that replaces the bracket with vertical lines.” I cannot recommend this text for independent study. It requires a guided presentation with amplification of key details and, in some cases, important clarifications.
The scope of content is ambitious in several ways for a textbook at this level. The author has brought in much material not in comparable textbooks. But this makes more glaring a key exception: there is a nearly complete absence of a set theoretic basis or motivation at any point. For instance, it seems that this awkward wording is the result of this avoidance: for the “Definition of the solution of equation” [sic], the author says that “A solution of equation is the numbers that produce true statement for the equation [sic].”
This is one of the rare textbooks I see that, especially at this level, introduces the complex plane along with complex numbers. Why, however, this is done a chapter ahead of introducing the rectangular coordinate system based on the reals I find elusive. In that chapter on the Cartesian coordinate system, distance formula makes an appearance, a chapter ahead of the Pythagorean Theorem. In my experience, going the other way around succeeds better with students.
Tom Schulte presents carefully curated classroom capsules to community college students outside of Detroit, Michigan.
• Dedication
• Preface
• Acknowledgments
• Chapter 1. Basic Concepts in Arithmetic
• Introduction
• 1.1. Basic arithmetic
• 1.1. Exercises
• 1.2. Decimals
• 1.2. Exercises
• 1.3. Percents (%)
• 1.3. Exercises
• Chapter 1 Review exercises
• Chapter 2. Introduction to Algebra
• Introduction
• 2.1. Introduction to algebra
• 2.1. Exercises
• 2.2. Addition, subtraction, multiplication, and division of monomials
• 2.2. Exercises
• 2.3. Ratio, proportion, and variation
• 2.3. Exercises
• Chapter 2 Review exercises
• Chapter 3. Equations, Inequalities, and Modeling
• Introduction
• 3.1. Equations
• 3.1. Exercises
• 3.2. Inequality equations and intervals
• 3.2. Exercises
• 3.3. Complex numbers
• 3.3. Exercises
• Chapter 3 Review exercises
• Chapter 4. Graphs and Functions
• Introduction
• 4.1. Graphs
• 4.1. Exercises
• 4.2. Functions
• 4.2. Exercises
• Chapter 4 Review exercises
• Chapter 5. Measurement
• Introduction
• 5.1. Length measurements
• 5.1. Exercises
• 5.2. Mass and weight measurements
• 5.2. Exercises
• 5.3. Capacity measurements of liquid
• 5.3. Exercises
• 5.4. Time measurements
• 5.4. Exercises
• 5.5. Temperature (T) measurement
• 5.5. Exercises
• 5.6. Derived units
• 5.6. Exercises
• Chapter 5 Review exercises
• Chapter 6. Geometry
• Introduction
• 6.1. Basic concepts in geometry
• 6.1. Exercises
• 6.2. Angle measurement and triangles
• 6.2. Exercises
• 6.3. Perimeter and circumference in geometry
• 6.3. Exercises
• 6.4. Area in geometry
• 6.4. Exercises
• 6.5. Volume in geometry
• 6.5. Exercises
• Chapter 6 Review exercises
• Chapter 7. Trigonometry
• Introduction
• 7.1. Units of angles
• 7.1. Exercises
• 7.2. Types of angles and right triangle
• 7.2. Exercises
• 7.3. Trigonometric functions
• 7.3. Exercises
• 7.4. Unit circle and oblique triangles
• 7.4. Exercises
• Chapter 7 Review exercises
• Chapter 8. Matrices, Determinants, and Vectors
• Introduction
• 8.1. Matrices
• 8.1. Exercises
• 8.2. Determinants
• 8.2. Exercises
• 8.3. Vectors
• 8.3. Exercises
• Chapter 8 Review exercises
• Appendix A. Maple
• Appendix B. MATLAB
• Appendix C. Solution Manual
• Glossary
• Index | 2021-01-18 05:28:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3943776786327362, "perplexity": 6689.176471952031}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703514121.8/warc/CC-MAIN-20210118030549-20210118060549-00700.warc.gz"} |
https://www.physicsforums.com/threads/polynomial-functions.545954/ | # Polynomial Functions.
f:]1,+∞[→]2,+∞[
x→ 2x/(x-1)
## Homework Equations
Prove that f is injective and serjective.
## The Attempt at a Solution
I already proved that it's injective by stating the injectivity law:
for every (a,b)ε]1,+∞[: f(a)=f(b) implies a=b
so: 2a/(a-1)=2b/(b-1) entails: 2ab-2b=2ab-2a entails -2b=-2a entails a=b
Can anyone please tell me how to prove that its serjective?
Related Calculus and Beyond Homework Help News on Phys.org
LCKurtz
Homework Helper
Gold Member
Sure. If y is in ]2,+∞[, calculate what x in ]1,+∞[ gives f(x) = y.
Ray Vickson
Homework Helper
Dearly Missed
f:]1,+∞[→]2,+∞[
x→ 2x/(x-1)
## Homework Equations
Prove that f is injective and serjective.
## The Attempt at a Solution
I already proved that it's injective by stating the injectivity law:
for every (a,b)ε]1,+∞[: f(a)=f(b) implies a=b
so: 2a/(a-1)=2b/(b-1) entails: 2ab-2b=2ab-2a entails -2b=-2a entails a=b
Can anyone please tell me how to prove that its serjective?
In your own words, what does it mean to say that f is surjective? (That is sUrjective, not sErjective!) Turn that verbal statement into an equation and then work on the equation, to see what conclusions you can make, or else use some known, general properties to get a conclusion.
RGV
i got it
f(x)=y
y=2x/x-1 equivalence y(x-1)=2x equivalence yx-2x-y=0
now we find Δ
Δ=4+4y^2
since Δ≥0 therefore there is some solution to this equation and therefore f is serjective.
Ray Vickson
Homework Helper
Dearly Missed
What does $\Delta = 4 + 4y^2$ have to do with anything here? Anyway, you are still spelling surjective incorrectly.
RGV
What does $\Delta = 4 + 4y^2$ have to do with anything here? Anyway, you are still spelling surjective incorrectly.
RGV
Well my teacher stated that if we find that Δ≥0 then therefore f is surjective and btw my first language is english, but i'm learning overseas in Morocco and all the lessons here are in Arabic, so that's probably the reason why i spelled it wrong.
Deveno
What does $\Delta = 4 + 4y^2$ have to do with anything here? Anyway, you are still spelling surjective incorrectly.
RGV
i believe he's taking the discriminant of a quadratic.
Ray Vickson | 2020-03-31 11:29:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7011598348617554, "perplexity": 2777.32572101477}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370500426.22/warc/CC-MAIN-20200331084941-20200331114941-00529.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/80503-hypergeomdist-correct-choice.html | # Thread: is hypergeomdist a correct choice?
1. ## is hypergeomdist a correct choice?
I want to find the odds of something:
I have 60 objects, of which I am going to select seven. Of these objects, 15 are the same (call them broken). What are the odds of me having at least three broken objects in my selection of seven?
$\displaystyle 1-hypergeomdist(0, 7, 15, 60)$$\displaystyle -hypergeomdist(1, 7, 15, 60)-hypergeomdist(2, 7, 15, 60)$
Second question:
Say 15 are broken in one way (A) and 7 are broken in another way (B). I now select ten of them. What are the odds of having at least two broken in one way and three broken in the other?
Do I multiply these odds (assuming I'm setting the problems up correctly in the first place) just like I would simpler probabilities? At this point i'm assuming that none of the objects are broken in both ways at the same time.
Thanks. Also, I've looked for sites dealing with the second question online and found little to nothing. Any sites you can direct me to so I can find my own answers. Not that I don't like giving you something to do. . .
2. The first one is correct AND this a hypergeo as long as we are selecting without relacement.
As for the second question, think of it as 3 groups.
(Just brains, no sites.)
You are selecting at least 2 from A, 3 from B, rest from C.
To do that exactly, well thats...
$\displaystyle { {15\choose 2} {7\choose 3} {38\choose 5}\over {60\choose 10}}$
Next figure out all the possibilities, 3 from A, 3 from B, 4 from C....
$\displaystyle { {15\choose 3} {7\choose 3} {38\choose 4}\over {60\choose 10}}$.
FIND all of these and add them together. The complement here is a pain.
3. Originally Posted by matheagle
The first one is correct AND this a hypergeo as long as we are selecting without relacement.
As for the second question, think of it as 3 groups.
(Just brains, no sites.)
You are selecting at least 2 from A, 3 from B, rest from C.
To do that exactly, well thats...
$\displaystyle { {15\choose 2} {7\choose 3} {38\choose 5}\over {60\choose 10}}$
Next figure out all the possibilities, 3 from A, 3 from B, 4 from C....
$\displaystyle { {15\choose 3} {7\choose 3} {38\choose 4}\over {60\choose 10}}$.
FIND all of these and add them together. The complement here is a pain.
Can we subtract all other choices instead?
$\displaystyle 1-({{15\choose 0}{7\choose0}{38\choose10}\over{60\choose10}}+{{15 \choose 1}{7\choose0}{38\choose9}\over{60\choose10}}+{{15\ choose 1}{7\choose1}{38\choose8}\over{60\choose10}}...)$
and is there anyway to do it multiplying our options of hyper geometric distributions, or am i just hoping here? cuz if we can do it that way, it would seem easier . . .
4. I wouldn't look at the complement. If you did you'd have a mess.
You can have 0 from group A, then 0 thru 7 from group B
You can have 1 from group B, then 0 thru 7 from group B....
5. Originally Posted by matheagle
I wouldn't look at the complement. If you did you'd have a mess.
You can have 0 from group A, then 0 thru 7 from group B
You can have 1 from group B, then 0 thru 7 from group B....
Both seems almost as bad as the other, but both are decent to set up in excel, i think. I'll mess around with it. My goal here would be to create something that would allow me to fluctuate both the total number of A and B broken in the set as well as a set number of acceptable of from both in the given selection.
Food for thought, certainly, and thank you. | 2018-03-22 18:38:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8884638547897339, "perplexity": 1245.740634147269}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647901.79/warc/CC-MAIN-20180322170754-20180322190754-00211.warc.gz"} |
https://discuss.codechef.com/questions/64923/hints-for-puzzle-pls-dont-mind-the-map | ×
# Hints for puzzle pls: "Don't mind the map"
Need help with the following problem please; don't even know where/how to begin.
After the trauma of Dr. Boolean's lab, the rabbits are eager to get back to their normal lives in a well-connected community, where they can visit each other frequently. Fortunately, the rabbits learned something about engineering as part of their escape from the lab. To get around their new warren fast, they built an elaborate subway system to connect their holes. Each station has the same number of outgoing subway lines (outgoing tracks), which are numbered.
Unfortunately, sections of warrens are very similar, so they can't tell where they are in the subway system. Their stations have system maps, but not an indicator showing which station the map is in. Needless to say, rabbits get lost in the subway system often. The rabbits adopted an interesting custom to account for this: Whenever they are lost, they take the subway lines in a particular order, and end up at a known station.
For example, say there were three stations A, B, and C, with two outgoing directions, and the stations were connected as follows
Line 1 from A, goes to B. Line 2 from A goes to C.
Line 1 from B, goes to A. Line 2 from B goes to C.
Line 1 from C, goes to B. Line 2 from C goes to A.
Now, suppose you are lost at one of the stations A, B, or C. Independent of where you are, if you take line 2, and then line 1, you always end up at station B. Having a path that takes everyone to the same place is called a meeting path.
We are interested in finding a meeting path which consists of a fixed set of instructions like, 'take line 1, then line 2,' etc. It is possible that you might visit a station multiple times. It is also possible that such a path might not exist. However, subway stations periodically close for maintenance. If a station is closed, then the paths that would normally go to that station, go to the next station in the same direction. As a special case, if the track still goes to the closed station after that rule, then it comes back to the originating station. Closing a station might allow for a meeting path where previously none existed. That is, if you have
A -> B -> C
and station B closes, then you'll have
A -> C
Alternately, if it was
A -> B -> B
then closing station B yields
A -> A
Write a function answer(subway) that returns one of:
-1 (minus one): If there is a meeting path without closing a station
The least index of the station to close that allows for a meeting path or
-2 (minus two): If even with closing 1 station, there is no meeting path.
subway will be a list of lists of integers such that subway[station][direction] = destination_station.
That is, the subway stations are numbered 0, 1, 2, and so on. The k^th element of subway (counting from 0) will give the list of stations directly reachable from station k.
The outgoing lines are numbered 0, 1, 2... The r^th element of the list for station k, gives the number of the station directly reachable by taking line r from station k.
Each element of subway will have the same number of elements (so, each station has the same number of outgoing lines), which will be between 1 and 5.
There will be at least 1 and no more than 50 stations.
For example, if subway = [[2, 1], [2, 0], [3, 1], [1, 0]] Then one could take the path [1, 0]. That is, from the starting station, take the second direction, then the first. If the first direction was the red line, and the second was the green line, you could phrase this as: if you are lost, take the green line for 1 stop, then the red line for 1 stop. So, consider following the directions starting at each station.
0 -> 1 -> 2.
1 -> 0 -> 2.
2 -> 1 -> 2.
3 -> 0 -> 2.
So, no matter the starting station, the path leads to station 2. Thus, for this subway, answer should return -1.
If
subway = [[1], [0]]
then no matter what path you take, you will always be at a different station than if you started elsewhere. If station 0 closed, that would leave you with
subway = [[0]]
So, in this case, answer would return 0 because there is no meeting path until you close station 0.
To illustrate closing stations,
subway = [[1,1],[2,2],[0,2]]
If station 2 is closed, then station 1 direction 0 will follow station 2 direction 0 to station 0, which will then be its new destination. station 1 direction 1 will follow station 2 direction 1 to station 2, but that station is closed, so it will get routed back to station 1, which will be its new destination. This yields
subway = [[1,1],[0,1]]
# Test cases
Inputs: (int) subway = [[2, 1], [2, 0], [3, 1], [1, 0]] Output: (int) -1
Inputs: (int) subway = [[1, 2], [1, 1], [2, 2]] Output: (int) 1
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question asked: 19 Feb '15, 13:46
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last updated: 19 Feb '15, 14:27 | 2019-01-20 13:40:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44187259674072266, "perplexity": 1464.4889147463468}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583716358.66/warc/CC-MAIN-20190120123138-20190120145138-00449.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/a-kite-flying-height-60-m-above-ground-string-attached-kite-tied-ground-it-makes-angle-60-ground-assuming-that-string-straight-heights-distances_50402 | # A Kite is Flying at a Height of 60 M Above the Ground. the String Attached to the Kite is Tied at the Ground. It Makes an Angle of 60° with the Ground. Assuming that the String is Straight, - Geometry
#### Question
A kite is flying at a height of 60 m above the ground. The string attached to the kite is tied at the ground. It makes an angle of 60° with the ground. Assuming that the string is straight, find the length of the string.
$\left( \sqrt{3} = 1 . 73 \right)$
#### Solution
Let AB be the height of kite above the ground and C be the position of the string attached to the kite which is tied at the ground.
Suppose the length of the string be x m.
Here, AB = 60 m and ∠ACB = 60º
In right ∆ABC,
$\sin60^\circ = \frac{AB}{AC}$
$\Rightarrow \frac{\sqrt{3}}{2} = \frac{60}{x}$
$\Rightarrow x = \frac{120}{\sqrt{3}} = 40\sqrt{3}$
$\Rightarrow x = 40 \times 1 . 73 = 69 . 2 m$
Thus, the length of the string is 69.2 m.
Is there an error in this question or solution?
#### APPEARS IN
Balbharati Mathematics 2 Geometry 10th Standard SSC Maharashtra State Board
Chapter 6 Trigonometry
Practice Set 6.2 | Q 6 | Page 137
#### Video TutorialsVIEW ALL [4]
A Kite is Flying at a Height of 60 M Above the Ground. the String Attached to the Kite is Tied at the Ground. It Makes an Angle of 60° with the Ground. Assuming that the String is Straight, Concept: Heights and Distances. | 2020-11-29 17:25:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4601008892059326, "perplexity": 357.61237868267585}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141201836.36/warc/CC-MAIN-20201129153900-20201129183900-00693.warc.gz"} |
https://www.hydrol-earth-syst-sci.net/24/227/2020/ | Journal topic
Hydrol. Earth Syst. Sci., 24, 227–248, 2020
https://doi.org/10.5194/hess-24-227-2020
Hydrol. Earth Syst. Sci., 24, 227–248, 2020
https://doi.org/10.5194/hess-24-227-2020
Research article 16 Jan 2020
Research article | 16 Jan 2020
A framework for deriving drought indicators from the Gravity Recovery and Climate Experiment (GRACE)
A framework for deriving drought indicators from the Gravity Recovery and Climate Experiment (GRACE)
Helena Gerdener, Olga Engels, and Jürgen Kusche Helena Gerdener et al.
• Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany
Correspondence: Helena Gerdener (gerdener@geod.uni-bonn.de)
Abstract
Identifying and quantifying drought in retrospective is a necessity for better understanding drought conditions and the propagation of drought through the hydrological cycle and eventually for developing forecast systems. Hydrological droughts refer to water deficits in surface and subsurface storage, and since these are difficult to monitor at larger scales, several studies have suggested exploiting total water storage data from the GRACE (Gravity Recovery and Climate Experiment) satellite gravity mission to analyze them. This has led to the development of GRACE-based drought indicators. However, it is unclear how the ubiquitous presence of climate-related or anthropogenic water storage trends found within GRACE analyses masks drought signals. Thus, this study aims to better understand how drought signals propagate through GRACE drought indicators in the presence of linear trends, constant accelerations, and GRACE-specific spatial noise. Synthetic data are constructed and existing indicators are modified to possibly improve drought detection. Our results indicate that while the choice of the indicator should be application-dependent, large differences in robustness can be observed. We found a modified, temporally accumulated version of the indicator particularly robust under realistic simulations. We show that linear trends and constant accelerations seen in GRACE data tend to mask drought signals in indicators and that different spatial averaging methods required to suppress the spatially correlated GRACE noise affect the outcome. Finally, we identify and analyze two droughts in South Africa using real GRACE data and the modified indicators.
Please read the corrigendum first before continuing.
1 Introduction
Droughts are recurrent natural hazards that affect the environment and economy with potentially catastrophic consequences. Drought impacts range from reduced streamflow, water scarcity, and reduced water quality to increased wildfires, soil erosion, and increased quantities of dust, crop failure, and large-scale famine. With climate change and population growth, the frequency and impact of droughts are projected to increase for many regions of the world (IPCC2013). Drought types can be distinguished depending on their effect on the hydrological cycle (e.g., Changnon1987; Mishra and Singh2010). In this study we focus on hydrological drought, a multiscale problem which may last weeks or many years and which may affect local or continental regions. For example, the severe drought between mid-2011 and mid-2012 affected millions of people in the entire eastern Africa region (Somalia, Djibouti, Ethiopia, and Kenya) and led to famine with an estimated 258 000 deaths . From 2012 to 2016, the US state of California experienced a historical drought that adversely affected groundwater levels, forests, crops, and fish populations and led to widespread land subsidence . In contrast, European droughts, for example in 2018, typically last a few months in exceptionally dry summers. For South Africa, due to a complex rainfall regime, areas and the percentage of land surface affected by drought can vary strongly .
Hydrological drought refers to a deficit of accessible water, i.e., water in natural and man-made surface reservoirs and subsurface storage, with respect to normal conditions. The propagation of drought through the hydrological cycle typically begins with a lack of precipitation, leading to runoff and soil moisture deficit, followed by decreasing streamflow and groundwater levels (Changnon1987). However, no unique standard procedures exist for measuring the deficit of each of these factors and for defining normal conditions. In order to arrive at operational definitions, which are required for triggering a response according to drought class for example, a large variety of drought indicators has been defined which typically seek to extract certain sub-signals from observable fields . Reviews of hydrological drought indicators are contained in , , , and . Streamflow is the most frequently used observable measurement in these studies.
Drought detection is mostly restricted to single fluxes (precipitation or streamflow) or storage (surface soil moisture or reservoir levels) that are easy to measure. Much fewer measurements are available to assess water content in deeper soil layers and groundwater storage deficit or the total of all storages. The NASA and German Aerospace Center (DLR) Gravity Recovery and Climate Experiment (GRACE) satellite mission, launched in 2002, has changed this situation since GRACE-derived monthly gravity field models can be converted to total water storage changes (TWSCs; ). GRACE consisted of two spacecraft following each other, which were linked together by an ultra-precise microwave ranging instrument; these ranges are routinely processed to provide monthly gravity models and thus maps of mass change. Since other mass transports in the atmosphere and ocean are removed during the processing, GRACE indeed provides quantitative measure of surface and subsurface water storages . Meanwhile, GRACE has been continued with the GRACE-FO (Follow-On) mission from which the first data are now available.
Studies of drought detection with GRACE-TWSC can be summarized in three groups: (i) using monthly maps of TWSC directly, (ii) partitioning TWSC time series into sub-signals that include drought signatures, or (iii) using indicators. For example, investigated the 2003 heat wave over seven central European basins using GRACE time series; they found a good agreement between TWSC and the combination of net precipitation and evaporation. Other studies focused on drought detection using TWSC sub-signals, e.g., trends were used to identify drought in central Europe and for the region encompassing the Tigris, Euphrates, and western Iran . After decomposing GRACE-TWSC into a seasonal and non-seasonal signals, were able to detect the 2005 drought in the central Amazon river basin while, identified two droughts in 2006 and 2011 in the Yangtze river basin. In the latter study, the El Niño–Southern Oscillation (ENSO) was identified as a possible driver for drought events in the Yangtze river basin. However, neither GRACE nor GRACE-FO enable one to separate different storage compartments, such as groundwater storage, without utilizing additional (e.g., compartment-specific) observations or model outputs, and their spatial and temporal resolutions (about 300 km and nominally 1 month respectively for GRACE) are limited. Several efforts are therefore focused on assimilating GRACE-TWSC maps into hydrological or land surface models .
Thus, perhaps not surprisingly, a number of GRACE-based drought indicators have been suggested , typically either based on normalization or percentile rank methods. However, a comprehensive comparison and assessment of these indicators is still missing, particularly in the presence of (1) trend signals as picked up by GRACE in many regions that may reflect non-stationary “normal” conditions, (2) correlated spatial noise that is related to the peculiar GRACE orbital pattern, and (3) the inevitable spatial averaging applied to GRACE, which results in smoothing out noise . From a water balance perspective, GRACE-TWSC variability mainly represents monthly total precipitation anomalies (e.g., Chen et al.2010; Frappart et al.2013). It is thus obvious that GRACE drought indicators will contain signatures that are visible in meteorological drought indicators, yet the difference should explain the magnitude of other contributions (e.g., increased evapotranspiration due to radiation) to hydrological drought.
Figure 1 shows a time series of region-averaged, detrended and deseasoned GRACE water storage changes over eastern Brazil (Ceará state) compared to the region-averaged 6-month Standardized Precipitation Index (SPI) to illustrate the potential of GRACE-TWSC for drought monitoring. As can be expected, TWSC and 6-month SPI appear moderately similar (correlation 0.43), characterized by positive peaks, for example at the beginning of 2004 and at the end of 2009, and negative peaks at the beginning of 2013. We also found correlations between TWSC and 6-month SPI in regions with different hydro-climatic conditions for the Missouri river basin (0.31), Maharashtra in western India (0.46), and South Africa (0.45) among other regions. This motivates us to modify common GRACE indicators to account for accumulation periods of input data, e.g., used with 6-month SPI but also for periods that are based on differences of input data. To our knowledge, this is the first study where (modified) indicators are tested in a synthetic framework based on a realistic signal that includes a hypothetical drought. We hypothesize that in this way we can (i) assess indicator robustness, with respect to identifying a “true” drought of given duration and magnitude, and (ii) understand how trend signals and spatial noise propagate into indicators and mask drought detection. In addition, we investigate to what extent the spatial averaging that is required for analyzing GRACE data affects indicators. For this, we compare spatially averaged gridded indicators to indicators derived from spatially averaged TWSC.
Figure 1Detrended and deseasoned GRACE-TWSC [mm] (orange) and the SPI [–] for 6 months of accumulated precipitation (blue), spatially averaged for Ceará, Brazil.
This contribution is organized as follows: in Sect. 2 we will review three GRACE-based drought indicators and modify them to accommodate either multi-month accumulation or differencing, while in Sect. 3 our framework for testing GRACE indicators in a realistic simulation environment will be explained. Then, Sect. 4 will provide simulation results and finally the results from real GRACE data. A discussion and conclusion will complete the paper.
2 Indicators for hydrological drought
Hydrological drought indicators are mostly based on observations of single water storages or fluxes, e.g., for precipitation, snowpack, streamflow, or groundwater. In general, indicator definitions can be arranged into four categories: (1) data normalization, (2) threshold-based, (3) quantile scores, and (4) probability-based .
Since total water storage deficit may be viewed as a more comprehensive information source on drought, the advent of GRACE total water storage change data has led to new indicators being developed. For example, developed a drought indicator based on yearly minima of water storage and a method for standardization, and computed recurrence times of yearly minima through the generalized extreme value theory. Other indicators explore the monthly resolution of GRACE, e.g., the Total Storage Deficit Index (TSDI; ), the GRACE-based Hydrological Drought Index (GHDI; ), the Drought Severity Index (DSI; ), and the drought index (DI; ). Further, presented a water storage deficit approach to detect drought magnitude, duration, and severity based on GRACE-derived TWSC. To our knowledge, only the , , and methods are able to detect drought events from monthly GRACE data without any additional information. Therefore, these three indicators will be discussed further.
In order to stress the link between GRACE-based and meteorological indicators, we first describe the relation of TWSC and precipitation. Assuming evapotranspiration (E) and runoff (Q) vary more regularly as compared to precipitation (i.e., ΔE=0 and ΔQ=0), the monthly GRACE-TWSC ($\stackrel{\mathrm{‾}}{\mathrm{\Delta }s}$) corresponds to precipitation anomalies (ΔP) accumulated since the GRACE storage monitoring began.
$\begin{array}{}\text{(1)}& \stackrel{\mathrm{‾}}{\mathrm{\Delta }s}\left(t\right)=\mathrm{\Delta }t\sum _{{t}_{\mathrm{0}}}^{t}\mathrm{\Delta }P,\end{array}$
where Δt is the time from t0 to t1. In contrast to Eq. (1), the difference between GRACE months as in
$\begin{array}{}\text{(2)}& \stackrel{\mathrm{‾}}{\mathrm{\Delta }s}\left({t}_{\mathrm{2}}\right)-\stackrel{\mathrm{‾}}{\mathrm{\Delta }s}\left({t}_{\mathrm{1}}\right)=\mathrm{\Delta }t\sum _{{t}_{\mathrm{1}}}^{{t}_{\mathrm{2}}}\mathrm{\Delta }P,\end{array}$
which corresponds to the precipitation anomaly accumulated between these months. Accumulated monthly TWSC thus corresponds to an iterative summation over the precipitation anomalies described by
$\begin{array}{}\text{(3)}& \sum _{{t}_{\mathrm{0}}}^{t}\stackrel{\mathrm{‾}}{\mathrm{\Delta }s}\left(t\right)=\mathrm{\Delta }t\sum _{\mathit{\tau }={t}_{\mathrm{0}}}^{t}\sum _{{t}_{\mathrm{0}}}^{\mathit{\tau }}\mathrm{\Delta }P.\end{array}$
In the following, we will discuss and extend the definition of , , and GRACE-based indicators, which are hence referred to as the Zhao method, Houborg method, and Thomas method, respectively.
2.1 Zhao method
In the approach of , one considers GRACE-derived monthly gridded TWSC for n years as in
$\begin{array}{}\text{(4)}& {x}_{i,j}=\stackrel{\mathrm{‾}}{\mathrm{\Delta }s}\left({t}_{i,j}\right),\end{array}$
with
$\begin{array}{}\text{(5)}& {t}_{i,j}=i+\left(j-\frac{\mathrm{1}}{\mathrm{2}}\right)\frac{\mathrm{1}}{\mathrm{12}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}i=\mathrm{1},\phantom{\rule{0.125em}{0ex}}\mathrm{\dots },\phantom{\rule{0.125em}{0ex}}n\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}j=\mathrm{1},\phantom{\rule{0.125em}{0ex}}\mathrm{\dots },\phantom{\rule{0.125em}{0ex}}\mathrm{12}.\end{array}$
Let us define the monthly climatology, i.e., mean monthly TWSC, ${\stackrel{\mathrm{̃}}{x}}_{j}$ with j=1, …, 12 and the standard deviation ${\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{j}$ of the anomalies in month j with respect to the climatological value as
$\begin{array}{}\text{(6)}& & {\stackrel{\mathrm{̃}}{x}}_{j}=\frac{\mathrm{1}}{n}\sum _{i=\mathrm{1}}^{n}{x}_{i,j},\text{(7)}& & {\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{j}={\left(\frac{\mathrm{1}}{n}\sum _{i=\mathrm{1}}^{n}{\left({x}_{i,j}-{\stackrel{\mathrm{̃}}{x}}_{j}\right)}^{\mathrm{2}}\right)}^{\mathrm{1}/\mathrm{2}}.\end{array}$
define their drought severity index (GRACE-DSI) as the standardized anomaly
$\begin{array}{}\text{(8)}& \mathrm{TWSC}\text{-}{\mathrm{DSI}}_{i,j}=\frac{{x}_{i,j}-{\stackrel{\mathrm{̃}}{x}}_{j}}{{\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{j}}\end{array}$
of a given month ti,j and provide a scale from −2.0 (exceptional drought) to +2.0 (exceptionally wet), as shown in Table 1. There is no particular probability distribution function (PDF) underlying the method; however if we assume the anomalies for a given month follow a Gaussian PDF, it is straightforward to compute the likelihood of a given month falling in one of the severity classes. For example, 2.1 % of months would be expected to turn out to be a period of exceptional drought and 2.1 % as exceptionally wet. This can be applied to any other PDF.
Table 1Drought severity level of TWSC-DSI . The values of TWSC-DSI are unitless.
Drought severity, however, should be related to the duration of a drought. For example showed how typical time scales of 3, 6, 12, 24, and 48 months of precipitation deficits are related to their impact on usable water sources. To account for the relation between severity and duration in the approach, we consider q months accumulated of TWSC, which is approximately related to precipitation in Eq. (3) as
$\begin{array}{}\text{(9)}& {x}_{i,j,q}^{+}=\sum _{k=\mathrm{1}}^{q}\stackrel{\mathrm{‾}}{\mathrm{\Delta }s}\left({t}_{i,j+\mathrm{1}-q}\right),\end{array}$
with ${t}_{i,j+\mathrm{1}-q}={t}_{i-\mathrm{1},j+\mathrm{13}-q}$ for $j+\mathrm{1}-q<\mathrm{1}$ or equivalently written for q months of averaged TWSC as
$\begin{array}{}\text{(10)}& {x}_{i,j,q}^{+}=\frac{\mathrm{1}}{q}\sum _{k=\mathrm{1}}^{q}\stackrel{\mathrm{‾}}{\mathrm{\Delta }s}\left({t}_{i,j+\mathrm{1}-q}\right).\end{array}$
For example for q=3, we would look for the 3-month running mean for December–January–February, January–February–March, and so on. In the next step, one computes, for example, the climatology and anomalies as with the original method. On the other hand, we can relate hydrological to meteorological indicators using Eq. (2). To develop a TWSC indicator that can be compared to indicators based on accumulated precipitation, one should rather consider the q-month differenced TWSC
$\begin{array}{}\text{(11)}& {x}_{i,j,q}^{-}=\stackrel{\mathrm{‾}}{\mathrm{\Delta }s}\left({t}_{i,j}\right)-\stackrel{\mathrm{‾}}{\mathrm{\Delta }s}\left({t}_{i,j+\mathrm{1}-q}\right).\end{array}$
Thus, as with TWSC-DSIi,j in Eq. (8), we can define two new multi-month indicators (TWSC-DSIA and TWSC-DSID) through standardization by using accumulated (A) and differenced (D) TWSC (Eqs. 9 and 11) as
$\begin{array}{}\text{(12)}& \mathrm{TWSC}\text{-}{\mathrm{DSIA}}_{i,j,q}=\frac{{x}_{i,j,q}^{+}-{\stackrel{\mathrm{̃}}{x}}_{j,q}^{+}}{{\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{j,q}^{+}}\end{array}$
and
$\begin{array}{}\text{(13)}& \mathrm{TWSA}\text{-}{\mathrm{DSID}}_{i,j,q}=\frac{{x}_{i,j,q}^{-}-{\stackrel{\mathrm{̃}}{x}}_{j,q}^{-}}{{\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{j,q}^{-}}.\end{array}$
Finally, it is obvious that sampling the full climatological range of dry and wet months is not yet possible with the limited GRACE data period. Therefore, suggest applying a bias correction to avoid the under- or overestimation of drought events. This implies using TWSC from multi-decadal model runs, which is feasible but is not the focus of this study.
2.2 Houborg method
define the drought indicator GRACE-DI via the percentile of a given month, ti,j, with respect to the cumulative distribution function (CDF). The GRACE-DI is applied to TWSC by
$\begin{array}{}\text{(14)}& \mathrm{TWSC}\text{-}{\mathrm{DI}}_{i,j}=\frac{\sum _{i}\left({x}_{j}\le {x}_{i,j}\right)}{\sum _{i}{x}_{j}}\cdot \mathrm{100},\end{array}$
i.e., all years containing month j are counted for which TWSC is equal or lower than TWSC in month j and year i, and these are normalized by the number of years that contain month j. The indicator value is assigned to five severity classes as shown in Table 2. For example, exceptional droughts occur up to 2 % of the entire time period at any location.
Table 2Drought severity level of TWSC-DI . The values of TWSC-DI are given in %.
Again, to relate drought severity to duration, we proceed via multi-month accumulation (Eq. 9) and differences (Eq. 11) resulting in the definition of two new indicators based on TWSC-DIi,j in Eq. (14),
$\begin{array}{}\text{(15)}& & \mathrm{TWSC}\text{-}{\mathrm{DIA}}_{i,j,q}=\frac{\sum _{i}\left({x}_{j,q}^{+}\le {x}_{i,j,q}^{+}\right)}{\sum _{i}{x}_{j,q}^{+}}\cdot \mathrm{100},\text{(16)}& & \mathrm{TWSC}\text{-}{\mathrm{DID}}_{i,j,q}=\frac{\sum _{i}\left({x}_{j,q}^{-}\le {x}_{i,j,q}^{-}\right)}{\sum _{i}{x}_{j,q}^{-}}\cdot \mathrm{100}.\end{array}$
Assuming again that the CDF equals the cumulative Gaussian PDF, 0.6 % of the months would be detected as exceptionally dry and 9.5 % of the months as abnormally dry. applied the percentile approach separately to surface soil moisture, root zone soil moisture, and groundwater storage, which were derived by assimilating GRACE-derived TWSC into a hydrological model, and the CDFs were adjusted to a long-term model run. Here, we focus on a simulated TWSC environment for the GRACE period only, and, as explained in Sect. 2.1, we therefore disregard the bias correction.
2.3 Thomas method
define a drought by considering the number of consecutive months below a threshold of TWSC. Given TWSC observations xi,j and a threshold c, we can compute anomalies by
$\begin{array}{}\text{(17)}& \mathrm{\Delta }{x}_{i,j}=\left\{\begin{array}{ll}\mathrm{0}& \mathrm{for}\phantom{\rule{0.25em}{0ex}}{x}_{i,j}\ge c\\ {x}_{i,j}-{x}_{j}& \mathrm{for}\phantom{\rule{0.25em}{0ex}}{x}_{i,j}
While the threshold can be derived from different concepts, use the monthly climatology xj (Eq. 6). Here, we also consider using a fitted signal for defining the threshold. The signal is computed by
$\begin{array}{}\text{(18)}& \begin{array}{rl}x\left(t\right)& ={a}_{\mathrm{0}}+{a}_{\mathrm{1}}\left(t-{t}_{\mathrm{0}}\right)+{a}_{\mathrm{2}}\frac{\mathrm{1}}{\mathrm{2}}{\left(t-{t}_{\mathrm{0}}\right)}^{\mathrm{2}}+{b}_{\mathrm{1}}\mathrm{cos}\left(\mathit{\omega }t\right)\\ & +{b}_{\mathrm{2}}\mathrm{sin}\left(\mathit{\omega }t\right)+{c}_{\mathrm{1}}\mathrm{cos}\left(\mathrm{2}\mathit{\omega }t\right)+{c}_{\mathrm{2}}\mathrm{sin}\left(\mathrm{2}\mathit{\omega }t\right),\end{array}\end{array}$
with time t with a constant a0, a linear trend term a1, a constant acceleration term a2, annual signal terms b1 and b2, and similarly semi-annual signal terms c1 and c2. Trends and possible accelerations in GRACE-TWSC can result from many different hydrological processes. For example, accelerations can result from trends in the flux precipitation, evapotranspiration, and runoff (e.g., Eicker et al.2016). In the following, the linear trends are denoted as trends, and constant accelerations are denoted as accelerations. The Thomas method then identifies drought events through the computation of their magnitude, duration, and severity: the magnitude or water storage deficit is equal to Δxi,j (Eq. 17), and the duration di,j is given by the number of consecutive months where TWSC is below the threshold. propose a minimum number of 3 consecutive months required for the computation of drought duration. By using the deficit Δxi,j and the duration di,j, the severity si,j of the drought event can finally be computed by
$\begin{array}{}\text{(19)}& {s}_{i,j}=\mathrm{\Delta }{x}_{i,j}{d}_{i,j}.\end{array}$
Severity is therefore a measure of the combined impact of duration and magnitude of water storage deficit .
3 Framework to derive synthetic TWSC for computing drought indicators
3.1 Methods
In order to analyze the performance of drought indicators, we first construct a synthetic time series of “true” total water storage changes on a grid. We base our drought simulations on the GRACE data model
$\begin{array}{}\text{(20)}& \mathrm{\Delta }s\left(t\right)=x\left(t\right)+\mathit{\eta }\left(t\right)+\mathit{ϵ}\left(t\right),\end{array}$
including the introduced (in Sect. 2.3) signal x (which contains seasonality and a constant, linear, and time-varying trend; Eq. 18), an interannual signal η (which has been detrended and deseasoned and which will carry the simulated true drought signature), and a GRACE-specific noise term ϵ. To simulate the true signal as realistically as possible using Eq. (20), we first analyze real GRACE-TWSC data following the steps summarized in Fig. 2. We derive (1) the signal components, constant, trend, acceleration, annual, and semi-annual sine wave, (2) temporal correlations, (3) a representative drought signal quantified by strength and duration, and (4) spatially correlated noise from GRACE error covariance matrices. While the first three steps are generic and can be used for simulating other observables, step 4 is directly related to the measurement noise (in this case the GRACE noise).
Figure 2Concept of the synthetic framework to generate synthetic TWSC.
As an input to the simulation, GRACE-TWSC data are derived by mapping monthly ITSG-GRACE2016 gravity field solutions of degree and order 60, provided by the Graz University of Technology , to TWSC grids. As per standard practice, we add degree-1 spherical harmonic coefficients from and degree 2, order 0 coefficients from laser ranging solutions . Then, we remove the temporal mean field, apply DDK3 filtering to suppress excessive noise, and map coefficients to TWSC via spherical harmonic synthesis. We also remove the effect of ongoing glacial isostatic adjustment (GIA) following .
Droughts are a multiscale phenomenon, and for a realistic simulation we must first define the largest spatial scale to which we will apply the model of Eq. (20). In other words, we first need to identify coherent regions in the input data for which our approach is then applied at the grid scale prior to step 1. For this, we apply two consecutive steps: we first compute temporal signal correlations by fitting an autoregressive (AR) model (Appendix A; ) to detrended and deseasoned GRACE data. These TWSC residuals contain interannual and subseasonal signals including real drought information. Next, temporal correlation coefficients are used as an input for expectation maximization (EM) clustering because regions with similar residual TWSC correlation within the interannual and subseasonal signal are hypothesized here to be more likely affected by the same hydrological processes. The EM algorithm by Chen (2018) is modified to identify regional clusters. The EM algorithm alternates an expectation and a maximization step to maximize the likelihood of the data . More details about EM clustering are provided in Appendix B.
As a result of this procedure, we identified three clusters located in eastern Brazil (EB), southern Africa (SA), and western India (WI), which were indeed affected by droughts in the past . The location and shape of the three chosen clusters are shown in Fig. 3, and a global map of all clusters is provided in Fig. B1. Cluster delineations from the above procedure should not be confused with political boundaries or watersheds. The following simulation steps are then applied to each of these three clusters.
Figure 3AR(1) model coefficients (–) for global GRACE-TWSC. The polygons of the clusters of eastern Brazil, southern Africa, and western India are added in magenta.
In step 1 we estimate the signal coefficients according to Eq. (18) through least squares fit for each grid cell within the cluster. The coefficients are then spatially averaged to create a signal representative of the mean conditions within the region, and they are then used to create the constant, trends, and the seasonal parts of the synthetic time series. To simulate realistic temporal correlations at the regional scale (step 2), we use the AR model identified beforehand (Fig. 2) and again average AR model coefficients within the cluster. Then, we apply an AR model with the estimated optimal order and the averaged correlation coefficient (Eq. A1) to the synthetic time series to add temporal correlations.
Simulating realistic drought events in step 3 is challenging because, to our knowledge, no unique procedure to simulate realistic drought periods for TWSC exists. For this reason, we first perform a literature review to identify representative drought periods and magnitudes for selected regions. Among others, this includes the 2003 European drought and the drought in the Amazon basin in 2011 (e.g., Seitz et al.2008; Espinoza et al.2011, respectively). TWSC data within the identified drought period are then eliminated from the time series. In the next step, the parameters describing the constant, trend, acceleration, and seasonal signal components before and after the drought are used to “extrapolate” these signals during the drought period. By computing the difference of the original GRACE-TWSC time series and the continued signal in the drought period, we can separate non-seasonal variations from the data, which represent the drought magnitude. Our hypothesis is that the non-seasonal variations that we derive from the procedure possibly show a systematic behavior that can be parameterized. To extract this systematic behavior, all extracted droughts are transformed to a standard duration. To compare the different drought signals, a standard duration and a standard magnitude are arbitrarily set to 10 months and −100 mm, respectively. Finally, a synthetic drought signal η is generated by using the extracted knowledge of drought duration, drought magnitude, and systematic behavior, and it is added to the synthetically generated signal (Eq. 20).
In step 4 we add GRACE-specific spatially correlated and temporally varying noise ϵ (Eq. 20). First, for each month t we extract a full variance–covariance matrix Σ for the region grid cells from GRACE-TWSC. Then, whenever Σ is positive definite, we apply the Cholesky decomposition Σ=RTR, while if Σ is only positive semi-definite, we apply eigenvalue decomposition (Appendix C). Second, we generate a Gaussian noise series v of the length n, where n represents the number of grid cells within the cluster. Finally, spatial noise in month t is simulated through
$\begin{array}{}\text{(21)}& \mathbit{ϵ}={\mathbf{R}}^{T}\mathbit{v}.\end{array}$
The final synthetic signals for each grid cell within a cluster will thus exhibit the same constant, trend, acceleration, seasonal signal, temporal correlations, and drought signal, but it has spatially different and correlated noise. In the following, we will test the hypothesis that GRACE indicators depend on the presence of trend and random input signals using the generated synthetic time series.
We believe that our synthetic framework based on real GRACE data has multiple benefits: (i) we are able to identify the ability of an indicator by comparing the true drought duration and magnitude (step 3) to the indicator results; (ii) we are able to detect the influence of other typical GRACE signals on the drought detection; and (iii) the synthetic framework enables us to identify strengths and weaknesses of each analyzed indicator, and it thereby enables us to choose the most suitable indicator for a specific application.
3.2 Synthetic TWSC
Here, we will briefly discuss the TWSC simulation following methods described in the previous section.
When estimating AR models for detrended and deseasoned global GRACE data, we find that for more than 70 % of the global land TWSC grids are best represented by an AR(1) process (Fig. A1). Therefore, we apply the AR(1) model for each grid. Figure 3 shows the estimated AR model coefficients, which represent the temporal correlations, ranging from very low up to 0.3, e.g., over the Sahara or in southwestern Australia, up to about 0.8, e.g., in Brazil or in the southeastern US. EM clustering is then based on these coefficients.
The selected three clusters (Fig. 3) show differences between the signal coefficients of the functional model (step 1; Eq. 18), which are hence discussed for the linear trend. We find a mean linear trend for the eastern Brazil cluster of 1.0 mm TWSC per year, a higher trend of 5.0 mm per year in southern Africa, and for western India a trend of 56.3 mm per year (Table 3). The trends for eastern Brazil and southern Africa in GRACE-TWSC have been identified before (e.g., Humphrey et al.2016; Rodell et al.2018). We did not find confirmations for the strong linear trend in western India found, for example, by , who identified about 7 mm per year within this region. We assume that in this study the linear trend for western India is estimated as strong positive because we additionally identify a strong negative acceleration of −8.03 mm yr−2 in western India. However, our simulation will cover weak and strong trends. In fact, all coefficients show strong differences, which suggests that we cover different hydrological conditions when simulating TWSC for the three regions. In step 2 we identify correlations of 0.74 in eastern Brazil, 0.79 in western India, and 0.42 in southern Africa (Table 3).
Table 3Coefficients (a0 to c2 from Eq. 18 and ϕ1 from Eq. A1) for signals contained in GRACE-TWSC that were extracted within the clusters of eastern Brazil, southern Africa, and western India. These coefficients are used to simulate synthetic TWSC.
Figure 4Extracted drought periods from GRACE-TWSC for the droughts in (a) Europe in 2003, (b) the Amazon river basin in 2005, (c) the Amazon river basin in 2010, and (d) Texas in 2011. (e) All droughts from (a)(d) were transformed to standard severity and duration. Panel (f) is the same as (e) but after removing four time series with significant temporally different behavior.
Performing literature research for drought duration and magnitude (step 3) led to four droughts seen in GRACE-TWSC (Table 4): the 2005 and 2010 droughts in the Amazon (e.g., Chen et al.2009; Espinoza et al.2011), the 2011 drought in Texas (e.g., Long et al.2013), and the 2003 drought in Europe (e.g., Seitz et al.2008). To extract the drought duration, we compared drought onset and end identified in these and other papers. We found that different studies do not exactly match, with inconsistencies likely due to different methodologies used. Furthermore, some authors only specified the year of drought. Droughts extracted from the literature had a duration of 3 to 10 months (Fig. 4a–d). Unless otherwise specified, we decided to base our simulations on a duration of 9 months to represent a clear identifiable drought duration. Extracted drought magnitudes range from about −20 to −350 mm TWSC (Fig. 4a–d). Therefore, in order to simulate a drought magnitude that has a clear influence on the synthetic time series, we set the magnitude to −100 mm.
Table 4Drought events in Europe, the Amazon river basin, and Texas with corresponding duration taken from the literature.
As described in Sect. 3.1, we transform these water storage droughts to a standard duration and magnitude to understand whether a typical signature can be seen. However, Fig. 4e remains inconclusive as there are, in particular, four standardized droughts, which show a very different temporal behavior: Toulouse in 2003, Óbidos in 2010, and Houston and Dallas in 2011. When we remove those four time series (Fig. 4f), a systematic behavior can be identified and parameterized using a linear or quadratic temporal model. However, due to these difficulties, we decided to use the most simple TWSC drought model, i.e., a constant water storage deficit within a given time span.
Figure 5Synthetic TWSC (mm) without (light blue) and with spatial GRACE noise (dark blue) using average parameters for the clusters in eastern Brazil (EB), southern Africa (SA), and western India (WI). Light brown shows the simulated drought period.
In step 4, we project the simulation on a 0.5 grid and add spatially correlated GRACE noise. A few representative time series of the gridded synthetic total water storage change are shown in Fig. 5 for eastern Brazil, southern Africa, and western India for the GRACE time period from January 2003 to December 2016. The effect of realistic GRACE noise (dark blue vs. light blue) is clearly visible, particularly for the SA case with low annual amplitude. The synthetic drought period is placed from January to September 2005 (light brown) in all three regions. Synthetic TWSC variability includes considerable (semi-)annual variations for EB based on Table 3. Furthermore, a strong negative acceleration is contained in the synthesized time series for eastern Brazil (Table 3) leading to strong negative TWSC towards the end of the time series. For western India a strong positive trend leads to low TWSC at the begin of the time series.
Figure 6A representative example of the synthetic DSI, DSIA, and DSID (–) for the eastern Brazil (EB), southern Africa (SA), and western India (WI) cluster over the periods of 3, 6, 12, and 24 months. Light brown shows the synthetic constructed drought period.
4 Indicator-based drought identification with synthetic and real GRACE data
4.1 Synthetic TWSC: masking effect of trend and seasonality
Here, we analyze how non-drought signals, such as a linear or accelerated water storage trend and the ubiquitous seasonal signal, propagate through the Zhao, Houborg, and Thomas GRACE indicators (Sect. 2) and potentially mask a drought. To this end, we select representative time series from each of the three synthetic grids of total water storage changes for eastern Brazil, southern Africa, and western India and apply the three methods. Since all results are based on TWSC, we refer to TWSC-DSIA, TWSC-DSID, TWSC-DIA, and TWSC-DID as DSIA, DSID, DIA, and DID, respectively (again, with accumulated (A) and differenced (D) variants).
We first assess the temporal characteristics of the Zhao method (Sect. 2.1). Figure 6 (left) shows time series for the DSI and DSIA (with 3, 6, 12, or 24 months of accumulated TWSC). It is obvious that trend and acceleration propagate into both DSI and DSIA (see eastern Brazil and western India). Resulting indicator values (e.g., for the years 2015 and 2016) are lower than those compared to a small trend (southern Africa) and this may lead to misinterpretations because a severe-to-mild drought is identified (−2 to −0.5), while none is actually simulated. In contrast, the actual simulated drought in 2005 is only identified as a moderate drought (values up to −1.0) for EB.
In the presence of a small trend (5.0 mm yr−1) and acceleration (−0.38 mm yr−2; Table 3, SA), we do identify an exceptional drought (Fig. 6 DSIA for southern Africa). This shows that the drought strength that we chose does indeed lead to a correct identification of exceptional drought if no masking occurs (but in the presence of GRACE noise), so at this point we can determine that exceptional drought represents the true drought severity class. As expected, a trend and/or an acceleration signal that are frequently observed in GRACE analyses can lead to misinterpretations in the indicators. However, the influence of the trend or acceleration also depends on the timing of the drought period within the analysis window. For example, assuming we simulate the time series with the same trend or acceleration but the drought were to occur in 2014, the drought detection would not have been influenced as much. Therefore, we decided to set up an additional experiment and discuss the influence of different trend strengths for the drought detection (Sect. 4.3).
Figure 7A representative example of the synthetic DI, DIA, and DID (%) for the eastern Brazil (EB), southern Africa (SA), and western India (WI) cluster over the periods of 3, 6, 12, and 24 months. Light brown shows the synthetic constructed drought period.
The analysis reveals that DSI and DSIA indicators are sensitive with respect to trends, while they are less sensitive to the annual and semi-annual signal. The seasonal signal is clearly dampened (e.g., compare Fig. 5 to the DSIA in Fig. 6). This is caused by removing the climatology within the Zhao method (Eq. 8). Comparing DSIA3, DSIA6, DSIA12, and DSIA24, e.g., for eastern Brazil, suggests that with a longer accumulation period, indicator time series are increasingly smoothed, and less severe droughts are identified (Fig. 6, left). Furthermore, the drought period appears shifted in time, and its duration is prolonged. This can lead to missing a drought identification if a trend or an acceleration is contained in the analyzed time series, for example for the 24-month DSIA for eastern Brazil. We find that all DSIA data are able to unambiguously detect a drought close to 2005, assuming that neither trend nor acceleration is apparent (Fig. 6 DSIA for southern Africa). Particularly, the 3- and 6-month DSIA data identify the drought close to 2005 for southern Africa, and its computation appears to dampen the temporal noise that is present in the DSI.
In contrast we find that the 3-, 6-, 12-, and 24-month TWSC-differencing DSID data exhibit stronger temporal noise as compared to the DSIA and the DSI. This can be seen in the light of Eq. (2) – these indicators are closer to meteorological indicators and thus do not inherit the integrating property of TWSC. The DSID does not propagate a trend and acceleration, annual signal, or semi-annual signal. All DSID time series, for example for eastern Brazil (Fig. 6, right), show a strong negative peak within the drought period, but this peak does not cover the entire drought period for the 3-, and 6-month differenced DSID. The negative peak within the drought period is always followed by a strong positive peak; when we consider Eq. (2), this lends to the interpretation that a pronounced drought period is normally followed by a very wet event to return to “normal” water storage condition. Despite higher noise and the positive peak and contrary to the DSIA, all DSID data (DSID3, DSID6, DSID12, and DSID24) correctly identify the drought within 2005 to be exceptionally dry for eastern Brazil and southern Africa. All different DSID time series for WI identify at least a moderate drought.
Analysis of the Houborg method shows a broadly similar behavior as compared to the Zhao method: the sensitivity of drought detection to an included trend or acceleration depends on the indicator type. Using the DIA we can confirm the large influence of the trend or acceleration on the indicator value, which is not the case for DID (e.g., Fig. 7 DIA and DID for eastern Brazil). Annual and semi-annual water storage signals are all considerably weakened in the Houborg method because they are effectively removed when computing the empirical distribution for each month of the year. Differences to the Zhao method appear when comparing more general properties; e.g., we find that DI is more noisy and the range of output values is restricted to about 7 % to 100 % (Fig. 7). This restriction is caused by the length of the time series; e.g., assuming we strive to identify an event with exceptional dry values (≤2 %), we would need at least 50 years of monthly observations. Yet, with GRACE we only have about 14 years of good monthly observations, so the simulation was also restricted to this period. If we then take the driest value that might occur only once, we can compute the minimum value of DI to be 7.14 %. Hence the detection of a period of exceptional or extreme drought is not possible when referring to the duration of the GRACE-TWSC time series. As mentioned in Sect. 2.2, applied a bias correction to the empirical CDF to mitigate this restriction. We do not follow Houborg's approach here in order to focus on the synthetic environment instead of the availability of model outputs.
Figure 8Drought magnitude (mm), duration (months) and severity (mm month−1) for the cluster in eastern Brazil (EB) using TWSC with the removed climatology (dark blue) and TWSC with removed trend and seasonal signal (red). The minimum duration (MD) is set to 3 months (blue and red) or 6 months (green). Light brown shows the synthetic constructed drought period.
The Thomas method is applied to simulated TWSC data to derive the magnitude, duration, and severity of a drought, which we show in Fig. 8 for the EB region. We find that the linear trend and acceleration propagate into the magnitude (Fig. 8, top) when using TWSC deficits with climatology removed (blue, Eq. 6) compared to using TWSC deficits with removed trends, accelerations, and seasonality (red, Eq. 18). When using non-climatological TWSC (blue), we identify a strong deficit in 2015 and 2016 (Fig. 8, top), which suggests a duration of up to 38 months (Fig. 8, center) and a severity of about −4000 mm months (Fig. 8, bottom). Using the detrended and deseasoned TWSC (red), drought is mainly detected in the true drought period (2005) and not at the end of the time series. Thus we conclude that a trend or acceleration indeed modifies the drought detection.
Results so far were derived by imposing a minimum duration of 3 months (blue and red). When moving to a minimum duration of 6 consecutive months (green, Fig. 8, middle and bottom) we find this would lead to a decrease in identified severity by half, and the beginning of the drought period shifts 3 months in time. This is in line with . The same findings are made for southern Africa and western India.
4.2 Synthetic TWSC: effect of spatially correlated GRACE errors
Here, we investigate how robust the Zhao, Houborg, and Thomas indicators are with respect to the spatially correlated and time-variable GRACE errors. However, any analysis must take into account that GRACE results cannot be evaluated directly at grid resolution.
Figure 9DSI and DI average in southern Africa (SA; a, b), severity average for the Thomas method (SA; c), and DI average in eastern Brazil (EB, d) by applying two different methods: the average of the indicators for all grids (light blue) and the indicators of averaged TWSC (dark blue). The grey shaded area represents the bandwidth for all grids. Light brown shows the synthetic constructed drought period.
In our first analysis, indicators based on (synthetic) TWSC grids are thus spatially averaged through two different methods (Sect. 3.1). We find that regional-scale DSI and DI indicators, as well as the outputs derived by the Thomas method for southern Africa computed from averaging TWSC first (dark blue Fig. 9), are indeed different to the averaging indicators computed at grid scale from TWSC (light blue, Fig. 9). These differences can be explained by the inherent non-linearity of the indicators. Since the synthetic data have been constructed from the same constants, trends, seasonal signal, temporal correlations, and drought signal, we isolate the effect of GRACE noise on regional-scale indicators here. Outside of the drought period we conclude that the sequence in which we spatially average causes larger differences for DI as compared to DSI. For southern Africa, the range of averaged DI is about 7 %–100 %, while the range of the DI of averaged TWSC is about 7 %–80 %. Within the drought period the DI exhibits little difference between both averaging methods. The DSI from averaged TWSC does suggest a weaker severity in the drought period compared to averaged DSI. In this case, both indicator averages identify the same (exceptional) drought severity class. Yet we find that for both DSI and DI the identification of drought severity is not sensitive to the choice of the averaging method for this cluster. However, for other cases these differences can be more significant. These may lead to misinterpretation (e.g., May and July 2005 for the DI for eastern Brazil, Fig. 9). For the Thomas method, we cannot distinguish which result is more significant, since we have no comparable true severity amount for that indicator.
To determine the influence of the GRACE-specific spatial noise on the detected drought severity, a second analysis is applied. This analysis computes the share of the area for each time step for which a given drought severity class is identified (Fig. 10). Since different grid cells for one time step only differ in their spatial noise, it is important to understand that identifying more than one severity class is directly related to the noise. Only one class of drought would be detected for one epoch, assuming the grid cells have no or exactly the same noise. For example, we identify all classes of droughts (abnormal to exceptional) in December 2015 by using DSI for the eastern Brazil cluster (Fig. 10, top left). Thus, the spatial noise has a large influence on the drought detection. To establish which indicator is most affected, the indicators are compared with each other.
Figure 10Drought-affected area of the DSI, DSIA, and DSID (%) considering the different drought severity classes within the clusters of eastern Brazil (EB) and southern Africa (SA).
We note that large differences are found between DSI, the 6-month accumulated DSIA, and the 6-month differenced DSID within the given drought period for the eastern Brazil region (Fig. 10, left). All three indicators manage to identify the drought, but they also do so with a different duration and percentage of the affected area. Within the simulated drought period, the DSI indicator identified no more than 14 % of all grid cells as being affected by exceptional drought where it should be 100 %. On the other hand, the DSIA does not detect exceptional drought in any grid cell. It is apparent that this indicator misses the exceptional dry event because of the included trend and acceleration.
When comparing the DSIA of eastern Brazil to the DSIA of southern Africa (Fig. 10, center), we find that DSIA is able to detect the drought strength correctly when there is a small trend or acceleration present. However, DSIA appears more robust against spatial noise, since it identifies severe drought or drier in more than 90 % of grid cells, while the DSI indicator identifies only about 60 %. As described in Sect. 4.1, longer accumulation periods lead to smoother and thus more robust indicators. We find that the DSID is more successful in detecting exceptional drought: more than 80 % of the DSID grid cells show exceptional drought, but the indicator appears more noisy than DSIA. Finally, with regard to the drought duration, we find that only DSI detects the true period correctly. When identified via DSIA, the duration appears longer, and when identified in DSID, the period was found shorter as compared to the true drought period.
Overall, we find that the different indicators of DSI, DSIA, and DSID all come with advantages and disadvantages regarding the presence of spatial and temporal noise. The same findings were made for the indicators of the Houborg method (results not shown). This analysis is not applied to the Thomas method, because the method does not refer to severity classes (Sect. 2.3).
4.3 Synthetic TWSC: experiments with variable trend, drought duration, and severity
Two experiments were additionally constructed to examine the influence of trends and drought parameters on the indicator capability. First, we consider how strong a linear trend in total water storage must be to mask drought in the indicators. For this, we test different trends from −10 to 10 mm yr−1 for DSI, DSIA, DI, DIA, and the Thomas method in the western India region (since these indicators were identified as being affected by trends; Sect. 4.1). No acceleration is included for these tests. We find that trends between −1 and 1 mm yr−1 cause no influence on all indicators, while differences start to appear when simulating a trend higher than 2 mm yr−1. This propagates into the DSI, DSIA, DI, and DIA indicators but did not affect the drought period.
A question we must ask is what would be the largest trend magnitude that does not affect the correct detection of drought duration and drought severity, and how can we verify this. An obvious influence within the drought period in 2005 is found when simulating a trend of −7 mm or lower per year. It is important at this point to understand that there is a relation between the timing of the drought and the sign of the trend, i.e., whether the trend is positive or negative. Assuming that a positive trend exists and the drought occurs closer to the end of the time series, the trend may lead to a drought that is identified as more dry than the true drought. But if the trend is negative, the drought is identified more easily.
Other factors, e.g., the length of the time series, have an influence on the masking by the trend and, as a result, affect drought detection. The longer the input time series, the more sensitive the drought detection is to the trend. At the same time, the magnitude of the trend needs to be considered relative to the variability or range of TWSC. For example, a −6 mm yr−1 trend has a larger influence on the drought detection if the range of TWSC is −50 to 50 mm compared to −200 to 200 mm. As a reference, the synthetic time series for western India, without any trend or acceleration signal, ranges from about −323 to 87 mm. So, deriving a general quantity for these dependencies is difficult.
Figure 11Percentage of drought-affected area for the 6-month DSIA (–) considering the different drought severity classes. Application on real GRACE-TWSC over South Africa from 2003 to 2016.
In a second experiment, we assess which input drought duration and magnitude would at least be visually recognized in the indicators. We choose 3, 6, 9, 12, and 24 months for the simulated duration and −40, −60, −80, −100, and −120 mm for the drought magnitude and apply both the Zhao and the Houborg methods. We compare the changes for one indicator time series for the eastern Brazil region. The drought always begins in January 2005 for the first tests. In general, we found that the identification of the severity class is less sensitive to changes in the drought duration, since a drought duration of 3, 6, 9, 12, and 24 months mostly results in equal drought severity classes, for example, a drought magnitude of 120 mm. Thus, we concentrate our analysis on changes in drought magnitude.
Exceptional drought is only classified by the Zhao method for eastern Brazil for a simulated drought magnitude of 120 mm; this is related to the trend and acceleration signal contained in the simulated TWSC and was already found in Sect. 4.1. For the Zhao method, extreme drought is identified when simulating a drought magnitude of at least −100 mm, while only a period of severe and moderate drought is identified when simulating a magnitude of −80 and −60 mm. The Houborg method fails to identify extreme and exceptional drought, as described in Sect. 4.1. Thus, simulating a magnitude of −100 and −120 mm is identified as severe drought for all simulated drought periods (3 to 24 months), while simulating a lower magnitude (−80 and −60 mm) causes moderate or abnormal dry events to be identified. We find that both methods are not able to clearly detect a drought that has a magnitude of −40 mm or weaker if the duration is between 3 and 24 months. This experiment supports our findings in Sect. 3.2.
4.4 Application to real GRACE data: droughts in South Africa
For South Africa, droughts are a recurrent climatic phenomenon. The complex rainfall regime has led to multiple occurrences of drought events in the past, for example to a strong drought in 1983 . These past droughts appeared in varying climate regions, at different times of the year, and with a different severity. Since 1960, many of them were linked to El Niño .
Based on the simulation results, we chose the 6-month accumulated DSIA to identify droughts for (the administrative area of) South Africa (GADM2018) in the GRACE total water storage data. DSIA has proven to be more robust with respect to the peculiar, GRACE-typical spatial and temporal noise as compared to the other tested indicators (Sect. 4.1 and 4.2).
GRACE-DSIA6 suggests two drought periods, from mid-2003 to mid-2006 and from 2015 to 2016 (Fig. 11). The first drought event is identified to affect at least 70 % of the area of South Africa. While 2003 was indeed a year of abnormal-to-severe dry conditions, extreme drought occurred during the period of 2004 to mid-2006. Figure 11 reveals that a small area (about 7976 km2, close to Lesotho) even experienced exceptional drought during 2004. This period is confirmed by the Emergency Events Database (EM-DAT2018) recording of a drought event in 2004 (e.g., Masih et al.2014). Extreme drought in 2004 mainly occurred in central and southeastern South Africa; this is exemplified in Fig. 12a for April 2004. Another confirmation is found in , who identified a drought period from 2003 to 2007 by using the SPI.
Figure 12DSIA6 (–) for real GRACE-TWSC data within South Africa (black line; ) for (a) April 2004 and (b) March 2016.
Despite affecting less area (about 50 % to 70 %; Fig. 11), the second drought in 2015 and 2016 is perceived as more intense than the drought from 2003 to 2006. Based on the GRACE-DSIA6 data, we conclude that in 2016 at least 30 % of South Africa was affected by extreme drought and about 20 % experienced an exceptional drought. The 2016 drought occurred in the northeastern part of South Africa (Fig. 12b). For comparison, the EM-DAT database similarly identified 2015 as a drought event, but it did not classify 2016 as such. We speculate that the differences are due to the drought criteria of EM-DAT (disasters are included when, for example, 10 or more people died or 100 or more people were affected). However, EM-DAT lists 2016 as a year of extreme temperature, which might be related to our detected drought. Furthermore, we can confirm the 2015–2016 drought is marked by a lower maximum precipitation in these years than in other years (about 65 mm) and by meteorological indicators indicating a period of severe-to-extreme drought (SPI; Standardized Precipitation Evapotranspiration Index; ; Weighted Anomaly Standardized Index; ).
5 Discussion
The framework developed in this study enables us to simulate GRACE-TWSC data with realistic signal and noise properties and thus to assess the ability of GRACE drought indicators to detect drought events in a controlled environment with known truth. This will be extended to GRACE-FO in the near future. GRACE studies have often been based on simplified noise models (e.g., Zaitchik et al.2008; Girotto et al.2016), where the GRACE noise model is not derived from the used GRACE data but, for example, from literature and assumed to be spatially uniform and uncorrelated. However, it is important to account for realistic error and signal correlation (e.g., Eicker et al.2014), in particular for drought studies where one will push the limits of GRACE spatial resolution. This signal correlation includes information about, for example, the geographic latitude, the density of the satellite orbits, the time dependencies of mission periods or north–south dependencies.
However, identifying a drought signal from real GRACE-TWSC data is indeed challenging since we do not know in advance what the signature of a drought looks like; a parametric drought model does not yet exist, and our experiment (Sect. 3.2) to extract such a model from TWSC data and known droughts did not lead to conclusive results. Still we believe that this first – to our knowledge – approach, despite being based on a small number of drought periods, identified a similar systematic behavior of different drought periods and should be pursued further. Based on literature and our own experiments (Sect. 4.3) we chose to define our “box”-like GRACE drought model as an immediate and constant water storage deficit.
When analyzing the Zhao, Houborg, and Thomas methods, we find that trends and accelerations in GRACE water storage maps tend to bias not only DSI, DI, and the Thomas indicator (which use non-climatological TWSC) but also DSIA and DIA (which use accumulated TWSC). The indicators DSID and DID, which utilize time-differenced TWSC, were not found to be biased by trends and accelerations; the same goes for the Thomas method when based on detrended and deseasoned TWSC. When we simulated smaller trends or accelerations, all indicators were able to detect drought, but they identified a different timing, duration, and strength, for example for the SA cluster (trend of 4.98 mm yr−1, acceleration of −0.38 mm yr−2). This suggests removing the trend in GRACE data first, but this must be done with care, since it can also influence the detection of, for example, long-term droughts. The same is true for removing the trend and seasonal signal prior of applying the Thomas method, although in this study we found that the removal of these signals simplified the correct drought detection (Sect. 4.1).
An experiment was then set up to understand the influence of the trend on the detected drought duration and severity. Several factors play a role here, e.g., the length of the time series, the TWSC range in relation to the trend magnitude, and the sign of the trend. We found that providing a general rule appears nearly impossible.
As expected, we find time series for the modified time-differencing GRACE indicators DSID and DID as much noisier when compared to the time-accumulating indicators DSIA and DIA; this can be linked to precipitation (Sect. 2) driving total water storage. The drought period was identified to be shorter than the true simulated drought period, e.g., for DSID3 and DSID6. After these drought periods, strongly wet periods were detected. Regarding future applications, we suggest a direct comparison of the DSID and meteorological indicators, in particular for confirming or rejecting drought duration and the following wet periods.
On the other hand, computing accumulated indicators implies a temporal smoothing causing the drought period to appear lagged in time; however for accumulation periods of 3 and 6 months the lag was found to be insignificant. DSIA and DIA are thus more robust against temporal and spatial GRACE noise as compared to DSID and DID, and again we would suggest utilizing 3 or 6 months accumulation periods. In general, we found the Zhao and Thomas indicators performed better in detecting the correct drought strength than the Houborg method, at least for the limited duration of the GRACE time series that we have at the time of writing.
By simulating the effect of spatial noise on drought detection, we found that some indicators appear less robust. Analysis of the percentage of the drought-affected area showed that the GRACE spatial noise limits correct drought detection. Again, DSIA was identified to be more robust compared to DSI and DSID – it was the only indicator that identified exceptional drought in nearly all grid cells. A second experiment was conducted to examine if the influence of the spatial noise can be reduced by using spatial averages. We found that spatially averaging DSI and DI appears less robust against the spatial noise compared to computing the indicator of the averaged TWSC. At this point we therefore suggest to compute the indicator from the spatially averaged TWSC. Since DI showed stronger difference between both averaging methods than DSI, we conclude that DI is generally less robust against spatial noise than DSI. In our real-data case study, due to these findings, the DSIA6 was thus applied to GRACE-TWSC, and it identified two drought periods: mid-2003 to mid-2006 in central and southeastern South Africa and 2015 to 2016 in northeastern South Africa.
6 Conclusions and outlook
A framework has been developed that enables a better understanding of the masking of drought signals when applying the methods of , , and . Four new GRACE-based indicators (DSIA, DSID, DIA, and DID) were derived and tested; these are modifications of the above mentioned approaches based on time-accumulated and time-differenced GRACE data. We found that indeed most indicators were mainly sensitive to water storage trends and to the GRACE-typical spatial noise.
Among these various indicators, we identified the DSIA6 as particularly well-performing; i.e., it is less sensitive to GRACE noise and is well capable of identifying the correct severity of drought, at least in absence of trends. However, the choice of the indicator should always be made in the context of the application.
We see ample possibilities to extend our framework. Future work should focus on better defining the onset and end of a drought and developing a signature for a TWSC drought. One should also consider other observable measurements in the simulation, such as groundwater for example, which can be derived from GRACE and by removing other storage contributions from direct modeling or through data assimilation.
In the GRACE community, efforts are currently being made to “bridge” the GRACE time series to the beginning of the GRACE-FO data period (e.g., Jäggi et al.2016; Lück et al.2018). These gap-filling data will inevitably have much higher noise and spatial correlations that may be very different from GRACE data, and drought detection capability should be investigated through simulation first. On the other hand, GRACE-FO is supposed to provide more precise measurements, and thus less influence of spatial noise on the drought detection may be expected. The combination of GRACE-FO data and a thorough understanding and “tuning” of GRACE drought identification methods, possibly through this framework, might then enable us to identify water storage droughts more precisely.
Appendix A: AR model coefficients computations
To extract temporal correlations from the GRACE total water storage changes we apply an autoregressive model, which is described by
$\begin{array}{}\text{(A1)}& X\left(t\right)={\mathit{\varphi }}_{\mathrm{1}}X\left(t-\mathrm{1}\right)+\mathrm{\dots }+{\mathit{\varphi }}_{p}X\left(t-p\right)+{\mathit{ϵ}}_{t},\end{array}$
where X represents the observed process at time t, p is the model order, ϕ is the correlation parameters, and ϵ is a white-noise process (Akaike1969). Here, detrended and deseasoned TWSC are used as the observed process X(t) because the remaining residuals contain interannual and subseasonal signal data as the drought information, which we want to extract with this approach. The approach is then applied for different model orders. The optimal order of the AR model is adjusted by means of the information criteria, for example the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Then, by using the optimal order, the AR model coefficients ϕ, which represent the temporal correlations, can be computed using a least squares adjustment.
The results for the optimal order of interannual and subseasonal TWSC is shown in Fig. A1. Most of the global land grids of detrended and deseasoned TWSC show an optimal order of 1 (about 70 %).
Figure A1Histogram of the optimal order of an AR model for global detrended and deseasoned GRACE-TWSC on land grids.
Appendix B: EM clustering
Expectation maximization represents a popular iterative algorithm that is widely used for clustering data. EM partitions data into clusters of different sizes and aims at finding the maximum likelihood of parameters of a predefined probability distribution . In the case of a Gaussian distribution the EM algorithm maximizes the Gaussian mixture parameters, which are the Gaussian mean μk, covariance Σk, and mixing coefficients πk (Szeliski2010). The algorithm then iteratively applies two consecutive steps to maximize the parameters: the expectation step (E step) and the maximization step (M step). Within the E step we estimate the likelihood that a data point xt is generated from the kth Gaussian mixture by
$\begin{array}{}\text{(B1)}& {z}_{ik}=\frac{\mathrm{1}}{{Z}_{i}}{\mathit{\pi }}_{k}\mathcal{N}\left(x|{\mathit{\mu }}_{k},{\mathrm{\Sigma }}_{k}\right).\end{array}$
The M step then re-estimates the parameters for each Gaussian mixture as follows:
$\begin{array}{}\text{(B2)}& {\mathit{\mu }}_{k}=\frac{\mathrm{1}}{{N}_{k}}\sum _{i}{z}_{ik}{x}_{i},\end{array}$
$\begin{array}{}\text{(B3)}& & {\mathrm{\Sigma }}_{k}=\frac{\mathrm{1}}{{N}_{k}}\sum _{i}{z}_{ik}\left({x}_{i}-{\mathit{\mu }}_{k}\right){\left({x}_{i}-{\mathit{\mu }}_{k}\right)}^{T},\text{(B4)}& & {\mathit{\pi }}_{k}=\frac{{N}_{k}}{N}\end{array}$
by using the number of points assigned to each cluster via
$\begin{array}{}\text{(B5)}& {N}_{k}=\sum _{i}{z}_{ik}.\end{array}$
Using the maximized parameters, EM assigns each data point to a cluster. The final global distributed clusters of the AR parameters (Fig. 3) are shown in Fig. B1. These clusters were derived by modifying and applying an EM algorithm provided by Chen (2018).
Figure B1Clusters based on EM clustering applied to the global AR model coefficients.
Appendix C: Eigenvalue decomposition
The decomposition of the variance–covariance matrix Σ by using Cholesky decomposition fails when Σ is positive semi-definite. To still be able to decompose the matrix, we can use eigenvalue decomposition, but this is accompanied by a loss of information due to the rank deficiency. The decomposition is then examined by Σ=UDUT, where U is a matrix with the eigenvectors of Σ in each column and D is a diagonal matrix of the eigenvalues. In this case, a decomposed matrix can be related to RT introduced in Sect. 3.1. RT can be computed by $\mathbf{U}\sqrt{\mathbf{D}}$. In Sect. 3.1, we multiply RT with a normal distributed noise time series of the same length as the rows of Σ. In this case, the number of normal distributed noise time series n is then replaced by the rank of Σ.
Data availability
Data availability.
GRACE-TWSCs are freely available at http://skylab.itg.uni-bonn.de/data_and_models/grace/hydrology/total_water_storage/ . Information about the postprocessing of the data can be found at https://www.apmg.uni-bonn.de/daten-und-modelle/grace-monthly-solutions .
Author contributions
Author contributions.
HG, OE, and JK designed all computations, and HG carried them out. HG prepared the paper with contributions from OE and JK.
Competing interests
Competing interests.
The authors declare that they have no conflict of interest.
Acknowledgements
Acknowledgements.
We acknowledge the funding from the German Federal Ministry of Education and Research (BMBF) for the “GlobeDrought” project through its funding measure Global Resource Water (GRoW).
Financial support
Financial support.
This research has been supported by the Bundesministerium für Bildung und Forschung (grant no. 02WGR1457A).
Review statement
Review statement.
This paper was edited by Bettina Schaefli and reviewed by two anonymous referees.
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http://math.stackexchange.com/questions/6532/centralizer-of-m-otimes-m | # Centralizer of $M \otimes M$
Let $k$ be a field, $V$ a finite-dimensional $k$-vectorspace and $M \in End(V)$. How can I determine $Z$, the centralizer of $M \otimes M$ in $End(V) \otimes End(V)$?
For example, if $$M=[[1,0],[0,2]],$$ then $M$ is 6-dimensional, consisting of block matrices of shape 1,2,1.
I was confused at first, because this seems to be a contradiction to the fact that the centralizer of a subalgebra of the form $A \otimes B$ is just the tensor product of the centralizers of $A$ and $B$; but here we are considering only the element $M \otimes M$, not $A \otimes A$, where $A$ is the subalgebra generated by $M$.
-
This looks like it can get complicated. Generically, at least over an algebraically closed field, a matrix $M$ will have distinct eigenvalues $m_1,\ldots,m_n$, and generically $M\otimes M$ will have distinct eigenvalues $m_1^2,\ldots,m_n^2$ with multiplicity one, and $m_1m_2,m_1m_3,\ldots,m_{n-1}m_n$ with multiplicity two. Thus the centralizer will have dimension $n+4{n\choose 2}=2n^2-n$.
But there are many degenerate cases: for instance if $M$ has eigenvalues $1,a,\ldots,a^{n-1}$ then $M\otimes M$ will have eigenvalues $1,a,\ldots,a^{2n-2}$ with multiplicities $1,2,\ldots n-1,n,n-1,\ldots,1$. Things can get more complicated still.
Then $M$ might have non-trivial Jordan blocks, and then the real fun starts!
It's reasonably well understood (and in the literature in many places, eg in work by B. Srinivasan, circa 1956) how to caclulate the Jordan blocks of $M \otimes M$ from that of $M.$ The famous (at least among group theorists) Hall-Higman paper (PLMS,1956) computes the centralizer of an element a single eigenvalue $1$ but arbitrary Jordan normal form. This suffices to do the general case ( as Robin suggests, the answer is far from pretty). – Geoff Robinson Jan 4 '12 at 5:45 | 2014-12-20 18:11:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9002665877342224, "perplexity": 121.93872795834154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802770071.75/warc/CC-MAIN-20141217075250-00170-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://codereview.stackexchange.com/questions/37901/reading-in-a-file-and-performing-string-manipulation/37919 | Reading in a file and performing string manipulation
In a question I answered I posted the following code:
#include <iostream>
#include <fstream>
#include <algorithm>
#include <sstream>
int main()
{
fstream iFile("names.txt", ios::in);
// This does not work so replacing with code that does
// iFile >> file;
// This is not the best way.
// But the closest to the intention of the original post.
// Get size of file.
ifile.seekg(0,std::ios::end);
std::streampos length = ifile.tellg();
ifile.seekg(0,std::ios::beg);
// Copy file into string.
std::string file(length, '\0');
// Original code continues.
std::istringstream ss(file);
std::string token;
std::vector<std::string> names;
while(std::getline(ss, token, ',')) {
names.push_back(token);
}
for (unsigned int i = 0; i < names.size(); i++) {
auto it = std::remove_if(names[i].begin(), names[i].end(), [&] (char c) { return c == '"'; });
names[i] = std::string(names[i].begin(), it);
}
for (unsigned int i = 0; i < names.size(); i++) {
std::cout << "names["<<i<<"]: " << names[i] << std::endl;
}
}
Would this be inefficient for larger files? Would it be better if I read the file into one string, did all the manipulation on that, and then put it into a vector?
• Is the code all inside main()? You can't have free-floating code like that in C++. Dec 22, 2013 at 10:00
• This code obviously does not work without main. So unless people have objects I will add main() around all the code so we are at least commenting on the code in the same way. Dec 23, 2013 at 21:42
• It still obviously does not work because iFile >> file; that is not doing very much. It reads a single space separated word. So I am going to change that to read the whole file into a string. Dec 23, 2013 at 21:43
• Now it works (as best it can). Dec 23, 2013 at 21:51
This code looks a little wild. I'd tame it like this:
#include <algorithm> // for std::remove
#include <fstream> // for std::ifstream
#include <string> // for std::string and std::getline
#include <utility> // for std::move
#include <vector> // for std::vector
std::ifstream infile("thefile.txt"); // we don't really care where you got the name from
std::vector<std::string> names;
for (std::string line; std::getline(infile, line, ','); )
{
line.erase(std::remove(line.begin(), line.end(), '"'), line.end());
names.push_back(std::move(line));
}
// done
A few points to note:
• Keep it simple. Don't overthink.
• Use ready-made algorithms, don't reinvent wheels (std::remove removes values).
• Include what you use (std::string, std::getline).
• Use algorithms, don't hand-roll your own loops. (erase/remove)
• Understand what algorithms can do for you (don't hand-write your own "erase"). Get familiar with the standard library.
• Don't leak scopes. The line string is only needed within the loop, and no further.
• Efficiency tip: You can move from the line string that you no longer need and save yourself a copy. Also, we process everything in one step; no need to loop repeatedly.
Furthermore, I suggest factoring this logic into a file:
std::vector<std::string> tokenize_file(std::string const & filename, char delimiter)
{
std::vector<std::string> result;
// ...
return result;
}
Usage:
std::vector<std::string> names = tokenize_file("thefile.txt", ',');
Instead of constructing an fstream, reading it all into a string (which is misleadingly named file, by the way), and creating an istringstream from that, why not just create an ifstream and call getline on it directly?
You appear to be trying to parse a CSV file where the fields may be double quoted. If that is your intention, then simply deleting all " characters is the wrong behaviour. See RFC 4180 for commonly accepted quoting rules for CSV documents. Consider using a CSV parsing library instead of rolling your own CSV parser.
Your code would be clearer if you unquoted each token before adding it to names.
• Totally agree on using CSV parsing library. CSV is one of those badly (not though through enough) defined formats that has a billion edge cases that you would never think about yourself. The simple case is trivial but doing it correctly for all cases is exceptionally hard. Dec 23, 2013 at 21:38 | 2022-05-20 03:57:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27465757727622986, "perplexity": 5223.032475667557}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662531352.50/warc/CC-MAIN-20220520030533-20220520060533-00532.warc.gz"} |
https://www.opuscula.agh.edu.pl/om-vol40iss4art7 | Opuscula Math. 40, no. 4 (2020), 509-516
https://doi.org/10.7494/OpMath.2020.40.4.509
Opuscula Mathematica
# Decompositions of complete 3-uniform hypergraphs into cycles of constant prime length
R. Lakshmi
T. Poovaragavan
Abstract. A complete $$3$$-uniform hypergraph of order $$n$$ has vertex set $$V$$ with $$|V|=n$$ and the set of all $$3$$-subsets of $$V$$ as its edge set. A $$t$$-cycle in this hypergraph is $$v_1, e_1, v_2, e_2,\dots, v_t, e_t, v_1$$ where $$v_1, v_2,\dots, v_t$$ are distinct vertices and $$e_1, e_2,\dots, e_t$$ are distinct edges such that $$v_i, v_{i+1}\in e_i$$ for $$i \in \{1, 2,\dots, t-1\}$$ and $$v_t, v_1 \in e_t$$. A decomposition of a hypergraph is a partition of its edge set into edge-disjoint subsets. In this paper, we give necessary and sufficient conditions for a decomposition of the complete $$3$$-uniform hypergraph of order $$n$$ into $$p$$-cycles, whenever $$p$$ is prime.
Keywords: uniform hypergraph, cycle decomposition.
Mathematics Subject Classification: 05C65, 05C85.
Full text (pdf)
1. C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1979.
2. J.C. Bermond, Hamiltonian decompositions of graphs, directed graphs and hypergraphs, Ann. Discrete Math. 3 (1978), 21-28.
3. J.C. Bermond, A. Germa, M.C. Heydemann, D. Sotteau, Hypergraphes hamiltoniens, [in:] Problémes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS, vol. 260, CNRS, Paris, 1978, 39-43.
4. D. Bryant, S. Herke, B. Maenhaut, W. Wannasit, Decompositions of complete $$3$$-uniform hypergraphs into small $$3$$-uniform hypergraphs, Australas. J. Combin. 60 (2014) 2, 227-254.
5. H. Jordon, G. Newkirk, $$4$$-cycle decompositions of complete $$3$$-uniform hypergraphs, Australas. J. Combin. 71 (2018) 2, 312-323.
6. D. Kühn, D. Osthus, Decompositions of complete uniform hypergraphs into Hamilton Berge cycles, J. Combin. Theory Ser. A 126 (2014), 128-135.
7. R. Lakshmi, T. Poovaragavan, $$6$$-Cycle decompositions of complete $$3$$-uniform hypergraphs, (submitted).
8. P. Petecki, On cyclic hamiltonian decompositions of complete $$k$$-uniform hypergraphs, Discrete Math. 325 (2014), 74-76.
9. M. Truszczyński, Note on the decomposition of $$\lambda K_{m,n} (\lambda K_{m,n}^{*})$$ into paths, Discrete Math. 55 (1985), 89-96.
10. H. Verrall, Hamilton decompositions of complete $$3$$-uniform hypergraphs, Discrete Math. 132 (1994), 333-348.
• R. Lakshmi (corresponding author)
• https://orcid.org/0000-0001-9633-7676
• Annamalai University, Department of Mathematics, Annamalainagar-608 002, India
• Dharumapuram Gnanambigai Government Arts College for Women, Department of Mathematics, Mayiladuthurai-609 001, India
• Communicated by Andrzej Żak.
• Accepted: 2020-06-05.
• Published online: 2020-07-09. | 2020-10-31 01:01:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7136167287826538, "perplexity": 2839.759600847397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107912593.62/warc/CC-MAIN-20201031002758-20201031032758-00525.warc.gz"} |
https://docs.chemaxon.com/display/docs/Theory+of+aqueous+solubility+prediction | ##### Page tree
This page summarizes the theoretical background behind ChemAxon's Aqeous Solubility (logS) Predictor. To find more information on the technical/usage side of the predictor, see the following page.
# Introduction
Aqueous solubility is one of the most important physico-chemical properties in modern drug discovery. It has impact on ADME-related properties like drug uptake, distribution and even oral bioavailability. Solubility can also be a relevant descriptor for property-based computational screening methods in the drug discovery process. Hence there is a significant interest in fast, reliable, structure-based methods for predicting solubility in water for promising drug candidates.
ChemAxon's Solubility Predictor is able to calculate two types of solubility: intrinsic and pH-dependent solubility.
On the logS unit
The logS is a common unit for measuring solubility. This unit is the 10-based logarithm of the solubility measured in mol/l unit, that is logS = log (solubility measured in mol/l).
On the temperature of the solubility prediction
The Solubility Predictor predicts solubility values at 25 °C.
On the result of the prediction
The predictor can provide quantitative results, giving the solubility in logS, mg/mL or mol/L units. The predictive accuracy of the plugin is considered to be 1 logS unit. In case only an estimation about how well soluble the compound is needed, the plugin can give a solubility category as a qualitative measure.
# Intrinsic solubility
The intrinsic solubility (usually denoted as logS0) of an ionizable compound is the solubility that can be measured after the equilibrium of solvation between the dissolved and the solid state is reached at a pH where the compound is fully neutral.
## Example
The intrinsic solubility of phenol can be measured at pH 6. Phenol is a weak acid with a pKa value of 10.02, which means that at pH 6 the molecule will be present in its neutral form, and the equilibrium can be measured only between the solid and the dissolved neutral form.
Fig. 1. Solvation equilibrium of phenol at pH 6
Our predictor uses a fragment-based method that identifies different structural fragments in the molecule and assigns an intrinsic solubility contribution to them. The contributions then are summed up to get the intrinsic solubility value. The implementation is based on the article of Hou et al.
The figure below shows a molecule split up into fragments that are used in the intrinsic solubility prediction.
Fig. 2. A molecule split into fragments to predict its intrinsic solubility
# pH-dependent solubility
The pH of the solution determines the ionization of the dissolved compound, which greatly affects the solvation equilibrium. With increasing ionization solubility increases compared to the intrinsic solubility.
## Example
The solubility of aniline in an acidic environment will be greater than its intrinsic solubility as the protonation of the compound shifts the equilibrium between the pure liquid aniline and its dissolved form to the right.
Fig. 3. Solvation equilibrium of aniline in an acidic environment
The pH-dependent solubility (usually denoted as logSpH) can be derived from the Henderson-Hasselbalch equation and the above definition of intrinsic and pH-dependent solubility.
In case of a weak acid the formula is the following:
$\log{S_{pH}} = \log{{S_0}} + \log(1 + 10^{(pH-pKa)})$
Considering a general case (based on the derivations for mono- and diprotic acids and bases and ampholytes) this formula can be transformed into the following form:
$\log S_{pH} = \log S_0 + \log(1 + \alpha)$ , where $\alpha = \frac{ \sum_i \alpha_{cA_i}} { \sum_j \alpha_{nA_j}}$
In this formula $\alpha_{cA_i}$ is the % of distribution of the i-th charged microspecies at the given pH, while $\alpha_{nA_j}$ is the % of distribution of the j-th neutral microspecies at the given pH.
Example of calculating pH-dependent solubility
Let's calculate the solubility of L-tyrosine at pH=9.2.
Zwitterionic molecules are the least soluble around their isoelectric point. The predicted isoelectric point of L-tyrosine is 5.5, which means that we can expect better solubility at pH 9.2 than at pH 5.5.
To get the pH-dependent solubility we first need the instrisic solubility of L-tyrosine. The logS Predictor gives -0.98 logS as intrinsic solubility.
To take ionization into account we have to calculate the microspecies distribution of L-tyrosine at pH=9.2. To do this we will use the pKa calculator, which can calculate the microspecies distribution based on the calculated pKa values. The following image shows the calculated distributions, with the highlighted row showing the distribution at pH 9.2.
From the image above we can read that the % distribution of the charged microspecies are (with the charges shown):
23.21 (-1), 0.0 (+1), 11.51 (-1, 1), 21.78 (-1, -1, +1)
The % distribution of the neutral microspecies:
43.50 (zwitterionic species)
Using these we can easily calculate the $\log(1 + \alpha)$ correction:
$$\log(1 + \alpha) = \log(1 + \frac{23.21 + 0.0 + 11.51 + 21.78}{43.5}) = 0.362$$
From this we get that the solubility at pH 9.2 is -0.618 logS.
The following image shows the whole pH-logS curve of the tyrosine with the calculated solubility at pH 9.2.
## Cut-off of the pH-dependent solubility curve
To put practical limits to the pH-dependent solubility curve and describe the fact that the solution reaches a certain saturation, a cut-off is applied to better match the real (experimental) pH-dependent solubility curve.
In our logS Predictor we apply the following cut-off (the following solubility values are all expressed in logS unit):
• if the predicted logS0 > -2, the applied cut-off will be +2, which means that the pH-dependent logS curve will be "cut off" at logS0 + 2. This means that the pH-dependent logS values won't increase above logS0 + 2.
• if the predicted logS0 < -2, the predicted pH-dependent solubility curve will not be allowed to rise above 0, so the cut-off will be at 0.
### Cut-off examples
1. The predicted intrinsic solubility (to which value the pH-dependent logS curve converges) is -1.0. Therefore the pH-dependent curve is cut off at +1.0.
2. The predicted intrinsic solubility (to which the pH-dependent logS curve converges) is -3.0. Therefore the pH-dependent curve is cut off at 0.
## Examples of predicted pH-dependent logS curves
In this section you will find some examples of predicted vs. experimental pH-dependent logS curves for specific drug molecules. The experimental values and the detailed analysis of the experimental logS curves can be found in the referenced literature.
### Example 1
The predicted pH-dependent logS curve of the HCl salt of ticlopidine shows a good correlation with the experimental curve, meaning that the HH-equation quite accurately describes the pH-dependence of the solubility. This result also comes from that ticlopidine is a monoprotic base, so the HH-equation works well.
The predicted logSis -3.47, while the experimental logS0 is -4.25. The absolute difference between the experimental and predicted intrinsic solubility is 0.78 logS unit.
### Example 2
The predicted pH-dependent logS curve of the hydrogen fumarate salt of the diprotic base quetiapine shows good correlation with the experimental curve. Its predicted logS0 is -4.27, while the experimental logS-2.84. The absolute difference between the experimental and the predicted instrinsic solubility is 1.42.
### Example 3
The predicted pH-dependent logS curve of the hydrogen fumarate salt of the ampholytic desvenlafaxine shows difference to the experimental curve below pH 6, which is due to salt formation. The predicted intrinsic logS is -2.16, while the experimental is -3.25. The absolute difference between these two values is 1.09 logS unit.
# Results of testing the model
The accuracy of the model was tested using fragment contributions calculated from the training set with a linear regression model. The obtained contributions were then used for calculating solubility for the training and the test set. The results are summarised on the following two charts:
Tests for pH-logS profile were also run. The two plots below show calculated and experimental pH-logS profiles for different acidic, basic and zwitter-ionic compounds:
Future goals
The Solubility Predictor will be developed further in the future. Among our future goals we have extending the prediction with a descriptor-based method and adding training features.
# References
1. Hou, T. J.; Xia, K.; Zhang, W.; Xu, X. J. ADME Evaluation in Drug Discovery. 4. Prediction of Aqueous Solubility Based on Atom Contribuition Approach. J. Chem. Inf. Comput. Sci. 200444, 266-275
2. Völgyi, G.; Baka, E. et al. Study of pH-dependent solubility of organic basis. Revisit of the Henderson-Hasselbalch relationship, Analytica Chimica Acta, 2010, 673, 40-46
3. Avdeef, A. et al. Equilibrium solubility measurement of ionizable drugs - consensus recommendations for improving data quality, ADMET & DMPK 4(2), 2016, 117-178
4. Shoghi, E.; Fuguet, E.; Bosch, E.; Rafols, C. Solubility-pH profiles of some acidic, basic and amphoteric drugs, European Journal of Pharmaceutical Sciences 201348, 291-300 | 2019-08-19 22:46:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7601224780082703, "perplexity": 4604.9288182767705}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315132.71/warc/CC-MAIN-20190819221806-20190820003806-00399.warc.gz"} |
https://mathematica.stackexchange.com/questions/45362/gray-background-in-export-pdf | # Gray background in export pdf
I am trying to output a simple plot with a legend and the output keeps coming out with a gray background. A simple example is the following:
Plot[{Sin[x], Cos[x]}, {x, 0, 2*3.14}, PlotStyle -> {Red, Blue},
PlotLegends -> SwatchLegend[{Red, Blue}, {"sin(x)", "cos(x)"}]]
Export["trig.pdf", %]
which outputs (notice the gray background!),
$\hspace{1cm}$
I have tried setting "Background->White" in different places, but only managed to have certain regions to get a white background. How can I get an export the image with a white background (and keeping it in pdf format)?
I am using Mathematica V9 on Linux.
Edit:
1. I just upgraded to V10 with hopes that it would fix the issue and the problem still persists.
2. pdf and eps both show the gray background while image formats such as jpg don't. Oddly enough svg also doesn't show the gray background.
3. I'm using Ubuntu 14.04 with Intel 4th generation graphics card.
4. I tried using different pdf viewers and they all showed the same gray background.
• Doesn't happen with M9.0.1 on OSX – halirutan Apr 4 '14 at 13:46
• I cannot reproduce this with M9.0.1 on Linux. Are you using a non-default style sheet in your notebook? – Szabolcs Apr 4 '14 at 14:02
• I sometimes have problems with pdf exports. Not the same. Have you tried exporting it to another format and get the same problem? – Mockup Dungeon Apr 4 '14 at 14:39
• FWIW, I have the same problem with MMA 9.0.0 on Win7 64bit. Shows grey in both SumatraPDF and Adobe Reader. I assumed it might be fixed with 9.0.1... – tkott Apr 4 '14 at 16:57
• @JeffDror If you are around, would you mind joining Mathematica Chat to discuss this problem? – halirutan Jul 15 '14 at 2:16
After a rather long debugging session in our chat we could determine the reason of the problem and come up with a workaround.
In short, we first tried whether the issue appears for the most basic Graphics[], which it didn't. As it turned out the gray background is introduced by using PlotLegends as in the example above. We went further by comparing AbsoluteOptions of the created graphics and the Options[SwatchLegend] settings. Everything looked the same on the OP's and my machine.
I then kind of remembered (guessed, whatever) that graphics export is done with the printing style sheet and we tried to set Format => Screen Environment to Printout. This was the first success, because this turned the graphics indeed gray.
Now, we knew the reason, but we couldn't find the exact source. Making a complete diff between our Core.nb stylesheet files showed nothing at all. I guess that this setting can be found somewhere else, although the OP stated that this issue persists for over 2 years and has survived even complete operating system changes.
The workaround is as simple as it is intuitive. Open the Preferences and then the Options Inspector and set the PrintingStyleEnvironment to Printout Gray (Is this awesome? It is, isn't it?)
If this issue would appear on my machine, I would probably take a closer look at my
FileNameJoin[{\$InstallationDirectory,"SystemFiles","FrontEnd","StyleSheets"}]
directory and grep through the Default.nb and the other things to find where this is set. I hope the workaround and the information help someone to track this down.
• I have exactly the same problem, and also only with PlotLegends.. However I don't get your workaround to work, the only three options for PrintingStyleEnvironment are "SlideShow", "Working", and "Printout". None of these three solve the problem, neither does manually typing "Printout Gray". Did I need to install another stylesheet first to get that to work? – freddieknets Dec 4 '17 at 10:58
• @freddieknets It is really hard to say what is happening on your machine. You don't need to install a style-sheet. The problem should be in your existing one. What exact operating system and Mathematica version are you using? – halirutan Dec 4 '17 at 12:56
• I am using Mathematica 11.1.0.0 on macOS 10.12.6 – freddieknets Dec 4 '17 at 14:15
• Unfortunately, I have only 11.1.1.0 on my OSX here. Otherwise, if we had the exact same versions, you could send my your stylesheet directory packed as zip and I would try to make a complete diff of the files. Do you have the possibility to update to 11.2 or 11.1.1? – halirutan Dec 4 '17 at 14:29
• Unfortunately I don't have premier service, so no (unless I am misunderstanding conditions to upgrade). – freddieknets Dec 5 '17 at 19:27
I recently encountered this problem, and none of the above solutions solved my problems (on version 10.3.1.0). However, I have found a solution which worked for me and should be independant of version.
On my version, the problem specifically occurred when using the Report stylesheet, but vanished with the Default one. The trick was to make a custom stylesheet that includes all of the style definitions from Report except the offending Graphics cell definition.
To achieve this practically, I switched to the Default stylesheet on the target notebook, opened a new mathematica notebook with a Report stylesheet , went to Format > Edit stylesheet on both and clicked Report/StandardReport.nb on the latter to load the Report's default cell definitions. Then I copied all the Report's cell definitions over to the Private Style Definitions for XXX.nb notebook and then went in and removed the offending local definition for graphics in Styles for Input and Output Cells as well as changing the background color for Output to white (select the cell then Format > Background Color > White).
Graphics now have a white background for both the image and text labels (an issue raised in another StackMMA question).
This new stylesheet can then be saved Default by following these instructions. | 2021-06-21 20:25:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26020270586013794, "perplexity": 2089.882037706792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488289268.76/warc/CC-MAIN-20210621181810-20210621211810-00574.warc.gz"} |
https://mathoverflow.net/questions/109420/is-this-a-known-solvable-problem-system-of-algebraic-equations | # Is this a known/solvable problem? (System of algebraic equations)
Hi there,
I am trying to find complex solutions with positive real part $\{t_j \;|\;{\rm Re}\;t_j>0, j = 1, 2, 3, \dots, n\}$ of the system of equations $$0 = 1 + \sum_j \left(t_j^{2l+1} + {t_j^*}^{2l+1}\right),\; l = 1,2,3,\dots m.$$ Where for a given $n$ I would like to make $m$ as large as possible. Since, this system is non-analytic and thus for $n=m$ most likely under-constrained, I had the idea to just fix the magnitude of all solutions to 1: $t_j = e^{i\phi_j}$ with $-{\pi\over 2} < \phi_1 \le \phi_2 \le \dots \le \phi_n <{\pi\over 2}$. In terms of these the system becomes: $$0 = 1 + 2\sum_j \cos{\left[\phi_j(2l+1)\right]},\; l = 1,2,3,\dots n.$$ This definitely has solutions up to $n=m=2$, but already for $n=3$, my naive attempt at numerically solving this (Mathematica's NSolve) is taking quite long. Is there some better way to find or at least confirm the existence of such solutions?
Thanks, Nik
-
What is $t_j^*$? Is that the complex conjugate? – Gerry Myerson Oct 11 '12 at 22:34
So letting $\phi_0=1$ and $\phi_{-i}=-\phi_i$ you want that for $0 \le \ell \le m$ the values $(e^{i\phi_k})^{2\ell+1}$ for $-n \le k \le n$ to have average real part $0$. If they were, in each case, equally distributed around the unit circle, that would suffice. – Aaron Meyerowitz Oct 12 '12 at 0:42
Gerry, yes that is correct, I should have said that! – nik Oct 12 '12 at 15:15
Aaron, yes, but is it possible to use those considerations to construct solutions with ${-\pi\over2} < \phi_j < {\pi\over 2}$? I.e. they must lie in the positive half-plane. – nik Oct 12 '12 at 15:22
I would try to get rid of the trigonometric functions, and rather rewrite the system as a polynomial system. If $x_j=\cos(\phi_j)$, then $\cos(\phi_j(2\ell+1))=T_{2\ell+1}(x_j)$, where $T_k$ is the $k$-th Chebychev polynomial of degree $k$. So your system of equations is $$0=1+2\sum_j T_{2\ell+1}(x_j),\;\;l=1,2,\dots,m,$$ with the requirement that $0<x_j\le 1$. Such a system can be discussed via Groebner bases, see here how to do that with Sage. For $m=n\le4$ there are only finitely many solutions. Among them pick those which fit your inequalities. For instance, for $m=n=4$, an approximation of a solution seems to be \begin{align*} x_1 &= 0.963494595276259\\ x_2 &= 0.852773246361416\\ x_3 &= 0.600336170417163\\ x_4 &= 0.058262327046178 \end{align*} Of course, you can use this technique also to handle the original case, where $t_j$ need not have length $1$. For instance, for $n=2$ one can show that $m\le4$, and and approximate solution for $m=4$ is $t_1=0.466916296430820 + 0.717248344919154i$, $t_2=0.856453001234213 + 0.445264622297009i$.
One cannot expect simple expressions for the $t_j$. For instance, the absolute values of the $t_j$ are roots of an irreducible (over the rationals) polynomial of degree $60$.
-
Thanks! That is very helpful! – nik Oct 12 '12 at 15:25
So what does the final solution for $m=n=4$ come out to be? – Aaron Meyerowitz Oct 13 '12 at 4:57
@Aaron: Not sure what you mean. We have $t_j=e^{i\phi_j}=\cos(\phi_j)+i\sin(\phi_j)=x_j\pm i\sqrt{1-x_j^2}$. Since nik adds $t_j^{2\ell+1}$ and its complex conjugate, it does not matter which sign you choose in $\pm i\sqrt{1-x_j^2}$ for each $j$. – Peter Mueller Oct 13 '12 at 9:10
Your problem may be understood as solving a system of equations and inequalities over $\mathbb{R}$. It is solvable in the sense that there exists an algorithm to find solutions (e.g. to find a point in every connected component specified by the system just mentioned).
Such algorithms are desribed, e.g., in this book: "Algorithms in Real Algebraic Geometry" by Saugata Basu, Richard Pollack, Marie-Françoise Roy.
How practical they are at present, is another question.
-
Consider the case $m=n=4.$ If we take $t_j=\cos(\frac{2j\pi}{9})+I\sin(\frac{2j\pi}{9})$ then $\sum_{j=0}^{8}t_j^q= 1 + \sum_{j=1}^{4} \left(t_j^{q} + {t_j^*}^{q}\right)=0$ for $1 \le q \le 8.$ This is because the set of values $t_j^q$ is just the nine $9$th roots of unity (or in two cases, the third roots of unity taken three times). Admittedly, this is not exactly what you wanted.
If you take just the real parts for $j=1,2,3,4$ you get
• $t_1=t_1^*=.766044443118979$
• $t_2=t_2^*=.173648177666934$
• $t_3=t_3^*=-.500000000000$
• $t_4=t_4^*=-0.939692620785905$
It can be seen that this makes $1 + \sum_{j=1}^4 \left(t_j^{q} + {t_j^*}^{q}\right)=0$ correct for $q=1,3,5,7.$
-
Hmm, maybe I am missing something, but please see my response to your comment above. – nik Oct 12 '12 at 15:24 | 2016-02-13 11:34:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9801374673843384, "perplexity": 204.34271066429778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701166570.91/warc/CC-MAIN-20160205193926-00319-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Congruent_number | # Congruent number
Triangle with the area 6, a congruent number.
In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides.[1][2] A more general definition includes all positive rational numbers with this property.[3]
The sequence of (integer) congruent numbers starts with
5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, ... (sequence A003273 in the OEIS)
Congruent number table: n ≤ 120
Congruent number table: n ≤ 120
—: non-Congruent number
C: square-free Congruent number
S: Congruent number with square factor
n 1 2 3 4 5 6 7 8
C C C
n 9 10 11 12 13 14 15 16
C C C
n 17 18 19 20 21 22 23 24
S C C C S
n 25 26 27 28 29 30 31 32
S C C C
n 33 34 35 36 37 38 39 40
C C C C
n 41 42 43 44 45 46 47 48
C S C C
n 49 50 51 52 53 54 55 56
S C S C S
n 57 58 59 60 61 62 63 64
S C C S
n 65 66 67 68 69 70 71 72
C C C C
n 73 74 75 76 77 78 79 80
C C C S
n 81 82 83 84 85 86 87 88
S C C C S
n 89 90 91 92 93 94 95 96
S C C C S
n 97 98 99 100 101 102 103 104
C C C
n 105 106 107 108 109 110 111 112
C C C S
n 113 114 115 116 117 118 119 120
S S C C S
For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle. 3 and 4 are not congruent numbers.
If q is a congruent number then s2q is also a congruent number for any natural number s (just by multiplying each side of the triangle by s), and vice versa. This leads to the observation that whether a nonzero rational number q is a congruent number depends only on its residue in the group
${\displaystyle \mathbb {Q} ^{*}/\mathbb {Q} ^{*2}}$,
where ${\displaystyle \mathbb {Q} ^{*}}$ is the set of nonzero rational numbers.
Every residue class in this group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers, when speaking about congruent numbers.
## Congruent number problem
The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2019) been brought to a successful resolution. Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.
Fermat's right triangle theorem, named after Pierre de Fermat, states that no square number can be a congruent number. However, in the form that every congruum (the difference between consecutive elements in an arithmetic progression of three squares) is non-square, it was already known (without proof) to Fibonacci.[4] Every congruum is a congruent number, and every congruent number is a product of a congruum and the square of a rational number.[5] However, determining whether a number is a congruum is much easier than determining whether it is congruent, because there is a parameterized formula for congrua for which only finitely many parameter values need to be tested.[6]
## Solutions
n is a congruent number if and only if the system
${\displaystyle x^{2}-ny^{2}=u^{2}}$, ${\displaystyle x^{2}+ny^{2}=v^{2}}$
has a solution where ${\displaystyle x,y,u}$, and ${\displaystyle v}$ are integers.[7]
Given a solution, the three numbers ${\displaystyle u^{2}}$, ${\displaystyle x^{2}}$, and ${\displaystyle v^{2}}$ will be in an arithmetic progression with common difference ${\displaystyle ny^{2}}$.
Furthermore, if there is one solution (where the right-hand sides are squares), then there are infinitely many: given any solution ${\displaystyle (x,y)}$, another solution ${\displaystyle (x',y')}$ can be computed from[8]
${\displaystyle x'=(xu)^{2}+n(yv)^{2}}$,
${\displaystyle y'=2xyuv}$.
For example, with ${\displaystyle n=6}$, the equations are:
${\displaystyle x^{2}-6y^{2}=u^{2}}$,
${\displaystyle x^{2}+6y^{2}=v^{2}}$.
One solution is ${\displaystyle x=5,y=2}$ (so that ${\displaystyle u=1,v=7}$). Another solution is
${\displaystyle x'=(5\cdot 1)^{2}+6(2\cdot 7)^{2}=1201}$,
${\displaystyle y'=2\cdot 5\cdot 2\cdot 1\cdot 7=140}$.
With this new ${\displaystyle x}$ and ${\displaystyle y}$, the right-hand sides are still both squares:
${\displaystyle u^{2}=1201^{2}-6\cdot 140^{2}=1324801=1151^{2}}$
${\displaystyle v^{2}=1201^{2}+6\cdot 140^{2}=1560001=1249^{2}}$.
Given ${\displaystyle x,y,u}$, and ${\displaystyle v}$, one can obtain ${\displaystyle a,b}$, and ${\displaystyle c}$ such that
${\displaystyle a^{2}+b^{2}=c^{2}}$, and ${\displaystyle {\frac {ab}{2}}=n}$
from
${\displaystyle a={\frac {v-u}{y}}}$, ${\displaystyle b={\frac {v+u}{y}}}$, ${\displaystyle c={\frac {2x}{y}}}$.
Then ${\displaystyle a,b}$ and ${\displaystyle c}$ are the legs and hypotenuse of a right triangle with area ${\displaystyle n}$.
The above values ${\displaystyle (x,y,u,v)=(5,2,1,7)}$ produce ${\displaystyle (a,b,c)=(3,4,5)}$. The values ${\displaystyle (1201,140,1151,1249)}$ give ${\displaystyle (a,b,c)=(7/10,120/7,1201/70)}$. Both of these right triangles have area ${\displaystyle n=6}$.
## Relation to elliptic curves
The question of whether a given number is congruent turns out to be equivalent to the condition that a certain elliptic curve has positive rank.[3] An alternative approach to the idea is presented below (as can essentially also be found in the introduction to Tunnell's paper).
Suppose a, b, c are numbers (not necessarily positive or rational) which satisfy the following two equations:
${\displaystyle {\begin{matrix}a^{2}+b^{2}&=&c^{2},\\{\tfrac {1}{2}}ab&=&n.\end{matrix}}}$
Then set x = n(a+c)/b and y = 2n2(a+c)/b2. A calculation shows
${\displaystyle y^{2}=x^{3}-n^{2}x}$
and y is not 0 (if y = 0 then a = -c, so b = 0, but (12)ab = n is nonzero, a contradiction).
Conversely, if x and y are numbers which satisfy the above equation and y is not 0, set a = (x2 - n2)/y, b = 2nx/y, and c = (x2 + n2)/y. A calculation shows these three numbers satisfy the two equations for a, b, and c above.
These two correspondences between (a,b,c) and (x,y) are inverses of each other, so we have a one-to-one correspondence between any solution of the two equations in a, b, and c and any solution of the equation in x and y with y nonzero. In particular, from the formulas in the two correspondences, for rational n we see that a, b, and c are rational if and only if the corresponding x and y are rational, and vice versa. (We also have that a, b, and c are all positive if and only if x and y are all positive; from the equation y2 = x3 - xn2 = x(x2 - n2) we see that if x and y are positive then x2 - n2 must be positive, so the formula for a above is positive.)
Thus a positive rational number n is congruent if and only if the equation y2 = x3 - n2x has a rational point with y not equal to 0. It can be shown (as an application of Dirichlet's theorem on primes in arithmetic progression) that the only torsion points on this elliptic curve are those with y equal to 0, hence the existence of a rational point with y nonzero is equivalent to saying the elliptic curve has positive rank.
Another approach to solving is to start with integer value of n denoted as N and solve
${\displaystyle N^{2}=ed^{2}+e^{2}}$
where
${\displaystyle {\begin{matrix}c&=&n^{2}/e+e\\a&=&2n\\b&=&n^{2}/e-e\end{matrix}}}$
## Smallest solutions
David Goldberg has computed congruent square-free numbers less than 104, along with the corresponding a and b values.[9]
## Current progress
Much work has been done classifying congruent numbers.
For example, it is known[10] that for a prime number p, the following holds:
• if p ≡ 3 (mod 8), then p is not a congruent number, but 2p is a congruent number.
• if p ≡ 5 (mod 8), then p is a congruent number.
• if p ≡ 7 (mod 8), then p and 2p are congruent numbers.
It is also known[11] that in each of the congruence classes 5, 6, 7 (mod 8), for any given k there are infinitely many square-free congruent numbers with k prime factors.
## Notes
1. ^ Weisstein, Eric W. "Congruent Number". MathWorld.
2. ^ Guy, Richard K. (2004). Unsolved problems in number theory ([3rd ed.] ed.). New York: Springer. pp. 195–197. ISBN 0-387-20860-7. OCLC 54611248.
3. ^ a b Koblitz, Neal (1993), Introduction to Elliptic Curves and Modular Forms, New York: Springer-Verlag, p. 3, ISBN 0-387-97966-2
4. ^ Ore, Øystein (2012), Number Theory and Its History, Courier Dover Corporation, pp. 202–203, ISBN 978-0-486-13643-1.
5. ^ Conrad, Keith (Fall 2008), "The congruent number problem" (PDF), Harvard College Mathematical Review, 2 (2): 58–73, archived from the original (PDF) on 2013-01-20.
6. ^ Darling, David (2004), The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes, John Wiley & Sons, p. 77, ISBN 978-0-471-66700-1.
7. ^ Uspensky, J. V.; Heaslet, M. A. (1939). Elementary Number Theory. Vol. 2. McGraw Hill. p. 419.
8. ^ Dickson, Leonard Eugene (1966). History of the Theory of Numbers. Vol. 2. Chelsea. pp. 468–469.
9. ^ Goldberg, David (7 June 2021). "Triangle Sides for Congruent Numbers less than 10,000". arXiv:2106.07373 [math.NT].
10. ^ Paul Monsky (1990), "Mock Heegner Points and Congruent Numbers", Mathematische Zeitschrift, 204 (1): 45–67, doi:10.1007/BF02570859
11. ^ Tian, Ye (2014), "Congruent numbers and Heegner points", Cambridge Journal of Mathematics, 2 (1): 117–161, arXiv:1210.8231, doi:10.4310/CJM.2014.v2.n1.a4, MR 3272014. | 2023-01-30 12:37:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 46, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7289194464683533, "perplexity": 290.2067851401785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499816.79/warc/CC-MAIN-20230130101912-20230130131912-00205.warc.gz"} |
https://deustotech.github.io/DyCon-Blog/tutorial/wp01/P0010 | # The interplay of control and deep learning
Author: - 30 April 2020
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It is superfluous to state the impact deep learning has had on modern technology. From a mathematical point of view however, a large number of the employed models remain rather ad hoc.
When formulated mathematically, deep supervised learning roughly consists in solving an optimal control problem subject to a nonlinear discrete-time dynamical system, called an artificial neural network.
## 1. Setup
Deep supervised learning [1, 3] (and more generally, supervised machine learning), can be summarized by the following scheme. We are interested in approximating a function $f: \R^d \rightarrow \R^m$, of some class, which is unknown a priori. We have data: its values $\{ \vec{y}_i \}_{i=1}^S \in \R^{m\times S}$ at $S$ distinct points $\{ \vec{x}_i \}_{i=1}^S \in \R^{d\times S}$ (possibly noisy). We generally split the $S$ data points into training data $\{ \vec{x}_i, \vec{y}_i \}_{i=1}^N$ and testing data $\{ \vec{x}_i, \vec{y}_i \}_{i=N+1}^S$. In practice, $N\gg S-N-1$.
“Learning” generally consists in:
1. Proposing a candidate approximation $f_\Theta( \cdot): \R^d \rightarrow \R^m$, depending on tunable parameters $\Theta$ and fixed hyper-parameters $L\geq 1$, ${ d_k}$;
2. Tune $\Theta$ as to minimize the empirical risk: $\sum_{i=1}^N \ell(f_\Theta(\vec{x}_i), \vec{y}_i),$ where $\ell \geq 0$, $\ell(x, x) = 0$ (e.g. $\ell(x, y) = |x-y|^2$). This is called training.
3. A posteriori analysis: check if test error $\sum_{i=N+1}^{S} \ell(f_\Theta(\vec{x}_i), \vec{y}_i)$ is small. This is called generalization.
Remark:
• There are two types of tasks in supervised learning: classification ($\vec{y}_i \in {-1,1}^m$ or more generally, a discrete set), and regression ($\vec{y}_i \in \R^m$). We will henceforth only present examples of binary classification, namely $\vec{y}_i \in {-1, 1}$, for simple presentation purposes.
• Point 3 is inherently linked with the size of the control parameters $\Theta$. Namely, a penalisation of the empirical risk in theory provides better generalisation.
Figure 1. Underfitting, good generalization, and overfitting. We wish to recover the function $f(x) = \cos(\frac32 \pi x)$ (blue) on $(0, 1)$ from $S=20$ noisy data samples. Constructed approximations using $N=12$ training data, while the remaining $8$ samples are used for testing the results. The most complicated model (right) is not necessarily the best (Occam's razor). This is related to the Runge phenomenon.
Remark: It is at the point of generalisation where the objective of supervised learning differs slightly from classical optimisation/optimal control. Indeed, whilst in deep learning one too is interested in “matching” the labels $\vec{y}_i$ of the training set, one also needs to guarantee satisfactory performance on points oustide of the training set.
## 2. Artificial Neural Networks
There exists an entire jungle (see [1, 3]) of specific neural networks used in practice and studied in theory. For the sake of presentation, we will discuss two of the most simple examples.
### 2.1. Multi-layer perceptron
We henceforth assume that we are given a training dataset $\{\vec{x}_i, \vec{y}_i\}_{i=1}^N \subset \R^{d\times N}\times \R^{m\times N}$.
Definition (Neural network): Let $L\geq 1$ and $\{ d_k \}_{k=1}^L \in \N^L$ be given. Set $d_0 := d$ and $d_{L+1} :=m$. A neural network with $L$ hidden layers is a map
where $z^L = z^L_i \in \R^m$ being given by the scheme
Here $\Theta = \{A^k, b^k\}_{k=0}^{L}$ are given parameters such that $A^k \in \R^{d_{k+1}\times d_k}$ and $b^k \in \R^{d_k}$, and $\sigma \in C^{0, 1}(\R)$ is a fixed, non-decreasing function.
Remark: Several remarks are in order:
• The above-defined neural network is usually referred to as the multi-layer perceptron (see Multilayer_perceptron)
• Observe that $z^k \in \R^{d_k}$ for $k \in \{0, \ldots, L\}$.
• The function $\sigma$ is always nonlinear in practice (as otherwise the optimisation problem roughly coincides with least squares for a linear regression). It is called the activation function.
• Generally, $\sigma(x) = \max(x, 0)$ or $\sigma(x) = \tanh(x)$ (but others work too).
• Deep learning means optimisation subject to a multi-layered neural net: $L\geq 2$ at least.
Let us denote $\Lambda_k x :=A^k x + b^k$.
Figure 2. The commonly used graph representation for a neural net. This figure essentially represents the discrete-time dynamics of a single datum from the training set through the nonlinear scheme.
An MLP scheme with $\sigma(x) = \tanh(x)$ can be coded in pytorch more or less as follows:
import torch.nn as nn
class OneBlock(nn.Module):
def __init__(self, d1, d2):
super(OneBlock, self).__init__()
self.d1 = d1
self.d2 = d2
self.mlp = nn.Sequential(
nn.Linear(d1, d2),
nn.Tanh()
)
def forward(self, x):
return self.mlp(x)
class Perceptron(nn.Module):
def __init__(self, dimensions, num_layers):
super(Perceptron, self).__init__()
self.dimensions = dimensions
_ = \
[OneBlock(self.dimensions[k], self.dimensions[k-1]) for k in range(1, num_layers-1)]
self.blocks = nn.Sequential(*_)
self.projector = nn.Linear(self.dimensions[-2], self.dimensions[-1])
def forward(self, x)_
return self.projector(self.blocks(x))
We can save this class in a file called model.py. Then one may create an instance of a Perceptron model for practical usage as follows:
import torch
device = torch.device('cpu')
from model import Perceptron
model = Perceptron(device, dimensions=[1,3,4,3,1], num_layers=5)
How it works.
We now see, in some overly-simplified scenarios, how the forward propagation of the inputs in the context of binary classification.
Figure 3. Here $A^0 \in R^{2\times 1}$ and $b^0 \in \R^2$. We see how the neural net essentially generates a nonlinear transform, such that the originally mixed points are now linearly separable.
An interesting blog on visualising the transitions is the following: https://colah.github.io/posts/2014-03-NN-Manifolds-Topology/.
### 2.2. Residual neural networks
We now impose that the dimensions of the iterations and the parameters stay fixed over each step. This will allow us to add an addendum term in the scheme.
Definition (Residual neural network [2]): Let $L\geq 1$ and $\{ d_k \}_{k=1}^L \in \N^L$ be given. Set $d_0 := d$ and $d_{L+1} :=m$. A residual neural network (ResNet) with $L$ hidden layers is a map
where $z^L = z^L_i \in \R^d$ being given by the scheme
Here $\Theta = {A^k, b^k}_{k=0}^{L}$ are given parameters such that $A^k \in \R^{d\times d}$ and $b^k \in \R^{d}$ for $k<L$ and $A^L \in \R^{m\times d}, b^L \in \R^m$, and $\sigma \in C^{0, 1}(\R)$ is a fixed, non-decreasing function.
## 3. Training
Training consists in solving the optimization problem:
• $\epsilon>0$ is a penalization parameter.
• It is a non-convex optimization problem because of the non-linearity of $f_L$.
• Existence of a minimizer may be shown by the direct method.
Once training is done, and we have a minimizer $\widehat{\Theta}$, we consider $f_{\widehat{\Theta}}(\cdot)$ and use it on other points of interest $\vec{x} \in \R^d$ outside the training set.
### 3.1. Computing the minimizer
The functional to be minimized is of the form
We could do gradient descent:
$\eta$ is step-size. But often $N \gg 1$ ($N=10^3$ or much more).
Stochastic gradient descent: (Robbins-Monro [7], Bottou et al [8]):
1. pick $i \in {1, \ldots, N}$ uniformly at random
2. $\Theta^{n+1} := \Theta^n - \eta \nabla J_{i}(\Theta^n)$
• Mini-batch GD: can also be considered (pick a subset of data instead of just one point)
• Use chain rule and adjoints to compute these gradients (“backpropagation”)
• Issues: might not converge to global minimizer; also how does one initialize the weights in the iteration (usually done at random)?
We come back to the file model.py. Here is a snippet on how to call training modules.
# Given generated data
# Coded a function def optimize() witin
# a class Trainer() which
# does the optimization of paramterss
epochs = input()
optimizer = torch.optim.SGD(perceptron.parameters(), lr=1e-3, weight_decay=1e-3)
trainer = Trainer(perceptron, optimizer, device)
trainer.train(data, epochs)
### 3.2. Continuous-time optimal control problem
We begin by visualising how the flow map of a continuous-time neural net separates the training data in a linear fashion at time $1$ (even before in fact).
Figure 4. The time-steps play the role of layers. We are working with an ode on $(0, 1)$. We see that the points are linearly separable at the final time, so the flow map defines a "linearisation" of the training dataset.
Recall that often $\varphi \equiv \sigma$ (classification) or $\varphi(x) = x$ (regression).
It can be advantageous to consider the continuous-time optimal control problem:
where $z = z_i$ solves
Remark:
The ResNet neural network can then be seen as the forward Euler discretisation with $\Delta t = 1$. The relevance of this scale when passing from continuous to discrete has not been addressed in the litearature.
Thus, in the continuous-time limit, deep supervised learning for ResNets can be seen as an optimal control problem for a parametrised, high-dimensional ODE.
This idea of viewing deep learning as finite dimensional optimal control was (mathematically) formulated in [12], and subsequently investigated from a theoretical and computational viewpoint in [11, 10, 5, 6, 13, 14], among others.
Remark:
There are many tricks which can be used in the above ODE to improve performance.
• For instance, we may embed the initial data in a higher dimensional space at the beginning, and consider the system in an even bigger dimension. It is rather intuitive that the bigger the dimension where the system evolves is, the easier it is to separate the points by a hyperplane.
• However, characterising the optimal dimension where one needs to consider the evolution of the neural net in terms of the topology of the training data is, up to the best of our knowledge, an open problem. The choice in practice is done by cross-validation.
Figure 5. Analogous scenario as in Figure 4, this time in dimension 3.
## References:
[1] Ian Goodfellow and Yoshua Bengio and Aaron Courville. (2016). Deep Learning, MIT Press.
[2] He, K., Zhang, X., Ren, S., and Sun, J. (2016). Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778.
[3] LeCun, Y., Bengio, Y., and Hinton, G. (2015). Deep learning. Nature, 521(7553):436–444.
[4] LeCun, Y., Boser, B. E., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, W. E., and Jackel, L. D. (1990). Handwritten digit recognition with a back-propagation network. In Advances in neural information processing systems, pages 396–404.
[5] Chen, T. Q., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. (2018). Neural ordinary differential equations. In Advances in neural information processing systems, pages 6571– 6583.
[6] Dupont, E., Doucet, A., and Teh, Y. W. (2019). Augmented neural odes. In Advances in Neural Information Processing Systems, pages 3134–3144.
[7] Herbert Robbins and Sutton Monro. A Stochastic Approximation Method. The Annals of Mathematical Statistics, 22(3):400–407, 1951
[8] Léon Bottou, Frank E. Curtis and Jorge Nocedal: Optimization Methods for Large-Scale Machine Learning, Siam Review, 60(2):223-311, 2018.
[9] Matthew Thorpe and Yves van Gennip. Deep limits of residual neural networks. arXiv preprint arXiv:1810.11741, 2018.
[10] Weinan, E., Han, J., and Li, Q. (2019). A mean-field optimal control formulation of deep learning. Research in the Mathematical Sciences, 6(1):10.
[11] Li, Q., Chen, L., Tai, C., and Weinan, E. (2017). Maximum principle based algorithms for deep learning. The Journal of Machine Learning Research, 18(1):5998–6026.
[12] Weinan, E. (2017). A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics, 5(1):1–11. | 2021-04-20 12:37:26 | {"extraction_info": {"found_math": true, "script_math_tex": 13, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8433355689048767, "perplexity": 1249.1279043226227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039398307.76/warc/CC-MAIN-20210420122023-20210420152023-00106.warc.gz"} |
https://socratic.org/questions/what-is-the-slope-of-the-line-perpendicular-to-2y-6x-8 | # What is the slope of the line perpendicular to 2y= -6x +8?
Jul 2, 2016
$+ \frac{1}{3}$
#### Explanation:
Change the equation into standard form first (÷2):
$y = - 3 x + 4$
The gradient is -3 - better written as $- \frac{3}{1}$
The gradients of perpendicular lines are negative reciprocals of each other.
The required gradient is $+ \frac{1}{3}$
A check is that their product should be -1
${m}_{1} \times {m}_{2} = - \frac{3}{1} \times \frac{1}{3} = - 1$ | 2020-12-03 19:24:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6384950280189514, "perplexity": 1209.4367404176915}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141732696.67/warc/CC-MAIN-20201203190021-20201203220021-00278.warc.gz"} |
http://mathhelpforum.com/statistics/204206-premiation.html | # Math Help - Premiation
1. ## Premiation
How many different man-woman couples can be formed at a dance where there are 12 women and 8 men?
1. 12C1 *8C1 or 12P1 *8P1
2. 202
3. 20 * 8
4. None of the above
We think the answer is 96. However, both answers in number 1 equal 96. But, can be premiation and combination but together in one answer?
2. ## Re: Premiation
Answer lies in basic counting principle.
3. ## Re: Premiation
So that would be 12 * 8 if we used basic counting principle? So "none of the above?"
4. ## Re: Premiation
${12 \choose 1}\cdot{8 \choose 1 }=12\cdot 8$
5. ## Re: Premiation
That is what I understood. I knew the answer was 96. I just dont ever see premiation and combination together. I did not know if those TWO could be an answer. I do not think they do, so my answer will be none of the above since 96 is not listed. | 2016-07-27 10:12:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.708607017993927, "perplexity": 1876.1130012362223}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257826759.85/warc/CC-MAIN-20160723071026-00326-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://iris.sissa.it/handle/20.500.11767/4816 | We have presented a newly developed Davidson-like algorithm to compute selected \textit{interior} eigenstates of the Liouvillian super-operator, and a recently introduced \textit{pseudo-Hermitian} variant of the Liouville-Lanczos approach to time-dependent density-functional theory (TDDFT). The new algorithms have been released as the new version of \texttt{turboTDDFT} in the \textsc{Quantum ESPRESSO} package together with an implementation of hybrid functionals. Our implementation has been thoroughly validated against benchmark calculations performed on a few moleucles using both the original \texttt{turboTDDFT} and \texttt{Gaussian09} code. Then we have applied the new algorithms to carry on a systematic study of anthocyanins' optical properties. We have found that the Oxygen-containing side groups on the phenyl ring of these molecules play important roles to modify their optical absorption spectra, which can be understood by a so-call \textit{double-pole approximation}. We have also applied a recently proposed explicit solvent model to study the solvent effects for these molecules, using a combination of Ab-initio molecular dynamics (AIMD) and TDDFT. We have found that PBE produces too red-shifted spectra, instead B3LYP gives the spectra and the simulated colors in very good agreement with the experiment.
Seeing colors with TDDFT: theoretical modeling of the optical properties of natural dyes / Ge, Xiaochuan. - (2013 Nov 18).
### Seeing colors with TDDFT: theoretical modeling of the optical properties of natural dyes
#### Abstract
We have presented a newly developed Davidson-like algorithm to compute selected \textit{interior} eigenstates of the Liouvillian super-operator, and a recently introduced \textit{pseudo-Hermitian} variant of the Liouville-Lanczos approach to time-dependent density-functional theory (TDDFT). The new algorithms have been released as the new version of \texttt{turboTDDFT} in the \textsc{Quantum ESPRESSO} package together with an implementation of hybrid functionals. Our implementation has been thoroughly validated against benchmark calculations performed on a few moleucles using both the original \texttt{turboTDDFT} and \texttt{Gaussian09} code. Then we have applied the new algorithms to carry on a systematic study of anthocyanins' optical properties. We have found that the Oxygen-containing side groups on the phenyl ring of these molecules play important roles to modify their optical absorption spectra, which can be understood by a so-call \textit{double-pole approximation}. We have also applied a recently proposed explicit solvent model to study the solvent effects for these molecules, using a combination of Ab-initio molecular dynamics (AIMD) and TDDFT. We have found that PBE produces too red-shifted spectra, instead B3LYP gives the spectra and the simulated colors in very good agreement with the experiment.
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Baroni, Stefano
Calzolari, Arrigo
Ge, Xiaochuan
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/4816 | 2022-06-25 08:39:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24856506288051605, "perplexity": 3365.148632015099}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103034877.9/warc/CC-MAIN-20220625065404-20220625095404-00088.warc.gz"} |
https://www.esaral.com/q/in-the-figure-below-p-and-q-59101/ | In the figure below, P and Q
Question:
In the figure below, $\mathrm{P}$ and $\mathrm{Q}$ are two equally intense coherent sources emitting radiation of wavelength $20 \mathrm{~m}$. The separation between $P$ and $Q$ is $5 \mathrm{~m}$ and the phase of $P$ is ahead of that of Q by $90^{\circ} . \mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are three distinct points of observation, each equidistant from the midpoint of PQ. The intensities of radiation at $\mathrm{A}, \mathrm{B}, \mathrm{C}$ will be in the ratio :
1. $0: 1: 4$
2. $2: 1: 0$
3. $0: 1: 2$
4. $4: 1: 0$
Correct Option: , 2
Solution:
(2) For (A)
$x_{P}-x_{Q}=(d+2.5)-(d-2.5)=5 \mathrm{~m}$
Phase difference $\Delta \phi$ due to path difference
$=\frac{2 \pi}{\lambda}(\Delta x)=\frac{2 \pi}{20}(5)=\frac{\pi}{2}$
At $A, Q$ is ahead of $P$ by path, as wave emitted by $Q$ reaches before wave emitted by $P$.
$\therefore$ Total phase difference at $A \frac{\pi}{2}-\frac{\pi}{2}=0$
(due to $P$ being ahead of $Q$ by $90^{\circ}$ )
$I_{A}=I_{1}+I_{2}+2 \sqrt{I_{1}} \sqrt{I_{2}} \cos \Delta \phi$
$=I+I+2 \sqrt{I} \sqrt{I} \cos (0)=4 I$
For $C$,
Path difference, $x_{Q}-x_{P}=5 \mathrm{~m}$
Phase difference $\Delta \phi$ due to path difference
$=\frac{2 \pi}{\lambda}(\Delta x)=\frac{2 \pi}{20}(5)=\frac{\pi}{2}$
Total phase difference at $C=\frac{\pi}{2}+\frac{\pi}{2}=\pi$
$I_{\text {net }}=I_{1}+I_{2}+2 \sqrt{I_{1}} \sqrt{I_{2}} \cos (\Delta \phi)$
$=I+I+2 \sqrt{I} \sqrt{I} \cos (\pi)=0$
For $B$,
Path difference, $x_{P}-x_{Q}=0$
Phase difference, $\Delta \phi=\frac{\pi}{2}$
(due to $P$ being ahead of $Q$ by $90^{\circ}$ )
$I_{B}=I+I+2 \sqrt{I} \sqrt{I} \cos \frac{\pi}{2}=2 I$
Therefore intensities of radiation at $A, B$ and $C$ will be in the ratio
$I_{A}: I_{B}: I_{C}=4 I: 2 I: 0=2: 1: 0$ | 2022-05-16 12:34:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7815203070640564, "perplexity": 793.0837164866134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510117.12/warc/CC-MAIN-20220516104933-20220516134933-00634.warc.gz"} |
https://aakinshin.net/posts/qrde-hd/ | # Quantile-respectful density estimation based on the Harrell-Davis quantile estimator
The idea of this post was born when I was working on a presentation for my recent DotNext talk. It had a slide with a density plot like this:
Here we can see a density plot based on a sample with highlighted decile locations that split the plot into 10 equal parts. Before the conference, I have been reviewed by @VladimirSitnikv. He raised a reasonable concern: it doesn’t look like all the density plot segments are equal and contain exactly 10% of the whole plot. And he was right!
However, I didn’t make any miscalculations. I generated a real sample with 61 elements. Next, I build a density plot with the kernel density estimation (KDE) using the Sheather & Jones method and the normal kernel. Next, I calculated decile values using the Harrell-Davis quantile estimator. Although both the density plot and the decile values are calculated correctly and consistent with the sample, they are not consistent with each other! Indeed, such a density plot is just an estimation of the underlying distribution. It has its own decile values, which are not equal to the sample decile values regardless of the used quantile estimator. This problem is common for different kinds of visualization that presents density and quantiles at the same time (e.g., violin plots)
It leads us to a question: how should we present the shape of our data together with quantile values without confusing inconsistency in the final image? Today I will present a good solution: we should use the quantile-respectful density estimation based on the Harrell-Davis quantile estimator! I know the title is a bit long, but it’s not so complicated as it sounds. In this post, I will show how to build such plots. Also I will compare them to the classic histograms and kernel density estimations. As a bonus, I will demonstrate how awesome these plots are for multimodality detection.
### The problem
To understand the problem better, consider a sample with three elements: $$x = \{ 3, 4, 7 \}$$. Let’s build a probability density function (PDF) based on kernel density estimation using the Sheather & Jones method and the normal kernel. Let’s also calculate the median and the $$95^{\textrm{th}}$$ percentile using three different methods:
• Type 7 quantile estimator
It’s the most popular quantile estimator which is used by default in R, Julia, NumPy, Excel (PERCENTILE, PERCENTILE.INC), Python (inclusive method). We call it “Type 7” according to notation from [Hyndman1996], where Rob J. Hyndman and Yanan Fan described nine quantile algorithms which are used in statistical computer packages.
• The Harrell-Davis quantile estimator
A quantile estimator that is described in [Harrell1982]. It’s more robust, and it provides more reliable estimations.
• KDE-based quantile estimator
Quantile values that are obtained from the kernel density estimation instead of the original sample.
Here is the density plot with highlighted quantiles:
As you can see, all three quantile estimators produced different values. However, only the KDE-based quantile estimator is consistent with the density plot. For example, the KDE-based median estimation splits the density plot into two equal parts while two other estimators produce other ratios. The problem becomes more obvious if we look at the $$95^{\textrm{th}}$$ percentile. As we can see, the KDE-based value (7.73) is bigger than the maximum sample element $$x_{\max} = 7$$. It’s an expected situation: the kernel density estimation estimates the whole underlying distribution. If we use the normal kernel, some parts of the density plot will always be between $$x_{\max}$$ and positive infinity. It means that some of the high quantiles will always be bigger than $$x_{\max}$$. However, we can’t say the same about the Type 7 and Harrell-Davis quantile estimators. For a sample-based estimator, no quantile estimation can exceed $$x_{\max}$$.
Thus, it’s not a good idea to present sample-based quantile estimation with the kernel density estimation because they are not consistent with each other. To make it consistent, we have two possible solutions:
• Present KDE-based quantile values that are consistent with the existing density plot
• Present another density plot that is consistent with the existing quantile values
The first option may confuse people. Indeed, let’s say we want to present the $$99.9^{\textrm{th}}$$ percentile. In the case of the KDE-based quantiles, this value may be extremely high or even tend to infinity. Most people will expect to see a value based on the existing data rather than on the KDE.
Let’s try the second option, where we try to build a density plot using the estimated quantile values.
### Quantile-respectful density estimation
Let’s introduce a new term: quantile-respectful density estimation1 (QRDE). It’s a density estimation that matches the given quantile values. Obviously, it highly depends on the used quantile estimator. In this section, we compare two different variations of QRDE that are based on the Type 7 quantile estimator (QRDE-T7) and on the HarrellDavis quantile estimator (QRDE-HD).
From the computational point of view, the easiest way to evaluate QRDE is to present it as a step function based on several quantile values. It’s convenient to consider the minimum and the maximum values as additional quantile values (although it’s not correct from the formal point of view). It’s also convenient to think about this step function as a density histogram. For each pair of consecutive quantiles values, we should draw a bin. The left and the right borders of the bin are equal to the quantile values. We want to make the QRDE consistent with the quantile values, so each bin’s area should be equal to $$1 / p$$ when we work with $$p$$-quantiles. Since we know the width and the area of each bin, it’s easy to calculate its height:
$h_i = \frac{\textrm{Area}_i}{\textrm{Width}_i} = \frac{1/p}{q_{i + 1} - q_i} = \frac{1}{p(q_{i + 1} - q_i)}.$
Now let’s look at some examples. For simplification, we start with deciles. They split our distribution into 10 equal sizes. Thus, we should get 10 bins, the area of each bin is 0.1. For the above sample $$x = \{ 3, 4, 7 \}$$, we have the following plots:
The Type 7 quantile estimator presents a nice visualization of this concept. The median of the sample (which equals 4 with the given estimator) splits the QRDE into two equal parts. Since this quantile estimator is based on linear interpolation, we have a flat area in each part.
The Harrell-Davis quantile estimator gives us more smoothness. At the start, you may be confused by the spikes at the ends of the plot. However, if you think about it a little bit, this phenomenon becomes pretty obvious and natural. Indeed, the sample’s corner elements are “magic” points, where a huge portion of density arises from nowhere. We have zero density on one side and a high positive density on another side. You may also observe such spikes with the KDE if you cut down parts before the minimum element and after the maximum element.
Now let’s increase the number of elements and consider a 500-element sample from the normal distribution:
We can recognize the bell-shape of the normal distribution, but it’s pretty rough. Also, we do not see a significant difference between the presented quantile estimators. The difference becomes more obvious if we reduce the quantization step value and switch to percentiles:
Now we can see that the Harrell-Davis gives us a smoother version of the QRDE. The Type7-based QRDE looks too spiky, which makes it not so useful. With a very small quantization step, it becomes completely useless because in most cases, we will observe only a few bins that have the maximum density:
Meanwhile, the Harrell-Davis-based QRDE keeps its smooth form:
Of course, it’s not as smooth as the KDE. We have such a wobbly plot because it describes our real data instead of oversmoothed estimation of the underlying distribution. If we don’t know the actual distribution and we want to just explore the data in the collected sample, the Harrell-Davis-based QRDE is one of the best ways to do it.
The most important fact about this plot is that it’s consistent with the quantile values. The estimated value of median splits this plot into two equal parts, the estimated decile values split this plot into ten equal parts, and so on.
### QRDE and multimodal distributions
The true power of the QRDE manifests itself when you start working with multimodal distributions. Let’s check out some advantages of the QRDE against classic histograms and the KDE.
It highlights multimodality without parameter tuning.
Here we have a distribution with four modes (a combination of $$\mathcal{N}(0, 4)$$, $$\mathcal{N}(0, 8)$$, $$\mathcal{N}(0, 12)$$, $$\mathcal{N}(0, 16)$$; 30 elements from each). For the classic histogram, we should choose the offset and the bandwidth, which may be a problem. For KDE, we should choose the bandwidth and the kernel, which also may be problem. With Harrell-Davis-based QRDE, we don’t have any parameters to tune: the plot is uniquely defined, and it almost always presents what we want to see.
It highlights multimodality even if two modes are close to each other.
Here we have a bimodal distribution with background noise (1000 elements from $$\mathcal{U}(0; 10)$$, 100 elements from $$\mathcal{N}(5, 0.1^2)$$, 100 elements from $$\mathcal{N}(5.5, 0.1^2)$$). It’s always impossible to see signs of multimodality on the classic histograms or the KDE plots. Meanwhile, the Harrell-Davis-based QRDE solves highlights multimodality without any problems. When two modes are clearly expressed, we will always see it on such a plot regardless of the total range, the distance between modes, and the noise pattern.
It highlights multimodality even there are too many modes.
Here we have a multimodal distribution with 20 modes (a combination of $$\mathcal{N}(4i, 1^2)$$ for $$i \in \{ 1..20\}$$; 100 elements from each). It’s completely impossible to detect such multimodality using the KDE; these plots are too smooth for this problem. It’s also impossible to distinguish such a situation from regular noise using classic histograms (another example of the bandwidth problem). In contrast, the Harrell-Davis-based QRDE doesn’t have a limitation on the number of modes: it always able to detect all the modes while they are clearly expressed.
### Conclusion
Currently, the QRDE-HD quantile estimator is my favorite way to present the collected sample’s raw shape. I want to highlight one more time two main advantages that I described in this post:
• Quantile-consistency
It’s safe to show quantile values on such a plot: it will always be consistent with the presented density.
• Multimodality detection
It will not hide multimodality phenomena from you, which makes it a convenient tool to explore samples from non-parametric distributions. In future posts, I will present an algorithm that detects locations of all existing modes.
### References
1. In the first version of this post, I used term “empirical probability density function.” After some thoughts, I decided to rename it to “quantile-respectful density estimation.” The new term is not so confusing (the suggested function is significantly different from the empirical function) and it describes the underlying concept much better. Naming is hard. ↩︎ | 2020-12-05 18:30:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7931217551231384, "perplexity": 696.0604533954772}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141748276.94/warc/CC-MAIN-20201205165649-20201205195649-00650.warc.gz"} |
https://www.academic-quant-news.com/2020/06/research-articles-for-2020-06-01.html | # Research articles for the 2020-06-01
A Comparative Study of Stock Screening Methodologies in Stock Exchanges of Bangladesh and Malaysia and Lessons to Be Learnt
Jalil, Md. Abdul
SSRN
The study aims to compare and critically evaluate the stock screening practices between Bangladesh and Malaysia. The specific objectives determined to fulfill the aims are: (i) To review some of the Islamic equity market norms along with juristic views (ii) To review the prevalent practices of stock screening methods used by international index providers (iii) To evaluate critically and compare the stock screening methodology used by Dhaka Stock Exchange (DSE), Chittagong Stock Exchange (CSE), and Bursa Malaysia. The study is descriptive in nature. Secondary data is utilized and collected from the books, standards, journal articles and relevant publications. AAOIFI standards, OIC Fiqh Academy resolutions etc. are referenced as needed.DSE, CSE and Bursa differs in formulating ratios, denominators, numerators and in determining benchmark. The study finds that Bursa uses two thresholds to measure shari’ah non-compliance whereas DSE and CSE use single benchmark, 5% and 4% respectively. For financial screening, DSE uses market value of equity as numerator whereas CSE and Bursa use total assets as numerator. Because of the differences, one company may be included in Shari’ah Index in Bangladesh, but not in Malaysia and vice versa.
A Theory of 'Auction as a Search' in speculative markets
Sudhanshu Pani
arXiv
The tatonnement process in high frequency order driven markets is modeled as a search by buyers for sellers and vice-versa. We propose a total order book model, comprising limit orders and latent orders, in the absence of a market maker. A zero intelligence approach of agents is employed using a diffusion-drift-reaction model, to explain the trading through continuous auctions (price and volume). The search (levy or brownian) for transaction price is the primary diffusion mechanism with other behavioural dynamics in the model inspired from foraging, chemotaxis and robotic search. Analytic and asymptotic analysis is provided for several scenarios and examples. Numerical simulation of the model extends our understanding of the relative performance between brownian, superdiffusive and ballistic search in the model.
Analysis of the Deposit Resources’ Regional Allocation in Ukraine
SSRN
The intermediary function of financial corporations is embodied in the redistribution of temporarily free funds raised from certain institutional units to those units that have a need for them. A significant part of such borrowings takes the form of deposits placed with commercial banks. Amounts accumulated in the form of deposits are an important component of the resource base of banking institutions. This allows them to conduct assets banking operations to lend the population and enterprises of the economy's real sector and contribute to economic growth in the country.The purpose of the study is to perform a statistical analysis of the regional allocation of deposits held with Ukrainian commercial banks over the period 2010-2018.The conceptual basis of the study is formed by the approaches of the System of National Accounts and Monetary Statistics. The study is based on the National Bank of Ukraine (NBU) information. The results obtained indicate a high degree of heterogeneity of the regions of Ukraine in terms of deposits attracted by commercial banks. Regional variation in deposits amounts is too big. Three regions can be classified as outliers. These are the capital region â€" Kyiv region and the most industrially developed Dnepropetrovsk and Donetsk region. This conclusion is confirmed by the analysis of variation and the form of regions' distribution. The most significant changes in the distribution of regions occurred in 2014 due to the dramatic events of modern Ukrainian history.We also conclude the growing level of deposits' concentration based on the concentration index-3, concentration index-5, Herfindahl-Hirschman Index and Gini coefficient. Since geographical diversification is a necessary prerequisite for reducing the riskiness of deposit operations, appropriate managerial decisions must be taken.
Are the Largest Banking Organizations Operationally More Risky?
Curti, Filippo,Frame, W. Scott, Mihov, Atanas
SSRN
This study demonstrates that, among large U.S. bank holding companies (BHCs), the largest ones are exposed to more operational risk. Specifically, they have higher operational losses per dollar of total assets, a result largely driven by the BHCs' failure to meet professional obligations to clients and/or faulty product design. Operational risk at the largest U.S. institutions is also found to: (i) be particularly persistent, (ii) have a counter-cyclical component (higher losses occur during economic downturns) and (iii) materialize through more frequent tail-risk events. We illustrate two plausible channels of BHC size that contribute to operational risk â€" institutional complexity and moral hazard incentives arising from “too-big-to-fail." Our findings have important implications for large banking organization performance, risk and supervision.
Asset Pricing with Ambiguous Signals: An Experiment
Bao, Te,Duffy, John,Zhu, Jiahua
SSRN
This paper explores how ambiguous signals and ambiguity aversion influence individuals' expectations and the pricing of asset in experimental financial markets. In line with the theory of Epstein and Schneider (2008) we find that subjects' degree of ambiguity aversion is positively correlated with their expectations about the variance of ambiguous signals. These signals matter for the determination of asset prices. We find that the distribution of the excess return of the asset exhibits negative skewness, and that price volatility is significantly larger under ambiguous signals. Our findings provide evidence in support of the idea that ambiguous information and ambiguity aversion may be a source of negative skewness and excess volatility in financial markets.
Bankruptcy Process for Sale
Ayotte, Kenneth,Ellias, Jared A.
SSRN
The lenders that fund Chapter 11 reorganizations exert significant influence over the bankruptcy process through the contract associated with the debtor-in-possession (“DIPâ€) loan. In this Article, we study a large sample of DIP loan contracts and document a trend: over the past three decades, DIP lenders have steadily increased their contractual control of Chapter 11. In fact, today’s DIP loan agreements routinely go so far as to dictate the very outcome of the restructuring process. When managers sell control over the bankruptcy case to a subset of the creditors in exchange for compensation, we call this transaction a “bankruptcy process sale.†We model two situations where process sales raise bankruptcy policy concerns: (1) when a senior creditor leverages the debtor’s need for financing to lock in a preferred outcome at the outset of the case (“plan protectionâ€); and (2) when a senior creditor steers the case to protect its claim against litigation (“entitlement protectionâ€). We show that both scenarios can lead to bankruptcy outcomes that fail to maximize the value of the firm for creditors as a whole. We study a new dataset that uses the text of 1.5 million court documents to identify creditor conflict over process sales, and our analysis offers evidence consistent with the predictions of the model.
CEO Influence on Funds from Operations (FFO) Adjustment for Real Estate Investment Trusts (REITs)
Feng, Zhilan,Lin, Zhilu,Wu, Wentao
SSRN
This paper investigates non-GAAP performance measures of the REIT industry, specifically the difference (FFO adjustment) between concurrent FFO and Net Income (NI). Using the U.S. Equity REIT data from 1993 to 2018, we first find evidence that both NI and FFO are associated with REIT market-adjusted stock returns, suggesting both contain information that is valuable to investors. Second, we document a significant and positive relationship between contemporaneous FFO adjustment and NI, indicating a possible “selective†and “intentional†inclusion and/or omission of the “good†vs. “bad†news in the FFO reporting. Third, we find direct evidence that more powerful CEOs are indeed associated with higher FFO adjustments, suggesting CEOs’ involvement in hiding subpar performance. Finally, we focus our attention in a more recent period, when the National Association of Real Estate Investment Trusts (NAREIT) provides additional clarifications and guidelines, and the U.S. Securities and Exchange Commission (SEC) increases scrutiny on FFO reporting. Our results show a diminished “manipulation†for the majority of the REITs, suggesting these guidelines and scrutiny have achieved the intended purposes. While non-GAAP performance measures might supply additional information to investors, our results indicate that providing continuous guidance and monitoring is essential.
Capital Market Liberalization and Equity Market Interdependence
Fry-McKibbin, Renee,Yan, Ziyu
SSRN
This paper uses tests drawn from the literature on financial market contagion measured by changes in higher-order comoments to establish the patterns in the interdependence between equity markets in Shanghai and Shenzhen with Hong Kong as mainland China liberalized their capital market. On the announcement of the opening of the Shanghai market correlations rise, but subside by the launch. Following the launch changes in coskewness, cokurtosis and covolatility emerge. The liberalization process is complete by mid-September 2016.
Central Bank Digital Currency: Central Banking for All?
Fernandez-Villaverde, Jesus,Sanches, Daniel,Uhlig, Harald
SSRN
The introduction of a central bank digital currency (CBDC) allows the central bank to engage in large-scale intermediation by competing with private ï¬nancial interme-diaries for deposits. Yet, since a central bank is not an investment expert, it cannot invest in long-term projects itself, but relies on investment banks to do so. We derive an equivalence result that shows that absent a banking panic, the set of allocations achieved with private ï¬nancial intermediation will also be achieved with a CBDC. Dur-ing a panic, however, we show that the rigidity of the central bank’s contract with the investment banks has the capacity to deter runs. Thus, the central bank is more stable than the commercial banking sector. Depositors internalize this feature ex-ante, and the central bank arises as a deposit monopolist, attracting all deposits away from the commercial banking sector. This monopoly might endanger maturity transformation.
Changes in Household Net Financial Assets After the Great Recession: Did Financial Planners Make a Difference?
Joseph W. Goetz,Lance Palmer,Lini Zhang,Swarn Chatterjee
arXiv
This study utilized the 2007-2009 Survey of Consumer Finances (SCF) panel dataset to examine the impact of financial planner use on household net financial asset level during the Great recession. Data included 3,862 respondents who completed the SCF survey and a follow up interview. The results indicated that starting to use a financial planner during the Great Recession had a positive impact on preserving and increasing the value of households' net financial assets, while curtailing the use of a financial planner during this time had a negative impact on preserving the value of households' financial assets. Thus, study findings indicated that the benefit of using a financial planner maybe particularly high during a major financial downturn.
Contingent Convertible Obligations and Financial Stability
Zachary Feinstein,T. R. Hurd
arXiv
This paper investigates whether a financial system can be made more stable if financial institutions share risk by exchanging contingent convertible (CoCo) debt obligations. The question is framed in a financial network model of debt and equity interlinkages with the addition of a variant of the CoCo that converts continuously when a bank's equity-debt ratio drops to a trigger level. The main theoretical result is a complete characterization of the clearing problem for the interbank debt and equity at the maturity of the obligations. We then consider a simple setting in which introducing contingent convertible bonds improves financial stability, as well as specific networks for which contingent convertible bonds do not provide uniformly improved system performance. To return to the main question, we examine the EU financial network at the time of the 2011 EBA stress test to do comparative statics to study the implications of CoCo debt on financial stability. It is found that by replacing all unsecured interbank debt by standardized CoCo interbank debt securities, systemic risk in the EU will decrease and bank shareholder value will increase.
Does Spurious Mean Reversion in Basis Changes Still Exist After the Introduction of Exchange Traded Funds
Richie, Nivine,Muthuswamy, Jayaram,Segara, Reuben,Webb, Robert I.
SSRN
In their seminal Journal of Finance article, Miller, Muthuswamy, and Whaley (MMW) [1994] document that the observed mean reversion of changes in the basis of cash and stock index futures prices is likely illusory. MMW use a simple time-series model to suggest that the apparent mean-reversion in the basis is a spurious artifact of non-synchronous prices between index futures and cash markets â€" rather than an indication of exploitable weak-form market inefficiency. Because the MMW effect is predominantly driven by liquidity differentials between cash and futures prices, the question naturally arises as to whether one would observe the same MMW phenomenon in the behaviour of the “basis†or difference between more actively traded ETF and cash market prices. This study attempts to answer that question by examining the “basis†behavior of the Standard and Poor’s Depository Receipt (SPDR) ETF traded on the American Stock Exchange. Overall, we find that the MMW phenomenon still persists strongly after the advent of Exchange Traded Funds. Moreover, an examination of the spread or “basis†between cash and ETF prices and the spread or “basis†between futures and ETF prices shows that the apparent mean reversion in both is even more pronounced than in the basis between cash and futures prices. This demonstrates that the MMW effect is extremely robust and unlikely to “go-away†soon.
Doubling Down on the Safe(ty) Bet: Bailouts and Risk-Shifting at the Intensive Margin
Eufinger, Christian,Ye, Zhiqiang
SSRN
Banks have a significant funding-cost advantage since their liabilities are protected by various government safety nets. We construct a corporate finance-style model that shows that banks can exploit this funding-cost advantage by just intermediating funds between investors and ultimate borrowers, thereby earning the spread between their reduced funding rate and the competitive market rate. This mechanism leads to a crowding-out of direct market finance and real effects for bank borrowers through bank risk-shifting at the intensive margin. That is, banks induce their borrowers to leverage excessively, to overinvest, and to conduct inferior high-risk projects.
Egalitarian and Just Digital Currency Networks
Gal Shahaf,Ehud Shapiro,Nimrod Talmon
arXiv
Cryptocurrencies are a digital medium of exchange with decentralized control that renders the community operating the cryptocurrency its sovereign. Leading cryptocurrencies use proof-of-work or proof-of-stake to reach consensus, thus are inherently plutocratic. This plutocracy is reflected not only in control over execution, but also in the distribution of new wealth, giving rise to rich get richer'' phenomena. Here, we explore the possibility of an alternative digital currency that is egalitarian in control and just in the distribution of created wealth. Such currencies can form and grow in grassroots and sybil-resilient way. A single currency community can achieve distributive justice by egalitarian coin minting, where each member mints one coin at every time step. Egalitarian minting results, in the limit, in the dilution of any inherited assets and in each member having an equal share of the minted currency, adjusted by the relative productivity of the members. Our main theorem shows that a currency network, where agents can be members of more than one currency community, can achieve distributive justice globally across the network by \emph{joint egalitarian minting}, where each agent mints one coin in only one community at each timestep. Equality and distributive justice can be achieved among people that own the computational agents of a currency community provided that the agents are genuine (unique and singular). We show that currency networks are sybil-resilient, in the sense that sybils (fake or duplicate agents) affect only the communities that harbour them, and not hamper the ability of genuine (sybil-free)communities in a network to achieve distributed justice.
Evolution of the Chinese Guarantee Network under Financial Crisis and Stimulus Program
Yingli Wang,Qingpeng Zhang,Xiaoguang Yang
arXiv
Our knowledge about the evolution of guarantee network in downturn period is limited due to the lack of comprehensive data of the whole credit system. Here we analyze the dynamic Chinese guarantee network constructed from a comprehensive bank loan dataset that accounts for nearly 80% total loans in China, during 01/2007-03/2012. The results show that, first, during the 2007-2008 global financial crisis, the guarantee network became smaller, less connected and more stable because of many bankruptcies; second, the stimulus program encouraged mutual guarantee behaviors, resulting in highly reciprocal and fragile network structure; third, the following monetary policy adjustment enhanced the resilience of the guarantee network by reducing mutual guarantees. Interestingly, our work reveals that the financial crisis made the network more resilient, and conversely, the government bailout degenerated network resilience. These counterintuitive findings can provide new insight into the resilience of real-world credit system under external shocks or rescues.
How and Why Do Managers Use Public Forecasts to Guide the Market?
Charoenwong, Ben,Kimura, Yosuke,Kwan, Alan
SSRN
We compare publicly disclosed forecasts and internal forecasts collected by confidential government surveys using a sample of publicly-listed Japanese firms. Both forecasts are mandatory and meaningfully predict corporate policy but on average public forecasts are pessimistic relative to internal forecasts. Firms with greater shareholder pressure and bonus-related compensation are more pessimistic. Public pessimism likely guides market beliefs down, predicting higher future stock returns, earnings surprises, and executive, but not rank-and-file, compensation. However, it flips to optimism when firms are financially constrained, consistent with an inter-temporal trade-off between benefits from meeting managerial goalposts versus maintaining financial flexibility.
Implied Dividend Yield as a New Stock Market Valuation Measure
Taran Grove,Michael Reyes,Andrey Sarantsev
arXiv
Long-run total real returns of the USA stock market are approximately equal to long-run real earnings growth plus average dividend yield. However, earnings can be distributed to shareholders in various ways: dividends, stock buybacks, debt payments. Thus the total returns minus earnings growth can be considered as implied dividend yield. This quantity must be stable in the long run. If the converse is true: this quantity is abnormally high in the last few years, then the market is overpriced. A measure of this is (detrended) cumulative sum of differences. We regress next year's implied dividend yield upon this current valuation measure. We simulate future returns, starting from the current market conditions. We reject the conventional wisdom that currently the market is overpriced. In our model the current market is undervalued and is likely to grow faster than historically. We show that this measure has better predictive power than the P/E and CAPE ratios.
Long-range memory test by the burst and inter-burst duration distribution
Vygintas Gontis
arXiv
It is empirically established that order flow in the financial markets is positively auto-correlated and can serve as an example of a social system with long-range memory. Nevertheless, widely used long-range memory estimators give varying values of the Hurst exponent. We propose the burst and inter-burst duration statistical analysis as one more test of long-range memory and implement it with the limit order book data comparing it with other widely used estimators. This method gives a more reliable evaluation of the Hurst exponent independent of the stock in consideration or time definition used. Results strengthen the expectation that burst and inter-burst duration analysis can serve as a better method to investigate the property of long-range memory.
Machine Learning Fund Categorizations
arXiv
Given the surge in popularity of mutual funds (including exchange-traded funds (ETFs)) as a diversified financial investment, a vast variety of mutual funds from various investment management firms and diversification strategies have become available in the market. Identifying similar mutual funds among such a wide landscape of mutual funds has become more important than ever because of many applications ranging from sales and marketing to portfolio replication, portfolio diversification and tax loss harvesting. The current best method is data-vendor provided categorization which usually relies on curation by human experts with the help of available data. In this work, we establish that an industry wide well-regarded categorization system is learnable using machine learning and largely reproducible, and in turn constructing a truly data-driven categorization. We discuss the intellectual challenges in learning this man-made system, our results and their implications.
Macro-Finance Decoupling: Robust Evaluations of Macro Asset Pricing Models
Cheng, Xu,Dou, Winston Wei,Liao, Zhipeng
SSRN
This paper shows that robust inference under weak identiï¬cation is important to the evaluation of many influential macro asset pricing models, including long-run risk models, disaster risk models, and multifactor linear asset pricing models. Building on recent developments in the conditional inference literature, we provide a new speciï¬cation test by simulating the critical value conditional on a sufficient statistic. This sufficient statistic can be intuitively interpreted as a measure capturing the macroeconomic information decoupled from the underlying content of asset pricing theories. Macro-ï¬nance decoupling is an effective way to improve the power of our speciï¬cation test when asset pricing theories are difficult to refute due to an imbalance in the information content about the key model parameters between macroeconomic moment restrictions and asset pricing cross-equation restrictions.
Measuring and Visualizing Place-Based Space-Time Job Accessibility
Yujie Hu,Joni Downs
arXiv
Place-based accessibility measures, such as the gravity-based model, are widely applied to study the spatial accessibility of workers to job opportunities in cities. However, gravity-based measures often suffer from three main limitations: (1) they are sensitive to the spatial configuration and scale of the units of analysis, which are not specifically designed for capturing job accessibility patterns and are often too coarse; (2) they omit the temporal dynamics of job opportunities and workers in the calculation, instead assuming that they remain stable over time; and (3) they do not lend themselves to dynamic geovisualization techniques. In this paper, a new methodological framework for measuring and visualizing place-based job accessibility in space and time is presented that overcomes these three limitations. First, discretization and dasymetric mapping approaches are used to disaggregate counts of jobs and workers over specific time intervals to a fine-scale grid. Second, Shen (1998) gravity-based accessibility measure is modified to account for temporal fluctuations in the spatial distributions of the supply of jobs and the demand of workers and is used to estimate hourly job accessibility at each cell. Third, a four-dimensional volumetric rendering approach is employed to integrate the hourly job access estimates into a space-time cube environment, which enables the users to interactively visualize the space-time job accessibility patterns. The integrated framework is demonstrated in the context of a case study of the Tampa Bay region of Florida. The findings demonstrate the value of the proposed methodology in job accessibility analysis and the policy-making process.
Mixing LSMC and PDE Methods to Price Bermudan Options
David Farahany,Kenneth Jackson,Sebastian Jaimungal
arXiv
We develop a mixed least squares Monte Carlo-partial differential equation (LSMC-PDE) method for pricing Bermudan style options on assets whose volatility is stochastic. The algorithm is formulated for an arbitrary number of assets and volatility processes and we prove the algorithm converges almost surely for a class of models. We also discuss two methods to improve the algorithm's computational complexity. Our numerical examples focus on the single ($2d$) and multi-dimensional ($4d$) Heston models and we compare our hybrid algorithm with classical LSMC approaches. In each case, we find that the hybrid algorithm outperforms standard LSMC in terms of estimating prices and optimal exercise boundaries.
Moment Approximations of Displaced Forward-LIBOR Rates with Application to Swaptions
van Appel, Jacques,McWalter, Thomas
SSRN
We present an algorithm to approximate moments for forward rates under a displaced lognormal forward-LIBOR model (DLFM). Since the joint distribution of rates is unknown, we use a multi-dimensional full weak order 2.0 Ito-Taylor expansion in combination with a second-order Delta method. This more accurately accounts for state dependence in the drift terms, improving upon previous approaches. To verify this improvement we conduct quasi-Monte Carlo simulations. We use the new mean approximation to provide an improved swaption volatility approximation, and compare this to the approaches of Rebonato, Hull-White and Kawai, adapted to price swaptions under the DLFM. Rebonato and Hull-White are found to be the least accurate. While Kawai is the most accurate, it is computationally inefficient. Numerical results show that our approach strikes a balance between accuracy and efficiency.
Mortality containment vs. economics opening: optimal policies in a SEIARD model
Andrea Aspri,Elena Beretta,Alberto Gandolfi,Etienne Wasmer
arXiv
We adapt a SEIRD differential model with asymptomatic population and Covid deaths, which we call SEAIRD, to simulate the evolution of COVID-19, and add a control function affecting both the diffusion of the virus and GDP, featuring all direct and indirect containment policies; to model feasibility, the control is assumed to be a piece-wise linear function satisfying additional constraints. We describe the joint dynamics of infection and the economy and discuss the trade-off between production and fatalities. In particular, we carefully study the conditions for the existence of the optimal policy response and its uniqueness. Uniqueness crucially depends on the marginal rate of substitution between the statistical value of a human life and GDP; we show an example with a phase transition: above a certain threshold, there is a unique optimal containment policy; below the threshold, it is optimal to abstain from any containment; and at the threshold itself there are two optimal policies. We then explore and evaluate various profiles of various control policies dependent on a small number of parameters.
On Policy Evaluation with Aggregate Time-Series Shocks
Dmitry Arkhangelsky,Vasily Korovkin
arXiv
We propose a general strategy for estimating treatment effects, in contexts where the only source of exogenous variation is a sequence of aggregate time-series shocks. We start by arguing that commonly used estimation procedures tend to ignore crucial time-series aspects of the data. Next, we develop a graphical tool and a formal test to illustrate the issues of the design using data from prominent studies in development economics and macroeconomic. Motivated by these studies, we construct a new IV estimator, which is based on the time-series model for the aggregate shock. We analyze the statistical properties of our estimator in a practically relevant case, where both cross-sectional and time-series dimensions are of similar size. Finally, to provide a causal interpretation for our estimator, we analyze a new causal model that allows taking into account both rich unobserved heterogeneity in potential outcomes and unobserved aggregate shocks.
On the optimality of joint periodic and extraordinary dividend strategies
Benjamin Avanzi,Hayden Lau,Bernard Wong
arXiv
In this paper, we model the cash surplus (or equity) of a risky business with a Brownian motion. Owners can take cash out of the surplus in the form of dividends'', subject to transaction costs. However, if the surplus hits 0 then ruin occurs and the business cannot operate any more.
We consider two types of dividend distributions: (i) periodic, regular ones (that is, dividends can be paid only at countable many points in time, according to a specific arrival process); and (ii) extraordinary dividend payments that can be made immediately at any time (that is, the dividend decision time space is continuous and matches that of the surplus process). Both types of dividends attract proportional transaction costs, and extraordinary distributions also attracts fixed transaction costs, a realistic feature. A dividend strategy that involves both types of distributions (periodic and extraordinary) is qualified as hybrid''.
We determine which strategies (either periodic, immediate, or hybrid) are optimal, that is, we show which are the strategies that maximise the expected present value of dividends paid until ruin, net of transaction costs. Sometimes, a liquidation strategy (which pays out all monies and stops the process) is optimal. Which strategy is optimal depends on the profitability of the business, and the level of (proportional and fixed) transaction costs. Results are illustrated.
Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems
Yu-Jui Huang,Zhenhua Wang
arXiv
We study an optimal stopping problem under non-exponential discounting, where the state process is a multi-dimensional continuous strong Markov process. The discount function is taken to be log sub-additive, capturing decreasing impatience in behavioral economics. On strength of probabilistic potential theory, we establish the existence of an optimal equilibrium among a sufficiently large collection of equilibria, consisting of finely closed equilibria satisfying a boundary condition. This generalizes the existence of optimal equilibria for one-dimensional stopping problems in prior literature.
Optimal Investing after Retirement Under Time-Varying Risk Capacity Constraint
Weidong Tian,Zimu Zhu
arXiv
This paper studies an optimal investing problem for a retiree facing longevity risk and living standard risk. We formulate the investing problem as a portfolio choice problem under a time-varying risk capacity constraint. We derive the optimal investment strategy under the specific condition on model parameters in terms of second-order ordinary differential equations. We demonstrate an endogenous number that measures the expected value to sustain the spending post-retirement. The optimal portfolio is nearly neutral to the stock market movement if the portfolio's value is higher than this number; but, if the portfolio is not worth enough to sustain the retirement spending, the retiree actively invests in the stock market for the higher expected return. Besides, we solve an optimal portfolio choice problem under a leverage constraint and show that the optimal portfolio would lose significantly in stressed markets. This paper shows that the time-varying risk capacity constraint has important implications for asset allocation in retirement.
Power Trades and Network Congestion Externalities
Nayara Aguiar,Indraneel Chakraborty,Vijay Gupta
arXiv
As power generation by renewable sources increases, power transmission patterns over the electric grid change. We show that due to physical laws, these new transmission patterns lead to non-intuitive grid congestion externalities. We derive the conditions under which network externalities due to power trades occur. Calibration shows that each additional unit of power traded between northern and western Europe reduces transmission capacity for the southern and eastern regions by 27% per unit traded. Given such externalities, new investments in the electric grid infrastructure cannot be made piecemeal. Power transit fares can help finance investment in regions facing network congestion externalities.
Reserves and Risk: Evidence from China
Fatum, Rasmus,Hattori, Takahiro,Yamamoto, Yohei
SSRN
We consider if the Chinese accumulation of reserves is associated with unintended consequences in the form of increased private sector risk taking. Using sovereign credit default swap spreads and stock index prices as indicators of risk taking, we provide evidence to suggest that as reserve holdings increase, so does the willingness of the private sector to take on more risk. This is an important finding that adds credence to the suggestion that insurance through costly reserves, to be used in the event of a crisis, may lead to private sector actions that in and of themselves make it more likely that this insurance will be used.
Scenes from a Monopoly: Renewable Resources and Quickest Detection of Regime Shifts
Neha Deopa,Daniele Rinaldo
arXiv
We study the stochastic dynamics of a renewable resource harvested by a monopolist facing a downward sloping demand curve. We introduce a framework where harvesting sequentially affects the resource's potential to regenerate, resulting in an endogenous ecological regime shift. In a multi-period setting, the firm's objective is to find the profit-maximizing harvesting policy while simultaneously detecting in the quickest time possible the change in regime. Solving analytically, we show that a negative regime shift induces an aggressive extraction behaviour due to shorter detection periods, creating a sense of urgency, and higher markup in prices. Precautionary behaviour can result due to decreasing resource rent. We study the probability of extinction and show the emergence of catastrophe risk which can be both reversible and irreversible.
Shadow Banking in China Compared to Other Countries
Allen, Franklin,Gu, Xian
SSRN
China’s shadow banking has been rising rapidly in the last decade, mainly driven by regulations for banks, the Fiscal Stimulus Plan in 2008, and credit constraints in restrictive industries. This sector has continued growing although the regulators repeatedly attempted to impose new regulations on banks and non-banks. The existence of shadow banking fulfills the high demand for funding. The standard view is that it poses risks to financial stability. However, in China this is not necessarily the case. Entrusted loans, implicit guarantees from non-banks, banks or government may provide a second-best arrangement in funding risky projects and improving welfare.
Shining a Light in a Dark Corner: EDGAR Search Activity Reveals the Strategically Leaked Plans of Activist Investors
Flugum, Ryan,Lee, Choonsik,Souther, Matthew E.
SSRN
We document a network of information flow between activists and other investors during the 10 days prior to the announcement of a campaign. We use EDGAR search activity matched to institutional investor IP addresses to identify investors who persistently download information on an individual activist’s campaign targets in the days prior to that activist’s 13D disclosures. This pattern of informed EDGAR access suggests leaked information from the activist to the unaffiliated institutional investor, who is not named in the 13D filing. We find that the presence of these informed investors is associated with higher pre-13D turnover, higher post-13D returns, and an increased likelihood of the activist pursuing and winning a proxy fight.
Sig-SDEs model for quantitative finance
Imanol Perez Arribas,Cristopher Salvi,Lukasz Szpruch
arXiv
Mathematical models, calibrated to data, have become ubiquitous to make key decision processes in modern quantitative finance. In this work, we propose a novel framework for data-driven model selection by integrating a classical quantitative setup with a generative modelling approach. Leveraging the properties of the signature, a well-known path-transform from stochastic analysis that recently emerged as leading machine learning technology for learning time-series data, we develop the Sig-SDE model. Sig-SDE provides a new perspective on neural SDEs and can be calibrated to exotic financial products that depend, in a non-linear way, on the whole trajectory of asset prices. Furthermore, we our approach enables to consistently calibrate under the pricing measure $\mathbb Q$ and real-world measure $\mathbb P$. Finally, we demonstrate the ability of Sig-SDE to simulate future possible market scenarios needed for computing risk profiles or hedging strategies. Importantly, this new model is underpinned by rigorous mathematical analysis, that under appropriate conditions provides theoretical guarantees for convergence of the presented algorithms.
Stress Testing as a Tool for Monitoring and Modelling the Dynamics of Business Activity of Manufacturing Enterprises in Russia in the Face of Market Shocks: Short-term Scenarios of Industry Tendencies
Lola, Inna,Manukov, Anton,Bakeev, Murat
SSRN
The article proposes a methodology for using macro-level stress testing based on the results of business tendency surveys to study possible scenarios for the development of crisis dynamics triggered by external unforeseen supply and demand shocks, as in the case of the COVID-19 pandemic, as well as a review of existing approaches in the field of stress testing and building stress indices with an emphasis on methods based on vector autoregressive models and their various modifications.The basis for empirical calculations is data from business tendency surveys of the leaders of Russian manufacturing enterprises, reflecting their combined estimates of the current state of business activity. Based on the results of business tendency surveys, four composite indices were formed reflecting various aspects of business activity of enterprises: demand index, production index, finance index and employment index. Index values calculated monthly from 2008 to March 2020 were used to build the Bayesian vector autoregressive model (BVAR). This model was used to predict the dynamics of indices under the condition of four possible shock scenarios: short-term shock, V-shaped shock, W-shaped shock and U-shaped shock. Moreover, for each of the scenarios, cases of a shock of demand, a shock of production, and a simultaneous shock of demand and production were separately considered.The results indicated the key role of demand in the dynamics of all the indices under consideration, the W-shaped shock, as the worst of the considered scenarios, as well as the relatively greater sensitivity of the employment index to the demand index and the finance index to the production index.
Tail Granger Causalities and Where to Find Them: Extreme Risk Spillovers vs Spurious Linkages
Mazzarisi, Piero,Zaoli, Silvia,Campajola, Carlo,Lillo, Fabrizio
SSRN
Identifying risk spillovers in financial markets is of great importance for assessing systemic risk and portfolio management. Granger causality in tail (or in risk) tests whether past extreme events of a time series help predicting future extreme events of another time series. The topology and connectedness of networks built with Granger causality in tail can be used to measure systemic risk and to identify risk transmitters. Here we introduce a novel test of Granger causality in tail which adopts the likelihood ratio statistic and is based on the multivariate generalization of a discrete autoregressive process for binary time series describing the sequence of extreme events of the underlying price dynamics. The proposed test has very good size and power in finite samples, especially for large sample size, allows inferring the correct time scale at which the causal interaction takes place, and it is flexible enough for multivariate extension when more than two time series are considered in order to decrease false detections as spurious effect of neglected variables. An extensive simulation study shows the performances of the proposed method with a large variety of data generating processes and it introduces also the comparison with the test of Granger causality in tail by [Hong et al., 2009]. We report both advantages and drawbacks of the different approaches, pointing out some crucial aspects related to the false detections of Granger causality for tail events. An empirical application to high frequency data of a portfolio of US stocks highlights the merits of our novel approach.
Tail Risk Transmission: A Study of Iran Food Industry
Mojtahedi, Fatemeh,Mojaverian, Seyed Mojtaba,Ahelegbey, Daniel Felix,Giudici, Paolo
SSRN
This paper extends the extreme downside correlations and hedge (EDC and EDH) methodology of Harris et al. (2019) to model the tail risk co-movement of financial assets under severe firm-level and market conditions. The model is applied to analyze both systematic and systemic exposures in the Iranian food industry. The empirical application address the following questions: 1) which food company is the safest for investors to diversify their investment, and 2) which companies are the risk transmitters'' and receivers'', especially in turbulent times. To this end, we sampled the time series of 11 manufacturing companies and proxy the market indicator with the food industry index, all of which are publicly listed on the Tehran Stock Exchange (TSE). The data covers daily close prices from October 5, 2015, to January 15, 2020. The systematic analysis reveals a positive and statistically significant relationship between the tail risk of the companies and the market index. The centrality analysis of the systemic exposures reveals Mahram Manufacturing as the safest and Behshahr Industries as the riskiest company. We also find evidence that W.Azar.Pegah is the main transmitter'' of tail risk, while Pegah.Fars.Co is the main receiver'' of risk.
The Effects of Board Structure on Corporate Performance: Evidence from East African Frontier Markets
Guney, Yilmaz,Karpuz, Ahmet,Komba, Gabriel
SSRN
The effectiveness of the well-known corporate governance practices may not be universal due to fundamental differences in the environments under which firms operate. By using hand-collected data from all the non-financial firms listed on the unexplored East African frontier markets (i.e., Kenya, Tanzania and Uganda), we examine the effect of board characteristics on the performance of firms. Our results show that board size has a negative and significant effect on firm performance. The presences of foreigners and civil servants on the board play positive roles on financial performance, where the agency and resource dependence theories apply. Further, we find that board members with higher education also contribute to firm performance. These findings still hold when we consider the 2008-2009 financial crisis period. Overall, we show that in a business climate where ownership is largely dominated by few shareholders, the conventional governance mechanisms do not work effectively.
The Hyperbolic Geometry of Financial Networks
Keller-Ressel, Martin,Nargang, Stephanie
SSRN
Based on data from the European banking stress tests of 2014, 2016 and the transparency exercise of 2018 we demonstrate for the first time that the latent geometry of financial networks can be well-represented by geometry of negative curvature, i.e., by hyperbolic geometry. This allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos et al., which is based on the Poincaré disc model of hyperbolic geometry. We show that the latent dimensions of popularity' and similarity' in this model are strongly associated to systemic importance and to geographic subdivisions of the banking system. In a longitudinal analysis over the time span from 2014 to 2018 we find that the systemic importance of individual banks has remained rather stable, while the peripheral community structure exhibits more (but still moderate) variability.
The Impact of M&A on Performance: Alternative Measures of Rates of Profit
Meeks, Geoff,Meeks, Jaqueline
SSRN
Previous studies of the impact of M&A on performance have employed a range of measures of “profitability†or “rate of returnâ€. Sometimes they have provided little in the way of rationalization; and sometimes the most appropriate measures have not been deployed for testing the chosen hypothesis, or supporting the final inferences. Here we explore a range of measures, their relation one to another, and caveats to their use in assessing operating efficiency or in monitoring the gains to shareholders. We discuss the profit margin, the return on net assets, the return on equity, earnings per share, and the total shareholder return.We show that the reported change in margin following M&A will, other things equal, overstate any improvement in operating performance where the M&A increases vertical integration. We analyse the difference between the accounting return on net assets and the accounting return on equity, examining the impact of associated changes in capital structure and/or tax arrangements: we report a consequent tendency for the return on equity (and earnings per share, but not the return on net assets) to overstate any improvement in operating performance after M&A. We describe the difference between the accounting return on equity and the total shareholder return, exploring the impact on the latter of changes in stock market perceptions of the combination’s prospects.
The Impact of Naked Short Selling on the Securities Lending and Equity Market
Lecce, Steven,Lepone, Andrew,McKenzie, Michael D.,Segara, Reuben
SSRN
This paper examines the impact of naked short selling on equity markets where it is restricted to securities on an approved list. Consistent with Miller's (1977) intuition, stocks with the highest dispersion of opinions and short sale constraints are the only stocks to exhibit significant and negative abnormal returns in the post-event period. We also find slightly higher stock return volatility and a small reduction in liquidity when naked short sales are allowed. Overall, it impairs market quality (liquidity and volatility), although there appears to be some improvement in price efficiency in stocks with high short sale constraints.
The Impact of Trading Halts on Liquidity and Price Volatility: Evidence From the Australian Stock Exchange
Frino, Alex,Lecce, Steven,Segara, Reuben
SSRN
This study examines market behaviour around trading halts associated with information releases on the Australian Stock Exchange, which operates an open electronic limit order book. Using the Lee, Ready and Seguin (1994) pseudo-halt methodology, we find trading halts increase both volume and price volatility. Trading halts also increase bid-ask spreads and reduce market depth at the best-quotes in the immediate post-halt period. The results of this study imply that trading halts impair rather than improve market quality in markets that operate open electronic limit order books.
The Importance of Cognitive Domains and the Returns to Schooling in South Africa: Evidence from Two Labor Surveys
Plamen Nikolov,Nusrat Jimi
arXiv
Numerous studies have considered the important role of cognition in estimating the returns to schooling. How cognitive abilities affect schooling may have important policy implications, especially in developing countries during periods of increasing educational attainment. Using two longitudinal labor surveys that collect direct proxy measures of cognitive skills, we study the importance of specific cognitive domains for the returns to schooling in two samples. We instrument for schooling levels and we find that each additional year of schooling leads to an increase in earnings by approximately 18-20 percent. Furthermore, we estimate and demonstrate the importance of specific cognitive domains in the classical Mincer equation. We find that executive functioning skills are important drivers of earnings in the rural sample, whereas higher-order cognitive skills are more important for determining earnings in the urban sample.
The Information Content of NAV Estimates
Chacon, Ryan,French, Dan,Pukthuanthong, Kuntara
SSRN
This paper investigates whether analysts’ estimates of firm fundamental value transmit unique information to security markets. Previous work has not studied analyst value estimates because of the scarcity of the release of such data. This study circumvents that limitation by considering the one type of firm for which a large sample of value estimates, known as the net asset value (NAV), exists: Real Estate Investment Trusts (REITs). Using a sample of 200 Equity REITs from 2001 to 2015, we document significant abnormal returns and share turnover on the announcement date of NAV revisions. This response is consistent with market reactions to announcements of other types of analysts’ estimates: earnings forecasts, price targets, and buy/sell recommendations. Our findings remain significant after controlling for these, suggesting the information contained in NAV revisions is incremental to that contained in other analyst estimates. Consistent with efficient information transmission, the market absorbs this new information quickly and completely.
The Sectoral Effects of Value-Added Tax: Evidence from UAE Stock Markets
Gopakumar, Anagha Ann
SSRN
This paper investigates the impact of 19 announcements relating to the introduction of value-added tax (VAT) in the United Arab Emirates (UAE), on the equities listed on Abu Dhabi Stock Exchange (ADX). Using a well-established event study methodology applied on daily data over the period from 2016 to 2018, a sector-wise assessment of the value constructiveness or destructiveness of these announcements is conducted. Significant sectoral differences in abnormal returns are expected, with industries like insurance and retail showing higher sensitivity. Some announcements are also expected to have a more substantial impact than the rest, irrespective of sectors. These results are highly valuable as a source of information regarding the sectoral effects of VAT in the UAE, as well as a reference for the possible impact of VAT introduction in other countries in the Gulf region and the rest of the world.
The impact of COVID-19 on the UK fresh food supply chain
Rebecca Mitchell,Roger Maull,Simon Pearson,Steve Brewer,Martin Collison
arXiv
The resilience of the food supply chain is a matter of critical importance, both for national security and broader societal well bring. COVID19 has presented a test to the current system, as well as means by which to explore whether the UK's food supply chain will be resilient to future disruptions. In the face of a growing need to ensure that food supply is more environmentally sustainable and socially just, COVOD19 also represents an opportunity to consider the ability of the system to innovative, and its capacity for change. The purpose of this case based study is to explore the response and resilience of the UK fruit and vegetable food supply chain to COVID19, and to assess this empirical evidence in the context of a resilience framework based on the adaptive cycle. To achieve this we reviewed secondary data associated with changes to retail demand, conducted interviews with 23 organisations associated with supply to this market, and conducted four video workshops with 80 organisations representing half of the UK fresh produce community. The results highlight that, despite significant disruption, the retail dominated fresh food supply chain has demonstrated a high degree of resilience. In the context of the adaptive cycle, the system has shown signs of being stuck in a rigidity trap, as yet unable to exploit more radical innovations that may also assist in addressing other drivers for change. This has highlighted the significant role that innovation and R&D communities will need to play in enabling the supply chain to imagine and implement alternative future states post COVID.
The impacts of asymmetry on modeling and forecasting realized volatility in Japanese stock markets
Daiki Maki,Yasushi Ota
arXiv
This study investigates the impacts of asymmetry on the modeling and forecasting of realized volatility in the Japanese futures and spot stock markets. We employ heterogeneous autoregressive (HAR) models allowing for three types of asymmetry: positive and negative realized semivariance (RSV), asymmetric jumps, and leverage effects. The estimation results show that leverage effects clearly influence the modeling of realized volatility models. Leverage effects exist for both the spot and futures markets in the Nikkei 225. Although realized semivariance aids better modeling, the estimations of RSV models depend on whether these models have leverage effects. Asymmetric jump components do not have a clear influence on realized volatility models. While leverage effects and realized semivariance also improve the out-of-sample forecast performance of volatility models, asymmetric jumps are not useful for predictive ability. The empirical results of this study indicate that asymmetric information, in particular, leverage effects and realized semivariance, yield better modeling and more accurate forecast performance. Accordingly, asymmetric information should be included when we model and forecast the realized volatility of Japanese stock markets.
Toxic Hedging | 2023-01-27 07:17:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32311177253723145, "perplexity": 2940.127884891682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494974.98/warc/CC-MAIN-20230127065356-20230127095356-00426.warc.gz"} |
https://elteoremadecuales.com/cantors-theorem/ | # Cantor's theorem
Cantor's theorem For other theorems bearing Cantor's name, see Cantor's theorem (disambiguation). The cardinality of the set {x, y, z}, is three, while there are eight elements in its power set (3 < 23 = 8), here ordered by inclusion. This article contains special characters. Without proper rendering support, you may see question marks, boxes, or other symbols. In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set {displaystyle A} , the set of all subsets of {displaystyle A,} the power set of {displaystyle A,} has a strictly greater cardinality than {displaystyle A} itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with {displaystyle n} elements has a total of {displaystyle 2^{n}} subsets, and the theorem holds because {displaystyle 2^{n}>n} for all non-negative integers.
Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also. As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details.
The theorem is named for German mathematician Georg Cantor, who first stated and proved it at the end of the 19th century. Cantor's theorem had immediate and important consequences for the philosophy of mathematics. For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest cardinal number (colloquially, "there's no largest infinity").
Contents 1 Proof 2 When A is countably infinite 3 Related paradoxes 4 History 5 Generalizations 6 See also 7 References 8 External links Proof Cantor's argument is elegant and remarkably simple. The complete proof is presented below, with detailed explanations to follow. | 2022-11-30 19:38:02 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9197060465812683, "perplexity": 338.4541962125368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710771.39/warc/CC-MAIN-20221130192708-20221130222708-00694.warc.gz"} |
https://gamedev.stackexchange.com/questions/86098/are-there-any-hex-tile-sizes-where-both-width-and-height-are-integers | # Are there any hex tile sizes where both width and height are integers?
I'm trying to figure out the optimal width and height (in pixels) to start building hex tiles for game development. My preference is for "flat topped" hex grids, but the math is similar for both.
I am looking for an "optimal" tile size that allows both the width and the height of the tile to be a rounded pixel number, based on the fact that height = sqrt(3)/2 * width.
My math skills being virtually nonexistent, I just ran a brute force script that ran through widths from 1 to 1024 and did not come up with a single value for w where h was an integer. Is this really the case? How does anyone create pixel-perfect hex tiles if there's no even width & height size that can accommodate a perfect hex aspect ratio?
• This is not important to gameplay. It is a form of procrastination. If it very important to you, look for the closest match instead of an actual fit. Oct 19 '14 at 6:25
• You said "pixel", right? So you're talking about programming? Internally, you would work with ints to say which cell you're in (there should be online resources about hex grids), and the drawing of the lines will be done by the computer. (Think: You can't draw a circle, either.) Oct 19 '14 at 20:24
• If you're a curious type then by all means read this where it says "Proof by infinite descent". Just Ctrl + f to find it. Oct 20 '14 at 10:27
• @Zehelvion haha and NOW I know what you mean by "procrastination" - I just spend the last 2 hours shaving the irrational numbers yak, and NOT creating a hex-tile based game. Oct 20 '14 at 15:49
• That must be quite a yak, since its fleece go on and on when represented decimally and never repeat the same pattern (really) . I didn't remember that reference from Ren & Stimpy; it's good to know. :) Oct 21 '14 at 3:40
No. √3 is an irrational number, and by definition an irrational number can not be used as a ratio between two natural numbers (integers) such as pixel counts.
However, there is no rule that says you have to use ideal hexagons in your game tiles. If you approximate it closely and avoid any miscalculations that may result, which you should be able to do with integer math anyway, you can get a good-looking product while working with easy numbers behind the scenes (if you can call 100 and 173 easy to work with).
• Nice, but √3 is the irrational number sqrt(3)=1.7320508075688772 is (say) a double, and can certainly be expressed as a ratio of integers (138907099/80198051). Oct 19 '14 at 23:10
• @SeanD Any number represented as a double or a float is a rational number. I don't see where you are going with this? Oct 20 '14 at 10:30
• NaNs are doubles, but they aren't rational. The answer claims "sqrt(3) is an irrational number" which is false in the context of programming, I was trying to draw a distinction between computer numbers and the real numbers. Oct 20 '14 at 10:47
• @SeanD Good point, computers store a close rational approximation of irrational numbers. In fact, for most rational numbers, computers store a close rational approximation also. So you could have a "perfect" hexagon in terms of the limited computer precision. We can only store 2^(numOfBits) of possible numbers within memory and there is an infinite amount of rational numbers between 0 .. 1, let alone irrational numbers of which there is greater infinite amount. Oct 20 '14 at 10:48
• Thanks for locating the √ character for me; I'll incorporate it into my answer so we don't need to argue about floating point precision. Oct 20 '14 at 13:49
Just in case anyone is interested:
Lets assume sqrt(3) is rational:
1. Therefore, there must be two integral numbers a and b such that a/b = sqrt(3)
2. We assume these numbers are coprime, if they have a common factor, we divide by it producing a coprime pair, a and b
3. We know that (a/b)^2 = 3 and therefore a^2 = 3 * b^2.
4. 3 * b^2 is devisible by 3 as b^2 is integral and therefore a^2 is also devisible by 3.
5. There are not integral numbers square is devisible by 3, but they are not. so it follows that a itself is devisible by 3. Lets define k = a/3.
6. a^2 = (3k)^2 = 3 * b^2 => 9 * k^2 = 3 * b^2 = > 3 * k^2 = b^2 which means that b is also devisible by 3.
7. This contradicts the base assumption that they are coprime integers.
Credits to wikipedie for refreshing my memory.
• Show off! ;-) +1 for refreshing my memoey Oct 21 '14 at 3:28
• @PieterGeerkens :) thanks, I managed to remember half of it (from Calculus 1) but then found it was explained real well well in wiki. Oct 21 '14 at 3:30
Lots of complex answers here. If you are looking for a 'Close enough' answer, try 7x8. Not a perfect hexagon, but close enough that most people will not notice the difference. | 2021-12-02 07:23:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5538705587387085, "perplexity": 745.1618157156779}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964361169.72/warc/CC-MAIN-20211202054457-20211202084457-00522.warc.gz"} |
https://mathematica.stackexchange.com/questions/111714/definite-integral-closed-form-expression | # Definite integral closed-form expression
Is there a way to get Mathematica yield a closed-form expression (in terms of special functions) for the integral: $$\int_{0}^{\infty} e^{-a t}\log(t)\log(1+t)\,dt,$$ where $a>0$? The obvious
Integrate[Exp[-a*t]*(Log[t]*Log[t + 1]), {t, 0, Infinity}, Assumptions -> a > 0]
doesn't give anything useful.
• LaplaceTransform[Log[t] Log[1 + t], t, a] does not work either. It seems likely that Mathematica doesn't know anything about it. – J. M.'s ennui Apr 2 '16 at 11:46
• I tried that too, and also didn't get anything. Yet apparently there is a way to get Mathematica recognize the integral: math.stackexchange.com/questions/1723844/… – Alex Apr 2 '16 at 13:07
• Did you try to do it analytically? Integrate by parts first to get two integrals with one log each, and then differentiation over a+ integration by parts for each of the two integrals. – Alexey Bobrick Apr 2 '16 at 13:37
• Differentiation wrt a? Could you be more specific? – Alex Apr 2 '16 at 13:39
Integrating by parts we have:
F[t_] := Exp[-a*t];
G[t_] := Log[t]*Log[1 + t];
HoldForm[Integrate[F[t]*G'[t], t] = F[t]*G[t] - Integrate[F'[t]*G[t], t]]
integral = Integrate[F[t]*G'[t], t] == F[t]*G[t] - Integrate[F'[t]*G[t], t]
First@Expand@Solve[integral, \[Integral]E^(-a t) Log[t] Log[1 + t] \[DifferentialD]t]
$$\int F(t) G'(t) \, dt=F(t) G(t)-\int F'(t) G(t) \, dt$$ $$\int e^{-a t} \left(\frac{\log (t)}{t+1}+\frac{\log (t+1)}{t}\right) \, dt=e^{-a t} \log (t) \log (t+1)+a \int e^{-a t} \log (t) \log (t+1) \, dt$$ $$\int e^{-a t} \log (t) \log (t+1) \, dt= \frac{\int e^{-a t} \left(\frac{\log (t)}{t+1}+\frac{\log (t+1)}{t}\right) \, dt}{a}-\frac{e^{-a t} \log (t) \log (t+1)}{a}$$
define integral is then:
Limit[-Exp[-a*t]*Log[t]*Log[1 + t]/a, t -> Infinity, Assumptions -> a > 0] -
Limit[-Exp[-a*t]*Log[t]*Log[1 + t]/a, t -> 0, Assumptions -> a > 0] +
Integrate[Log[1 + t]/(Exp[a*t]*t), {t, 0, Infinity},Assumptions -> a > 0]/a +
Integrate[Log[t]/(Exp[a*t]*(1 + t)), {t, 0, Infinity}, Assumptions -> a > 0]/a
$$\frac{G_{2,3}^{3,1}\left(a\left| \begin{array}{c} 0,1 \\ 0,0,0 \\ \end{array} \right.\right)-e^a \left(G_{2,3}^{3,0}\left(a\left| \begin{array}{c} 1,1 \\ 0,0,0 \\ \end{array} \right.\right)+(\log (a)+\gamma ) \Gamma (0,a)\right)}{a}$$
Numerical check for: a=2
a = 2;
NIntegrate[Exp[-a*t]*(Log[t]*Log[t + 1]), {t, 0, Infinity}]
(*-0.0730318*)
1/a (-E^a (Gamma[0, a] (EulerGamma + Log[a]) + MeijerG[{{}, {1, 1}}, {{0, 0, 0}, {}}, a]) + MeijerG[{{0}, {1}}, {{0, 0, 0}, {}}, a])//N
(*-0.0730318*)
• Great, thank you. – Alex Apr 3 '16 at 18:55
EDIT
I have now confirmed my closed form expression numerically. This was possible by helping Mathematica to calculate the numerical value of mixed partial derivatives of HypergeometricU[a,b,z] with respect to a and b.
Original post
We derive a closed form by another procedure. The procedure seems to be valid but Mathematica has difficulties with the numeric values of the derivatives of HypergeometricU[a,b,z] with respect to a and b.
Nevertheless it might be useful to have a look at these short developments the results of which resemble those of Mariusz.
$Version (* Out[7]= "10.1.0 for Microsoft Windows (64-bit) (March 24, 2015)" *) Consider the related integral f = Integrate[Exp[-a t ] t^p (1 + t)^q, {t, 0, \[Infinity]}, Assumptions -> {a > 0, p > -1}] (* Out[21]= Gamma[1 + p] HypergeometricU[1 + p, 2 + p + q, a] *) Now we retrieve the original integral by twofold differentiation D[f,p]/.p->0 $$U^{(0,1,0)}(1,q+2,a)+U^{(1,0,0)}(1,q+2,a)+\gamma e^a \left(-a^{-q-1}\right) \Gamma (q+1,a)$$ D[%, q] /. q -> 0 $$-\frac{\gamma e^a \left(G_{1,2}^{2,0}\left(a\left| \begin{array}{c} 1 \\ 0,0 \\ \end{array} \right.\right)+e^{-a} \log (a)\right)}{a}+U^{(0,2,0)}(1,2,a)+U^{(1,1,0)}(1,2,a)+\frac{\gamma \log (a)}{a}$$ FunctionExpand[%] $$U^{(0,2,0)}(1,2,a)+U^{(1,1,0)}(1,2,a)-\frac{\gamma e^a \left(-\text{Ei}(-a)+\frac{1}{2} \left(\log (-a)-\log \left(-\frac{1}{a}\right)\right)+e^{-a} \log (a)-\log (a)\right)}{a}+\frac{\gamma \log (a)}{a}$$ Following the comment of Alex we finally impose a>0 and find ff1= Simplify[ff, a > 0] $$U^{(0,2,0)}(1,2,a)+U^{(1,1,0)}(1,2,a)+\frac{\gamma e^a \text{Ei}(-a)}{a}$$ This is the closed form expression of the integral in terms of special functions as requested in the OP. Numerical check (part of EDIT) We now check the numerical agreement. As mentioned before Mathematica refuses to evaluate the expression. In order to see in more detail which part makes the problem we split the expression into a list ff2 = List @@ ff1 $$\left\{\frac{\gamma e^a \text{Ei}(-a)}{a},U^{(0,2,0)}(1,2,a),U^{(1,1,0)}(1,2,a)\right\}$$ % /. a -> 1. $$\left\{-0.344221,0.531931,U^{(1,1,0)}(1,2,1.)\right\}$$ As Mathematica does not evaluate the mixed derivative numerically we give it a little help defining the partial derivative explicitly as a quotient of diffences in a sufficiently small interval ff13[a_] := ff13[a_] := With[{d = 10^(-3)}, (1/ d)*(Derivative[1, 0, 0][HypergeometricU][p, q + d/2, a] - Derivative[1, 0, 0][HypergeometricU][p, q - d/2, a]) /. {p -> 1, q -> 2}] Where smaller d does not change the result. The explicit expression of the integral which can also be evaluated numerically is given by ff1N[a_] = ff1[[1]] + ff1[[2]] + ff13[a]; Taking the same value as Mariusz did we find ff1N[2.] (* Out[191]= -0.0730318 *) For comparison the numerical integral is fi[a_] := NIntegrate[Exp[-a t] Log[t] Log[1 + t], {t, 0, \[Infinity]}] There is excellent agreement over a greater range of "a" as can be checked with this plot (to be done by the reader as it just shows a line y = 1) Plot[{fi[a] ff1N[a]}, {a, 0, 2}, PlotRange -> {0.99, 1.01}] (* plot not shown *)] • Thank you. The only thing though is that your code appears to be ignoring the assumption that$a>0$, because otherwise$\log(-a)$and$\log(-1/a)\$ wouldn't be present in the expression you obtained. – Alex Apr 3 '16 at 18:55
• @Alex Thanks for the hint. After taking a>0 into account the result simplifies. See my completed answer. – Dr. Wolfgang Hintze Apr 4 '16 at 9:13 | 2021-05-08 11:12:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7786325812339783, "perplexity": 3051.159644991496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988858.72/warc/CC-MAIN-20210508091446-20210508121446-00605.warc.gz"} |
http://bakingandmath.com/ | 27 Jul
That’s really all I have to say about this. It’s incredible. So well written. It’s the best piece I’ve read that explains math and doing math to non-mathers. We talk a lot about analogies (caves or dark rooms with light switches or knives etc.) but this one just goes straight to it (the Devil’s game). I loved this quote and also it made me tear up a bit as I was sitting in my windowless office with the door closed taking a break from a problem:
As a group, the people drawn to mathematics tend to value certainty and logic and a neatness of outcome, so this game becomes a special kind of torture. And yet this is what any would-be mathematician must summon the courage to face down: weeks, months, years on a problem that may or may not even be possible to unlock. You find yourself sitting in a room without doors or windows, and you can shout and carry on all you want, but no one is listening.
Unfortunately, in the print edition it’s not very well formatted (too much wall of text, which the NYT Magazine has been doing lately). But the online version looks great.
Anyway, go read it. And if you want to know more about his childhood and see the way Australians write the word “pediatric,” read this one too. The NYT one is a better piece of writing, but The Age one covers different ground.
Regular (very long) math post coming up on Thursday! Left orderable groups!
## Easy ga kho gung (Vietnamese braised chicken with ginger) [based on my mom’s recipe]
23 Jul
Slow-cooked chicken with ginger, garlic, and onions. I love this recipe. It’s one of those classic home-cooking recipes that you aren’t likely to see at a restaurant, but every family makes it. In fact, when I was in Vietnam several years ago I made some friends and visited their village for one night. They showed me how to grow rice in their paddy, and we walked around the village, and practiced driving a motor scooter while trying to avoid the water buffalo that hung out on the roads. At night their mom made us a big feast for dinner, consisting of rice from said paddy, rau muong xao toi, and this braised chicken (freshly killed from their neighbor). Very traditional, very delicious.
I also ask my mom for it every time I make it, so I thought I’d blog it so I could stop bugging her. I’ll tell you how I do it and also make notes for where my mom takes more time and makes it more delicious than I do. Also, it’s made with items that you probably have in your pantry (we buy garlic from Costco and always have onions and ginger and frozen chicken parts).
This is a RAW file. I’m just kidding it’s a jpeg.
If you don’t have fish sauce in your pantry and you’re interested in making Vietnamese food ever, then you should buy a bottle. If you aren’t, then I’m not sure why you’re reading this post. I like using coconut water/juice (I always keep it around because it’s all I drink when I’m sick), but water or chicken broth work great too.
My mom always soaks chicken in salted water for half an hour before cooking it, “to get rid of the smell.” Brining does keep the chicken super moist, but I’m always too lazy to do it. It’s good if you feel like it though!
Also, traditionally the chicken parts are chopped up into bite-size pieces for this dish. Part of that is frugality and part of it is flavor- more surface area to soak in more of the sauce. Plus it’s fun to bust out your cleaver! I generally make the pieces baby-fist sized (so three or four bites) because I am lazy. You could also not chop them.
I guess they had to give that suburban sitcom star a nickname instead of just calling him by his last name. Then it’d sound like a serial killer sitcom instead of a family one: Leave it to Cleaver!
Next, chop up some garlic, onion, and ginger. A few thoughts on this: for our wedding someone gave us a mortar and pestle, and it is AWESOME for garlic. I don’t even peel or smash the cloves, I just throw them in and smash them a couple times. The paper falls off and you can pick it out. This isn’t great if you care about uniform sizes, but if you want a ton of garlic quickly smashed into smallish pieces, this is definitely the way to go.
If the actress from Young Frankenstein comes up to you and wants to fight, try to walk away. You’ll get Teri Garr-licked in no time.
I am a total sucker for those stupid “17 life hacks that will change the way you sit on a couch!!!” articles. I’ve seen “one weird trick” a few times for peeling ginger: use a spoon. Unfortunately, this one actually works! Especially if you have a fairly smooth/not-too-knobby piece of ginger. Just push the spoon tip in at one end of peel, eating side facing the ginger, and pull down while pushing into the ginger. I can’t believe this worked and now I’ll go nail polish my keys so I don’t mix them up and save my bread bag close-things to label cords.
I actually have naked ginger in my house a lot (my baby is a redhead!)
You’ll want diced onion, smashed pieces of garlic, and matchsticks of ginger. Throw that in with your chicken (if you brined it, toss the brine), along with sugar and salt, and let it marinate for at least 15 minutes.
Just like in the human world, in horse races there are far more males than females. I’ve been to exactly one horse race in my life, and it was almost all stallions, but there was one lane with a female. There was a mare in eight. [I just told my husband this caption, and his response: “our kid is going to love you.” Not even a chuckle from him!]
My mom first browns the chicken in a little bit of oil, then adds in the garlic, ginger, onion. I actually marinated it in the pot, and just put the pot on the stove and turned it on. Like I said, super easy. Put it on high, add some fish sauce for flavor, and then your liquid (I used coconut water). Bring it to a boil, then turn it down and simmer for as long as you have. The longer you simmer, the richer the flavor. The chicken will be cooked after about 15 minutes so if you’re in a rush just eat it then.
How cute would it be if we called every adult animal like we call chickens? Kittenens? Puppyens? PUPPY YENS?!?!
A few minutes before you want to finish it, I like to add some cornstarch to thicken it. To avoid lumps, put the cornstarch in a small bowl/ramekin and spoon some of the hot liquid into it, then whisk that til smooth. Add the mix to your pot, and stir. Then bring it back to a boil.
‘don, sob’, some’, shiratak': these are all rame’kin.
Serve this with rice and plain boiled vegetables to soak up as much of the sauce as you can.
Easy ga kho gung:
Bone-in chicken pieces (I like thighs, but drumsticks or a whole chicken are also great) [Enough for the number of people you are serving]
1/2 head of garlic (just a lot of garlic. Like 7 cloves at least)
1 thumb-sized piece of garlic per 4 servings
1/2-1 onion
2 TB sugar
2 TB fish sauce (nuoc mam; we always get Three Crabs brand)
salt, pepper
2 C water or chicken broth or coconut juice
2 TB corn starch
Chop chicken into pieces. If desired, brine in salt water for half an hour.
Dice onion, smash garlic, and peel and matchstick ginger. Add to chicken (drain brine, if using) with sugar, and salt and pepper to taste (about 1 TB of salt should be fine, you can always add more fish sauce later). Stir, and marinate for at least 15 minutes and up to overnight.
If desired, heat 2 TB of oil over medium-high, and brown chicken pieces, 5 minutes. Then add marinade, and proceed.
If you didn’t brown, just cook the whole thing over medium high. Add fish sauce and liquid of choice, bring to a boil, stirring a few times (at least two or three times). Lower heat and simmer at least 15 minutes, or for an hour.
Five minutes before you want to eat, place corn starch into a small bowl. Spoon in some of the hot liquid, and whisk until smooth. Add corn starch mix to pot, and incorporate and bring back to a boil. Boil for one minute while stirring, then turn off stove.
Serve with rice and boiled vegetables.
## Efficient geodesics in the curve complex
15 Jul
I have a not-secret love affair with blogging the curve complex: I (intro), II (dead ends), III (connected). I’m surprised I didn’t blog the surprising and cute and wonderful proof that the curve complex is hyperbolic, which came out two years ago. Maybe I’ll do that next math post (but I have a large backlog of math I want to blog). Anyways, I was idly scrolling through arXiv (where mathematicians put their papers before they’re published) and saw a new paper by the two who did the dead ends paper, plus a new co-author. So I thought I’d tell you about it!
If you don’t remember or know what the curve complex is, you’d better check out that blog post I (intro) above (it is also here in case you didn’t want to reread the last paragraph). Remember that we look at curves (loops) up to homotopy, or wriggling. In this post we’ll also talk about arcs, which have two different endpoints (so they’re lines instead of loops), still defined up to homotopy.
The main thing we’ll be looking at in this post are geodesics, which are the shortest path between two points in a space. There might be more than one geodesic between two spaces, like in the taxicab metric. In fact, in the curve complex there are infinitely many geodesics between any two points.
It’s easy to get metrics messed up, but the taxicab metric is pretty straightforward- there are lots of geodesics between the red star and the starting point. I guess if you’re an alien crossed with a UFO crossed with a taxi then maybe the metric is difficult (butI totally nailed portraits of UFO-taxi-aliens)
Infinity is sort of a lot, so we’ll be considering specific types of geodesics instead. First we need a little bit more vocabulary. Let’s say I give you an arc and a simple (doesn’t self intersect) closed curve (loop) in a surface, and you wriggle them around up to homotopy. If you give me a drawing of the two of them, I’ll tell you that they’re in minimal position if the drawings you give me intersect the least number of times of all such drawings.
All three toruses have the same red and green homotopy classes of curves, but only the top right is in minimal position – you can homotope the red curve in the other two pictures to decrease the number of times red and green intersect. I just couldn’t make a picture w/out a cute blushing square.
If you have three curves a, b, c all in minimal position with each other, then a reference arc for a,b,c is an arc which is in minimal position with b, and whose interior is disjoint from both and c.
Green is a reference arc for red, orange, yellow: its interior doesn’t hit red or yellow, and it intersects orange once. Notice that it starts and ends in different points, unlike the loops. (This picture is on a torus) [Also red and yellow aren’t actually in minimal position; why not?]
Now if you give me a series of curves on a surface, I can hop over to the curve complex of that surface and see that series as a path. If the path $v_0,v_1,\ldots,v_n$ is geodesic, then we say it is initially efficient if any choice of reference arc for $v_0,v_1,v_n$ intersects $v_1$ at most n-1 times.
The geodesic $v_0,v_1,\ldots,v_n$ is an efficient geodesic if all of these geodesics are initially efficient: $(v_0,\ldots, v_n), (v_1,\ldots,v_n),\ldots,(v_{n-3},\ldots,v_n)$. In this paper, Birman, Margalit, and Menasco prove that efficient geodesics always exist if $v_0,v_n$ have distance at least three.
Note that there are a bunch of choices for reference arcs, even in the picture above, and at first glance that “bunch” looks like “infinitely many,” which sort of puts us back where we started (infinity is a lot). Turns out that there’s only finitely many reference arcs we have to consider as long as $d(v_0,v_n)\geq 3$. Remember, if you’ve got two curves that are distance three from each other, they have to fill the surface: that means if you cut along both of them, you’ll end up with a big pile of topological disks. In this case, they take this pile and make them actual polygons with straight sides labeled by the cut curves. A bit more topology shows that you only end up with finitely many reference arcs that matter (essentially, there’s only finitely many interesting polygons, and then there are only so many ways to draw lines across a polygon).
So the main theorem of the paper is that efficient geodesics exist. The reason why we’d care about them is the second part of the theorem: that there are at most $n^{6g-6}$ many curves that can appear as the first vertex in such a geodesic, which means that there are finitely many efficient geodesics between any two vertices where they exist.
I DID NOT MAKE THIS PICTURE IT IS FROM BIRMAN, MARGALIT, MENASCO. But look at how cool it is!!!
Look at this picture! The red curve and blue curve are both vertices in the curve complex, and they have distance 4 in the curve complex, and here they are on a surface! So pretty!
If you feel like wikipedia-ing, check out one of the authors on this paper. Birman got her Ph.D. when she was 41 and is still active today (she’s 88 and a badass and I want to be as cool as she is when I grow up).
## Mmmm rummmm cake
9 Jul
When I started this blog I couldn’t imagine the directions my life would take. Two and a half years ago, I did a post on a semi-homemade rum cake and titled it “Shame on me” for being semi-homemade instead of from scratch. Rereading it, I feel like that’s the cake I should’ve made the other night. I was feeling bad because I’d found a fundamental problem in my research (still unresolved…) after wasting away my morning and before picking up the baby late from daycare because I’d forgotten an umbrella. So after putting the baby down and eating dinner, I decided to make a rum cake. During this time I was chatting with my husband and offhandedly asked him the last time he was proud of me. He said “I’m proud of your right now, you’re making that cake. You’re so capable.”
Hell yes I am capable! My research is stagnant; last time I timed myself it took me 13.5 minutes to run a mile; I keep not getting the oil changed on my car. BUT I’m trying research, I am active, and my baby is still alive. AND I made a rum cake.
Some days, you need pep talks from those who love you. Some days, you need to bake a cake.
Semi-homemade doesn’t hurt the eggsecution of this dish
I didn’t have a Bundt pan (we don’t know what happened to that old one but rum cake is the only thing I make in a Bundt pan so it seems silly to buy one), so I used a 13 x 9 pyrex instead. Generously grease and flour it, then sprinkle with nuts (I used peanuts do not do that. Use pecans or walnuts).
Peanuts, equality, and granite- doesn’t quite have the same ring to it as liberté, egalité, et fraternité. I guess France can keep its flag-they’d be nuts to adopt this one, even though it rocks.
Then you make your doctored cake by adding pudding mix, an extra egg, and rum. Yum. This makes the cake a bit lighter and bouncier while somehow super moist (that’s the pudding mix).
When you whisk upon a star, you’ll probably burn up no matter who you are. Because stars are really hot and why would you be whisking anyway anything you wanted to bake would already be burned to a crisp/nothingness
I thought this label was hilarious on the box:
IF I PUT THINGS IN ALL CAPS YOU WILL READ THEM. (google translate:) Si pongo COSAS EN MAYÚSCULAS USTED puedan leerlas.
So then I took this picture:
Breaking all the rules! That’s me! Or illiterate! O analfebetos!
Anyways, while that’s baking you can doodle around the house for half an hour, then make the glaze- it’s just butter + sugar, boil it, then add rum.
I ran out of white sugar so I substituted in brown for the remainder. Insert appropriate racial joke here? It’s hard because multiple ethnic groups claim “brown” which doesn’t make sense because isn’t everyone some hue of brown? I’m staring at my white husband’s skin right now to try to figure out the color. Pinky-peach but not like a sunset. He just held up a piece of paper but white people aren’t all albino. I don’t know. I’m not good with colors. That’s why I subbed in the sugar.
After the butter and sugar have boiled, turn off the heat. Make sure you use a deep sided pot, not a pan, because when you add the rum things get exciting and it fizzes up.
This isn’t a zero-rum game
After the rum
After the cake comes out of the oven, poke it all over with toothpicks or skewers, as much as you can until you get bored (so maybe 40 times?). Then drizzle half the glaze all over it. Wait a bit, invert the cake onto a plate or baking pan, and poke again + drizzle again. It’s great warm, but it’s awesome about 12 hours later, when the rum has soaked in a lot and everything is moist and rummy.
Buttery rum cake, adapted from allrecipes
cake part:
1 c chopped nuts (I prefer pecans)
1 box yellow cake mix
1 box vanilla pudding mix
4 eggs
1/2 c each water, vegetable oil, rum
glaze part:
1/2 c each butter, rum, white sugar, brown sugar
1/4 c water
cake part: grease and flour a pan (Bundt if you have it, 13×9 if you don’t, two 8″ or 9″ rounds if you don’t have that either, if you don’t have those I’m not sure why you’re looking at a cake recipe) Sprinkle with nuts (toast if you so desire)
Whisk together cake mix and pudding, then add remaining cake ingredients and mix. Pour into pan, bake at 325 for one hour.
glaze part: fifteen minutes before cake is done, melt butter in a pot with sugars and water. Bring to a boil, constantly stirring. Take off the heat and stir in rum BE CAREFUL IT BUBBLES UP.
When cake is done, poke holes all over it. Dribble half the glaze over it. Carefully loosen sides of the cake from pan, then cover with a serving plate and invert the cake. Poke holes all over the nutted top half, and dribble remaining glaze all over it.
## Not a sociologist or ethnographer, but I am a curious person (about gender and race)
2 Jul
Inspiration for this post: this tweet.
So I’ve written before about being a woman in math, and this will not be my last post on the subject either. First, some background. One really, really awesome thing about my field (geometric group theory) is its webpage. Some time ago, a great professor at UCSB made this website which includes a list of all active geometric group theorists in the world (self-reported), a list of all departments in the world with said people, lists of publishers and interesting links/software, and most importantly for me, a list of all conferences in the area.
Long aside: said professor once gave me some great advice which I have since forgotten/warped in my memory to mean: do what you want to do. This is probably not what he said, but he did use this amazing website as an example: at the time, people said that making the site was a waste of his time, and now its a treasured resource for researchers around the world. Everyone in GGT knows this site (because they or their advisor is on it!) So that’s part of the reason I have this blog, and started that women in math conference- it’s maybe a “waste” of my time, but it’s something I want to do and now people are starting to know me for it. At both the Cornell and the MSRI programs I went to these past two months, a graduate student has come up to me and told me she reads my blog, so yay! I love you, readers! Also, side note in this aside: the video lectures from the summer graduate school in geometric group theory are already posted (in the schedule part of this link), so if you like videos and GGT I’d recommend them. Lots of first and second year graduate students in the audience, so they’re relatively approachable.
Back to topic: I went through the list of conferences that had occurred so far this year and “ran some numbers,” by which I mean I divided. I did this because I noticed that at the past few conferences I’ve attended, there seem to be disproportionately many female speakers (in a good way). For instance, at this summer school I counted 12/60 female students (though later someone said there are 14 of us so don’t rely on my counting) and 1/4 female speakers. But the numbers at that level are so low that the data is essentially meaningless: 25% vs. 20% isn’t that meaningful when the other choices are 0, 50, 75, or 100% female speakers. But if you collect enough data, it probably becomes meaningful. See my table below.*
If I were a sociologist or ethnographer, I would do this for all the conferences and interview a random sample of attendees and organizers in order to come to some data-backed conclusions about the phenomena here. I’m not, so I’ll just make some guesses. It looks like American conferences artificially inject more gender diversity into their invited speakers lists, while foreign ones don’t (YGGT in Spa a notable exception). I’d also guess that conferences that target graduate students have more women speakers than conferences that don’t.
Three things that support my “artificial diversity” theory: to attend an MSRI summer school, graduate students are nominated by their schools. Schools can nominate two students, and a third if she is a woman or an underrepresented minority. The NSF, which is a huge source of funding for American conferences, is really into “Broadening Participation”, which means including participants who are women, African-American, Native American, Hispanic, or disabled. And, as seen in table above, the percentage of female domestic speakers is twice that of foreign speakers.
I think this is great! It’s much easier to do something if you see someone who looks like you/has gone through similar struggles doing so.
A response to myself from a few years ago, when I felt feelings about the burden of representing all women at a table full of men: I felt bad recently for wanting to ask a Hispanic female graduate student what she thought about increasing numbers of Hispanic women in math, because I thought I was placing this exact burden on her. I was expecting her to speak for all Hispanic women. But another graduate student solved this conundrum for me- her experience is invaluable in trying to understand the plight of her demographic, but we shouldn’t be too hasty to generalize from it. And more importantly, someone needs to ask these questions. My discomfort is relatively stupid and small compared to the issue at hand- we should try to solve these problems together and respectfully, but there’s bound to be missteps along the way, and that’s OK.
I don’t have solutions, and I’ve barely stated the problem or why we should care about it, but at least I’m trying to ask questions.
## Universal acylindrical actions
25 Jun
I’m at a fantastic summer graduate school at MSRI (the Mathematical Sciences Research Institute, a.k.a. “math heaven”) right now and re-met a friend I’d seen at a few earlier conferences. I saw that she’d posted a preprint up on arXiv recently, so I thought I’d try to blog about it!
Remember that a group is a collection of elements paired with some kind of operation between them (the integers with addition, rational numbers with multiplication, symmetries of a square with composition). For that operation, you put in two group elements and get another group element out. You can imagine different functions with different inputs and outputs. Like you might have a function where you put in Yen and late night, and it outputs pumpkin. Or you could put one group element in, and a location, and get a different location [like if you put in the group element -2 to the location (3,3), maybe you get (1,1)]. More precisely, a group action on a space is a homomorphism* which takes in a group element and a point in the space and outputs a (possibly different) point on that space. For instance, you can give an action of the integers on the circle by saying that rotates the circle by $n/2\pi$.
Each integer rotates the circle by pi/2 times the integer. Looks like circle is getting a little sick of the action…
In the picture above, if you input the integer 2 and the original purple dot, you get the new location of the dot (180 degrees from its original location, aka pi away). If you say the original purple dot is location and the new location is y, the notation is that 2.x=y. A homomorphism is a function that respects this: f(xy)=f(x)f(y).
We say a space is hyperbolic if it locally “looks like” hyperbolic space (there’s a particularly nice function between it and hyperbolic space). The title of Carolyn’s paper is “Not all acylindrically hyperbolic groups have universal acylindrical actions,” so we need to learn what “acylindrical” means (look, we’ve already learned what groups and actions are, and we know the words “not”,”all”,and “have”! We’re doing great!)
Here’s the precise definition, and then I’ll break it down:
An action of a group on a hyperbolic space is called acylindrical if, for any $\epsilon >0,$ there exist numbers M,N>0 such that for every pair of points x,y with distance d(x,y)>M, the number of group elements that translate both x,y by less than epsilon is bounded by N: $|\{g: d(x,g.x)\leq \epsilon, d(y,g.y)\leq \epsilon\}| \leq N$.
Here’s the non math-y intuition for this: if you have a pool noodle and you spin one end around, the other one generally will fly away from where it used to be.
Here’s the math-y intuition for this: choose two points that are M-far apart. Make a little $\epsilon$-circle around each, then connect the two with a cylinder. The condition says that only a few group elements preserve the cylinder (that means that when acts on all the points in the cylinder, it maps them back into other points in the cylinder). So if you have a bunch (perhaps infinitely many) elements that preserve one circle, most of them send the other circle/rest of the cylinder away.
A group is called acylindrically hyperbolic if you can find a hyperbolic space on which the group acts acylindrically. In practice, such groups actually act on a whole bunch of different spaces acylindrically.
Now suppose that you’ve got an element in G and you want to see how that particular element acts. We say is loxodromic if you can find a space and a point in it so that the map $\mathbb{Z}\to X$ that sends an integer to the orbit of the point $n\mapsto g^n.s$ is a quasi-isometry– roughly, if you draw all the points that gets mapped to if you apply over and over again, you get something that looks like a line.
The older tree is the same as the younger tree up to scaling (multiplication) and adding some constants (the leaves). This is an example of a quasi-isomeTREE. [Also pretend both trees go on forever.]
Just for fun here’s a picture of something that’s not a quasi-isometry:
The ribbon on the right goes on forever in both directions, so it’s not quasi-isometric to the tree
You might’ve noticed above that we say an element is loxodromic if we can find space on which it acts in this particular way. But we also said that a group can act on several different spaces. So even if an element acts loxodromically on one space, that doesn’t necessarily mean it acts loxodromically on another space (even if the group acts on that other space). We actually call an element generalized loxodromic if there exists some space on which it acts loxodromically. Then if you can find an action so that all generalized loxodromic actions are, in fact, loxodromic, you’ve found a universal acylindrical action. So this paper gives an example of an acylindrically hyperbolic group that doesn’t have such an action.
Blog notes: For the summer I’m going to blog every Thursday (day was chosen arbitrarily). Also, I went back and tagged all the gluten-free recipes as gluten-free. And you should know that whenever I mention a person in this blog by name or link to them, that means that I admire them/am inspired by them.
## Soda-unders? Nope, popovers!
17 Jun
I wonder if the author of The Circle is really into baking. He must get that a lot.
When baby was four months old I was really itching to start baking again, but I needed things that required very little time/effort and preferably had lots of reward. Turning to my trusty Moosewood Cookbook (affiliate link), I paged through until I saw a super easy recipe that included the word “or” in the ingredient list. I don’t think I’ve said enough how much I love this cookbook- the recipes are so good and so forgiving and I actually follow them. I love that this recipe says “2 or 3 or 4 eggs.” And despite my love of excess amounts of butter (see super easy french toast souffle), Moosewood in general uses little butter to great effect.
So these are popovers, which I hadn’t had before but I guess are a thing. A DELICIOUS thing! First, butter some muffin tins- I have these little ceramic ramekins and I microwaved two tablespoons for thirty seconds and stirred it up. Also, we have pastry brushes now (thanks, Crate and Barrel wedding gift cards!)
I haven’t greased pans a lot in this way- guess I have to BRUSH up on my skills
Butter, flour, eggs, milk, salt. That’s it for these crisp-on-the-outside, custardy-on-the-inside soft rolls. And it’s just one bowl.
I feel like I’m running out of puns… guess I should milk my head for what I can
After whisking up the eggs, add the flour and salt and beat that. Then pour into your buttered muffin tins.
My lovely ladle lumps (I realize that is a whisk but I am tired let’s just pretend it’s a ladle and that ladle sounds like lady)
I used to always get confused when people said “in” after a “y” sound because I thought it sounded like my name (Minnesota accent?). I wonder if that famous blind musician got confused a lot when people baked muffins and said oh look, an ar-RAY of muffins
That’s it! Half an hour from start to finish, even faster than lime pie! These are so good fresh and hot with nothing, or with a bit of jam or butter on them. After they come out, poke the top sides with a fork so the steam can escape.
Moosewood Popovers
2 TB melted butter
4 eggs (moosewood says 2 or 3 but I love eggs)
1 1/4 c milk
1 1/4 c flour
1/2 tsp salt
Preheat the oven to 375, and grease a muffin tin with the melted butter. Whisk the eggs with the milk until mixed, then add the flour and salt and whisk together. Pour into the muffin tins and bake for 35 minutes until the top looks dry. Prick with a fork and eat! | 2015-07-29 21:58:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 13, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43064847588539124, "perplexity": 2744.3520823982462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986646.29/warc/CC-MAIN-20150728002306-00197-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://www.openproblemgarden.org/comment/reply/46837 | Importance: Medium ✭✭
Author(s): Tuza, Zsolt
Subject: Graph Theory » Extremal Graph Theory
Keywords:
Recomm. for undergrads: no
Posted by: fhavet on: March 6th, 2013
Conjecture If has at most edge-disjoint triangles, then there is a set of edges whose deletion destroys every triangle.
This conjecture may be rephrased in terms of packing and edge-transversal. A triangle packing is a set of pairwise edge-disjoint triangles. A triangle edge-tranversal is a set of edges meeting all triangles. Denote the maximum size of a triangle packing in by and the minimum size of a triangle edge-transversal of by . Clearly . The conjecture translates in .
This conjecture, if true, is best possible as can be seen by taking, say or . Trivially, , since the set of edges of a maximum triangle packing is a triangle edge-transversal. Haxell [H] proved that edges whose deletion destroys every triangle.
As usual, one can define fractional packing and fractional transversal. Let be the set of triangles of . A fractional triangle packing is a function such that for every edge . A fractional triangle edge-transversal is a function such that for every triangle . We denote by the maximum of over all fractional triangle packing and by the minimum of over all fractional edge-transversals. By duality of linear programming . Krivelevich [K] proved two fractional versions of the conjecture:
and .
## Bibliography
[H] P.Haxell, Packing and covering triangles in graphs, Discrete Mathematics 195 (1999), no. 1–3, 251–254.
[K] M. Krivelevich, On a conjecture of Tuza about packing and covering of triangles Discrete Mathematics 142 (1995), 281-286.
*[T] Z. Tuza, A conjecture on triangles of graphs. Graphs Combin. 6 (1990), 373-380.
* indicates original appearance(s) of problem. | 2020-01-22 12:17:25 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8780933022499084, "perplexity": 1456.7831650663518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250606975.49/warc/CC-MAIN-20200122101729-20200122130729-00126.warc.gz"} |
https://log.lain.li/blog/algebras-in-list-monad/ | Posted on November 15, 2018
Tags: explanation Categories: category theory
I’m currently watching Dr. Bartosz Milewski’s video lecture series on category theory on YouTube. In this lecture Category Theory III 4.2, Monad algebras part 3, he stated an interesting fact that in an algebra that is compatible with the list monad is a monoid.
In his video lecture he explained it very briefly and drawn the conclusion quickly before I can follow to convince myself on the fact. After that, in comment, I saw someone else had the similar questions at the ambiguous notation Dr. Milewski uses. So I derived the theorem myself to clarify my understanding. I think the outcome is kind of interesting that worths a post about it.
I want to write this post in a very beginner-friendly manner, to explain the concepts to people whom knew only some basic category theory. Hopefully it’s gonna also help me to clear things out.
## Monoid
Monoid captures the generalized idea of multiplication. A monoid on a set $$S$$ is made of an element $$\eta\in S$$ called unit and binary operation $$\mu: S \times S \to S$$ called multiplication that satisfies these laws:
• left identity law: $$\mu(\eta, a) = a$$, for $$a \in S$$
• right identity law: $$\mu(a, \eta) = a$$, for $$a \in S$$
• associativity law: $$\mu(a,\mu(b,c)) = \mu(\mu(a,b),c)$$, for $$a,b,c\in S$$
Here are some common examples of monoids:
• list monoid on list, where $$\eta$$ is empty list and $$\mu$$ is the append operator (++ in Haskell)
• additive monoid on integer, where $$\eta$$ is $$0$$ and $$\mu$$ is $$+$$
• multiplicative monoid on integer, where $$\eta$$ is $$1$$ and $$\mu$$ is $$\times$$
A monad is defined as an endofunctor $$T$$ along with two natural transformations $$\eta: a \to T a$$ called unit and $$\mu: T^2 a\to T a$$ called multiplication that satisified these laws:
• identity law:
$\begin{CD} T a @>\eta>> T^2 a \\ @VVT\mu V @V\mu VV \\ T^2 a @>\mu >> T a \end{CD}$
where the $$T a$$ at top-left is equal to the $$T a$$ at bottom right.
• mutiplication law:
$\begin{CD} T^3 a @>T \mu>> T^2 a \\ @VV\mu V @VV\mu V \\ T^2 a @>\mu >> T a \end{CD}$
These laws are essentially just monoid laws (left/right identity law and associativity law) on the category of endofunctors.
The list functor is a monad with $$\eta$$ and $$\mu$$ defined as following:
η x = [x]
μ xs = concat xs
where $$\eta$$ sends a value to a singleton list containing that value, and $$\mu$$ is concatenation function of a list of lists.
It’s easy show the monad laws for this list monad hold, since it’s not today’s focus, I’ll skip it.
## Algebra
An algebra on an endofunctor $$F: C\to D$$ is given by a tuple $$(a, \sigma)$$, where $$a$$ is an object in $$C$$ and $$\sigma$$ is an endofunction $$F a \to a$$. It’s worth noting that an algebra is not a natural transformation as it seems.
A natural transformation has no knowledge on its component, therefore must be a polymorphic function. This restriction is, however, not required for an algebra. In an algebra, $$\sigma$$ is bound to a specific object $$a: C$$, thus it can do transformations on $$a$$ or generate an $$a$$ from nowhere.
An algebra can be viewed as a map to evaluate a functor on values (e.g. algebraic expression) into an single value. Here are some examples of algebras:
• sum on list of additive numbers
• length on polymorphic list
• foo (x:_) = x; foo [] = 1 on list of integers
• eval: ExprF a -> a on an expression of type a
In an algebra the functor plays the role to form an expression, and the $$\sigma$$ function evaluates it.
## Category of algebras
Algebras on an given endofunctor $$F:C\to D$$ can form a category, where the objects are the algebras $$(a, \sigma)$$, and the morphisms from $$(a,\sigma)$$ to $$(b,\tau)$$ can be defined as morphisms in $$C(a,b)$$. We now show that the morphisms are composible:
$\begin{CD} F a @>F f>> F b \\ @VV\sigma V @VV\tau V \\ a @>f>> b \end{CD}$
Since $$F$$ is a functor, this diagram automatically commutes.
Given an endofunctor $$T$$, A monad algebra on $$T$$ is a monad on $$T$$ along with an compatible algebra on $$T$$. A monad algebra contains all the operations from its monad part and its algebra part:
• $$\eta: a \to T a$$
• $$\mu: T^2 a \to T a$$
• $$\sigma: T a \to a$$
Be noted that a specific algebra can have a specific $$a$$.
To make the algebra compatible with the monad, we need to impose these two conditions:
• with unit, $$(\sigma \circ \eta) a = a$$
• with multiplication, the diagram below should commute
$\begin{CD} T^2 a @>\mu>> T a \\ @VV T\sigma V @VV\sigma V \\ T a @>\sigma>> a \end{CD}$
These two conditions are strong. Not all algebras on $$T$$ are compatible with a given monad on $$T$$. For example, in the list monad of integers, the condition requires η [x] = x; this eliminates all other algebras that don’t satisfy this property, like the length algebra.
Now we finally get to the interesting one. There are many monad-compatible algebras on list, for example: sum, product, concat etc. These algebras do various of operations but there’s one thing in common: they all seems to related to some monoid. In fact they indeed do. We will now prove it.
First we see what properties do algebras on list monad hold. By the compatibility we discussed eariler, we always have:
• η [x] = x and,
• (σ∘Tσ) x = (σ∘μ) x, where $$\mu$$ is the concat operator for list
Let σ [] = e and σ [x,y] = x <> y, we now show e is an unit and x <> y is the multiplication operator in a monoid.
We now prove the left identity law for the monoid. We prove this by evaluting (σ∘Tσ) [[], [x]] in two ways. On the left we got (σ∘Tσ) [[], [x]] = σ [e, x] = e <> x, on the right we got (σ∘Tσ) [[], [x]] = (σ∘μ) [[], [x]] = σ [x] = x. This shows e <> x = x. The right identity law can be proved in a similar fashion.
Now the associativity law, first we get (σ∘Tσ) [[x,y],z] = [x <> y, z] = (x <> y) <> z, and (σ∘Tσ) [x,[y,z]] = [x, y<>z] = x <> (y <> z). We also no that they both equal to (σ∘μ) [[x,y],z] = (σ∘μ) [x,[y,z]] = σ [x,y,z]. For consistency, this means σ [x,y,z] must be defined as (x <> y) <> z or x <> (y <> z), and they are equal.
Now we have proved that the algebra must gives rise to a monoid, and $$\sigma$$ is the mconcat function. | 2022-01-19 11:40:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8605585098266602, "perplexity": 1020.8032889854203}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301309.22/warc/CC-MAIN-20220119094810-20220119124810-00388.warc.gz"} |
https://d2mvzyuse3lwjc.cloudfront.net/pdfs/NAG26/Manual/html/g05/g05khc.html | # NAG Library Function Document
## 1Purpose
nag_rand_leap_frog (g05khc) allows for the generation of multiple, independent, sequences of pseudorandom numbers using the leap-frog method.
## 2Specification
#include #include
void nag_rand_leap_frog (Integer n, Integer k, Integer state[], NagError *fail)
## 3Description
nag_rand_leap_frog (g05khc) adjusts a base generator to allow multiple, independent, sequences of pseudorandom numbers to be generated via the leap-frog method (see the g05 Chapter Introduction for details).
If, prior to calling nag_rand_leap_frog (g05khc) the base generator defined by state would produce random numbers ${x}_{1},{x}_{2},{x}_{3},\dots$, then after calling nag_rand_leap_frog (g05khc) the generator will produce random numbers ${x}_{k},{x}_{k+n},{x}_{k+2n},{x}_{k+3n},\dots$.
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_leap_frog (g05khc).
The leap-frog algorithm can be used in conjunction with the NAG basic generator, both the Wichmann–Hill I and Wichmann–Hill II generators, the Mersenne Twister and L'Ecuyer.
## 4References
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley
## 5Arguments
1: $\mathbf{n}$IntegerInput
On entry: $n$, the total number of sequences required.
Constraint: ${\mathbf{n}}>0$.
2: $\mathbf{k}$IntegerInput
On entry: $k$, the number of the current sequence.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
3: $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
4: $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
## 6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
NE_INT_ARRAY
On entry, cannot use leap-frog with the base generator defined by state.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
Not applicable.
## 8Parallelism and Performance
nag_rand_leap_frog (g05khc) is not threaded in any implementation.
The leap-frog method tends to be less efficient than other methods of producing multiple, independent sequences. See the g05 Chapter Introduction for alternative choices.
## 10Example
This example creates three independent sequences using nag_rand_leap_frog (g05khc), after initialization by nag_rand_init_repeatable (g05kfc). Five variates from a uniform distribution are then generated from each sequence using nag_rand_basic (g05sac).
### 10.1Program Text
Program Text (g05khce.c)
None.
### 10.3Program Results
Program Results (g05khce.r)
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017 | 2022-01-27 20:04:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.905610978603363, "perplexity": 3296.668327693874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305288.57/warc/CC-MAIN-20220127193303-20220127223303-00304.warc.gz"} |
https://stats.stackexchange.com/questions/120237/regression-model-for-ordinal-dependent-variable-and-categorical-independent-vari | # Regression model for ordinal dependent variable and categorical independent variables
If I'm using R, which regression model should I use for my dataset? (I need to get the R-squared value.) I have 1 dependent variable and 6 independent variables as follows:
1 dependent variable:
• concern {-2, -1, 0, 1, 2}
6 independent variables:
• org { scl_msg, scl_pg, fin}
• type_d { prsnl, activ, log}
• type_f { x-t, user-x, t-x}
• gender { male, female}
• age { 18-25, 26-30 , 31-35, 36-40, 40+}
• awareness { fully-aware , partially-aware, not-aware}
• Are you saying that your dv can only take the values -2, -1, 0, 1, 2? Oct 16 '14 at 2:33
• yes it is only {-2, -1, 0, 1, 2}
– sdj
Oct 16 '14 at 14:25
• Then your dependent variable isn't continuous. Eg, you can't have a 1.5 or a -3, etc. Do you have good reason to believe that the difference between 2 & 1 is the same as -1 & -2? These look like ordinal data to me. Oct 16 '14 at 15:27
• it is representing concern levels , so -2 means extremly concened, 0 means neutral and 2 means not concerned. in this case if it ordinal do you think logistic regression model should be the corect model to use ? or do you have another seggastions ?
– sdj
Oct 16 '14 at 15:56
You will be best off using ordinal logistic regression. There are at least four ways to do this in R (meaning different functions in different packages). The uniformly excellent UCLA statistics help site has a fairly comprehensive tutorial (albeit using only polr in MASS) here. There is a nice overview of the different possibilities here (it is primarily code you can run, with less explanation).
Note that there isn't really such a thing as R-squared for generalized linear models such as ordinal logistic regression. There are a number of so-called pseudo R-squareds, but it is important to understand what each one measures (there is a nice guide here), and their value is debatable (for an overview of the issues, see this excellent CV thread: Which pseudo $$R^2$$ is the one to report for logistic regression). | 2022-01-24 18:02:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7478549480438232, "perplexity": 1409.623045508545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304572.73/warc/CC-MAIN-20220124155118-20220124185118-00673.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-concepts-through-functions-a-unit-circle-approach-to-trigonometry-3rd-edition/chapter-12-counting-and-probability-section-12-2-permutations-and-combinations-12-2-assess-your-understanding-page-875/33 | ## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)
$8$
The generalized basic counting principle says that an event $e_1$ can be performed in $n_1$ ways and an event $e_2$ can be performed in $n_2$ ways, then there are $n_1n_2$ ways of performing them together. This can easily be extended to $n$ events. Here we have $2$ choices for all $3$ digits, thus using the generalized basic counting principle the number ways: $2^3=8$ | 2020-03-28 12:48:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7184756994247437, "perplexity": 258.86159748033486}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370491857.4/warc/CC-MAIN-20200328104722-20200328134722-00462.warc.gz"} |
https://www.nature.com/articles/s41598-021-03586-0?error=cookies_not_supported&code=5d3d4f35-acfb-425c-a007-15205dccc069 | Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
# Quantum computing formulation of some classical Hadamard matrix searching methods and its implementation on a quantum computer
## Abstract
Finding a Hadamard matrix (H-matrix) among all possible binary matrices of corresponding order is a hard problem that can be solved by a quantum computer. Due to the limitation on the number of qubits and connections in current quantum processors, only low order H-matrix search of orders 2 and 4 were implementable by previous method. In this paper, we show that by adopting classical searching techniques of the H-matrices, we can formulate new quantum computing methods for finding higher order ones. We present some results of finding H-matrices of order up to more than one hundred and a prototypical experiment of the classical-quantum resource balancing method that yields a 92-order H-matrix previously found by Jet Propulsion Laboratory researchers in 1961 using a mainframe computer. Since the exactness of the solutions can be verified by an orthogonality test performed in polynomial time; which is untypical for optimization of hard problems, the proposed method can potentially be used for demonstrating practical quantum supremacy in the near future.
## Introduction
### Background
A Hadamard matrix (H-matrix) is a binary orthogonal matrix with $$\{-1, +1\}$$ elements whose any distinct pair of its columns (or rows) are orthogonal to each other. Such a matrix only exists when it is square and the length of its column (row) is equal to 1, 2 or a multiple of four; i.e., for an $$M \times M$$ dimension H-matrix, then $$M=1,2$$ or $$M=4k$$ for a positive integer k. The reversed statement that for any positive integer k there is a H-matrix is also believed to be true, although neither a mathematical proof nor disproof yet exists. This is a long standing problem of the Hadamard Matrix Conjecture.
The H-matrix has been a subject of scientific and practical interests. First discovered and described by Sylverster in 18671, it is further studied by Hadamard concerning its relationship with the determinant problem2. The orthogonal property and binaryness of its elements make it widely used in information processing and digital communications. The CDMA (Code Division Multiple Access) system employs Hadamard-Walsh code to reduce interference among their users, so that the capacity of the communication system is not badly deteriorated by the increasing number of its users3,4. The H-matrix was also used by Mariner 9 space-craft as its ECC (Error Correcting Code) for sending images of Mars to a receiving station located on Earth, thanks to its capability for long error correction4,5.
Some particular kinds of H-matrices can be found (constructed) easily, while others need huge computational resource to do. An H-matrix of size $$M\times M$$ is also called an M-order H-matrix. When M follows a particular pattern of $$M=2^n$$, where n is a positive integer, the matrix can be easily constructed by the Sylvester’s method of tensor product. Hadamard2 constructed the H-matrices of order 12 and 20, whose orders do not follow the $$2^n$$ pattern. It indicates that other orders than prescribed by the Sylvester’s method do exist. Paley showed the construction of H-matrix of order $$M=4k$$ where $$k \equiv 1 \mod 4$$ and $$k \equiv 3 \mod 4$$, which are known as the Paley Type I and Type II H-matrices, respectively6. In the formulation, he employed the method of quadratic residues in a Galois field GF(q), where q is a power of an odd prime number.
Various kinds of construction methods have also been proposed. A cocyclic technique, which is based on a group development over a finite group G modified by the action of a cocycle defined on $$G \times G$$, has been introduced by De Launey and Horadam7,8. The Hadamard matrices can be generated by this scheme when it is applied to binary matrices. A general introduction on the cocyclic methods are described by Horadam9 and recent progress are presented; among others by Alvarez et al.10,11. In developing a quantum computing based H-matrix searching method, we found that a simple and straight forward method will be a good starting point. Our methods described in this paper have been based on earlier techniques proposed by Williamson12, Baumert–Hall13, and Turyn14, which is suitable for this purpose. These three methods involve searching of particular binary sequences as an essential stage. In this paper, we will refer these methods to as classical H-matrix searching methods.
Although at a glance it looks simple, finding a H-matrix is actually a challenging task. To find a H-matrix of order 92, in 1961 three JPL (Jet Propulsion Laboratory) researchers employed a state-of -the-art computer at that time, i.e. the IBM/7090 Mainframe15. For matrix order under 1000, the most recent unknown H-matrix successfully found is the one with order 428, which was discovered in 2005 by using computer search of particular binary sequences16. The method described in the paper is of particular interest because the next unknown H-matrices, such as the one with order 668, possibly can be found by using the same method. The main reason they have not been found at this time is because of the huge computational resource needed to find such matrices, which grows exponentially by the order of the matrix.
Finding a H-matrix of order M among all of $$O(2^{M^2})$$ binary matrices, which we refer to as H-SEARCH, is a hard problem. We have proposed to find such a matrix by using a quantum computer considering its capability in solving hard problems17. Theoretically, a quantum computer will need $$O(M^2)$$ qubits in superposition to solve such a problem. However, in the existing quantum annealing processor, we need $$O(M^3)$$ due to extra ancillary qubits required to translate k-body terms into 2-body Ising Hamiltonian model. In this paper, we show that by adopting the classical searching methods, we can reduce the required computing resource, which for a quantum annealing processor implementing the Ising model, will become $$O(M^2)$$. We describe how to formulate the corresponding Hamiltonians related to the classical methods and show some results of order up to more than one hundred. We also describe how to further develop this technique to find higher order matrices, by managing the classical and quantum computing resources. In such a classical-quantum hybridized algorithm, the complexity of the classical part still grows exponentially, but the quantum part grows polynomially. We shows that this algorithm extends the capability of a pure quantum method with limited number of qubits, so that a few higher order of H-matrices can be found, compared to the pure quantum method that cannot be implemented on present days quantum computer.
Usually, solving an optimization problem by annealing or heuristic methods yields only an approximate solution, i.e., we can not sure that it is actually the optimal point, unless all of possible solutions are enumerated. However, enumeration of all possible solutions of a hard problem is an extremely laborious task. In contrast, the correctness of a solution in H-SEARCH can be verified easily in polynomial time; i.e., by evaluating the orthogonality of the found matrix (solution). If we consider the solution as a certificate, H-SEARCH behaves like an NP-complete problem because finding the solution is hard, but checking its correctness is easy. In this particular point of view, H-SEARCH is an interesting hard problem worth to consider in addressing practical quantum supremacy.
### A brief on quantum computing and finding H-matrices using quantum computers
Quantum computers are expected to have computational capability beyond their classical counterparts; a feature which is well known as quantum speedup18 or even quantum supremacy19. An important progress regarding this issue is the achievement of the Google researchers in 2019, who claimed that their Sycamore quantum processor needs only about 200 s to do a particular computational task; which is sampling random quantum circuits in this case, where a classical supercomputer would take about 10,000 years to perform20. In the next step, a capability of solving a real-life problems, where classical computers cannot do in a reasonable time, is desired. Creative thinking of building algorithms that can demonstrate such practical supremacy are needed.
The working principle of QAM computers are based on quantum annealing (QA)22,23, which is a quantum analog to the classical (thermal) annealing (CA). Whereas the CA works by gradually decreasing temperature with sometimes allowing the system to jump over higher energy, the QA seeks for the solution by quantum tunneling through the energy barrier. Energy landscape of the H-SEARCH problem’s Hamiltonian are degenerates; i.e., there are many equivalent binary matrices that have identical energy. Illustration of the potential energy landscapes (PEL) for 2-order and 4-order binary matrices are given in Fig. 1. Considering the PELs, quantum annealing approach is suitable to find the solution. We expect that the speed-up comes from the process of finding the minimum energy by quantum tunneling. Further analysis on how quantum computing can speed up a search algorithm is described by Farhi and Gutmann24. A comprehensive review on quantum annealing and analog quantum computation has been given by Das and Chakrabarti25.
In general, existing quantum computers can be categorized into the universal quantum gate (QGM-Quantum Gate Machine) and quantum annealer (QAM-Quantum Annealing Machine). Regardless some issues related to noise and other non-ideal conditions, both of these types of quantum processors have been built and are accessible by public users through the Internet. The implementation scheme of the proposed methods for both of these kinds of quantum computers are illustrated in Fig. 2. The direct method; which works for QAM that has been described in our previous paper17, will be used as a reference. Three main proposed quantum computing methods are derived from non-quantum computing/classical H-matrix construction/searching methods, which we will referred to as the Williamson, Baumert–Hall, and Turyn methods.
The QAM processor, such as the D-Wave, only accepts problems in the form of a 2-body Hamiltonian that generally can be expressed by
\begin{aligned} {\hat{H}}_{pot} \left( {\hat{\sigma }} \right) \equiv -\sum _{i\ne j} J_{ij} {\hat{\sigma }}_i^z {\hat{\sigma }}_j^z -\sum _i h_i {\hat{\sigma }}_i^z \end{aligned}
(1)
which is a Hamiltonian of an Ising system, where $$J_{ij}$$ is a coupling constant or interaction strength between a spin at site i with a spin at site j, $$h_j$$ is magnetic strength at site j, and $$\{{\hat{\sigma }}_i^{\alpha }\}$$ are Pauli’s matrices of directions $$\alpha =\{x,y,z\}$$ at site-i. The processor performs quantum annealing by introducing a transverse field given by
\begin{aligned} {\hat{H}}_{kin}\left( {\hat{\sigma }} \right) \equiv -\Gamma \sum _i {\hat{\sigma }}_i^x \end{aligned}
(2)
which is evolved over time according to the following equation
\begin{aligned} {\hat{H}}_{QA}\left( {\hat{\sigma }}, t \right) =\left( 1-\frac{t}{\tau } \right) {\hat{H}}_{kin} \left( {\hat{\sigma }} \right) + \frac{t}{\tau }{\hat{H}}_{pot}\left( {\hat{\sigma }} \right) \end{aligned}
(3)
where $$t \in [0,\tau ]$$ denotes time22,26. The problem to solve should be encoded in $${\hat{H}}_{pot}$$, which is represented by the Ising’s coefficient $$J_{ij}$$ and $$h_i$$ for each of the problem. Some optimization problems have been solved by the quantum annealing methods; among others are: graph isomorphism27, wireless network optimization28, nurse scheduling problem29, hand written digit recognition30, computational biology31, and hydrologic inverse analysis32.
In a QAM, the formulation of the H-SEARCH is started by calculation of its energy function E(s) as a function of binary variables $$s \in \{-1,+1\}$$. For conciseness, we will represent the value of s by its signs $$\{-, +\}$$. In general, E(s) might contain high order k-body interaction terms so that we will denote it by $$E_k(s)$$, whereas the Ising model allows only up to 2-body terms in $$E_2(s)$$. To obtain the 2-body expression, and eventually a 2-body quantum Hamiltonian $${\hat{H}}_2\left( {\hat{\sigma }}\right)$$, a sequence of transforms given by the following construction diagram should be conducted17,
\begin{aligned} E_k(s) \rightarrow E_k(q) \rightarrow E_2(q) \rightarrow E_2(s)\rightarrow {\hat{H}}_2\left( {\hat{\sigma }}\right) \end{aligned}
(4)
where $$q\in \{0,1\}$$ is a Boolean variable. Actually both of s and q are binary variables, but with different values. For now on, we will refer $$s \in \{-1,+1\}$$ as spin variable and $$q \in \{0,1\}$$ as Boolean variable.
In the previous paper17, implementation of an M-order H-matrix on a QAM needs $$M^2$$ number of logical (binary) variables and additional $$M^2\times (M-1)/2$$ ancillary variables (ancillas) so that the overall complexity is $$O(M^3)$$. In this paper, by adopting classical H-matrix construction/searching methods, we can reduce the required number of variables significantly into $$O(M^2)$$ which enables the search of higher order H-matrices than before. In the followings, we will address three quantum H-SEARCH methods, which are derived from the classical methods of Williamson, Baumert–Hall, and Turyn. For each of these methods, we derive their corresponding Hamiltonians based on some criteria that are specifics for each of the cases. Low order cases can be calculated by hand, while higher order ones should be calculated by a computer through symbolic computing due to the large number of terms and variables which are involved. The complete lists and expressions of the Hamiltonians are provided in the Supplementary Information section.
In the QGM quantum computing, we can employ QAOA (Quantum Approximate Optimization Algorithm)33, which is well-suited for solving an optimization problem on NISQ (Noisy Intermediate-Scale Quantum) processors. In principle, the general k-body Hamiltonian can directly be implemented on a QGM. Therefore, the required number of physical qubits will be about the same as the number of logical qubits. However, since the implementation needs direct connection to the actual machine, which is not available for us at this time, we will not address it in the current paper.
## Results
### Williamson based quantum computing method
The Williamson’s method builds a matrix W of size $$4k \times 4k$$ from four sub-matrices ABCD each of size $$k \times k$$4,12,34. Any pair of these sub-matrices are commutative. The orthogonality property of W will be satisfied when
\begin{aligned} V \equiv A^TA + B^TB + C^TC + D^TD = 4k I_k \end{aligned}
(5)
where $$V=I_k$$ is a $$k \times k$$ identity matrix. Then, the problem becomes choosing the elements of $$s_i \in \{-1,+1\}$$ in the sub-matrices that makes the orthogonality condition in Eq. (5) is satisfied. Further simplification and efficiency of the number of variables can be achieved when we choose the sub-matrices which are symmetric and circular.
By imposing the orthogonality conditions, the commutativity among the sub-matrices, and the non-negativity of the energy, we arrive to the following s-dependent energy function
\begin{aligned} E_k(s)=\sum _{i=0}^{k-1} \sum _{j=0}^{k-1} \left( v_{i,j}(s)-4k\delta _{i,j}\right) ^2 \end{aligned}
(6)
where $$v_{i,j}$$ denotes the element at row i and column j of the matrix V that consists of products of spin/binary variables $$s_i$$ given by Eq. (5) and $$\delta _{i,j}$$ is the Kronecker delta function. The orthogonality requirement of W will be satisfied when $$E_k(s)=0$$, which is the lowest value of the energy function of Eq. (6). For $$k=3$$ and a particular value of Boolean reduction factor $$\delta$$ (note that it was written as $$\delta _{ij}$$ in17), by expanding this equation and then following the construction diagram in Eq. (4), we will arrive to the following 2-body Hamiltonian
\begin{aligned}{\hat{H}}_2({\hat{\sigma }}^z)&= 13,728{\hat{\sigma }}_0^z + 13,728{\hat{\sigma }}_1^z\\ & \quad+ \cdots + 13,488{\hat{\sigma }}_0^z{\hat{\sigma }}_1^z + \cdots +192{\hat{\sigma }}_{10}^z{\hat{\sigma }}_{11}^z + 162,720 \end{aligned}
(7)
which can be encoded into a quantum annealing processor.
In the experiment, we extract the Ising coefficients $$\{J_{ij}, h_i\}$$ then submit them to the D-Wave. We observe that the magnitude of the coefficients in the Hamiltonian’s terms are quite large, however they will be normalized by the D-Wave system. Additionally, the constant term, such as 162, 720 in $${\hat{H}}_2({\hat{\sigma }}^z)$$ of Eq. (7), will also be removed. Consequently, instead of zero, the minimum of the energy will be a negative value. We have set the number of reads to 10,000 and obtain some solutions at minimum energy values. For $$k=3$$, which corresponds to H-matrix of order 12, the required number of logical qubits was 8 which translates into 36 physical qubits. We obtained the minimum energy at $$-45.988$$. The experimental results are displayed in Fig. 3a, where the bottom part shows the found H-matrix H on the left side and its indicator matrix $$D\equiv H^TH$$ on the right side, whereas the top parts show energy distribution of the solutions. Higher orders matrices, up to order 36 that needs 49 physical qubits to implement, have also been found successfully using the D-Wave. They are listed in the Supplementary Information section.
### Baumert–Hall based quantum computing method
The Baumert–Hall method works in a similar manner as the Williamson’s by first finding the ABCD block matrices, except that the construction of the H-matrix is given by a $$12\times 12$$ structure of block matrix13,34, which yields a $$12k\times 12k$$ matrix for particular values of positive integers k.
Experiments on finding Baumert–Hall matrices using D-Wave quantum processor indicates that the capability of the method is limited by the available number of qubits, the number of couplers, and the capability of the embedding tool35. We have successfully found a few of Hadamard matrices up to order 108 using this method. For the 108-order case; which corresponds to $$k=9$$, by following the construction diagram with particular value of the Boolean reduction factor $$\delta$$, we will obtain a 2-body Hamiltonian given by,
\begin{aligned} {\hat{H}}_2({\hat{\sigma }}^z)&= 10,555,200{\hat{\sigma }}^z_{0} + \cdots +2,636,352{\hat{\sigma }}^z_{0}{\hat{\sigma }}^z_{1} \nonumber \\&\quad +\cdots + 1,728{\hat{\sigma }}^z_{54}{\hat{\sigma }}^z_{59} + 316,483,200 \end{aligned}
(8)
After extracting the Ising parameters and submitting to the D-Wave, we obtain the solutions containing correct values of $$s_i$$ for building the H-matrices. Figure 3b shows a 108 order H-matrix, which was found by the Baumert–Hall based method and its corresponding energy statistics as output of the quantum computer. Other Baumert–Hall matrices found by this method, i.e. 36, 60 and 84, are listed in the Supplementary Information section.
### The Turyn based quantum computing Method
In this method, first we have to find a set of 4-sequences $$\{X,Y,Z,W\}$$ that has particular properties, then use them to construct a H-matrix based on Goethals-Seidel method14,16. We derive the energy function from the requirement of a valid TT-sequences given by,
\begin{aligned} N_X(r) +N_Y(r) +2N_Z(r) +2N_W(r)=0; r\ge 1 \end{aligned}
(9)
where $$N_X(r),N_Y(r),N_Z(r),N_W(r)$$ are non-periodic auto-correlation functions of the sequences $$\{X,Y,Z,W\}$$ calculated at lag-r, respectively. Since the value given by the left-hand side of Eq. (9) can be negative, whereas the annealing is performed to achieve a minimum value, we modify it into a non-negative energy function which are squared sum of the auto-correlation function at each lag $$r\ge 1$$ as follows,
\begin{aligned} E_k\equiv \sum _{r\ge 1} \left( N_X(r)+N_Y(r)+2N_Z(r)+2N_W(r)\right) ^2 \end{aligned}
(10)
We have inserted a k subscript to indicate that the energy may includes k-body interaction terms. The searching problem becomes finding a TT-sequence that satisfy this condition. We will represent the elements of $$\{X,Y,Z,W\}$$ as spin variables $$s_i$$ as before. As an example we will calculate the Hamiltonian for $$k=4$$. By considering normalized sequence for efficiently use variables16, we obtain the following expressions for TT(4)
\begin{aligned} X&= \left( 1,1,1,-1\right) ^T \nonumber \\ Y&= \left( 1,s_0,-s_0,-1\right) ^T \nonumber \\ Z&= \left( 1,s_1,s_2,1\right) ^T \nonumber \\ W&= \left( 1,s_3,s_4\right) ^T \end{aligned}
(11)
Then, the energy in Eq. (10) which after following construction diagram given by Eq. (4) with a particular value of Boolean reduction factor $$\delta$$, yields the following 2-body Hamiltonian
\begin{aligned} \begin{aligned} {\hat{H}}_2({\hat{\sigma }}^z) = 912{\hat{\sigma }}^z_{0} + 1376{\hat{\sigma }}^z_{1} + \cdots + 8{\hat{\sigma }}^z_{7}{\hat{\sigma }}^z_{10} + 8,448 \end{aligned} \end{aligned}
(12)
In the experiment, we encode the Hamiltonian to the D-Wave system. We have successfully found the lowest order H-matrix by the Turyn-based method shown in Fig. 3c. By setting $$k=6$$, we also found H-matrix of order 68 listed in the Supplementary Information section.
### Balancing the quantum and classical resources: extension of the Turyn based quantum computing method
Finding H-matrix by the Turyn’s method can be achieved by checking all possible binary vector that satisfy the TT-sequences $$\{X,Y,Z,W\}$$ requirements. Exhaustive enumeration of all $$(n, n, n, n-1)$$ TT-sequence needs $$2^{4n-1}$$ steps, which is an exponentially increasing task. For finding higher order H-matrices, we can explore the properties of the TT-sequence to reduce the number of binary sequence to enumerate16,36. In this method, instead of finding all $$\{X,Y,Z,W\}$$ at once, it will be more computationally realistic to start with filling some part of them, then subsequently imposing conditions and properties of the TT-sequence to limit the number of the sequences to check.
Partially filled sequences $$\{X^*, Y^*, Z^*, W^*\}$$ with m-elements on the left part and another m-elements on the right one, are given as follows
\begin{aligned} X^*&= (x_0, x_1, ..., x_{m-1},*,*,\ldots ,*,*, x_{n-m},\ldots ,x_{n-1})^T \nonumber \\ Y^*&= (y_0, y_1, ..., y_{m-1},*,*,\ldots ,*,*, y_{n-m},\ldots , y_{n-1})^T \nonumber \\ Z^*&= (z_0, z_1, ..., z_{m-1},*,*,\ldots ,*,*, z_{n-m},\ldots , z_{n-1})^T \nonumber \\ W^*&= (w_0, w_1, ..., w_{m-1},*,*,\ldots ,*,*, w_{n-m},\ldots , w_{n-2})^T \end{aligned}
The requirement of non-periodic auto-correlation sum for these sequences is now become
\begin{aligned} N_{X^*}(r) + N_{Y^*}(r) + 2N_{Z^*}(r) + 2N_{W^*}(r)=0; \ r\ge (n-m) \end{aligned}
(13)
We will refer all $$\{X^*, Y^*, Z^*, W^*\}$$ sequences satisfying condition given by Eq. (13) as solution prototypes. Then the energy function becomes
\begin{aligned} E_k\equiv \sum _{r\ge 1} \left( N_{X^*}(r)+N_{Y^*}(r)+2N_{Z^*}(r)+2N_{W^*}(r)\right) ^2 \end{aligned}
(14)
Figure 4 shows a block diagram of extended Turyn-based quantum computing method, involving both of classical and quantum computing parts. The generation of $$\{X^*,Y^*,Z^*,W^*\}$$ solution prototypes and their corresponding Hamiltonians are conducted in a classical computer. They are fetch one-by-one and processed by a quantum computer which deliver solutions. In the next step, the classical computer checks the orthogonality of the matrices. Notes that the simplest way to check the orthogonality of an $$M \times M$$ matrix H is by multiplication $$H^TH$$ which consisting of M times multiplications for each of all $$M^2$$ entries in the product followed by checking them whether the off diagonal are zeros and the diagonal entries are equal to $$M^2$$. Therefore, the orthogonality test can be done in $$O(M^3)$$.
Although increasing the value of m in Eq. (13) will reduce the number of sequence to check in the following steps, it also increases the number of the solution prototypes itself. There are about 2 millions prototypes for $$2m=12$$, which will increase into about 23 millions for $$2m=14$$16,36,37. It has been reported that a few TT-sequence of up to 40 can be found using classical computers, whereas higher order ones need more powerful computers which is impossible to be implemented at the moment. This is one of the main reasons that H-matrix of order 668 has not been found nor declared non-exists yet, assuming that such a matrix can be constructed by the Turyn’s method.
On the other hand, we can use the solution prototypes to reduce the number of required qubits when a quantum computer is involved in the searching process. For clarity, in the followings we illustrate this method by a simple case which is implementable on a current quantum processor. We will consider a (4, 4, 4, 3) solution prototype to find a (8, 8, 8, 7) TT-sequences by using quantum computing; therefore, it is a kind of finding higher order sequence by extending the lower one. The extended TT-sequences can be expressed by
\begin{aligned} X&= (x_0, x_1, s_0, s_1,s_2, s_3, x_2,x_3)^T \nonumber \\ Y&= (y_0, y_1, s_4, s_5,s_6, s_7, y_2,y_3)^T \nonumber \\ Z&= (z_0, z_1, s_8, s_9,s_{10}, s_{11}, z_2,z_3)^T \nonumber \\ W&= (w_0, w_1, s_{12}, s_{13},s_{14}, s_{15}, w_2)^T \end{aligned}
(15)
with known $$x_0,\ldots ,x_3, y_0,\ldots ,y_3,\ldots ,\ldots ,w_0, w_2$$ and unknown $$s_0,s_1,\ldots ,s_{15}$$.
To find the unknown values represented by $$s_i$$, we calculate the energy of the Turyn’s based method as before. Among all possible $$\{X^*,Y^*,Z^*,W^*\}$$ prototypes and the replacement of the unknowns with binary variables, we choose the following solution prototype as an example
\begin{aligned} X&= (1, 1, s_0, s_1, s_2, s_3, 1, -1)^T\nonumber \\ Y&= (1, -1, s_4, s_5, s_6, s_7, 1, -1 )^T \nonumber \\ Z&= (1, -1, s_8, s_9, s_{10}, s_{11}, -1, 1)^T\nonumber \\ W&= (1, -1, s_{12}, s_{13}, s_{14}, s_{15}, 1)^T \end{aligned}
(16)
Note that in the real case, we might have to check all of the solution prototypes.
Further calculation by applying the construction diagram with a particular value of Boolean reduction factor $$\delta$$ yields the following 2-body Hamiltonian,
\begin{aligned} {\hat{H}}_2({\hat{\sigma }}^z)&=197,860{\hat{\sigma }}^z_0 + \cdots + 16,484{\hat{\sigma }}^z_0{\hat{\sigma }}^z_1+ \cdots \nonumber \\&\quad +64{\hat{\sigma }}^z_{102}{\hat{\sigma }}^z_{107} + 4,551,232 \end{aligned}
(17)
We then encode the Hamiltonian into the D-Wave. Calculation shows that we need 108 physical qubits to implement, but embedding into the Chimera graph with the D-Wave provided embedding tools indicates that more qubits are required, which in this case is 860. After quantum annealing, we get among others, the following solution
\begin{aligned} X&= (1, 1,-1, 1,-1, 1, 1,-1)^T \nonumber \\ Y&= (1,-1, 1, 1, 1, 1, 1,-1^T \nonumber \\ Z&= (1,-1,-1, 1, 1, 1,-1,1^T \nonumber \\ W&= (1,-1,-1,-1,-1,-1,1)^T \end{aligned}
(18)
Among 10,000 of obtained results, we identified two correct solutions. One of the solution that has been constructed to a H-matrix; its corresponding indicator matrix, and solution statistics are displayed in Fig. 3d. This TT(8)-sequences yields a 92-order Hadamard matrix, which in 1961 was also found by JPL researchers in a search using IBM/7090 mainframe computer15.
## Discussions
Difficulties in finding a H-matrix by classical computing methods, due to the exponential grows of the complexity, can be overcome by quantum-computing based search such as by directly represents each elements of the matrix into binary variables, which is then translated into qubits17. However, the availability of quantum computing resource limits the implementation to only finding low order H-matrices. We have shown in the previous section that classical H-matrix searching methods can be adopted to efficiently use available quantum computing resource to solve larger problems, i.e., finding higher order H-matrices than previously achieved by the direct method17.
The data displayed in the top part of Table 1 shows required resource and results in the Williamson and Baumert–Hall based methods for each order of the H-matrix. Since both of them share the same ABCD block matrices, we put them side-by-side on the table. We observe in the table that the number of required logical qubits grows linearly by O(M) with respect to the order of the searched matrix, whereas the number of physical qubits grows quadratically as $$O(M^2)$$, which is caused by the ancillary qubits required to translate k-body into 2-body Hamiltonians. In the implementation, the physical qubits and their connections should be mapped to the topology of qubits’s connections in the quantum annealing processor; which is the Chimera graph in the DW-2000Q. We have used (default) embedding tool provided by D-Wave35 and the number of embedding qubits displayed in the table are taken from the output of the software. This mapping process, which is also called minor embedding, further increases the number of required qubits. In the following discussions, the number of required qubits after the embedding process will be labeled as the embedding qubits.
The Williamson and Baumert–Hall adopted methods can be implemented to all of matrix order as long as the embedding process is successful, which is up to 36 for the Williamson and up to 108 for the Baumert–Hall. We observe from the output of embedding tool that the highest order needs 1, 492 qubits to implement, which is more than 6 times of the required physical qubits. Observing that the trends of the embedding-to-physical qubits ratio; denoted by E/P ratio in the table, increases with the H-matrix order, by taking a moderate estimate of 7 times, the 300 physical qubits for the order of 132 matrix (in the Baumert–Hall based method) requires 2, 100 qubits to be implemented; which is more than currently available qubits in the DW-2000Q. We also observe from the experiment results, especially those displayed in the last column of Table 1, that the number of correct solutions among 10,000 number of reads tends to decrease with the increasing order of the matrix; i.e., it is about $$4\%$$ at the beginning then decreased to about $$0.2 \%$$ at order 108 for the Baumert–Hall. A possible explanation to this phenomena is that when the order of the matrix is increased, the magnitude of the coefficients in the Hamiltonian are also increased so that the difference between the largest to the smallest value becomes very large. Since they are normalized when fed into the D-Wave, with limited resolution to 1/256, the D-wave will set lower coefficients to zero. Accordingly, some of the terms will be discarded and the solutions become degenerate. It makes the percentage of the correct true solutions are reduced, as shown in the last column of the table. Since the number of reads in one run is limited by the D-Wave system to 10,000, several repeated runs on the quantum processor should be done to find higher order H-matrices. Figure 5 plots the probability of success against the order of Baumert–Hall matrices; it shows that in general higher order matrices are more difficult than the lower ones to find by the method. This also means that, for finding higher order H-matrices; assuming that the processor has a number of sufficient qubits to implement, what we have to do is by repeating the experiments many times.
Middle part of Table 1 shows required number of qubits and performance of the Turyn based quantum computing method. An H-matrix of order 44 and 68 have successfully been found, but higher order matrices have not. We observe that the E/P ratio grows faster than the similar case in the Williamson and Baumert–Hall based methods; i.e, it is now about 11 times at the order of 68. Assuming this factor stay the same, higher order matrices of 92, which needs 199 physical qubits, might require about 2, 189 embedding qubits. This is more than the currently available number of qubits in the DW-2000Q quantum processor, and therefore the search of order 92 H-matrix did not successful. We have proposed a solution for the limitation of quantum computer resource by the extended Turyn based method described previously.
Bottom part of Table 1 shows the required resource and performance of extended Turyn based method. For extending $$k=4$$ into $$k=8$$, we need 108 physical qubits; which is then increased into 860 embedding qubits. An important feature of this method is that, as long as the number of additional/extension $$\Delta k=4$$ is kept the same, the required qubits to solve extended problem will also stay the same, regardless the targeted order. However, this advantage should be paid by increasing number of solution prototypes, implying that more classical computing resources is needed and the frequency usage of the quantum processor will be increased. We expect to have an optimal point where the combination of the classical and quantum resources delivers the best solution and achieves highest order of the searched H-matrix.
At present, some of H-matrices of order under 1000; such as 668, 716, and 892, have not been found by any methods yet due to huge computational resource required to perform the computation by existing classical methods. When using the Turyn-based quantum computing method, even after extension, H-SEARCH for such orders still cannot be implemented. As an illustration, with a 12 pre-filled $$\{X^*, Y^*, Z^*, W^*\}$$, the required logical qubits for the 668 case will be $$4\cdot (56-12)= 176$$ which becomes $$176\cdot 175/2 = 15,400$$ physical qubits. Assuming the similar embedding performance as before at a factor of 8, the required number of qubits is 123, 200 which is beyond the capability of current quantum annealing processors.
Figure 6 shows the progress of available qubits in D-Wave quantum annealers [? ] and the decrease number of required qubits to implement H-matrix search of order 668 by solving the $$\{X^*, Y^*, Z^*, W^*\}$$ problem. The points in the graph shows actual number of qubits achieved in every year since 2011. We can see that the number of qubits doubled every two years; therefore, by using regression we get a linear line in a semi-logarithmic plot as shown by a dotted red curve. The middle dashed green horizontal line indicates the number of required qubits when no additional embedding qubits are required, which means that an ideal complete graph connection among the qubits is available. The top blue dashed dotted line indicates the number of required qubits with embedding factor of 8. Assuming that the connections among qubits are also improved substantially every year, we can expect the H-SEARCH of order 668 can be implemented between the year 2022 to 2029. Additionally, recent achievement of 64 qubits volume43 and the 1000 qubits milestone44 of QGM processor, the H-SEARCH implementation through QAOA is also very promising to explore.
## Methods
### Derivation of the Williamson based method
The Sylvester construction method builds a larger H-matrix $$H_{2^n}$$ from smaller ones $$H_{2^{n-1}}$$ by iteratively applying the following tensor product,
\begin{aligned} H_{2^n}=H_2 \otimes H_{2^{n-1}} = \begin{pmatrix} H_{2^{n-1}} &{} H_{2^{n-1}} \\ H_{2^{n-1}} &{} -H_{2^{n-1}} \end{pmatrix} \end{aligned}
where $$H_2=\begin{pmatrix} + &{} + \\ + &{} - \end{pmatrix}$$, i.e., it is a kind of plugging-in smaller H-matrices into a particular structure to obtain a larger H-matrix. Similarly, the Williamson method also builds a higher-order matrix from smaller ones, except that the smaller matrices are not necessarily orthogonal. In general, we can express the Williamson type H-matrices W by4,12,34
\begin{aligned} W= \begin{pmatrix} A &{} B &{} C &{} D\\ -B &{} A &{} -D &{} C\\ -C &{} D &{} A &{} -B\\ -D &{} -C &{} B &{} A \end{pmatrix} \end{aligned}
(19)
where ABCD are block matrices, whose any pair of them are commutative, i.e., $$[A,B]=[A,C]=[A,D]=[B,C]=[B,D]=[C,D]=0$$, with $$[A,B]=A^TB-B^TA, \ldots ,$$ etc expressed the commutativity of a pair of matrices $$A,B, \ldots$$ etc. The orthogonality property of W needs the following requirement to be satisfied,
\begin{aligned} A^TA + B^TB + C^TC + D^TD = 4k I_k \end{aligned}
(20)
where $$I_k$$ is a $$k\times k$$ identity matrix. We will use the properties of the Williamson matrix; especially the one given by Eq. (20), to formulate the Hamiltonian of Williamson-based quantum computing method. To further reduce the number of variables, we choose ABCD sub-matrices which are symmetric and circular.
For an illustration, consider $$k=3$$ which yields a $$4k=12$$-order H-matrix. The matrices can be expressed in terms of binary variables $$s_i \in \{-1,+1\}$$ by
\begin{aligned} A= & {} \begin{pmatrix} s_0 &{} s_1 &{} s_1\\ s_1 &{} s_0 &{} s_1\\ s_1 &{} s_1 &{} s_0 \end{pmatrix} , B= \begin{pmatrix} s_2 &{} s_3 &{} s_3\\ s_3 &{} s_2 &{} s_3\\ s_3 &{} s_3 &{} s_2 \end{pmatrix}, \nonumber \\ C= & {} \begin{pmatrix} s_4 &{} s_5 &{} s_5\\ s_5 &{} s_4 &{} s_5\\ s_5 &{} s_5 &{} s_4 \end{pmatrix}, D= \begin{pmatrix} s_6 &{} s_7 &{} s_7\\ s_7 &{} s_6 &{} s_7\\ s_7 &{} s_7 &{} s_6 \end{pmatrix}. \end{aligned}
(21)
Then, the requirement for Williamson matrix given by Eq. (20) for $$k=3$$ becomes
\begin{aligned} A^TA + B^TB + C^TC + D^TD= \begin{pmatrix} 12 &{} v &{} v \\ v &{} 12 &{} v \\ v &{} v &{} 12 \end{pmatrix} =12I_3 \end{aligned}
(22)
where $$v=4+2\left( s_0s_1+s_2s_3+s_4s_5+s_6s_7\right)$$. Suppose that $$V \equiv [v_{i,j}]=A^TA+B^TB+C^TC+D^TD$$. Naturally, we can define an s-dependent k-body energy function by
\begin{aligned} E_k(s)=\sum _{i=0}^2 \sum _{j=0}^2 \left( v_{i,j}(s)-12\delta _{i,j}\right) ^2 \end{aligned}
(23)
where $$\delta _{i,j}$$ is the Kronecker delta function. The orthogonality requirement of W will be satisfied when $$E_k(s)=0$$, which is the lowest value of the energy function in Eq. (23). In the $$k=3$$ case, the energy function $$E_k(s)$$ can be expanded into
\begin{aligned} E_k(s)=6\left( 4+2(s_0s_1+s_2s_3+s_4s_5+s_6s_7) \right) ^2 \end{aligned}
(24)
For implementing an energy function to a QAM processor; such as in the D-Wave, the k-body energy function $$E_k(s)$$ should be transformed into a 2-body energy function $$E_2(s)$$ using the steps given by the construction diagram in Eq. (4). In the process, we should choose a Boolean reduction factor $$\delta$$ to transform the k-body into 2-body function, that should be larger than the maximum value $$E_{max}$$ of the energy function45. By taking $$E_{max}=26,976$$, which is the maximum value of $$E_k(s)$$ by assuming all of $$s_i=+1$$, then setting $$\delta =2E_{max}$$, we obtain the following result
\begin{aligned} E_2(s)&= 13,728s_0 + 13,728s_1 + \cdots + 13,488s_0s_1 \nonumber \\&\quad + \cdots + 192s_{10}s_{11} + 162,720 \end{aligned}
(25)
This 2-body energy function gives the potential Hamiltonian $${\hat{H}}_{pot}({\hat{\sigma }}) \equiv {\hat{H}}_2({\hat{\sigma }}^z)$$ as follows,
\begin{aligned} {\hat{H}}_2({\hat{\sigma }}^z)&= 13,728{\hat{\sigma }}_0^z + 13,728{\hat{\sigma }}_1^z \nonumber \\&\quad + \cdots+13,488{\hat{\sigma }}_0^z{\hat{\sigma }}_1^z + \cdots +192{\hat{\sigma }}_{10}^z{\hat{\sigma }}_{11}^z + 162,720 \end{aligned}
(26)
Complete expressions for the equations can be found in Supplementary Information section.
### Derivation of the Baumert–Hall based method
In principle, the Baumert–Hall quantum computing method works in a similar manner as the Williamson’s by first finding the ABCD block matrices, except that the construction of the H-matrix is given by the following $$12\times 12$$ structure of block matrix13,34:
\begin{aligned} H= \left( \begin{array}{rrrr rrrr rrrr} A &{} A &{} A &{} B &{} -B &{} C &{} -C &{} -D &{} B &{} C &{} -D &{} -D\\ A &{} -A &{} B &{} -A &{} -B &{} -D &{} D &{} -C &{} -B &{} -D &{} -C &{} -C\\ A &{} -B &{} -A &{} A &{} -D &{} D &{} -B &{} B &{} -C &{} -D &{} C &{} -C\\ B &{} A &{} -A &{} -A &{} D &{} D &{} D &{} C &{} C &{} -B &{} -B &{} -C\\ B &{} -D &{} D &{} D &{} A &{} A &{} A &{} C &{} -C &{} B &{} -C &{} B\\ B &{} C &{} -D &{} D &{} A &{} -A &{} C &{} -A &{} -D &{} C &{} B &{} -B\\ D &{} -C &{} B &{} -B &{} A &{} -C &{} -A &{} A &{} B &{} C &{} D &{} -D\\ -C &{} -D &{} -C &{} -D &{} C &{} A &{} -A &{} -A &{} -D &{} B &{} -B &{} -B\\ D &{} -C &{} -B &{} -B &{} -B &{} C &{} C &{} -D &{} A &{} A &{} A &{} D\\ -D &{} -B &{} C &{} C &{} C &{} B &{} B &{} -D &{} A &{} -A &{} D &{} -A\\ C &{} -B &{} -C &{} C &{} D &{} -B &{} -D &{} -B &{} A &{} -D &{} -A &{} A\\ -C &{} -D &{} -D &{} C &{} -C &{} -B &{} B &{} B &{} D &{} A &{} -A &{} -A \end{array}\right) \end{aligned}
(27)
Considering the usage efficiency of the variables, ABCD are also chosen to be symmetric circulant block matrices identical to the Williamsons’s method described in the previous section. For a $$k\times k$$ size of the block matrices, Eq. (27) yields a $$12k \times 12k$$ dimension of the H-matrix. The formulation of the energy function also follows the Williamsons method described previously.
Experiments on finding Baumert–Hall matrices using D-Wave quantum processor indicates that the capability of the method is limited by the available number of qubits and the capability of the embedding tool35. We have successfully find the Hadamard matrix up to order 108 using this method. For the 108-order case, initial energy function $$E_k(s)$$ to find this matrix is given by the following
\begin{aligned} E_k(s)&= 432s_0s_2 + \cdots + 720s_{18}s_{19} \nonumber \\&\quad +\cdots+ 432s_{16}s_{17}s_{18}s_{19} + 5760 \end{aligned}
(28)
whose corresponding k-body Hamiltonian is given by
\begin{aligned} {\hat{H}}_k({\hat{\sigma }}^z)&= 432{\hat{\sigma }}^z_0{\hat{\sigma }}^z_2 +\cdots + 720{\hat{\sigma }}^z_{18}{\hat{\sigma }}^z_{19} \nonumber \\&\quad + \cdots+ 432{\hat{\sigma }}^z_{16}{\hat{\sigma }}^z_{17}{\hat{\sigma }}^z_{18}{\hat{\sigma }}^z_{19} + 5760 \end{aligned}
(29)
Then the 2-body Hamiltonian realized on the quantum annealing processor will be given by,
\begin{aligned} {\hat{H}}_2({\hat{\sigma }}^z)&= 10,555,200{\hat{\sigma }}^z_{0} + \cdots +2,636,352{\hat{\sigma }}^z_{0}{\hat{\sigma }}^z_{1} \nonumber \\&\quad +\cdots + 1,728{\hat{\sigma }}^z_{54}{\hat{\sigma }}^z_{59} + 316,483,200 \end{aligned}
(30)
Complete expressions for the equations can be found in Supplementary Information section.
### Derivation of the Turyn based method
In this method, first we find a set of 4-sequences $$\{X,Y,Z,W\}$$ that has particular properties, then use them to construct a H-matrix based on Goethals-Seidel method14,16. We translate the requirements into energy functions which then programmed into a quantum processor. In essence, the workflows of the Turyn based method are as follows
1. 1.
Find an $$(n, n, n, n-1)$$ Turyn-Type (TT) sequence $$\{X,Y,Z,W\}$$.
2. 2.
Construct base sequences $$\{A,B,C,D\}$$
3. 3.
Construct T-sequences $$\{ T_1,T_2,T_3,T_4\}$$
4. 4.
Construct seed sequences $$\{A_1,A_2,A_3,A_4\}$$
5. 5.
Construct block symmetric circular matrices $$\{X_{A1},X_{A2},X_{A3},X_{A4}\}$$
6. 6.
Construct Hadamard matrix, which is given by
\begin{aligned} H= \begin{pmatrix} X_{A1} &{} X_{A2}R &{} X_{A3}R &{} X_{A4}R \\ -X_{A2}R &{} X_{A1} &{} X_{A4}^TR &{} -X_{A3}^TR\\ -X_{A3}R &{} -X_{A4}^TR &{} X_{A1} &{} X_{A2}^TR\\ -X_{A4}R &{} X_{A3}^TR &{} -X_{A2}^TR &{} X_{A1} \end{pmatrix} \end{aligned}
(31)
where R is a back-diagonal identity matrix of size $$k \times k$$ as follows
\begin{aligned} R= \begin{pmatrix} 0 &{} 0 &{} \cdots &{} 0 &{} 1 \\ 0 &{} 0 &{} \cdots &{} 1 &{} 0 \\ \cdots &{} \cdots &{}\cdots &{} \cdots &{} \cdots \\ 0 &{} 1 &{} \cdots &{} 0 &{} 0 \\ 1 &{} 0 &{} \cdots &{} 0 &{} 0 \\ \end{pmatrix} \end{aligned}
(32)
We derive the energy function from the requirement of a valid TT-sequences given by,
\begin{aligned} N_X(r) +N_Y(r) +2N_Z(r) +2N_W(r)=0; r\ge 1 \end{aligned}
(33)
where $$N_X(r),N_Y(r),N_Z(r),N_W(r)$$ are non-periodic auto-correlation functions of the sequences $$\{X,Y,Z,W\}$$ calculated at lag-r, respectively. The non-periodic auto-correlation function of a sequence $$V=[v_0,v_1, \ldots , v_{n-1}]^T$$ is given by,
\begin{aligned} N_V(r)=\sum _{t=0}^{n-1-r} v_i v_{i+r} \end{aligned}
(34)
for $$r = 0, 1,\ldots , n-1$$ and $$N_V(r)=0$$ for $$r\ge n$$. Since the value given by the left-hand side of Eq. (33) can be negative, whereas the annealing is performed to achieve a minimum value, we adopt a non-negative energy function which are sum of squared value of the auto-correlation function at each lag $$r\ge 1$$ as follows,
\begin{aligned} E\equiv \sum _{r\ge 1} \left( N_X(r)+N_Y(r)+2N_Z(r)+2N_W(r)\right) ^2 \end{aligned}
(35)
To efficiently use available qubits in the quantum processor, it is important to reduce the number of variables encoded to the qubits as few as possible. We can achieve this by further employing the property of the TT-sequences. In this case, we can normalize the TT-sequences16 to obtain $$X^T=(x_0, x_1, \ldots x_{n-1})$$, $$Y^T=(y_0, y_1, \ldots y_{n-1})^T$$, $$Z^T=(z_0, z_1, \ldots z_{n-1})$$, and $$W^T=(w_0, w_1, \ldots w_{n-1})$$, which have the following properties
• $$x_0=y_0=z_0=w_0=1$$
• $$x_{n-1}=y_{n-1}=-1, z_{n-1}=1$$
• $$x_1=x_{n-2}=1, y_1y_{n-2}=-1$$
For clarity, in the followings we present an example of the Hamiltonian formulation for the lowest order of $$k=4$$ case. The first step as described previously is to find a TT(4) -sequences $$\{X,Y,Z,W\}$$. By representing the elements of the sequences as binary (spin) variables $$s_i\in \{-1,+1\}$$, and applying the properties of a normalized sequence explained previously, a TT(4) will be as follows,
\begin{aligned} X&= \left( 1,1,1,-1\right) ^T \nonumber \\ Y&= \left( 1,s_0,-s_0,-1\right) ^T \nonumber \\ Z&= \left( 1,s_1,s_2,1\right) ^T \nonumber \\ W&= \left( 1,s_3,s_4\right) ^T \end{aligned}
(36)
To determine the energy function, we have to calculate non-periodic auto-correlation functions $$N_X,N_Y,N_Z,N_W$$ given by Eq. (34). Since $$s_i^2=1$$, we get the following results after simplifications
\begin{aligned} N_X&= ( 4, 1, 0, -1)^T \nonumber \\ N_Y&= ( 4, 2s_0 - 1, -2s_0, -1)^T \nonumber \\ N_Z&= \left( 4, s_1 + s_2 + s_1s_2, s_1 + s_2, 1\right) ^T \nonumber \\ N_W&= \left( 3, s_3 + s_3s_4, s_4\right) ^T \end{aligned}
(37)
Therefore, the energy $$E \equiv E_k(s)$$ in Eq. (35), whose terms may contain products of k variables, is now given by
\begin{aligned} E_k(s)&= 2s_1 + 2s_2 + 2s_4 + 4s_0s_3 + 4s_1s_2 \nonumber \\&\quad - 4s_0s_4 + 2s_1s_3 + 2s_1s_4 + 2s_2s_3 + 2s_2s_4 \nonumber \\&\quad + 4s_0s_1s_2+ 2s_1s_2s_3 + 4s_0s_3s_4 + 2s_1s_3s_4 \nonumber \\&\quad + 2s_2s_3s_4 + 2s_1s_2s_3s_4 + 242 \end{aligned}
(38)
In the following steps, as described by the construction diagram in Eq. (4), the energy function should be transformed into a 2-body interacting Ising Hamiltonian. Therefore, we have to change the s-dependent energy function into q-dependent energy function $$E_k(q)$$. After simplification, this transform yields the following
\begin{aligned} E_k(q)&= - 16q_0 - 40q_1 - 40q_2 - 40q_3 - 24q_4 \nonumber \\&\quad + 16q_0q_1 + 16q_0q_2 + 32q_0q_3 + 48q_1q_2+ 32q_1q_3 \nonumber \\&\quad + 24q_1q_4 + 32q_2q_3 + 24q_2q_4 + 40q_3q_4- 32q_0q_1q_2 \nonumber \\&\quad - 32q_1q_2q_3 - 32q_0q_3q_4 - 16q_1q_2q_4 - 32q_1q_3q_4 \nonumber \\&\quad - 32q_2q_3q_4 + 32q_1q_2q_3q_4 + 276 \end{aligned}
(39)
The conversion into 2-body energy function requires a Boolean reduction factor $$\delta$$ set to be larger than the maximum value of the energy function $$E_{max}(k)$$. Assuming it is at least an absolute sum of the $$E_k(q)$$ coefficients as before, we have $$E_{max} = 908$$. By taking twice of this maximum value, we obtain $$\delta =1,816$$, which transforms Eq. (39) into
\begin{aligned} E_2(q)&= - 16q_0 - 40q_1 - 40q_2 - 40q_3 - 24q_4 \nonumber \\&\quad + 5,464q_5+ 5,480q_6 + 5,496q_7 + 5,480q_8 \nonumber \\&\quad + 5,480q_9 + 5,488q_{10} + 1,816q_0q_1 + 16q_0q_2 \nonumber \\&\quad + 1,816q_0q_3 + 1,816q_1q_2+ 1,816q_1q_3-3,632q_0q_5 \nonumber \\&\quad + 24q_1q_4 + 1,816q_2q_3- 3,632q_0q_6 - 3,632q_1q_5 \nonumber \\&\quad + 24q_2q_4 - 32q_2q_5 + 1,816q_3q_4 - 3,632q_1q_7 \nonumber \\&\quad - 3,632q_1q_8 - 3,632q_2q_7- 3,632q_3q_6 - 32q_3q_7\nonumber \\&\quad - 32q_4q_6 - 3,632q_2q_9- 3,632q_3q_8 - 16q_4q_7 \nonumber \\&\quad - 3632q_3q_9 - 32q_4q_8 - 3,632q_3q_{10} - 32q_4q_9 \nonumber \\&\quad - 3,632q_4q_{10} + 32q_7q_{10} + 276 \end{aligned}
(40)
Transforming back Eq. (40) to the s-domain yields the following expression,
\begin{aligned} E_2(s)&= 912s_{0} + 1376s_{1} + 926s_{2} + 1844s_{3} + 482s_{4} \nonumber \\&\quad - 908s_{5} - 916s_{6} - 928s_{7}- 916s_{8} - 916s_{9} \nonumber \\&\quad - 936s_{10} + 454s_{0}s_{1} + 4s_{0}s_{2} + 454s_{0}s_{3} + 454s_{1}s_{2} \nonumber \\&\quad + 454s_{1}s_{3} - 908s_{0}s_{5} + 6s_{1}s_{4} + 454s_{2}s_{3} - 908s_{0}s_{6} \nonumber \\&\quad - 908s_{1}s_{5} + 6s_{2}s_{4} - 8s_{2}s_{5} + 454s_{3}s_{4} - 908s_{1}s_{7} \nonumber \\&\quad - 908s_{1}s_{8} - 908s_{2}s_{7}- 908s_{3}s_{6} - 8s_{3}s_{7} - 8s_{4}s_{6} \nonumber \\&\quad - 908s_{2}s_{9} - 908s_{3}s_{8} - 4s_{4}s_{7} - 908s_{3}s_{9} - 8s_{4}s_{8} \nonumber \\&\quad - 908s_{3}s_{10} - 8s_{4}s_{9} - 908s_{4}s_{10} + 8s_{7}s_{10} + 8,448 \end{aligned}
(41)
which corresponds to the following 2-body Hamiltonian,
\begin{aligned} {\hat{H}}_2({\hat{\sigma }}^z)&= 912{\hat{\sigma }}^z_{0} + 1376{\hat{\sigma }}^z_{1} + 926{\hat{\sigma }}^z_{2} + 1844{\hat{\sigma }}^z_{3} \nonumber \\&\quad + 482{\hat{\sigma }}^z_{4} - 908{\hat{\sigma }}^z_{5}- 916{\hat{\sigma }}^z_{6} - 928{\hat{\sigma }}^z_{7} -916{\hat{\sigma }}^z_{8} \nonumber \\&\quad - 916{\hat{\sigma }}^z_{9} - 936{\hat{\sigma }}^z_{10} + 454{\hat{\sigma }}^z_{0}{\hat{\sigma }}^z_{1} + 4{\hat{\sigma }}^z_{0}{\hat{\sigma }}^z_{2} \nonumber \\&\quad + 454{\hat{\sigma }}^z_{0}{\hat{\sigma }}^z_{3} + 454{\hat{\sigma }}^z_{1}{\hat{\sigma }}^z_{2} + 454{\hat{\sigma }}^z_{1}{\hat{\sigma }}^z_{3}- 908{\hat{\sigma }}^z_{0}{\hat{\sigma }}^z_{5} \nonumber \\&\quad + 6{\hat{\sigma }}^z_{1}{\hat{\sigma }}^z_{4} + 454{\hat{\sigma }}^z_{2}{\hat{\sigma }}^z_{3} - 908{\hat{\sigma }}^z_{0}{\hat{\sigma }}^z_{6} - 908{\hat{\sigma }}^z_{1}{\hat{\sigma }}^z_{5} \nonumber \\&\quad + 6{\hat{\sigma }}^z_{2}{\hat{\sigma }}^z_{4} - 8{\hat{\sigma }}^z_{2}{\hat{\sigma }}^z_{5} + 454{\hat{\sigma }}^z_{3}{\hat{\sigma }}^z_{4} - 908{\hat{\sigma }}^z_{1}{\hat{\sigma }}^z_{7} \nonumber \\&\quad - 908{\hat{\sigma }}^z_{1}{\hat{\sigma }}^z_{8} - 908{\hat{\sigma }}^z_{2}{\hat{\sigma }}^z_{7} - 908{\hat{\sigma }}^z_{3}{\hat{\sigma }}^z_{6} - 8{\hat{\sigma }}^z_{3}{\hat{\sigma }}^z_{7} \nonumber \\&\quad - 8{\hat{\sigma }}^z_{4}{\hat{\sigma }}^z_{6} - 908{\hat{\sigma }}^z_{2}{\hat{\sigma }}^z_{9} - 908{\hat{\sigma }}^z_{3}{\hat{\sigma }}^z_{8} - 4{\hat{\sigma }}^z_{4}{\hat{\sigma }}^z_{7} \nonumber \\&\quad - 908{\hat{\sigma }}^z_{3}{\hat{\sigma }}^z_{9} - 8{\hat{\sigma }}^z_{4}{\hat{\sigma }}^z_{8} - 908{\hat{\sigma }}^z_{3}{\hat{\sigma }}^z_{10} - 8{\hat{\sigma }}^z_{4}{\hat{\sigma }}^z_{9} \nonumber \\&\quad - 908{\hat{\sigma }}^z_{4}{\hat{\sigma }}^z_{10} + 8{\hat{\sigma }}^z_{7}{\hat{\sigma }}^z_{10} + 8,448 \end{aligned}
(42)
### Complexity analysis
This subsection describes complexity analysis on the number of required qubits, especially the reduction from $$O(M^3)$$ in the direct method of17 to $$O(M^2)$$ proposed in this paper. In worst case condition, a brute force method of finding an $$M \times M$$ H-matrix should check all possible combinations of “-1” and “+1” in the $$M^2$$ entries of the matrix, i.e., we should perform orthogonality test to all of $$2^{M\times M}$$ matrices. In the SI (Supplementary Information) of17, we have showed that we need $$M^2$$ logical qubits if the machine capable to implement k-body interactions for any non negative integer k; which in this case is up to 4-body Hamiltonian terms (Eq. (S5)). When the machine is only capable of implementing 2-body Hamiltonian terms, additional ancillary qubits are required. In the sub section High-Order Case: The Needs of Symbolic Computing of the SI, we have showed that it will further increase the number of required qubits into $$M^2+M\times M(M-1)/2$$; i.e., an increase from $$O(M^2)$$ to $$O(M^3)$$.
Further reduction of the needed qubits is achieved through the usage of proposed methods described in this paper, such as the Turyn based method. As explained in the section Methods, subsection C. Derivation of the Turyn based Method, the (Turyn) Hadamard matrix can be constructed from an $$\left( n, n, n, n-1\right)$$ Turyn Type/TT-sequence. For a given $$\left( n, n, n, n-1\right)$$ TT-sequence, we can construct a $$4(4n-1)$$ order Hadamard matrix; i.e, to find an $$4(4n-1)$$ order H matrix, we only need to find a $$(4n-1)$$ length sequence. Therefore, in the Turyn-based method, the required logical qubits to find the $$M\times M$$ Hadamard matrix is O(M). The quadratic energy function given by Eq. (35) implies that there will be up to 4-body terms in the Hamiltonian. Again, when using D-Wave that can only accommodate up-to 2-body terms, additional ancillary qubits are needed. Accordingly, the final number of the required logical qubits will be $$O(M^2)$$.
## Data and code availability
All of codes and data will be provided upon direct request to the authors. Some parts of the codes can be found in https://github.com/suksmono.
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## Acknowledgements
This work has been supported partially by the WCR Program of Indonesian Ministry of Education (formerly Min. of Research and Higher Education), P3MI 2019 Program of ITB, and by the Blueqat Inc. (formerly MDR Inc.), Tokyo, Japan.
## Author information
Authors
### Contributions
A.B.S. formulated the theory, conducted the experiment(s), and analyzed the results. Y.M. assisting experiments in the quantum processor.
### Corresponding author
Correspondence to Andriyan Bayu Suksmono.
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Suksmono, A.B., Minato, Y. Quantum computing formulation of some classical Hadamard matrix searching methods and its implementation on a quantum computer. Sci Rep 12, 197 (2022). https://doi.org/10.1038/s41598-021-03586-0
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For Windows users, please install TeX Live.
### Build¶
To build a single PDF using LaTeX, use the following command:
jupyter-book build mybookname/ --builder pdflatex
or
jb build mybookname/ --builder pdflatex
Note
If you would just like to generate the latex file you may use:
jb build mybookname/ --builder latex
Individual PDF Files:
To build PDF files for each page of the project, you can specify the option --individualpages for --builder=pdflatex.
The individual PDF files will be available in the _build/latex build folder. These files will have the same name as the source file or, if nested in folders, will be named {folder}-{filename}.pdf.
Note
When specifying a page using the build command, the --individualpages will automatically be set to True.
In the future we intend for this to produce latex documents more suitable to single pages (see issue #904).
### Updating the name of the Global PDF file¶
To update the name of your PDF file you can set the following in _config.yml
latex:
latex_documents:
targetname: book.tex
This will act as an automatic override when Sphinx builds the latex_documents. It is typically inferred by Sphinx but when using jupyter-book naming the file in the _config.yml generally makes it easier to find.
### Using a different LaTeX engine¶
Some users may want to switch to using a different LaTeX engine to build the PDF files. For example, if your project contains Unicode you will need to use xelatex to build the PDF file.
To update the LaTeX engine to xelatex you can add the following to your _config.yml
latex:
latex_engine: xelatex
Note
We will be making xelatex the default in the near future, so this can be used to specify other builders such as pdflatex, or lualatex.
See the Sphinx documentation for available builders
### Other Sphinx LaTeX settings¶
Other LaTeX settings available to Sphinx can be passed through using the config section of Sphinx in the _config.yml file for your project.
For example, if you would like to set the latex_toplevel_sectioning option to use part instead of chapter you would use:
sphinx:
config:
latex_toplevel_sectioning: 'part' | 2020-11-29 09:50:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7238057851791382, "perplexity": 3047.5266173116547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141197593.33/warc/CC-MAIN-20201129093434-20201129123434-00657.warc.gz"} |
http://ecommons.library.cornell.edu/handle/1813/5861 | Please use this identifier to cite or link to this item: http://hdl.handle.net/1813/5861
Title: Abstract Identifiers and Textual Reference Authors: Allen, Stuart Keywords: computer sciencetechnical report Issue Date: 15-Nov-2002 Publisher: Cornell University Citation: http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR2002-1885 Abstract: Here are three proposals concerning the structure and maintenance of formal, inter-referential, digitally stored texts: (1) include abstract atomic identifiers in texts, (2) identify these identifiers with references to text objects, and (3) keep among the texts records of computationally substantiated claims about those texts. We use formal'' in a narrow sense approximating computer-checkable. We are informed by informal symbolic practices used in mathematical text and program source text, which we hope to enhance and exploit explicitly; the basic management problem is how to alter texts {\em rather} {\em freely} without ruining the bases for claims depending upon them, which becomes an issue of accounting for various dependencies between texts. We are {\em not} here proposing the use of abstract structured text; nonetheless, experience using it in Nuprl4 has led us to appreciate the benefits of distinguishing abstract form from concrete presentation, and also has shown us the cognitive and practical {\em un}importance of just which identifiers occur in abstract structured texts when texts are mediated by a system that realizes concrete presentation. Abstract treatment of identifiers involves concrete realization during communications between text servers'' and their clients. The benefit of treating identifiers abstractly is a radical avoidance of name collision, even at runtime, and is important for claims about texts that are based upon program execution. The notion of text collections and equivalence of text collections modulo change of identifiers is made precise by the second proposal. The complete identification of abstract identifiers with reference values is discussed, addressing the issues of dangling pointers, the association of ordinary symbolic identifiers with meaningful defining texts, the flatness'' of the pointer space, and the perhaps counterintuitive collapse of two abstract name spaces into one. The notion of certification system'' is introduced as a formalization of generic computationally defined claims about texts, emphasizing the diversity of clients who may not agree on a common logic''. The notion of a certificate whose computational meaning, in the context of the texts it refers to, is completely specified (although perhaps non-deterministic), and the notion of a certificate further being deterministic, are introduced and elaborated with regard to their epistemic value. What it means to give certificate texts the force of factual records, and mechanisms to accomplish this, are discussed. Scenarios for practically exploiting identifier abstractness and fully deterministic certificates are considered, involving the combination of partially independently developed texts and the experimental modification of texts in a collection. The importance of implementing multiple certification systems is articulated. URI: http://hdl.handle.net/1813/5861 Appears in Collections: Computer Science Technical Reports
Files in This Item:
File SizeFormat
2002-1885.ps257.65 kBPostscriptView/Open | 2014-09-19 13:54:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7460966110229492, "perplexity": 3255.025542738219}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657131376.7/warc/CC-MAIN-20140914011211-00144-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://socratic.org/questions/if-the-substrate-is-primary-can-we-rule-out-sn1-and-e1-entirely | # If the substrate is primary, can we rule out SN1 and E1 entirely?
Oct 31, 2015
No. You should assume that $E 1$ and ${S}_{N} 1$ may still happen to some extent, because they usually still do.
Many factors contribute to that occurring; for primary alkyl halides in particular, they may include:
• nucleophile strength
• nucleophile steric hindrance and bulkiness
• solvent steric hindrance and bulkiness
• solvent interactions (nonpolar, polar protic, polar aprotic)
• reaction temperature
• etc.
Let us take some examples that I'm pulling from one of my 2013 worksheets:
(The given product ratios were on the worksheet as-is.)
You can tell that the substrate each time is a primary alkyl halide, but clearly there are exceptions that allow for elimination to occur, even at normal temperatures.
In the first one, the phase-transferable (organic/aqueous solubility) tetrabutylammonium chloride (t-BuNCl) nucleophile is extremely strong and is not hindered by the organic polar aprotic solvent, promoting substitution. This has no $E 1$ or $E 2$, but has an unknown mixture of ${S}_{N} 1$ and ${S}_{N} 2$.
In the second one, the nucleophile is pretty strong but not extremely so, and interacting with an organic polar protic solvent via H-bonding decreases the nucleophilicity, decreasing the amount of substitution by a bit. This has ${S}_{N} 2$, some $E 2$, and little ${S}_{N} 1$ and $E 1$, but still some.
In the third one, the steric hindrance of the nucleophile promotes elimination, particularly $E 2$ and $E 1$, but also allows for some ${S}_{N} 1$. Likely no ${S}_{N} 2$ because the bulkiness on the nucleophile prevents it from finding the substrate quickly. | 2019-05-25 23:38:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 14, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7345322370529175, "perplexity": 3728.810202397289}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232258453.85/warc/CC-MAIN-20190525224929-20190526010929-00379.warc.gz"} |
http://codereview.stackexchange.com/questions/47090/similar-methods-using-loops | # Similar methods using loops
Is it recommended to combine two similar methods into one?
public void initialize_board(){
for(int y = 0; y < board.length; y++){
for(int x = 0; x < board.length; x++){
if(x == 0)
board[x][y] = (char)y;
else
board[x][y] = '~';
}
}
}
public void display_board(){
for(int y = 0; y < board.length; y++){
for(int x = 0; x < board.length; x++){
System.out.print(board[x][y]);
}
System.out.print("\n");
}
}
-
Use the camelCase for naming. – Dozortsev Anton Apr 13 at 21:47
Post rolled back. Please do not edit the original code based on answers; that will invalidate them. – Jamal Apr 14 at 14:18
Small trick: board[x][y] = (x == 0) ? (char) y : '~'; But you may want to stick to the original version for readability. – toto2 Apr 14 at 16:48
By the Single Responsibility Principle, they should remain separate methods. Furthermore, initialize_board() should probably be a constructor for a Board class, and display_board() should be its toString().
-
Indeed, the only reason I turned my constructor into a method was to ask this question. Also, toString() wouldn't work because it's an array of char – user89428 Apr 13 at 21:25
@user89428 What exactly prevents you from concatenating chars to a string? – Vogel612 Apr 13 at 21:32
I don't know, it says : "The method toString() in the type Object is not applicable for the argument (int)". I haven't been able to use toString() in Java successfully yet. – user89428 Apr 13 at 21:38
### Not so similar...
Just because the method bodies "look similar" doesn't make them similar. It's better to think in terms of purpose instead. The two methods are for completely different things. They are not similar, and good to be separated.
In functional languages you could use a "walk" method to iterate over all cells of the board and use a callback function to act on each cell. You could emulate that behavior in Java, but the code will get more complex, and I'm not sure it will be worth it in this example.
### Robustness
Do you intend to initialize the board more than once in the lifetime of the object? If not (probably), then move the initialization to the constructor.
### Writing style
You should use camelCase for naming, for example initializeBoard instead of initialize_board.
It's recommended to use braces with ifs, for example:
if (x == 0) {
board[x][y] = (char)y;
} else {
board[x][y] = '~';
}
It's better to print a newline using println:
System.out.println();
-
The two methods look similar because at some level they do the same thing : they traverse the board. What is repeated is the code to traverse the data structure. The gang of four identified isolating the traversal behaviour as the iterator pattern.
In object-oriented programming, the iterator pattern is a design pattern in which an iterator is used to traverse a container and access the container's elements.
(from wikipedia)
So you could apply the iterator pattern. If it's really just those two methods, it's probably overkill, but keep it in mind when you find yourself traversing the datastructure in other places as well.
-
1. I guess this does not do what you really want:
board[x][y] = (char)y;
It sets the first few ASCII control characters and display_board() shows weird results. I'd try this:
board[x][y] = Character.forDigit(y, 10);
System.out.print("\n");
you should use println() (as @janos already mentioned) but if you want to put a line feed manually you should use
System.out.print("%n");
instead. %n outputs the correct platform-specific line separator. | 2014-09-02 06:46:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1917358636856079, "perplexity": 2425.777481250654}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535921869.7/warc/CC-MAIN-20140901014521-00275-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/205082 | Infoscience
Conference paper
# Consistency of $\ell_1$-Regularized Maximum-Likelihood for Compressive Poisson Regression
We consider Poisson regression with the canonical link function. This regression model is widely used in regression analysis involving count data; one important application in electrical engineering is transmission tomography. In this paper, we establish the variable selection consistency and estimation consistency of the $\ell_1$-regularized maximum-likelihood estimator in this regression model, and characterize the asymptotic sample complexity that ensures consistency even under the compressive sensing setting (or the $n \ll p$ setting in high-dimensional statistics).
#### Reference
• EPFL-CONF-205082
Record created on 2015-02-16, modified on 2016-08-09 | 2016-12-04 09:05:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22034955024719238, "perplexity": 1380.4663925816853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541220.56/warc/CC-MAIN-20161202170901-00316-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://solvedlib.com/n/shouin-the-hgurc-a-charge-2e-authe-orieinandan-unknoyncharge,16569089 | # Shouin the hgurc a charge +2e Authe orieinandan unknoyncharge Terontre nmawhabintnemienatudeand Stenolthe unknoncharkuKutxeelectric (Orcesmall positive test chargeisGe d81ga charge +20unknown
###### Question:
shouin the hgurc a charge +2e Authe orieinandan unknoyncharge Terontre nmawhabintnemienatudeand Stenolthe unknoncharku Kutxe electric (Orce small positive test chargeis Ge d81ga charge +20 unknown charge If4 nm F0 0,022 1+4
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13 . According to a Pew Research Center survey in January 2017,59% of black internet users say they have experienced online harassment; with margin of error of 9% at a 95% level of confidence. Write an English sentence that correctly explains this finding in a way that would be clear to someone who ... | 2023-03-24 21:34:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6174864768981934, "perplexity": 12201.976147747118}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945289.9/warc/CC-MAIN-20230324211121-20230325001121-00442.warc.gz"} |
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# Weather-related Economics of Natural Gas Transport for Two Propulsion Plant Configurations
It has always been difficult to incorporate the effect of environmental conditions into an economical evaluation. This article describes the influence of weather conditions to the LNG shipping costs.
## Summary
This study shows that scenario simulations can be a valuable tool to get an idea about the uncertainties concerning power demand and roundtrip duration due to wind, waves and current. This article presents the procedure and results of an economical evaluation of an LNG carrier with two different propulsion plant configurations.
The evaluation is firstly performed without weather influences. Subsequently, the weather influences are incorporated by using scenario simulations.
## Nomenclature
• MMBtu – Million British thermal units. 1 MMBtu natural gas equals approximately 0,019 tonnes LNG;
• Tcf – Trillion cubic feet. 1 Tcf natural gas equals approximately 21 million tonnes LNG;
• mtpa – Million tonnes per annum;
• tonnes – Metric ton (1 000 kg).
Unless otherwise stated, all prices mentioned in this article are given at the price level of 2003.
## Introduction
In the last couple of years several studies have been made to analyse potential economic and environmental benefit of alternative propulsion systems as diesel-electric, gas turbine driven power plants (both with utilisation of boil-off gas) or low speed diesel engines in combination with reliquefaction plants.
A significant simplification in most feasibility studies is the fixed voyage duration and fixed propulsion load. The influence of varying weather conditions on a particular route are generally not taken into account. However, the constantly changing wind, waves and (tidal) currents play a major role in the reliability of the service as well as the fuel consumption and thereby influence the revenue-making potential of a new concept.
Maritime Research Institute Netherlands developed a new generation of scenario simulation tools that offer the opportunity to solve the above problem in a more detailed and realistic way. In these simulations several years of operational service are carefully mimicked and account for the weather on the adopted route, a scenario for the way the master sails the ship (including risk avoiding behaviour) and the hydrodynamic characteristics of the design.
In the present case study an LNG carrier equipped with a low speed diesel engine in combination with a reliquefaction plant is compared with a conventional steam turbine powered LNG carrier. The simulations are made on the Qatar – US East Coast trade where the Suez Canal passage is a complicating factor. The results of the study are presented in terms of transport capacity, reliability, required freight rate and rate of return.
## LNG Chain
### Exploration and Production
This stage covers the exploration, development and production of natural gas fields. Most of the time natural gas is discovered during the search for oil. LNG projects require large gas reserves to produce for at least 15 to 20 years. According to BP’s statistical review of world energy the proven reserves of natural gas are 5 501 Tcf (2003). The worldwide natural gas consumption has grown from 86,0 Tcf in 2000 to 91,4 Tcf in 2003.
For short distances natural gas is transported by pipeline. The turning point lies around 1 100 nautical miles for Offshore supply chain of Liquefied Natural Gasoffshore pipelines and 2100 nm for onshore pipelines over longer distances shipping LNG is generally cheaper.
### Shipping
In 2002, 12 countries shipped 5,4 Tcf natural gas (Picture 1). The largest LNG exporter is Indonesia (21 %) followed by Algeria (17 %), Malaysia (14 %) and Qatar (13 %).
According to LNG One World the current fleet consists of 164 LNG carriers with 81 vessels on order. The cargo capacity of most newbuildings is between 135 000 and 145 000 m3 LNG. The applied cargo containment systems are generally the spherical (Moss) design or the membrane design. At the moment two third of the all newbuildings are equipped with Cargo Containment Systems of LPG and LNGmembrane type cargo tanks. According to Fairplay, the newbuilding cost of an LNG carrier is roughly 150 to 190 million dollar.
LNG carriers have traditionally been propelled by steam turbine power plants, mainly because the cargo boil-off can be used as fuel and of course because of its reliability. A great disadvantage of steam turbine propulsion is the very low efficiency.
In general the transportation costs increase linearly with distance. The unit cost of transport is a function of the capital cost, the financing terms, the rate of return and the distance to be covered.
## Operational Profile and Design
According to EIA, it is expected that overall US gas demand will increase from 22,5 Tcf in 2002 to around 31,4 Tcf in 2025 (+40 %). The US domestic natural gas production and the Canadian pipeline imports are not likely to meet this increase. For this reason it is expected that LNG imports will increase significantly in the coming years. In 2003 most LNG imports originated from Trinidad & Tobago, Algeria and Nigeria (Picture 2). These exporting countries are preferred to Oman, Qatar and Malaysia mainly because of the distance advantage.
However, for the present study it is chosen to evaluate a relatively long route between Qatar and the northeast coast of the US. This route is selected for the following reasons:
• The reserves of Qatar are almost 15 % of the total proven reserves of the world. It is expected that export will increase significantly when the new liquefaction trains become operational in 2004-2006.
• Due to the increasing natural gas demand in the US prices will rise and it becomes feasible to import LNG from distant exporting terminals.
• The Suez Canal is an expensive and logistic barrier on this route, which makes it an interesting case.
• The weather conditions on the transatlantic crossing will require a more complete approach because it is expected that the reliability and fuel consumption will be influenced significantly by wind, waves and current.
In the figure below the four existing US importing terminals are denoted by the triangles. The numbered circles denote the planned import terminals. Everett is the most northern existing terminal. This terminal or one of the nearby planned terminals is chosen as importing terminal.
### Schedule
The schedule, which is adopted between Qatar and the northeast US coast, is given in Table 2. The total roundtrip time is 39 days which equals 9,23 roundtrips per year when 5 non-workable days per year are taken into account.
The route is cut in two by the Suez Canal which introduces two additional fixed arrival times. Arriving late at Suez or Port Said cause at best a considerable surcharge or at worst a day’s delay because there is only one departure from each side of the canal (LNG carriers which are not gas free have to take the first convoy in both directions).
In addition the arrival time at Qatar and starting loading LNG (both bold-printed in the schedule) should match when roundtrips are made on a regular basis.
At first glance it appears that there is not much spare time incorporated in the schedule, but the scheduled speed is 18,35 knots while the trial speed is around 21 knots at 90 % MCR. Accordingly it could be possible to catch up delays up to one day without missing the Suez Canal slot provided that the weather conditions are good. Furthermore, it is possible to arrive up to two hours late at Suez or Port Said if a surcharge is paid. The applied manoeuvring time of three hours per arrival are estimates only, and are not based on detailed information of the local situation.
### LNG Carrier Design
It is chosen to evaluate a conventional membrane type LNG carrier with a nominal cargo capacity of 145 000 m3. The main dimensions of the vessel are given in Table 1. The boil-off rate in loaded and ballast condition is approximated at 0,12 % respectively 0,06 % of nominal cargo capacity per day.
Table 1. Main dimensions
Length over all[m]286,00
Length perpendiculars[m]274,00
Depth to main deck[m]26,00
Draught[m]11,38
Gross tonnage[tonnes]98 261
Displacement[m3]102 170
Cargo capacity[m3]145 000
The design is evaluated for two propulsion plant configurations. It is assumed that the engine room length, and therefore the cargo capacity, is equal for both alternatives.
The first alternative is equipped with a steam turbine as prime mover. Two dual fuel boilers supply steam for the turbine (29,0 MW at 83 RPM) and the two turbo generators (2×3 500 kW). The adopted efficiencies are respectively 0,88, 0,35 and 0,25.
The second alternative is equipped with a 2-stroke diesel engine with a nominal output of 31 MW at 76 RPM. For the electrical power three 4-stroke generator sets are installed (output 3×2 800 kW). The adopted efficiencies are 0,51 (167 g/kWh) and 0,46 (183 g/kWh).
The differences in efficiency are striking, 0,31 against 0,51 for the prime mover and 0,22 against 0,46 for the electrical power supply.
The fuel consumption is calculated at the highest engine efficiency i. e. the increase in specific fuel oil consumption at partial load is not accounted for.
## Capital and Running Costs
### Capital Costs
The building cost of the design equipped with a steam turbine are estimated at $160 M. The building cost of the alternative design, equipped with a diesel engine, are more difficult to predict. By subtracting the costs of the steam turbine propulsion system and adding the costs of the two-stroke diesel engine, generator sets and Birth of the Reliquefaction, Design and Operation of the Reliquefaction LPG Plantreliquefaction plant a first estimate is obtained. The specific costs of the steam turbine propulsion system are estimated at 600$/kW (estimate based on land-based power plants). For the 29 MW installation the cost amount to $17,3 M. The specific costs of the two-stroke main engine and the generator sets are estimated at respectively 290$/kW and 580 $/kW. The corresponding costs are$9,0 M and $5,0 M. The costs of the onboard reliquefaction plant are related to the capital costs of BP’s Trinidad liquefaction train of 165$/mtpa. The reliquefaction capacity is based on a nominal boil-off capacity of 0,15 % per day and 100 % redundancy. The total costs are estimated at $11,1 M. The total buildings costs of the diesel engine alternative are estimated at$167,7 M (+4,8 %).
### Running Costs
The running costs include the crew, insurance, docking, special survey and management costs. Usually, the maintenance and lubrication oil costs are incorporated in the running costs as well. In the current analysis it is chosen to put these costs in the voyage costs because they are dependent on the engine load.
The crew costs – to the amount of $4,0 M per year – are based on a crew of 39 persons (24 officers and 15 crew). A first estimate of the insurance, docking, special survey and management costs, to the amount of$1,5 M per year.
## Voyage Costs
### Scenario Simulations
Previous studies show that key assumptions are made with respect to engine load and vessel speed. The fuel consumption and reliability is heavily dependent on both parameters. The environmental conditions change not only on an hourly basis but also per sea area. For this reason it is expected that assuming a constant engine load and speed will not give a representative estimate of the fuel consumption and reliability.
Simulating a large number of voyages during several years will avoid this difficulty. In these simulations the sustained speed as well as ship motions are calculated on the basis of the actual weather conditions (wind, waves and current). When the ship motions are not tolerable the speed will be reduced and, if possible, the delay occurred will be recovered in a later stage of the voyage.
The primary result is the distribution of fuel consumption and duration of the individual voyages. This approach leads to an improved starting point for the evaluation of both alternatives.
The flow chart per simulated voyage is given in Picture 4.
### Operational Scenario
Reliability is essential in LNG shipping, for this reason a just in time scenario including voluntary speed reduction is adopted. In this scenario it is always tried to arrive in time. Delays, caused by involuntary speed loss or voluntary speed reduction, are recovered when possible.
Involuntary speed loss denotes the speed loss due to the increased power demand in wind and waves. Voluntary speed reduction denotes speed reduction by the master due to intolerable ship behaviour.
In the simulations the relative velocity at the bow is the only criterion on which is effectively decided to reduce speed. The relative velocity at the bow is a good measure for the probability of keel and bow flare slamming. A threshold value of 9,0 m/s is adopted.
In the simulations not more than 90 % MCR is allowed and the minimum allowable engine load is 30 % MCR. The present scenario simulations do not include the issues listed below.
• Delays due to e. g. terminal congestion or engine failure.
• Engine load dependent specific fuel oil consumption.
• Added resistance due to steering and drift angle.
• Effect of shallow water on resistance and ship motions.
• Effect of seawater and air temperature.
• Fouling of hull and propeller.
### Results Simulations
1 050 roundtrips are simulated in a period of 4 years, which equals more than 100 years simulation time. If it is assumed that 4 years weather date is a representative sample of the climate, the simulations provide reliable “lifetime” statistics.
The average wave height on the Qatar to Suez route is 0,7 m with extremes of 2,6 m (this wave height is exceeded in 1 % of the time). The wind speed is around Beaufort 3. The schedule on this route is met at 100 % of the time.
However, the Port Said – US east coast leg show a less favourable climate. The average wave height is around 1 m higher and a wave height of 6 m is exceeded in 1 % of the time. On this route the sustained speed and brake power are highly variable depending on the local weather conditions.
In Picture 5 a time series of the sustained speed is given.
During the summer season (0,5, 1,5 and 2,5 year) the scheduled speed of 18,35 kn is maintained almost continuously. The points just below the 18,35 kn line denote the involuntary speed loss, the points between approximately 5 and 10 denote “voluntary” speed loss and the points above 18,35 kn denote the attempts to meet the schedule.
The reliability of the service is high. Nevertheless 15-20 % of the crossings from Port Said to the US east coast show a delay of more than the incorporated spare time of 2 hours (Picture 6). This delay has to be recovered on the return voyage to Suez. In only 1,2 % of the roundtrips it is not possible to meet the schedule of 39 days (i. e. out of the eighty roundtrips one roundtrip last 40 days).
It has to be noted that the differences in reliability between the diesel engine and steam turbine alternative are mainly caused by the higher power available of the diesel engine.
Due to the high reliability, it might be interesting to evaluate a tighter schedule. Saving one day at each side of the Suez Canal seems possible. However, based on the results from Port Said to the US east coast it can be concluded that the decrease in reliability is considerable (Table 3). For this reason the results are not presented in the article.
Table 3. Reliability in terms of fraction of in-time arrivals at import terminal US east coast
AlternativeTotalSummerWinter
Steam, 39 days79,7 %96,9 %60,9 %
Steam, 37 days48,9 %74,1 %21,2 %
Diesel, 39 days84,5 %98,0 %69,7 %
Diesel, 37 days67,6 %90,3 %42,7 %
### Voyage Costs – Steam Turbine
The voyage costs consist of harbour and canal costs, fuel costs and in the present study also of maintenance and lubrication oil costs. The maintenance costs of the steam turbine and boilers are negligible. The harbour dues are estimated at $90 000 per arrival. The transit costs of the Suez Canal are substantial and are based on Suez Canal Net Tonnage of the LNG carrier (about 84 100 SCNT). The transit costs are estimated at$420 000 in loaded and $360 000 in ballast condition. The electrical power demand: • Manoeuvring: 5 000 kW; • Loading: 3 900 kw; • Discharging: 6 000 kW; • At sea: 1 500 kW. The fuel consumption is calculated from the electrical power demand, the required power to the propeller and the amount of boil-off gas. The transport capacity is calculated on the basis of a cargo tank filling level of 98,5 % of the nominal capacity in loaded condition and 2,0 % in ballast condition. The boil-off is 2 370 tonnes LNG per roundtrip and the transport capacity 64 200 tonnes per roundtrip. Read also: Rules for Safe Transportation of Cargoes by Sea on the Cargo Ship Without weather influences the heavy fuel consumption is 1 930 tonnes per roundtrip. When adopting an HFO price of 150$/ton, the fuel costs amount to $0,29 M per roundtrip. When the weather conditions are taken into account the above figures are affected. The number of roundtrips per year is almost equal (from 9,230 to 9,218 roundtrips per year), and the boil-off costs hardly increase. Compared to the fuel consumption without weather influences the fuel costs are increased by 17 % to approximately$0.34 M per roundtrip. Picture 7 shows that the spread in fuel consumption is substantial. The increase in fuel consumption is not only caused by the influence of wind, waves and current but also by the pressure to meet the schedule.
### Voyage Costs – Diesel Engine
The harbour and canal costs are equal to the costs as given in paragraph above. The transport capacity, however, is larger because boil-off gas is reliquefied. The annual transport capacity is 607 000 tonnes (against 592 500 tonnes for the steam turbine alternative).
The electrical power demand is equal to the first design alternative apart from the additional power demand of the reliquefaction plant which is estimated at 920 W/kg*h.
Estimates of the maintenance costs and lubrication oil consumption and costs are given in the table below.
Table 4. Specific diesel engine costs
Main Engine
System oil72 kg/d700 $/tonnes Cylinder oil1,5 g/kWh800$/tonnes
Maintenance costs1,0 $/MWh Generator sets Lubrication oil1,0 g/kWh700 Maintenance costs2,5$/MWh
Without the effect of the weather, the lubrication and maintenance costs amount to $45 900 per roundtrip and the fuel costs amount to$0,45 M per roundtrip.
Including the weather effect, the average fuel costs per roundtrip are increased by 6 % to $0,51 M. The average maintenance and lubrication oil costs are$47 200 (+3 %).
Just as the steam turbine alternative, the spread in fuel costs of the diesel engine alternative is large. Statistically, the fuel costs per roundtrip exceed $0,54 M around one roundtrip per year. ### Economical Evaluation The following financing parameters are used in the evaluation from MPT consult and this book: • Lifetime 20 years; • A loan percentage of 60 % of the building costs at an interest rate of 6,5 %; • The building price is paid during the construction in several instalments. The total pre-financing costs are considered to be 3,5 % of the building price; • The residual value is estimated at 8 % of the building costs. The above figures show the cost distributions for both alternatives and are valid for year five. The insurance, docking, special survey and management costs belong to miscellaneous costs. The capital costs (redemption and interest) take around one third of the total costs. Furthermore, the Suez Canal costs are significant, taking almost 24 % of annual costs. It could be worthwhile to investigate into possibilities for fuel saving, because the fuel costs contribute substantially to the annual costs (19 % for the steam turbine alternative and almost 16 % for the diesel engine alternative). The annual transport capacity of the diesel engine alternative is 2,4 % higher than the steam turbine alternative (607 000 respectively 592 500 tonnes per year). The transport capacity is hardly influenced by taking into account the weather influences because it turns out that delays can easily be recovered during a roundtrip. The shipowner is basically interested in a high profit. Therefore, the required freight rate (RFR) is not the most suitable criterion to judge the design alternatives because the RFR takes only the costs into account. The internal rate of return (IRR) is more appropriate for this purpose as it stimulates the generation of additional income. The IRR is defined to be the discount rate that makes the net present value of the cash flows equal to zero. In Table 5 we present the required freight rates and the calculated rate of return, where the IRR is based on an estimated income of 1,35$/MMBtu. The values between brackets are referring to the results from the scenario without weather influences.
Table 5. Comparison of RFR and IRR for both alternatives with and without water influence
RFR [\$/MMBtu]IRR [%]
Steam0,923 (0,907)20,5 % (21,6 %)
Diesel0,895 (0,886)21,7 % (22,3 %)
What can be concluded from Table 5? The differences between both alternatives seem to be small. However, in the case of weather influences being included, the use of diesel seems to be 3,03 % cheaper than the use of steam. In the case of weather influences not being included that difference is only 2,3 %. Small differences per unit generate huge amounts on a yearly basis.
This can be translated into similar results while looking at the internal rate of return. Taking into account weather influences and using diesel engines will generate an IRR of 21,7 %, being 5,9 % higher than in the case of using steam. In the case of weather influences not included, the difference amounts to 3,2 %.
While interpreting these results we have to stress that the above mentioned results are very sensitive to the LNG/HFO price ratio.
## Conclusion
It has always been difficult to incorporate in this kind of scientific work the effect of environmental conditions into an economic evaluation. In most cases power demand was increased with a fixed proportion, or even omitted completely.
With this article we try to show that that scenario simulation could be a valuable methodology. The uncertainty concerning the information about power demand and trip duration due to wind, waves and current, can as such be included in the calculations.
From the simulations presented in this article we can calculate important differences concerning internal rate of return and required freight rates. The reliability is high. Out of the eighty roundtrips, only one roundtrip showed a delay of one day. However the reliability on the Port Said – US northeast coast leg can be a point of concern in the winter season.
The results presented in this article are interesting and invite for additional and even more detailed empirical research. One application could be to alter the schedule, increase the speed and as such also the transport capacity supplied. A consequence of this strategy will be a decrease of the reliability, especially in the winter season. Such an effect could be unfavourable because of the higher gas demand and the corresponding smaller buffer capacity in winter.
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http://math.stackexchange.com/questions/27633/distance-between-consecutive-primes-related-to-polignacs-conjecture | # distance between consecutive primes (related to Polignac's conjecture)
Is there an elementary(or not) proof that there are at least two consecutive primes which have difference $2n$ for every natural number $n$? i remind that Polignac's conjecture states that there should be infinite such pairs for every $n$ so it is a much more easy question to ask and of course should be valid.
In addition, for every $n\in\mathbb N$, is there any $m_n\in\mathbb N$ such that there are two natural numbers $k_1,k_2$ so that $k_2-k_1=2n$, $k_1, k_2$ are not divisible by $p_1,\dots p_{m_n}$, where $p_i$ is the $i$-th prime, and for every $k\in\{k_1+1,\dots k_2-1\}$ there exists a $p\in\{p_1,\dots p_{m_n}\}$ such that $p|k$? (elementary proof)
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I think your title is a little misleading, as Polignac's conjecture is only related to your question and not your actual question. Would you mind changing it to something that more accurately represents what you're after? – Mike Spivey Mar 17 '11 at 17:14
do you have something better to propose, if so feel free to edit the title – minasteris Mar 17 '11 at 17:39
@minasteris: How is this? (Feel free to edit or change back; this is your question, after all.) – Mike Spivey Mar 17 '11 at 17:42
The answer to the new question is no. Take $m=1$ and $n=2$. – Jonas Meyer Mar 18 '11 at 18:10
sorry wrong edit – minasteris Mar 18 '11 at 19:25
I don't know the state of the art on this question, but some searches indicate that it is not known whether every even number is a difference of two primes, let alone a difference of consecutive primes. Perhaps a little old, but this is mentioned in The new book of prime number records by Paulo Ribenboim, 1996, on page 250. (This problem is called the "Goldbach Variation" in Hofstadter's Gödel, Escher, Bach.)
The following source seems more recent, and mentions that Chen's work related to the Goldbach conjecture showed that every even number is a difference of a prime and a product of at most two primes: http://primes.utm.edu/notes/conjectures/
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where can i find chen's article? – minasteris Mar 18 '11 at 6:38
I'm not sure, but it might be this one: books.google.com/… – Jonas Meyer Mar 18 '11 at 7:05
@minasteris: Perhaps what you had in mind was related to the comments on Chris Caldwell's Prime Pages just below the item about Chen's work related to the Goldbach conjecture (see Jonas's last link in Answer). That next item concerns Polignac's conjecture and says (in part): "It is easy to show that for every positive integer m there is an even number 2n such that there are more than m pairs of consecutive primes with difference 2n." Note that n is not free here, but has an existence conditioned on m. – hardmath Mar 18 '11 at 19:48
@hardmath: my question is different – minasteris Mar 18 '11 at 20:41
The reworked addition to the second question has a positive answer. For any positive integer $n$ we can find an initial segment of the primes $p_1,p_2,...,p_m$ (where $m > 0$) and a positive number $k$ such that neither $k$ nor $k+2n$ is divisible by any of those primes, but each integer strictly between $k$ and $k+2n$ is divisible by at least one of them.
For motivation consider the special case in which $2n+1$ happens to be a prime. Then take $k=1$ and the primes to be those less than $2n+1$. Clearly $k$ and $k+2n$ are free from those small divisors, while every integer in between must be divisible by at least one of them.
Edited: Changed to make larger primes those above $2n$ (rather than above $n$)
For the general case we will use the CRT. It is simply a matter of choosing remainders for $k$ modulo the primes less than or equal to $2n$ so that neither $k$ nor $k + 2n$ is divisible, and then deploying the larger primes as needed to knock out any in-between integers not already divided by one of the smaller primes.
Obviously we want $k$ to be odd, so that $k+2n$ is odd as well. For odd primes $p_i$ up to (and if necessary) including $n$, there will always be at least one remainder for $k$ mod $p_i$ such that neither $k$ nor $k+2n$ is divisible by $p_i$. If $p_i$ divides $2n$, then this is really just a matter of choosing a nonzero remainder at $k$. If not, the fact that $p_i$ > 3 means there are only two "bad" choices for the remainder at $k$, so there's always at least one good choice.
Now consider any integers strictly between $k$ and $k+2n$ that are not already "sieved out" by those smaller primes. Then use the successive primes above $2n$ to get one-by-one a divisibility condition on those intervening numbers, which of course specifies a nonzero remainder for $k$ modulo each such prime. A crude estimate is that we need no more than $n$ primes greater than $2n$ to accomplish knockouts of all the entries unsieved by 2 (and hence by the collective set of smaller primes).
Finally determine a $k > 0$ having the specified remainders modulo the primes $p_1,...,p_m$ and we are done. None of them divide $k$ or $k+2n$, but at least one divides each of integers strictly in between.
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http://diversdream.cz/new-york-mbrxi/lyman-series-wavelength-equation-6fb013 | Lyman series (n l =1). The He II Lyman lines have almost exactly one-quarter the wavelength of their hydrogen equivalents: for example, He II Lyman-α is at 30.4 nm, and the corresponding Lyman limit is at 22.7 nm. According to Einsteins equation, the metal (s) which will emit photoelectrons for a radiation of wavelength 4100 ${{M}_{2}}$ is/are, The ratio of the dimensions of Plancks constant and that of the moment of inertia is the dimensions of, Application of a forward bias to a p-n junction, If a vector ${{L}_{2}}\,.\,{{M}_{1}}={{M}_{2}}$ is perpendicular to the vector ${{L}_{1}}\,.\,2{{L}_{2}}$ ok, then the value of a is, When an ideal monoatomic gas is heated at consant pressure, fraction of heat energy supplied which increases the internal energy of gas, is, The first line of the Lyman series in a hydrogen spectrum has a wavelength of $1210 Å$. All the wavelength of Lyman series falls in Ultraviolet band. Semiconductor Electronics: Materials Devices and Simple Circuits, The wavelength of Lyman series for first number is, The mass and diameter of a planet are twice those of earth. The series is named after its discoverer, Theodore Lyman, who discovered the spectral lines from 1906–1914. Time it out for real assessment and get your results instantly. NCERT P Bahadur IIT-JEE Previous Year Narendra Awasthi MS Chauhan. History. (Adapted from Tes) The wavelength is given by the Rydberg formula. Solution. Favorite Answer. The wavelengths (nm) in the Lyman series are all ultraviolet: Open App Continue with Mobile Browser. The $(\frac{1}{r})$ dependence of $|\vec{F}|$ can be understood in quantum theory as being due to the fact that the ‘particle’ of light (photon) is massless. The wavelength of Lyman series for first number is. In the spectrum of hydrogen, the ratio of the longest wavelength in the Lyman series to the longest wavelength in the Balmer series is. Test Yourself. The Balmer series in a hydrogen atom relates the possible electron transitions down to the n = 2 position to the wavelength of the emission that scientists observe.In quantum physics, when electrons transition between different energy levels around the atom (described by the principal quantum number, n ) they either release or absorb a photon. Pentachoron. How can I calculate wavelength in meters? 249 kPa and temperature $27^\circ\,C$. Biology. Calculate the longest wavelength in Lyman Series. Since the atomic number of Hydrogen is 1. (The Lyman series is a related sequence of wavelengths that describe electromagnetic energy given off by energized atoms in the ultraviolet region.) How do you calculate the wavelength from speed? 40 views. Identify the spectral regions to which these wavelengths correspond. To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line. Physics. Hydrogen exhibits several series of line spectra in different spectral regions. KEAM 2013: The wavelength (in cm) of second line in the Lyman series of hydrogen atomic spectrum is (Rydberg constant = R cm-1) (A) ((8R /9 )) (B) ((9 The term is also used to describe certain lines in the spectrum of singly ionized helium. In 1880, Rydberg worked on a formula describing the relation between the wavelengths in spectral lines of alkali metals. In the line spectra of hydrogen atom, difference between the largest and the shortest wavelengths of the Lyman series . The entire system is thermally insulated. wavelength, frequency, and velocity but these values will be significantly different numerically from water and rope waves. 1 Answer. For the first member of the Lyman series: since the electron is de-exited from #1(\text{st})# exited state (i.e #\text{n} = 2#) to ground state (i.e #text{n} = 1#) for first line of Lyman series. asked Dec 22, 2018 in Physics by Maryam (79.1k points) atoms; nuclei; neet +1 vote. The Wave Number in Series: The wavenumber of a photon is the number of waves of the photon in a unit length. How can I calculate wavelength of a photon? #1/lambda = \text{R}(1/(n_1)^2 - 1/(n_2)^2) * \text{Z}^2#, R = Rydbergs constant (Also written is #\text{R}_\text{H}#) The first emission line in the Lyman series corresponds to the electron dropping from #n = 2# to #n = 1#. If the same voltmeter is connected across a 6 V battery then the mass of copper deposited in 45 min would be, A network of four capacitors of capacities equal to $\sqrt{\frac{2h}{g}}$ and $\sqrt{\frac{h}{g}}$ are connected to a battery as shown in the figure. around the world, Calculations with wavelength and frequency. 1 answer. n= 2,3,4,...). AIIMS 2010: The wavelength of Lyman series for first number is (A) (4×1.097×107/3) m (B) (3/4×1.097×107) m (C) (4/3×1.097×107) m (D) (3/4)×1.09. The Balmer series, or Balmer lines in atomic physics, is one of a set of six named series describing the spectral line emissions of the hydrogen atom.The Balmer series is calculated using the Balmer formula, an empirical equation discovered by Johann Balmer in 1885.. #1/lambda = \text{R}(1/(1)^2 - 1/(2)^2) * 1^2#. $\lambda$ is the wavelength and R is the Rydberg constant. #1/λ = -"109 677 cm"^"-1" × (1/2^2 -1/1^2)#, #= "109 677" × 10^7color(white)(l) "m"^"-1" (1/4-1/1)#, #= -"109 677 cm"^"-1" × (-3/4) = "82 257.8 cm"^"-1"#, #λ = 1/("82 257.8 cm"^"-1") = "1.215 69" × 10^"-5"color(white)(l) "cm"#, #= "1.215 69" × 10^"-7"color(white)(l) "m"#, 100507 views The Lyman series is caused by electron jumps between the ground state and higher levels of the hydrogen atom.
Strategy: The Lyman series is given by the Balmer -Rydberg equation with and . For which one of the following, Bohr model is not valid? [Comptes rendus des séances de l'Académie des sciences. The corresponding line of a hydrogen- like atom of $Z = 11$ is equal to, The inverse square law in electrostatics is$\left|\vec{F}\right| = \frac{e^{2}}{\left(4\pi\varepsilon_{0}\right)\cdot r^{2}}$ for the force between an electron and a proton. Try this, The Lyman Series say the for the second is 121.6nm (nano metres) For the third it is 102.6 and the fourth is 97.3 all in Nano Metres which *10^-9. Relevance. Therefore, the lines seen in the image above are the wavelengths corresponding to n = 2 on th… The process is: A screw gauge has least count of 0.01 mm and there are 50 divisions in its circular scale. Test Series. The wavelength λ of the spectral line of Lyman series can be calculated using the following formula: 1 λ = R [ 1 1 2 − 1 n 2 2] The longest wavelength is the first line of the series for which n 2 = 2 He noticed that lines came in series and he found that he could simplify his calculations using the wavenumber (the number of waves occupying the unit length, equal to 1/λ, the inverse of the wavelength) as his unit of measurement. You can convert … If the wavelength of the first line of the Balmer series of hydrogen is $6561 \, Å$, the wavelength of the second line of the series should be, If $\upsilon_{1}$ is the frequency of the series limit of Lyman series, $\upsilon_{2}$ is the frequency of the first line of Lyman series and $\upsilon_{3}$ is the frequency of the series limit of the Balmer series, then. See all questions in Calculations with wavelength and frequency. For Lyman series,1λ=R1n12-1n2215R16=R112-1n2215R16R=n22-1n22 =15 n22 =16 n22-16 n22=16, n2=4 Previous Year Papers. The shortest-wavelength line occurs when is zero or when is infinitely large (i.e., if , then . #color(blue)(bar(ul(|color(white)(a/a) 1/λ = -R(1/n_f^2 -1/n_i^2)color(white)(a/a)|)))" "# where. Books. Série 2, Mécanique-physique, Chimie, Sciences de l'univers, Sciences de la Terre] -- 1983-04 -- periodiques (b) What value of n corresponds to a spectral line at The period of oscillation of pendulum on this planet will be (if it is a seconds pendulum on earth), A cord is used to lower vertically a block of mass M by a distance d with constant downward acceleration $m=\mu M=\frac{1}{2}\times 44=22g$ . Solution for Do the Balmer and Lyman series overlap? B is completely evacuated. If photons had a mass $m_p$, force would be modified to. This formula gives a wavelength of lines in the Lyman series of the hydrogen spectrum. A body weighs 72 N on the surface of the earth. What is the gravitational force on it, at a height equal to half the radius of the earth? Its density is :$(R = 8.3\,J\,mol^{-1}K^{-1}$). Lyman α emissions are weakly absorbed by the major components of the atmosphere—O, O 2, and N 2 —but they are absorbed readily by NO and have… Read More; line spectra How can I calculate wavelength of radio waves? Chemistry. How do you calculate the frequency of a wave? And it says that the reciprocal of the wavelength in the spectrum is Rydberg's constant times 1 over the final energy level squared minus the 1 over the initial energy level squared. In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1. LYMAN, série de: Angl. Different lines of Lyman series are . Therefore, the lines seen in the image above are the wavelengths corresponding to n=2 on the right, to n= on the left (there are infinitely many spectral lines, but they become very dense as they approach to n=, so only some of the first lines and the last one appear). #R =# the Rydberg constant (#"109 677 cm"^"-1"#) and #n_i# and #n_f# are the initial and final energy levels. Thus it is named after him. If the same force F is applied on the wire of the same material and radius 2r and length 2L, then the change in length of the other wire is, When a copper voltmeter is connected with a battery of emf 12 V. 2 g of copper is deposited in 30 min. Show that the (a) wavelength of 100 nm occurs within the Lyman series, that (b) wavelength of 500 nm occurs within the Balmer series, and that (c) wavelength of 1000 nm occurs within the Paschen series. The phase difference between displacement and acceleration of a particle in a simple harmonic motion is: A cylinder contains hydrogen gas at pressure of The first emission line in the Lyman series corresponds to the electron dropping from #n = 2# to #n = 1#. A contains an ideal gas at standard temperature and pressure. Therefore, longest wavelength (121.5 nm) emitted in the Lyman series is the electron transition from n=2 --> n=1, which also called the Lyman-alpha (Ly-α) line. : Lyman series. Your equation is looking at the frequency of a given transition. Answer Save. The ratio of the charges on $\sqrt{3}$ and $\frac{R}{\sqrt{3}}$ is, The work functions for metals A, B and C are respectively 1.92 eV, 2.0 eV and 5 eV. The relation between λ1:wavelength of series limit of Lyman series, λ2: the wavelength of the series limit of ← Prev Question Next Question → 0 votes . The version of the Rydberg formulawhich generated the Lyman series was: Where n is a natural number greater or equal than 2 (i.e. By doing the math, we get the wavelength as. Série de raies (U.V.) R = $1 . According to Bohr’s model, Lyman series is displayed when electron transition takes place from higher energy states(n h =2,3,4,5,6,…) to n l =1 energy state. How can I calculate the wavelength from energy? The Lyman series of the hydrogen spectrum can be represented by the equation: v=3.2881 x10^15 s^-1(1/1^2-1/n^2) (where n=2,3,...) Calculate the maximum and minimum wavelength lines, in nanometers, in this series. You can calculate the frequency (f), given the wavelength (λ), using the following equation: λ = v / f. where We can use the Rydberg equation (Equation \ref{1.5.1}) to calculate the wavelength: \[ \dfrac{1}{\lambda }=R_H \left ( \dfrac{1}{n_{1}^{2}} - \dfrac{1}{n_{2}^{2}}\right ) \nonumber$ A … 097 \times {10}^7\] m-1. Calculate the wavelength of the spectral line in Lyman series corresponding to n_(2) = 3 Doubtnut is better on App. Download Solved Question Papers Free for Offline Practice and view Solutions Online. The Lyman series of the hydrogen spectrum can be represented by the equation: v=3.2881 x10^15 s^-1(1/1^2-1/n^2) (where n=2,3,...) Calculate the maximum and minimum wavelength lines, in nanometers, in this series. The stop cock is suddenly opened. The Lyman series of the hydrogen spectrum can be represensted by the equation `v = 3.2881 xx 10^(15)s^(-1)[(1)/((1)^(2)) - (1)/((n)^(2))] [where n = 2,3,……. NCERT DC Pandey Sunil Batra HC Verma Pradeep Errorless. The frequency of light emitted at this wavelength is 2.47 × 10^15 hertz. Paiye sabhi sawalon ka Video solution sirf photo khinch kar . What is the ratio of the shortest wavelength of the Balmer series to the shortest wavelength of the Lyman series ? The Lyman series of the hydrogen spectrum can be represented by the equation ν = 3.2881 × 10 15 s − 1 (1 1 2 − 1 n 2) (where n = 2, 3, …) (a) Calculate the maximum and minimum wavelength lines, in nanometers, in this series. Main article: Lyman series. For a positive wavelength, we set the initial as #n = 1# and final as #n = 2# for an absorption instead. The solids which have negative temperature coefficient of resistance are : The energy equivalent of 0.5 g of a substance is: The Brewsters angle $i_b$ for an interface should be: Two cylinders A and B of equal capacity are connected to each other via a stop clock. Take Zigya Full and Sectional Test Series. The wavelength is given by the Rydberg formula, #color(blue)(bar(ul(|color(white)(a/a) 1/λ = -R(1/n_f^2 -1/n_i^2)color(white)(a/a)|)))" "#, #R =# the Rydberg constant (#"109 677 cm"^"-1"#) and This formula tells us what the wavelength is in the spectrum for a hydrogen-like atom; it's equation [30.13]. The series was discovered during the years 1906-1914, by Theodore Lyman. α line of Lyman series p = 1 and n = 2; α line of Lyman series p = 1 and n = 3; γ line of Lyman series p = 1 and n = 4; the longest line of Lyman series p = 1 and n = 2; the shortest line of Lyman series p = 1 and n = ∞ 1 decade ago. The refractive index of a particular material is 1.67 for blue light, 1.65 for yellow light and 1.63 for red light. Work done by the cord on the block is, A force F is applied on the wire of radius rand length L and change in the length of wire is I. What is the shortest wavelength (in nanometers) in the Lyman series of the hydrogen spectrum? Z = atomic number, Since the question is asking for #1^(st)# line of Lyman series therefore. 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Series was discovered during the years 1906-1914, by Theodore Lyman, who discovered spectral... | 2021-04-23 05:21:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8432583212852478, "perplexity": 1416.86795045546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039601956.95/warc/CC-MAIN-20210423041014-20210423071014-00190.warc.gz"} |
http://tex.stackexchange.com/questions/26707/one-column-equation-in-twocolumn-document-class/26711 | # One column equation in twocolumn document class
I am writing a paper (revtex4-1, reprint documentclass which I believe uses twocolumn) and need to have a long equation that needs to break the two column format. I've seen answers to this question using multicol but I can't use that class. So does anyone know how to do this from twocolumn class?
I have tried one way that makes the equation a figure but I can't get the alignment with the text correct.
Any help would be greatly appreciated.
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The widetext environment changes the formatting from two-column to one-column to better accommodate very long equations that are more easily read when typeset to the full width of the page:
\documentclass[twocolumn]{revtex4-1}
\usepackage{lipsum}
\begin{document}
\lipsum[1]
\begin{widetext}
$a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p + q$
\end{widetext}
\lipsum[1]
\end{document}
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thanks so much! this seems to work exactly how I want it! – BeauGeste Aug 28 '11 at 4:22
I tried to put this code in a Elsevier two-column article but it doesn't work. It adds some strange text above and below (as you can even see in this example itself) the equation and stay only within one column. Any suggestions? – Udita K. Jul 13 '13 at 3:07
@UditaK.: The package lipsum and its command \lipsum is just added to provide some dummy/filler text. – Speravir Mar 27 '14 at 3:52
If you write the command as
\documentclass[twocolumn]{revtex4-1}
\usepackage{lipsum}
\begin{document}
\begin{widetext}
$a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p + q$
\end{widetext}
The strange text will go away
\end{document}
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Could you, please, write more elaborate, what the relevant difference is to Gonzalo’s answer? What I mean: The package lipsum and its command \lipsum is just for dummy text, but I do not know, whether the added new lines are signifacant changes in REVTeX. – Speravir Mar 27 '14 at 3:36
OK, actually I have REVTeX installed, and running Gonzalo’s code shows no issue with the missing new lines. – Speravir Mar 27 '14 at 3:54 | 2015-07-04 11:55:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8241586089134216, "perplexity": 2126.1503530060368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375096706.9/warc/CC-MAIN-20150627031816-00102-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://proofwiki.org/wiki/Category:Abundant_Numbers | # Category:Abundant Numbers
This category contains results about Abundant Numbers.
Definitions specific to this category can be found in Definitions/Abundant Numbers.
$n$ is abundant if and only if it is smaller than its aliquot sum.
## Subcategories
This category has the following 4 subcategories, out of 4 total.
## Pages in category "Abundant Numbers"
The following 8 pages are in this category, out of 8 total. | 2019-07-24 09:23:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46297141909599304, "perplexity": 2680.0282965710903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195532251.99/warc/CC-MAIN-20190724082321-20190724104321-00330.warc.gz"} |
https://www.homebuiltairplanes.com/forums/threads/lilium-point-to-point-small-air-transport-reinvigorated.27553/page-3 | Lilium - Point to point small air transport reinvigorated?
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Well-Known Member
besides being electric/sexy/trendy I don't see what the benefit of this is over something like a robinson R44.. which is proven, carries 4 people and has basically the same specs of speed and range that this is aiming for. a new robinson is like $400k right?? I would highly doubt if one of these is made for less. I doubt it'll be significantly cheaper too. The difference is in the running cost. A helicopter is a mechanical geek's wet dream, but an accountants nightmare; so much mechanical stuff that moves that it's going to eat up complex expensive parts all the time and maintenance is measured in US$ cubed.
Electric has the potential to reduce this enormously.
I've talked to several owners of electric sailplanes. Most owned 2-stroke self-launchers before and happily paid more for an electric system. Dependent on who you ask, reliability (MTBF) is increased 10 to 100+ fold while running cost are decreased by about the same amount.
As long as battery lifetime (not 1000 but 10^6 charge cycles) can be reasonably solved, I don't see why you can't operate something like the Lilium far below the cost of running a normal car, assuming normal GA price levels for components etc.
Himat
Well-Known Member
Indeed, you are right on many points there. However, there is a large difference between a helicopter with large rotary wings and a strong tail rotor to align it's fuselage as it pleases - and this Lilium project. Helicopters have control over the wind because the airflow of the blades is stronger than the wind. In other words, a helicopter can overcome wind by pure motoric force. What do you think, would the same be possible with these tiny little fans? Could they ever compensate the force of the wind acting on the wing&canard in a strong windgust or a crosswind? Just multiply the disc area of one canard or wing (left or right) and simple physics might show that this aircraft would be tossed around like a plastic toy in a hurricane.
A little rhetorical, do “crosswind” even exists in the airplanes reference frame?
The question about stability in gusty wind, where a change in wind direction and the planes inertia make the plane heading not point into the oncoming air is valid. But is not this only a question about directional stability? If the airplane is made directional stable without a fin, the fin is superfluous. It look like it could be done, still I would not be surprised if a fin is added later on to get more directional stability in cruise mode.
Hovering a fin only would add to the side area that the gust could grab. Less side area should make the impact of a gust less, as the inertia to side area ratio then get larger.
DangerZone
Well-Known Member
HBA Supporter
A little rhetorical, do “crosswind” even exists in the airplanes reference frame?
The question about stability in gusty wind, where a change in wind direction and the planes inertia make the plane heading not point into the oncoming air is valid. But is not this only a question about directional stability? If the airplane is made directional stable without a fin, the fin is superfluous. It look like it could be done, still I would not be surprised if a fin is added later on to get more directional stability in cruise mode.
Hovering a fin only would add to the side area that the gust could grab. Less side area should make the impact of a gust less, as the inertia to side area ratio then get larger.
There's so much more than directional stability in question.
Actually, winds mean a bit more than people expect. Specially with a ducted fan design like the one designed by Lilium. Ducted fans rely on laminar flow into the duct, while a helicopter or propeller blade easily works in turbulent air. In other words, IF the wind creates turbulence and makes the pressure drop at the duct entry, there will be no laminar flow into the duct to feed the fan some air. Turbulence often prevents air from entering the duct like a crowd prevents easy entrance through a narrow gate if they all push forward at the same time. Or you could try breathing in low pressure at 9km high to get the idea what I'm writing about - without air flow most suffocate pretty quickly. Simple physics apply to airplanes and ducted fans too, so these pretty expensive drones would fall from the skies and drop like flies in moderate wind gusts or turbulence.
So, either the guys at Lilium are missing some Learnium (some people never learn and repeat the same mistakes over and over again) or they know about it all yet keep sucking on EU funds.
Well-Known Member
Sure dude. This is why ducted tail rotors lead to instant obliteration...
gtae07
Well-Known Member
Third, how many of these guys are pilots? I bet 99% of them are neither pilots nor airspace engineers.
So? I work at a company that makes "real" airplanes. I'd bet that maybe 1-2% of the people working there are pilots (not counting the people we employ as test pilots etc., I'm talking about the engineers/mechanics/HR/management/marketing/etc.). The same is probably true at most aircraft companies--heck, I'd bet that even most of the employees at Van's aren't pilots. And most non-pilots at these companies have no interest in it either--I know Toobuilder has related before that even when flying and instruction was dirt cheap at the company/base flying club, they had few takers.
And who cares what kind of engineering was listed on their diplomas? Our chief structures guy was a civil engineering major. Most of the guys in my department (systems) were mechanical or electrical majors. Your specialty gives you a leg up, but any competent engineer can probably slot into almost any other engineering role given time to learn the job. That's the real purpose of an engineering education, IMO--to teach you how to solve unknown problems by finding the material and figuring it out, from first principles if need be, not to focus you on one specific type of problem.
Fifth, such an aircraft would have no stability in high winds or if power goes out. Who would fly in such a safety neglecting aircraft?
Why does everyone keep assuming this won't be stable in high winds because it doesn't have a given aerodynamic control surface? It vectors its thrust to achieve the same result. You have fans running almost all the way down the length of the wing so that gives you plenty of yaw control.
Dependence on electricity for stability isn't a new thing. Look at fly-by-wire aircraft and helicopters. Some have really rudimentary means of manual control (e.g. I think the A320 family provides some means of rudder and elevator control) but many of them are completely dependent on some level of electric power to retain control. So they have multiple levels of redundancy--multiple generators, batteries, standby wind-driven turbines, etc.--to ensure they retain power. And when you look at aircraft like the F-16, F-22, F-35, Rafale, etc. if they lose all power for the flight controls, they go unstable very very quickly.
besides being electric/sexy/trendy I don't see what the benefit of this is over something like a robinson R44.. which is proven, carries 4 people and has basically the same specs of speed and range that this is aiming for. a new robinson is like \$400k right?? I would highly doubt if one of these is made for less.
As autoreply mentioned, it's operational cost. Something like this may well have fewer moving parts in the whole power and flight control system than that R-44 does on its tail rotor.
And if they're able to make it so that you don't need to be a pilot to operate it (get in, hit "go here", and sit back for the ride), and if a reasonable means of accommodating them in the airspace system can be found, you have a potential for massive sales and economy of scale. Imagine something like a Tesla factory churning these things out (though how you would actually do this in practice, with the massive legally-mandated paperwork burden that accompanies certified aircraft production, I have no idea).
also the test unit was missing some of it's core components listed on the web page... canard, winglets, retractable landing gear and pilots... this adds a LOT of weight and significantly changes the dynamics. given the efficient cruise mode claims they seemed to ditch that aspect to get it in the air with a 2x3 arrangement of the ducted fans instead of straight 6 along the canard as depicted...
Again... so? This is a proof-of-concept demonstrator. A true prototype, not a production test vehicle. A loose comparison would be the X-35B compared to the F-35B--the former had no mission systems, could carry no payload, and was a fair bit different in many respects from the aircraft that's in service. But that wasn't its job. Its purpose was to demonstrate the feasibility of the shaft-driven lift fan concept, which it did.
Or, it's like the XV-15, which had no payload capability, limited range, and no actual usefulness beyond demonstrating the feasibility of a tiltrotor.
This prototype is intended to prove out the arrangement of the fans, the power and control algorithms, and the mechanism of transition to and from hover. This is the hard part. To do so it doesn't need to carry people or indeed be full-scale at all--that would simply be a waste of time, money, and materials at this point. It need not have the full range of the intended production vehicle, only enough to fly the test program. Assuming things work more or less as intended, they will take the lessons they've learned from this prototype and build one much closer to an intended final production vehicle.
A lot of the work at this stage is validating the models they're using for development (which is also part of why flight testing of large civilian and military aircraft takes so long these days--they do a lot of flying gathering data to fine-tune their CFD, FEA, and performance models). If they can show their CFD and control models work on a smaller cheaper prototype, then they can be reasonably confident that they'll work on a larger, more expensive version.
DangerZone
Well-Known Member
HBA Supporter
Sure dude. This is why ducted tail rotors lead to instant obliteration...
Hopefully you do realize these tail rotors are shrouded props, not ducted fans?
DangerZone
Well-Known Member
HBA Supporter
So? I work at a company that makes "real" airplanes. I'd bet that maybe 1-2% of the people working there are pilots (not counting the people we employ as test pilots etc., I'm talking about the engineers/mechanics/HR/management/marketing/etc.). The same is probably true at most aircraft companies--heck, I'd bet that even most of the employees at Van's aren't pilots. And most non-pilots at these companies have no interest in it either--I know Toobuilder has related before that even when flying and instruction was dirt cheap at the company/base flying club, they had few takers.
This comes as a surprise non-engineers and non-pilots could successfully perform engineering jobs at Van's and most aircraft companies. I had the impression that people working on research, development and design needed some formal education to know what they are doing. Sure, assembly might hire uneducated workers (and marketing might use economists) but I was convinced most of the people working in engineering and design would need aviation specific competences. Besides Van's, which other companies hire non-engineers and non-pilots for aircraft design?
And who cares what kind of engineering was listed on their diplomas? Our chief structures guy was a civil engineering major. Most of the guys in my department (systems) were mechanical or electrical majors. Your specialty gives you a leg up, but any competent engineer can probably slot into almost any other engineering role given time to learn the job. That's the real purpose of an engineering education, IMO--to teach you how to solve unknown problems by finding the material and figuring it out, from first principles if need be, not to focus you on one specific type of problem.
True, agreed. As long as someone is competent to do the job, let them do it.
Why does everyone keep assuming this won't be stable in high winds because it doesn't have a given aerodynamic control surface? It vectors its thrust to achieve the same result. You have fans running almost all the way down the length of the wing so that gives you plenty of yaw control.
Because of physics?
Fans require airflow, laminar airflow into the duct. If a ducted fan can't provide lift and direction/thrust, magic ain't gonna help much. Only some other aerodynamic surfaces could. Like wings, winglets, elevators, rudders, parachutes, props and the likes. Without these, any aircraft is stalled. And we all know how that ends up - with gravity problems.
Dependence on electricity for stability isn't a new thing. Look at fly-by-wire aircraft and helicopters. Some have really rudimentary means of manual control (e.g. I think the A320 family provides some means of rudder and elevator control) but many of them are completely dependent on some level of electric power to retain control. So they have multiple levels of redundancy--multiple generators, batteries, standby wind-driven turbines, etc.--to ensure they retain power. And when you look at aircraft like the F-16, F-22, F-35, Rafale, etc. if they lose all power for the flight controls, they go unstable very very quickly.
It seems we missed each other by a mile. I am all pro-electricty for stability control. It works, it works great. Yet a couple of small electric ducted fans work horribly worse than one large ducted fan or a shrouded prop. Such small fans would not be redundancy, that would be poor engineering, lousy efficiency, running of five to 30 minutes max before battery cells run dead dry. Using batteries to the max equals fifty cycles (charging-discharging) tops. Discharge the batteries down to 50% and you might get 500 to 1000 cycles depending on the cell technology. Discharge down to only 70% of capacity and you'd get up to 2000 cycles before the battery capacity drops bellow upper top useful specs. A well designed aircraft of such size would use shrouded props for better efficiency and safety, not toy ducted fans. There is a reason why many electric motor producers write 'NOT FOR AIRCRAFT USE' on their products, because some incompetent schmuck might come to the idea to transport humans in such a thing. Sure, it could work if one knows what he is doing. But if you are right and only 1% to 2% of people working on airplane development & design are neither engineers not pilots - safety would be a problem. No matter how many toy gadgets they'd install.
gtae07
Well-Known Member
This comes as a surprise non-engineers and non-pilots could successfully perform engineering jobs at Van's and most aircraft companies. I had the impression that people working on research, development and design needed some formal education to know what they are doing. Sure, assembly might hire uneducated workers (and marketing might use economists) but I was convinced most of the people working in engineering and design would need aviation specific competences. Besides Van's, which other companies hire non-engineers and non-pilots for aircraft design?
...
But if you are right and only 1% to 2% of people working on airplane development & design are neither engineers not pilots - safety would be a problem.
Nowhere did I say that Van's or anyone else hired non-engineers to do design work. I said that the vast majority of all of the employees--engineers and everyone else--were not pilots. Big difference.
Occasionally you do see people without formal engineering educations working in certain engineering roles, but typically those people have extensive experience as mechanics or electricians and have worked their way up from the floor by demonstrating exceptional skill and ability. And for some roles you much prefer those people over some kid out of school with an engineering degree and no practical sense. And there are still roles equivalent to draftsmen where you're basically a CAD driver. But the overwhelming part of the design, calculation, and certification (and flight test analysis) is being done by people with formal engineering educations.
Because of physics?
Fans require airflow, laminar airflow into the duct. If a ducted fan can't provide lift and direction/thrust, magic ain't gonna help much. Only some other aerodynamic surfaces could. Like wings, winglets, elevators, rudders, parachutes, props and the likes. Without these, any aircraft is stalled. And we all know how that ends up - with gravity problems.
It seems we missed each other by a mile. I am all pro-electricty for stability control. It works, it works great. Yet a couple of small electric ducted fans work horribly worse than one large ducted fan or a shrouded prop. Such small fans would not be redundancy, that would be poor engineering, lousy efficiency, running of five to 30 minutes max before battery cells run dead dry. Using batteries to the max equals fifty cycles (charging-discharging) tops. Discharge the batteries down to 50% and you might get 500 to 1000 cycles depending on the cell technology. Discharge down to only 70% of capacity and you'd get up to 2000 cycles before the battery capacity drops bellow upper top useful specs. A well designed aircraft of such size would use shrouded props for better efficiency and safety, not toy ducted fans.
These aren't the only people looking at distributed propulsion with smaller ducted fans. Yes, there may be scaling issues. But maybe it's also possible they have something that works--and seeing as they're actually flying some hardware, something is working. They're doing a lot more than any of us sitting around here grousing on the interwebs.
There is a reason why many electric motor producers write 'NOT FOR AIRCRAFT USE' on their products, because some incompetent schmuck might come to the idea to transport humans in such a thing.
They put it there because someone might try to put it on a certified airplane and then someone's lawyer will come after them for selling airplane parts without a PMA or TSO. Not to mention the absurd liability problems.
The Facet cube pumps Van's recommends for boost pump usage, and which Spruce carries, also say "not for aircraft use" on them. People use them anyway.
BBerson
Light Plane Philosopher
HBA Supporter
I doubt they have any engineers with VTOL experience that understand the power density needed for tiny high disc loaded propulsion.
Jay Kempf
Curmudgeon in Training (CIT)
I am not sure why people are questioning the engineering talent on this team. There is actually quite a bit of press on this effort. They have quite a few ringers. That doesn't necessarily make a successful project but it doesn't hurt. High disc loading is a compromise to get redundancy while sacrificing most likely endurance. But the tilt wing aspect buys some of that back depending on how the mission profile is put together. I am curious to see if this thing gets to real trials and evolves as a design to have a reasonably low drag cruise. The Joby craft is moving along the same trajectory.
pictsidhe
Well-Known Member
The difficulties I see with Lilium:
Regulatory, loads of these flying around cities?
Range, getting a good L/D is going to be challenging with all those dinky DFs.
Cost, they are going to need a lot of high energy density batteries. Those just aren't particularly cheap, and won't be without new chemistry. Life is part of that.
BBerson
Light Plane Philosopher
HBA Supporter
I asked Joby about the thrust to weight ratio of his propulsor systems. He didn't have an answer.
Jay Kempf
Curmudgeon in Training (CIT)
I agree with the regulatory stuff. Just because they can build something they can test doesn't mean it is viable. Steep hill to climb there with any multirotor concept. But they are getting more prevalent.
L/D vs. range: Not sure whether this is clear or not to most. I see this as designing aero wings that work and being able to run the cruise power down low enough to save watts per mile. This project and Joby are optimizing around the same mission, short distance point to point, no runway transport of 1 or 2 humans. I actually really have hope for this mission. It's a start in the right direction. endurance can be expanded later. If this can work, it would be quite an accomplishment on it's own. A whole new category of aero craft.
Batteries are a problem for all electric craft. Nothing specific to this project there. Don't put this in with airplanes when doing comparisons. This is a hybrid mission, VTOL, tilt wing. And it hasn't even gotten through final design yet. That happens after testing a mock up. This thing get's lumped in with things like the Osprey and multi-copters.
cheapracer
Well-Known Member
Log Member
So what if it's a model?
Issue to me is nobody here, in other media outlets, or their Youtube video, is clear as to what it is.
A lot less bollshot on themselves, manhugz with feelgood music and a little more detail of what's going on technically would be of great benefit to most.
They're doing a lot more than any of us sitting around here grousing on the interwebs.
It's 11PM right now on a Saturday night and I am detailing my aileron linkage in 3D, I might get to bed before 3. I'll be at the factory before 10 am in the morning. Sunday morning.
I'm also sure a number of other members I could name have at least their minds on their work and or projects as well, if they are not actually hands on as we speak.
You need to stop apologizing for this group, their presentation/information is poor, it's they by themselves who have created the confusion and doubt, not "Us".
Well-Known Member
Issue to me is nobody here, in other media outlets, or their Youtube video, is clear as to what it is.
A lot less bollshot on themselves, manhugz with feelgood music and a little more detail of what's going on technically would be of great benefit to most.
But not to them. They have their funding. Only thing they need is more good engineers and a good image for future lobbying.
You or we are simply not the target audience.
pictsidhe
Well-Known Member
If they want engineers, they aren't doing a great job of attracting them. Looks more like a funding drive.
Himat
Well-Known Member
There's so much more than directional stability in question.
Actually, winds mean a bit more than people expect. Specially with a ducted fan design like the one designed by Lilium. Ducted fans rely on laminar flow into the duct, while a helicopter or propeller blade easily works in turbulent air. In other words, IF the wind creates turbulence and makes the pressure drop at the duct entry, there will be no laminar flow into the duct to feed the fan some air. Turbulence often prevents air from entering the duct like a crowd prevents easy entrance through a narrow gate if they all push forward at the same time. Or you could try breathing in low pressure at 9km high to get the idea what I'm writing about - without air flow most suffocate pretty quickly. Simple physics apply to airplanes and ducted fans too, so these pretty expensive drones would fall from the skies and drop like flies in moderate wind gusts or turbulence.
So, either the guys at Lilium are missing some Learnium (some people never learn and repeat the same mistakes over and over again) or they know about it all yet keep sucking on EU funds.
Ducted fans experience better performance and efficiency if the inlet flow is unobstructed and laminar. Still, they work reasonable without.
Just have a look at one of the early radio control model ducted fan installations. Especially the Byron F-16 comes to mind. Engine in pusher mode in front of fan, tuned pipe extending forward of fan into intake duct, fan deep in the fuselage, one small forward air inlet and the front wheel forward of the main air inlet, a “cheater” hole.
The Lilum ducted fan inlets look overly refined compared that old RC plane ducted fan inlet. Have a look at a Byron F-16 at RcGroups: https://www.rcgroups.com/forums/showthread.php?971166-Bryon-f16-e-dynamax-the-dynabyro-fan#post11177199
There is the main undercarriage in front of the fan too.
Himat
Well-Known Member
I am not sure why people are questioning the engineering talent on this team. There is actually quite a bit of press on this effort. They have quite a few ringers. That doesn't necessarily make a successful project but it doesn't hurt. High disc loading is a compromise to get redundancy while sacrificing most likely endurance. But the tilt wing aspect buys some of that back depending on how the mission profile is put together. I am curious to see if this thing gets to real trials and evolves as a design to have a reasonably low drag cruise. The Joby craft is moving along the same trajectory.
What is available is another possible reason for the selected ducted fan size. Selecting the largest commercial available electric ducted fan intended for model aircraft can speed up development of the vehicle while cutting cost. Then, as the vehicle is developed better propulsion units may be available. That is without the vehicle developer spending any money or time. Not so if they have to develop their own lift units.
If the performance differences between smaller or larger fans are a draw depending on the criteria, availability and cost often is the deciding factors. I must confess that I have designed some part bin specials in my day time job.:gig: | 2021-06-18 23:19:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3088861405849457, "perplexity": 2289.6642803872073}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487643354.47/warc/CC-MAIN-20210618230338-20210619020338-00135.warc.gz"} |
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#### A lower bound for linear approximate compaction
##### MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44233
Chaudhuri, Shiva
Algorithms and Complexity, MPI for Informatics, Max Planck Society;
##### Externe Ressourcen
Es sind keine Externen Ressourcen verfügbar
##### Volltexte (frei zugänglich)
MPI-I-93-146.pdf
(beliebiger Volltext), 10MB
##### Ergänzendes Material (frei zugänglich)
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##### Zitation
Chaudhuri, S.(1993). A lower bound for linear approximate compaction (MPI-I-93-146). Saarbrücken: Max-Planck-Institut für Informatik.
The {\em $\lambda$-approximate compaction} problem is: given an input array of $n$ values, each either 0 or 1, place each value in an output array so that all the 1's are in the first $(1+\lambda)k$ array locations, where $k$ is the number of 1's in the input. $\lambda$ is an accuracy parameter. This problem is of fundamental importance in parallel computation because of its applications to processor allocation and approximate counting. When $\lambda$ is a constant, the problem is called {\em Linear Approximate Compaction} (LAC). On the CRCW PRAM model, %there is an algorithm that solves approximate compaction in $\order{(\log\log n)^3}$ time for $\lambda = \frac{1}{\log\log n}$, using $\frac{n}{(\log\log n)^3}$ processors. Our main result shows that this is close to the best possible. Specifically, we prove that LAC requires %$\Omega(\log\log n)$ time using $\order{n}$ processors. We also give a tradeoff between $\lambda$ and the processing time. For $\epsilon < 1$, and $\lambda = n^{\epsilon}$, the time required is $\Omega(\log \frac{1}{\epsilon})$. | 2018-03-21 18:51:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9427796602249146, "perplexity": 2799.5772049742327}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647681.81/warc/CC-MAIN-20180321180325-20180321200325-00295.warc.gz"} |
http://mathoverflow.net/revisions/43906/list | 2 added 384 characters in body
Finding all the solutions is probably hard, if I am not mistaken any set containing $d\ZZ$ where $d$ is gcd $(A, B)$ is a solution, but this is far from optimal.
If you are looking for the the minimal $S$, just by looking over the general pattern, you are solving multiple higher order recurences at once (at each step the number of recurences increases).
You start with $x_0=1, x_1= A+B$ and at each step, given $x_0,..., x_{2^n}$ you try to figure out a new term $x_{??} = A x_{k}+ B x_{m}$ with $k,m \leq 2^n$.
In particular the solutions to the following recurences will always be in your set: $$x_1=1, x_{n+1}= (A+B) x_n \,.$$ $$x_1=1, x_2= A+B x_{n+1}= A x_n+ Bx_{n-1} \,.$$ $$x_1=1, x_2= A+B x_{n+1}= B x_n+ Ax_{n-1} \,.$$ but also you have things like
$$x_1, x_2, x_3 \in { 1, A+B, A+AA+B^2, A^2+AB+B , (A+B)^2 } x_{n+1}= A x_n+ Bx_{n-2} \,.$$ $$x_1, x_2, x_3 \in { A+AA+B^2, A^2+AB+B , (A+B)^2 } x_{n+1}= A x_{n-1}+ Bx_{n-2} \,.$$ $$x_1, x_2, x_3 \in { A+AA+B^2, A^2+AB+B , (A+B)^2 } x_{n+1}= B x_{n-1}+ Ax_{n-2} \,.$$ $$x_1, x_2, x_3 \in { A+AA+B^2, A^2+AB+B , (A+B)^2 } x_{n+1}= B x_{n}+ Ax_{n-2} \,.$$
and so on.
I migth be wrong, but if I am not mistaken, the Question you are asking is equivalent to the following:
For all the possible $k$ describe recursivelly the general solution to all the recurences of order $k$ of the type $x_{n+k} = A x_{n+m} + Bx_n$ and $x_{n+k} = B x_{n+m} + Ax_n$, where $x_1,..., x_{k-1}$ are solutions to a reccurence of this type of order at most $k-1$.
1
Finding all the solutions is probably hard, if I am not mistaken any set containing $d\ZZ$ where $d$ is gcd $(A, B)$ is a solution, but this is far from optimal.
If you are looking for the the minimal $S$, just by looking over the general pattern, you are solving multiple higher order recurences at once (at each step the number of recurences increases).
You start with $x_0=1, x_1= A+B$ and at each step, given $x_0,..., x_{2^n}$ you try to figure out a new term $x_{??} = A x_{k}+ B x_{m}$ with $k,m \leq 2^n$.
In particular the solutions to the following recurences will always be in your set: $$x_1=1, x_{n+1}= (A+B) x_n \,.$$ $$x_1=1, x_2= A+B x_{n+1}= A x_n+ Bx_{n-1} \,.$$ $$x_1=1, x_2= A+B x_{n+1}= B x_n+ Ax_{n-1} \,.$$ but also you have things like
$$x_1, x_2, x_3 \in { 1, A+B, A+AA+B^2, A^2+AB+B , (A+B)^2 } x_{n+1}= A x_n+ Bx_{n-2} \,.$$ $$x_1, x_2, x_3 \in { A+AA+B^2, A^2+AB+B , (A+B)^2 } x_{n+1}= A x_{n-1}+ Bx_{n-2} \,.$$ $$x_1, x_2, x_3 \in { A+AA+B^2, A^2+AB+B , (A+B)^2 } x_{n+1}= B x_{n-1}+ Ax_{n-2} \,.$$ $$x_1, x_2, x_3 \in { A+AA+B^2, A^2+AB+B , (A+B)^2 } x_{n+1}= B x_{n}+ Ax_{n-2} \,.$$
and so on. | 2013-05-25 06:12:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7606227993965149, "perplexity": 204.9204735711796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368705575935/warc/CC-MAIN-20130516115935-00065-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://vitalab.github.io/article/2018/10/25/Ynet.html | ## Method
The new method adds a reconstruction decoder to the classical encoder-decoder segmentation in order to align source and target encoder features.
## Ynet
The method wants to compensate the domain shift between source and target domain by introducing feature (distribution) similarity metrics. They are introducing a second decoder to the standard encoder-decoder setup which serves to reconstruct the input data which originates from both source and target domain. The complete architecture is trained end-to-end with the following loss function:
Where $$L_r$$ is the reconstruction loss function, in their case is a mean-squared error, $$\hat{x}^{s/t}$$ are reconstructions of the source/target inputs obtained by the auto-encoding sub-network.
The network is initially trained in an unsupervised fashion, after which the reconstruction decoder is discarded.
## Results
• FT: Finetuning baseline.
• MMD: Maximum mean discrepancy.
• Coral: Correlation difference.
• DANN: Distributions in an adversarial setup. | 2022-12-09 08:54:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4262022376060486, "perplexity": 2724.317682252306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711394.73/warc/CC-MAIN-20221209080025-20221209110025-00167.warc.gz"} |
https://www.striim.com/docs/archive/3103/en/avoiding-split-brain.html | # Striim 3.10.3 documentation
### Avoiding split brain
In a multi-server Striim cluster with the metadata repository hosted on Oracle or PostgreSQL, a network partition that splits the cluster into two subsets that cannot communicate with each other will cause both subsets to go into failover mode (commonly called split brain), resulting in an unpredictable variety of errors and eventually a crash.
To prevent this from happening, on each server:
1. Edit startUp.properties and set ClusterQuorumSize to just over half the number of servers in the cluster. For example, for a three-server cluster, set ClusterQuorumSize=2; for a four-node cluster, set ClusterQuorumSize=3.
2. By default, when the number of servers in the cluster drops below the quorum, each server will wait 60 seconds for communication to resume before shutting down. To change that timeout, set ClusterHeartBeatTimeout to the desired number of seconds.
Then restart all servers (see Starting and stopping Striim). | 2022-01-17 15:35:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5145385265350342, "perplexity": 3743.3242377585666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300574.19/warc/CC-MAIN-20220117151834-20220117181834-00454.warc.gz"} |
https://www.physicsforums.com/threads/qm-normalizing-a-wave-function.251025/ | # QM: Normalizing a wave function
## Homework Statement
Hi all.
I have a wave function given by
$$\Psi \left( {x,0} \right) = A\frac{x}{a}$$
I have to normalize it, which is OK. But in the solution to this problem, the teacher uses |A|2 when squaring A. Is there any particular reason for this? I mean, if you square the constant, then why bother with the signs?
I thought that it maybe because A is a complex constant, but still - I cannot see what difference it would make taking the absolute value of A before squaring.
Redbelly98
Staff Emeritus
Homework Helper
If A is not real, then |A|2 and A2 are different. Allowing for A to be complex is the only reason I can think of for including the absolute value signs.
nicksauce
Homework Helper
For complex numbers, |z|^2 is not the same as z^2. Suppose z = 1 + i. Then |z| = sqrt(2), so |z|^2 = 2. But z^2 = 1 + 2i + i^2 = 2i. |z|^2 always gives a nonnegative real number, which is required to interpret the wave function as a probability density.
Great, thanks to both of you.
I have another question related to this.
When I find the constant A, then am I finding the complex number A or the modulus of the complex number A, |A|?
Last edited:
Redbelly98
Staff Emeritus | 2021-11-27 00:13:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.857262134552002, "perplexity": 483.42438536962055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358074.14/warc/CC-MAIN-20211126224056-20211127014056-00408.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/76961-random-sample-continuous-type-dist.html | Math Help - Random sample from a continuous type dist
1. Random sample from a continuous type dist
Let X1,X2,,,,Xn be a random sample from a continuous type distribution
a) find P(X1<=X2),P(X1<=X2,X1<=X3),...,P(X1<=Xi, i=2,3,...,n).
(The answer for this is 1/n but I am not sure the reasoning behind of that.
Can anybody explain this?)
b) Suppose the sampling continues until X1 is no longer the smallest observation, (i.e., Xj < X1 <= Xi, i=2,3,...,j-1). Let Y equal the number of trials until X1 is no longer the smallest observation, (i.e., Y=j-1). Show that the distribution of Y is P(Y=y) = 1 / y(y+1), y=1,2,3,...
(????????)
c) Compute the mean and variance of Y if they exist.
(?????????)
2. Since $1=P(X>Y)+P(X.
I would think you just have to show that these probabilities are equal,
giving you .5.
As for (c), if P(Y=y) = 1 / y(y+1), y=1,2,3,..., then
$E(Y)=\sum_{y=1}^{\infty}{1\over y+1}=\sum_{n=2}^{\infty}{1\over n}=\infty$.
Thus the second moment and variance are also infinite. | 2014-10-24 16:28:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5186408758163452, "perplexity": 1447.3933518441359}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119646269.50/warc/CC-MAIN-20141024030046-00209-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/particle-on-a-ring.224421/ | # Particle on a ring
Consider a mass m confined to a ring of radius r. The potential everywhere on the ring is zero. In the ml = +3 state, identify the points on the ring where the probability of the particle existing is zero.
I was thinking that every point would be zero, because it's a wave, not a particle. It cannot exist at any one point, basically. Am I thinking about this wrong? | 2019-12-08 20:26:07 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8465201258659363, "perplexity": 252.5477717072955}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540514893.41/warc/CC-MAIN-20191208202454-20191208230454-00549.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=1799096 | MathSciNet bibliographic data MR1799096 11F70 (22E55) Kim, Henry H.; Shahidi, Freydoon Functorial products for $\rm GL_2\times GL_3$$\rm GL_2\times GL_3$ and functorial symmetric cube for $\rm GL_2$$\rm GL_2$. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 8, 599–604. Article
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https://zbmath.org/authors/?q=ai%3Ahenderson.johnny | # zbMATH — the first resource for mathematics
## Henderson, Johnny Lee
Compute Distance To:
Author ID: henderson.johnny Published as: Henderson, Johnny; Henderson, J.; Henderson, Johny; Henderson, Johnny L.; Henderson, J. L. Homepage: http://blogs.baylor.edu/johnnyhenderson/ External Links: MGP · ORCID · Google Scholar · dblp · GND
Documents Indexed: 498 Publications since 1981, including 12 Books Biographic References: 1 Publication
all top 5
all top 5
#### Serials
38 Communications on Applied Nonlinear Analysis 29 Journal of Difference Equations and Applications 24 Electronic Journal of Differential Equations (EJDE) 21 Journal of Mathematical Analysis and Applications 20 Electronic Journal of Qualitative Theory of Differential Equations 18 Computers & Mathematics with Applications 15 Dynamic Systems and Applications 15 Communications in Applied Analysis 14 Panamerican Mathematical Journal 12 Applicable Analysis 11 Journal of Differential Equations 11 Nonlinear Studies 10 Applied Mathematics Letters 9 Rocky Mountain Journal of Mathematics 9 Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 8 Fractional Calculus & Applied Analysis 7 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 7 Proceedings of the American Mathematical Society 7 Nonlinear Analysis. Theory, Methods & Applications 6 Applied Mathematics and Computation 6 International Journal of Mathematics and Mathematical Sciences 6 Boundary Value Problems 5 Mathematical and Computer Modelling 5 Georgian Mathematical Journal 4 Archivum Mathematicum 4 Dynamics of Continuous, Discrete and Impulsive Systems 4 Differential Equations and Dynamical Systems 4 Mathematical Modelling and Analysis 4 Differential Equations and Applications 4 International Electronic Journal of Pure and Applied Mathematics 4 Trends in Abstract and Applied Analysis 3 Journal of Computational and Applied Mathematics 3 Tamkang Journal of Mathematics 3 SIAM Journal on Mathematical Analysis 3 Abstract and Applied Analysis 3 Nonlinear Dynamics and Systems Theory 3 Mathematical Sciences Research Journal 3 Mediterranean Journal of Mathematics 3 Advances in Difference Equations 3 Communications in Mathematical Analysis 3 Involve 2 Journal of Mathematical and Physical Sciences 2 Mathematical Methods in the Applied Sciences 2 Commentationes Mathematicae Universitatis Carolinae 2 Mathematische Nachrichten 2 Topological Methods in Nonlinear Analysis 2 The Canadian Applied Mathematics Quarterly 2 Nonlinear Times and Digest 2 Discrete and Continuous Dynamical Systems 2 Discrete Dynamics in Nature and Society 2 International Journal of Applied Mathematics 2 JIPAM. Journal of Inequalities in Pure & Applied Mathematics 2 Nonlinear Analysis. Modelling and Control 2 International Journal of Pure and Applied Mathematics 2 Fixed Point Theory 2 The Australian Journal of Mathematical Analysis and Applications 2 International Journal of Applied Mathematical Sciences 2 De Gruyter Series in Nonlinear Analysis and Applications 2 Journal of Fixed Point Theory and Applications 2 International Journal of Difference Equations 2 ROMAI Journal 2 Mathematics in Engineering, Science and Aerospace MESA 2 Fractional Differential Calculus 2 Advances in the Theory of Nonlinear Analysis and its Applications 1 Lithuanian Mathematical Journal 1 Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 1 Annales Polonici Mathematici 1 Canadian Mathematical Bulletin 1 Dissertationes Mathematicae 1 Indian Journal of Mathematics 1 Journal of Approximation Theory 1 Mathematica Slovaca 1 Numerical Functional Analysis and Optimization 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Results in Mathematics 1 Zeitschrift für Analysis und ihre Anwendungen 1 Bulletin of the Korean Mathematical Society 1 Radovi Matematički 1 Differential and Integral Equations 1 Aequationes Mathematicae 1 Bulletin of the Institute of Mathematics. Academia Sinica 1 Analele Ştiinţifice ale Universităţii Al. I. Cuza din Iaşi. Serie Nouă. Matematică 1 Neural, Parallel & Scientific Computations 1 Journal of Mathematical Sciences (New York) 1 Nonlinear World 1 Turkish Journal of Mathematics 1 Memoirs on Differential Equations and Mathematical Physics 1 NoDEA. Nonlinear Differential Equations and Applications 1 Opuscula Mathematica 1 Journal of Inequalities and Applications 1 Mathematical Sciences Research Hot-Line 1 Mathematical Inequalities & Applications 1 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 1 Mathematica Moravica 1 Communications de la Faculté des Sciences de l’Université d’Ankara. Séries A1. Mathematics and Statistics 1 International Journal of Differential Equations and Applications 1 Journal of Nonlinear and Convex Analysis 1 Nonlinear Analysis. Real World Applications 1 Electronic Journal of Mathematical and Physical Sciences (EJMAPS) 1 Journal of Applied Mathematics and Computing ...and 23 more Serials
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#### Fields
420 Ordinary differential equations (34-XX) 121 Operator theory (47-XX) 98 Difference and functional equations (39-XX) 24 Real functions (26-XX) 16 Integral equations (45-XX) 15 Partial differential equations (35-XX) 8 Global analysis, analysis on manifolds (58-XX) 5 Calculus of variations and optimal control; optimization (49-XX) 4 General and overarching topics; collections (00-XX) 4 General topology (54-XX) 3 Numerical analysis (65-XX) 3 Systems theory; control (93-XX) 2 History and biography (01-XX) 1 Functional analysis (46-XX) 1 Probability theory and stochastic processes (60-XX) 1 Mechanics of deformable solids (74-XX) 1 Biology and other natural sciences (92-XX)
#### Citations contained in zbMATH Open
373 Publications have been cited 4,694 times in 2,651 Documents Cited by Year
Impulsive differential equations and inclusions. Zbl 1130.34003
Benchohra, M.; Henderson, J.; Ntouyas, S.
2006
Existence results for fractional order functional differential equations with infinite delay. Zbl 1209.34096
Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.
2008
Positive solutions for nonlinear eigenvalue problems. Zbl 0876.34023
Henderson, Johnny; Wang, Haiyan
1997
Multiple symmetric positive solutions for a second order boundary value problem. Zbl 0949.34016
Henderson, Johnny; Thompson, H. B.
2000
Fractional functional differential inclusions with finite delay. Zbl 1159.34010
Henderson, Johnny; Ouahab, Abdelghani
2009
Three symmetric positive solutions for a second-order boundary value problem. Zbl 0961.34014
Avery, R. I.; Henderson, J.
2000
Triple positive solutions and dependence on higher order derivatives. Zbl 0935.34020
Davis, John M.; Eloe, Paul W.; Henderson, Johnny
1999
Two positive fixed points of nonlinear operators on ordered Banach spaces. Zbl 1014.47025
Avery, Richard I.; Henderson, Johnny
2001
Twin solutions of boundary value problems for ordinary differential equations and finite difference equations. Zbl 1006.34022
Avery, R. I.; Chyan, Chuan Jen; Henderson, J.
2001
Positive solutions for $$(n-1,1)$$ conjugate boundary value problems. Zbl 0871.34015
Eloe, Paul W.; Henderson, Johnny
1997
Existence of multiple solutions for second-order discrete boundary value problems. Zbl 1005.39014
Henderson, J.; Thompson, H. B.
2002
Existence of three positive pseudo-symmetric solutions for a one dimensional $$p$$-Laplacian. Zbl 1028.34022
Avery, Richard; Henderson, Johnny
2003
Existence of solutions for a one dimensional $$p$$-Laplacian on time-scales. Zbl 1058.39010
Anderson, Douglas; Avery, Richard; Henderson, Johnny
2004
Upper and lower solution methods for fully nonlinear boundary value problems. Zbl 1019.34015
Ehme, Jeffrey; Eloe, Paul W.; Henderson, Johnny
2002
Impulsive differential inclusions with fractional order. Zbl 1200.34006
Henderson, Johnny; Ouahab, Abdelghani
2010
Existence of multiple solutions for second order boundary value problems. Zbl 1013.34017
Henderson, Johnny; Thompson, H. B.
2000
General Lidstone problems: Multiplicity and symmetry of solutions. Zbl 0966.34023
Davis, John M.; Henderson, Johnny; Wong, Patricia J. Y.
2000
Eigenvalue problems for nonlinear differential equations on a measure chain. Zbl 0953.34068
Chyan, Chuan Jen; Henderson, Johnny
2000
Positive solutions and nonlinear eigenvalue problems for third-order difference equations. Zbl 0933.39003
Agarwal, R. P.; Henderson, J.
1998
An exploration of combined dynamic derivatives on time scales and their applications. Zbl 1114.26004
Sheng, Q.; Fadag, M.; Henderson, J.; Davis, J. M.
2006
Singular nonlinear $$(k,n-k)$$ conjugate boundary value problems. Zbl 0870.34031
Eloe, Paul W.; Henderson, Johnny
1997
Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces. Zbl 0998.34064
Benchohra, M.; Henderson, J.; Ntouyas, S. K.
2001
Existence results for fractional functional differential inclusions with infinite delay and applications to control theory. Zbl 1149.26010
Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.
2008
On a system of fractional differential equations with coupled integral boundary conditions. Zbl 1315.34012
Henderson, Johnny; Luca, Rodica; Tudorache, Alexandru
2015
Positive solutions for a system of nonlocal fractional boundary value problems. Zbl 1312.34015
Henderson, Johnny; Luca, Rodica
2013
Implicit fractional differential and integral equations. Existence and stability. Zbl 1390.34002
Abbas, Saïd; Benchohra, Mouffak; Graef, John R.; Henderson, Johnny
2018
Singular nonlinear boundary value problems for higher order ordinary differential equations. Zbl 0731.34015
Eloe, Paul W.; Henderson, Johnny
1991
Positive solutions for a system of fractional differential equations with coupled integral boundary conditions. Zbl 1338.34062
Henderson, Johnny; Luca, Rodica
2014
Existence of solutions of right focal point boundary value problems for ordinary differential equations. Zbl 0468.34010
Henderson, Johnny
1981
Positive solutions for systems of nonlinear discrete boundary value problems. Zbl 1185.39003
Henderson, J.; Ntouyas, S. K.; Purnaras, I. K.
2009
Multiple solutions for $$2m$$th order Sturm-Liouville boundary value problems on a measure chain. Zbl 0965.39008
Henderson, Johnny
2000
Double solutions of impulsive dynamic boundary value problems on a time scale. Zbl 1003.39019
Henderson, Johnny
2002
Multiplicity of positive solutions for higher order Sturm-Liouville problems. Zbl 0989.34012
Davis, John M.; Erbe, Lynn H.; Henderson, Johnny
2001
Eigenvalue comparison for fractional boundary value problems with the Caputo derivative. Zbl 1310.34007
Henderson, Johnny; Kosmatov, Nickolai
2014
Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument. Zbl 1220.45011
Darwish, Mohamed Abdalla; Henderson, Johnny; O’Regan, Donal
2011
Positive solutions for systems of $$n$$th order three-point nonlocal boundary value problems. Zbl 1182.34029
Henderson, J.; Ntouyas, S. K.
2007
Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations. Zbl 1014.39012
Henderson, Johnny; Thompson, H. B.
2001
Existence of three positive pseudo-symmetric solutions for a one dimensional discrete $$p$$-Laplacian. Zbl 1053.39003
Avery, Richard; Henderson, Johnny
2004
On first order impulsive dynamic equations on time scales. Zbl 1054.39012
Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.
2004
Existence of solutions for three-point boundary value problems for second order equations. Zbl 1061.34009
Henderson, Johnny; Karna, Basant; Tisdell, Christopher C.
2005
Impulsive differential inclusions. A fixed point approach. Zbl 1285.34002
Graef, John R.; Henderson, Johnny; Ouahab, Abdelghani
2013
Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem. Zbl 0958.34020
Avery, Richard I.; Davis, John M.; Henderson, Johnny
2000
Positive solutions and nonlinear multipoint conjugate eigenvalue problems. Zbl 0888.34013
Eloe, Paul W.; Henderson, Johnny
1997
Nonexistence of positive solutions for a system of coupled fractional boundary value problems. Zbl 1341.34006
Henderson, Johnny; Luca, Rodica
2015
Positive solutions for systems of nonlinear boundary value problems. Zbl 1148.34016
Henderson, J.; Ntouyas, S. K.
2008
Existence and multiplicity of positive solutions for a system of fractional boundary value problems. Zbl 1307.34013
Henderson, Johnny; Luca, Rodica
2014
Impulsive functional differential equations with variable times. Zbl 1070.34108
Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.
2004
Uniqueness implies existence for fourth-order Lidstone boundary value problems. Zbl 0960.34011
Davis, John M.; Henderson, Johnny
1998
Topological transversality and boundary value problems on time scales. Zbl 1047.34014
Henderson, Johnny; Tisdell, Christopher C.
2004
Uniqueness of solutions of right focal point boundary value problems for ordinary differential equations. Zbl 0438.34015
Henderson, Johnny
1981
Singular boundary value problems for difference equations. Zbl 0761.39002
Henderson, Johnny
1992
Nonlinear eigenvalue problems for quasilinear systems. Zbl 1092.34517
Henderson, J.; Wang, Haiyan
2005
On the existence and uniqueness of solutions to boundary value problems on time scales. Zbl 1089.39005
Henderson, Johnny; Peterson, Allan; Tisdell, Christopher C.
2004
Boundary value problems for systems of differential, difference and fractional equations. Positive solutions. Zbl 1353.34002
Henderson, Johnny; Luca, Rodica
2016
Best interval lengths for boundary value problems for third order Lipschitz equations. Zbl 0668.34017
Henderson, Johnny
1987
Positive solutions for singular higher order nonlinear equations. Zbl 0873.34013
Chyan, Chuan Jen; Henderson, Johnny
1994
Existence theorems for boundary value problems for nth-order nonlinear difference equations. Zbl 0671.34017
Henderson, Johnny
1989
Positive solutions for systems of generalized three-point nonlinear boundary value problems. Zbl 1212.34058
Henderson, J.; Ntouyas, S. K.; Purnaras, I. K.
2008
Inequalities based on a generalization of concavity. Zbl 0868.34008
Eloe, Paul W.; Henderson, Johnny
1997
Positive solutions for a system of second-order multi-point boundary value problems. Zbl 1251.34039
Henderson, Johnny; Luca, Rodica
2012
Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations. Zbl 1120.34010
Clark, Stephen; Henderson, Johnny
2006
Positive solutions for nonlinear difference equations. Zbl 0883.39002
Henderson, J.
1997
Positive solutions for systems of $$M$$-point nonlinear boundary value problems. Zbl 1166.34014
Henderson, J.; Ntouyas, S. K.; Purnaras, I. K.
2008
Measure of noncompactness and fractional differential equations in Banach spaces. Zbl 1182.26007
Benchohra, Mouffak; Henderson, Johnny; Seba, Djamila
2008
Positive solutions and nonlinear $$(k,n-k)$$ conjugate eigenvalue problems. Zbl 1003.34018
Eloe, Paul W.; Henderson, Johnny
1998
Twin solutions of boundary value problems for differential equations on measure chains. Zbl 1134.39301
Chyan, Chuan Jen; Henderson, Johnny
2002
Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory. Zbl 1254.34026
Tian, Yu; Henderson, Johnny
2012
Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems. Zbl 1275.34037
Henderson, Johnny; Luca, Rodica
2013
Extremal points for impulsive Lidstone boundary value problems. Zbl 0963.34022
Eloe, P. W.; Henderson, J.; Thompson, H. B.
2000
Triple positive symmetric solutions for a Lidstone boundary value problem. Zbl 0981.34014
Davis, John M.; Henderson, Johnny
1999
Uniqueness, existence, and optimality for fouth-order Lipschitz equations. Zbl 0642.34005
Henderson, Johnny; McGwier, Robert W. jun.
1987
Comparison of eigenvalues for discrete Lidstone boundary value problems. Zbl 0941.39009
Davis, John M.; Eloe, Paul W.; Henderson, Johny
1999
Positive solutions of $$2m$$th-order boundary value problems. Zbl 1019.34019
Chyan, Chuan Jen; Henderson, J.
2002
Existence of positive solutions for a singular fractional boundary value problem. Zbl 1420.34017
Henderson, Johnny; Luca, Rodica
2017
Positive solutions of a nonlinear higher order boundary-value problem. Zbl 1117.34023
Graef, John R.; Henderson, Johnny; Yang, Bo
2007
Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for $$n$$th order differential equations. Zbl 1396.34011
Eloe, Paul W.; Henderson, Johnny
2007
Systems of Riemann-Liouville fractional equations with multi-point boundary conditions. Zbl 1411.34014
Henderson, Johnny; Luca, Rodica
2017
Eigenvalue intervals for nonlinear right focal problems. Zbl 1031.34083
Davis, John M.; Henderson, Johnny; Prasad, K. Rajendra; Yin, William K. C.
2000
Existence of positive solutions for a system of second-order multi-point discrete boundary value problems. Zbl 1286.39003
Henderson, Johnny; Luca, Rodica
2013
Nondensely defined evolution impulsive differential inclusions with nonlocal conditions. Zbl 1039.34056
Benchohra, M.; Gatsori, E. P.; Henderson, J.; Ntouyas, S. K.
2003
Positive solutions for systems of three-point nonlinear boundary value problems. Zbl 1177.34032
Henderson, J.; Ntouyas, S. K.
2008
Positive solutions for two-point boundary value problems. Zbl 0876.34016
Eloe, Paul W.; Henderson, Johnny; Wong, Patricia J. Y.
1996
Double solutions of three-point boundary-value problems for second-order differential equations. Zbl 1075.34013
Henderson, Johnny
2004
Nonlinear boundary value problems and a priori bounds on solutions. Zbl 0547.34015
Eloe, P. W.; Henderson, Johnny
1984
Positive solutions for a system of higher-order multi-point boundary value problems. Zbl 1236.34026
Henderson, Johnny; Luca, Rodica
2011
Fractional differential equations with anti-periodic boundary conditions. Zbl 1283.34003
Benchohra, Mouffak; Hamidi, Naima; Henderson, Johnny
2013
Uniqueness implies existence for three-point boundary value problems for second order differential equations. Zbl 1092.34507
Henderson, Johnny
2005
Positive solutions for higher order ordinary differential equations. Zbl 0814.34017
Eloe, Paul W.; Henderson, Johnny
1995
Singular $$(k,n-k)$$ boundary value problems between conjugate and right focal. Zbl 0901.34026
Henderson, Johnny; Yin, William
1998
Existence of solutions for some singular higher order boundary value problems. Zbl 0795.34016
Eloe, P.; Henderson, J.
1993
Singular boundary value problems for higher order difference equations. Zbl 0843.39002
Henderson, Johnny
1996
Superlinear and sublinear focal boundary value problems. Zbl 0872.34006
Agarwal, Ravi P.; Henderson, Johnny
1996
An existence result for first-order impulsive functional differential equations in Banach spaces. Zbl 1005.34069
Benchohra, M.; Henderson, J.; Ntouyas, S. K.
2001
Disconjugacy, disfocality, and differentiation with respect to boundary conditions. Zbl 0613.34014
Henderson, Johnny
1987
Focal boundary value problems for nonlinear difference equations. I. Zbl 0706.39001
Henderson, Johnny
1989
Right focal point boundary value problems for ordinary differential equations and variational equations. Zbl 0533.34015
Henderson, Johnny
1984
Existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions. Zbl 1412.34034
Khan, Rahmat Ali; Ur Rehman, Mujeeb; Henderson, Johnny
2011
Double solutions of boundary value problems for 2$$m^{\text{th}}$$-order differential equations and difference equations. Zbl 1070.34036
Graef, J. R.; Henderson, J.
2003
Existence of solutions for third-order boundary value problems on a time scale. Zbl 1057.39011
Henderson, J.; Yin, W. K. C.
2003
Positive solutions and conjugate points for multipoint boundary value problems. Zbl 0760.34022
Eloe, Paul W.; Hankerson, Darrel; Henderson, Johnny
1992
Coupled Hilfer and Hadamard random fractional differential systems with finite delay in generalized Banach spaces. Zbl 07332056
Abbas, Saïd; Al Arifi, Nassir; Benchohra, Mouffak; Henderson, Johnny
2020
Topological methods for differential equations and inclusions. Zbl 1411.34002
Graef, John R.; Henderson, Johnny; Ouahab, Abdelghani
2019
A fully Hadamard and Erdélyi-Kober-type integral boundary value problem of a coupled system of implicit differential equations. Zbl 1423.34007
Berrabah, Fatima Zohra; Hedia, Benaouda; Henderson, Johnny
2019
Positive solutions for a system of Neumann boundary value problems of second-order difference equations involving sign-changing nonlinearities. Zbl 1410.39004
Jiang, Jiqiang; Henderson, Johnny; Xu, Jiafa; Fu, Zhengqing
2019
Existence of positive solutions for a system of semipositone coupled discrete boundary value problems. Zbl 1422.39012
Henderson, Johnny; Luca, Rodica
2019
Existence and attractivity results for Hilfer fractional differential equations. Zbl 1433.34005
Abbas, S.; Benchohra, M.; Henderson, J.
2019
Caputo-Hadamard fractional differential Cauchy problem in Fréchet spaces. Zbl 1434.34005
Abbas, Saïd; Benchohra, Mouffak; Berhoun, Farida; Henderson, Johnny
2019
Existence of local solutions for fractional difference equations with Dirichlet boundary conditions. Zbl 1427.39002
Henderson, Johnny
2019
First extremal point comparison for a fractional boundary value problem with a fractional boundary condition. Zbl 1455.34006
Henderson, Johnny; Neugebauer, Jeffrey T.
2019
Existence and uniqueness results for nonlinear implicit fractional systems. Zbl 1458.93111
Abbas, Saïd; Benchohra, Mouffak; Bouriah, Soufyane; Henderson, Johnny
2019
Implicit fractional differential and integral equations. Existence and stability. Zbl 1390.34002
Abbas, Saïd; Benchohra, Mouffak; Graef, John R.; Henderson, Johnny
2018
Caputo-Hadamard fractional differential equations in Banach spaces. Zbl 1434.34006
Abbas, Saïd; Benchohra, Mouffak; Hamidi, Naima; Henderson, Johnny
2018
Positive solutions for a system of coupled fractional boundary value problems. Zbl 1394.34013
Henderson, Johnny; Luca, Rodica
2018
Nonlinear implicit Hadamard’s fractional differential equations with retarded and advanced arguments. Zbl 1400.34126
Benchohra, M.; Bouriah, S.; Henderson, J.
2018
Existence of positive solutions for a singular fractional boundary value problem. Zbl 1420.34017
Henderson, Johnny; Luca, Rodica
2017
Systems of Riemann-Liouville fractional equations with multi-point boundary conditions. Zbl 1411.34014
Henderson, Johnny; Luca, Rodica
2017
Existence of nonnegative solutions for a fractional integro-differential equation. Zbl 1386.45012
Henderson, Johnny; Luca, Rodica
2017
Variational approaches to $$p$$-Laplacian discrete problems of Kirchhoff-type. Zbl 1371.39002
Heidarkhani, Shapour; Afrouzi, Ghasem A.; Henderson, Johnny; Moradi, Shahin; Caristi, Giuseppe
2017
Multivalued versions of a Krasnosel’skii-type fixed point theorem. Zbl 1453.47008
Graef, John R.; Henderson, Johnny; Ouahab, Abdelghani
2017
Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete $$p(k)$$-Laplacian operator. Zbl 1375.39016
2017
Parameter dependence for existence, nonexistence and multiplicity of nontrivial solutions for an Atici-Eloe fractional difference Lidstone BVP. Zbl 1438.39005
Yang, Aijun; Henderson, Johnny; Wang, Helin
2017
Positive solutions for an impulsive second-order nonlinear boundary value problem. Zbl 1369.34039
Henderson, Johnny; Luca, Rodica
2017
Smallest eigenvalues for a fractional difference equation with right focal boundary conditions. Zbl 1387.39004
Henderson, Johnny; Neugebauer, Jeffrey T.
2017
Existence and nonexistence of positive solutions to a discrete boundary value problem. Zbl 1399.39004
Henderson, Johnny; Luca, Rodica; Tudorache, Alexandru
2017
Constructive existence results for solutions to systems of boundary value problems via general Lyapunov methods. Zbl 1365.34043
Henderson, J.; Sheng, Q.; Tisdell, C. C.
2017
Infinitely many solutions for a perturbed quasilinear two-point boundary value problem. Zbl 1399.34058
Heidarkhani, Shapour; Henderson, Johnny
2017
Boundary value problems for systems of differential, difference and fractional equations. Positive solutions. Zbl 1353.34002
Henderson, Johnny; Luca, Rodica
2016
Existence of positive solutions for a system of semipositone fractional boundary value problems. Zbl 1363.34007
Henderson, Johnny; Luca, Rodica
2016
Positive solutions for a system of semipositone coupled fractional boundary value problems. Zbl 1344.34014
Henderson, Johnny; Luca, Rodica
2016
Existence and Ulam stabilities for Hadamard fractional integral equations with random effects. Zbl 1334.34011
Abbas, Saïd; Albarakati, Wafaa A.; Benchohra, Mouffak; Henderson, Johnny
2016
Positive solutions for a system of difference equations with coupled multi-point boundary conditions. Zbl 1338.39007
Henderson, Johnny; Luca, Rodica
2016
On a system of Riemann-Liouville fractional boundary value problems. Zbl 1353.34007
Henderson, Johnny; Luca, Rodica
2016
An extension of the compression-expansion fixed point theorem of functional type. Zbl 1362.47040
Avery, Richard I.; Anderson, Douglas R.; Henderson, Johnny
2016
Boundary value problems for fractional differential inclusions with nonlocal conditions. Zbl 1354.34019
Hamani, Samira; Henderson, Johnny
2016
Sturm-Liouville BVPs with Carathéodory nonlinearities. Zbl 1357.34059
Benmezai, Abdelhamid; Esserhane, Wassila; Henderson, Johnny
2016
A variational approach to difference equations. Zbl 1375.39013
Heidarkhani, Shapour; Afrouzi, Ghasem A.; Caristi, Giuseppe; Henderson, Johnny; Moradi, Shahin
2016
Existence and nonexistence of positive solutions for coupled Riemann-Liouville fractional boundary value problems. Zbl 1417.34064
Henderson, Johnny; Luca, Rodica; Tudorache, Alexandru
2016
Existence of positive solutions for a system of fractional boundary value problems. Zbl 1357.34013
Henderson, Johnny; Luca, Rodica; Tudorache, Alexandru
2016
Nonlinear interpolation and boundary value problems. Zbl 1385.34003
Eloe, Paul W.; Henderson, Johnny
2016
Measure of noncompactness and neutral functional differential equations with state-dependent delay. Zbl 1382.34082
Benchohra, Mouffak; Henderson, Johnny; Medjadj, Imene
2016
On a system of fractional differential equations with coupled integral boundary conditions. Zbl 1315.34012
Henderson, Johnny; Luca, Rodica; Tudorache, Alexandru
2015
Nonexistence of positive solutions for a system of coupled fractional boundary value problems. Zbl 1341.34006
Henderson, Johnny; Luca, Rodica
2015
Existence and stability results for nonlinear implicit neutral fractional differential equations with finite delay and impulses. Zbl 1358.34088
Benchohra, Mouffak; Bouriah, Soufyane; Henderson, Johnny
2015
Extremal points for a higher-order fractional boundary-value problem. Zbl 1334.34023
Yang, Aijun; Henderson, Johnny; Nelms, Charles jun.
2015
Positive solutions for a fractional boundary value problem. Zbl 1320.34010
Henderson, Johnny; Luca, Rodica; Tudorache, Alexandru
2015
Fractional differential inclusions in the Almgren sense. Zbl 1316.34005
Graef, John R.; Henderson, Johnny; Ouahab, Abdelghani
2015
Positive solutions of discrete Neumann boundary value problems with sign-changing nonlinearities. Zbl 1329.39006
Bai, Dingyong; Henderson, Johnny; Zeng, Yunxia
2015
Boundary-value problems for third-order Lipschitz ordinary differential equations. Zbl 1355.34048
Graef, John R.; Henderson, Johnny; Luca, Rodrica; Tian, Yu
2015
Differentiation with respect to parameters of solutions of nonlocal boundary value problems for difference equations. Zbl 1328.39003
Henderson, Johnny; Jiang, Xuewei
2015
Omitted ray fixed point theorem. Zbl 1339.47069
Avery, Richard; Henderson, Johnny; Liu, Xueyan
2015
Fixed point theorems for positive maps and applications. Zbl 1348.47044
Benmezai, Abdelhamid; Mechrouk, Salima; Henderson, Johnny
2015
Positive solutions for a singular third order boundary value problem. Zbl 1337.34025
Henderson, Johnny; Luca, Rodica; Nelms, Charles jun.; Yang, Aijun
2015
Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Zbl 1341.34032
Luca, Rodica; Henderson, Johnny
2015
Positive solutions for a system of fractional differential equations with coupled integral boundary conditions. Zbl 1338.34062
Henderson, Johnny; Luca, Rodica
2014
Eigenvalue comparison for fractional boundary value problems with the Caputo derivative. Zbl 1310.34007
Henderson, Johnny; Kosmatov, Nickolai
2014
Existence and multiplicity of positive solutions for a system of fractional boundary value problems. Zbl 1307.34013
Henderson, Johnny; Luca, Rodica
2014
Infinitely many solutions for perturbed difference equations. Zbl 1291.39002
Moghadam, Mohsen Khaleghi; Heidarkhani, Shapour; Henderson, Johnny
2014
On a second-order nonlinear discrete multi-point eigenvalue problem. Zbl 1300.39001
Henderson, Johnny; Luca, Rodica
2014
Multiple positive solutions for a multi-point discrete boundary value problem. Zbl 1333.39004
Henderson, Johnny; Luca, Rodica; Tudorache, Alexandru
2014
Smoothness of solutions with respect to multi-strip integral boundary conditions for $$n$$th order ordinary differential equations. Zbl 1314.34042
Henderson, Johnny
2014
Positive solutions for systems of nonlinear second-order multipoint boundary value problems. Zbl 1314.34050
Henderson, Johnny; Luca, Rodica
2014
Critical point approaches to quasilinear second-order differential equations depending on a parameter. Zbl 1360.34043
Heidarkhani, Shapour; Henderson, Johnny
2014
Positive solutions for systems of multi-point nonlinear boundary value problems. Zbl 1320.34035
Henderson, Johnny; Luca, Rodica
2014
Nodal solutions for singular second-order boundary-value problems. Zbl 1300.34048
Benmezai, Abdelhamid; Esserhane, Wassila; Henderson, Johnny
2014
Three-point third-order problems with a sign-changing nonlinear term. Zbl 1300.34054
Henderson, Johnny; Kosmatov, Nickolai
2014
Positive solutions for a system of nonlocal fractional boundary value problems. Zbl 1312.34015
Henderson, Johnny; Luca, Rodica
2013
Impulsive differential inclusions. A fixed point approach. Zbl 1285.34002
Graef, John R.; Henderson, Johnny; Ouahab, Abdelghani
2013
Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems. Zbl 1275.34037
Henderson, Johnny; Luca, Rodica
2013
Existence of positive solutions for a system of second-order multi-point discrete boundary value problems. Zbl 1286.39003
Henderson, Johnny; Luca, Rodica
2013
Fractional differential equations with anti-periodic boundary conditions. Zbl 1283.34003
Benchohra, Mouffak; Hamidi, Naima; Henderson, Johnny
2013
Existence of solutions to second-order boundary-value problems with small perturbations of impulses. Zbl 1292.34028
Bonanno, Gabriele; di Bella, Beatrice; Henderson, Johnny
2013
On a multi-point discrete boundary value problem. Zbl 1262.39004
Henderson, Johnny; Luca, Rodica
2013
Anti-periodic solutions of higher order nonlinear difference equations: a variational approach. Zbl 1295.39001
Tian, Yu; Henderson, Johnny
2013
Positive solutions for singular systems of multi-point boundary value problems. Zbl 1271.34029
Henderson, Johnny; Luca, Rodica
2013
Infinitely many solutions for a boundary value problem with impulsive effects. Zbl 1291.34056
Bonanno, Gabriele; Di Bella, Beatrice; Henderson, Johnny
2013
Positive solutions for systems of second-order integral boundary value problems. Zbl 1340.34097
Luca, Rodica; Henderson, Johnny
2013
Existence of infinitely many anti-periodic solutions for second-order impulsive differential inclusions. Zbl 1295.34021
Heidarkhani, Shapour; Afrouzi, Ghasem A.; Hadjian, Armin; Henderson, Johnny
2013
Existence and multiplicity for positive solutions of a second-order multi-point discrete boundary value problem. Zbl 1262.39003
Henderson, Johnny; Luca, Rodica
2013
Asymptotic attractive nonlinear fractional order Riemann-Liouville integral equations in Banach algebras. Zbl 1305.45005
Abbas, Said; Benchohra, Mouffak; Henderson, Johnny
2013
Positive solutions for singular systems of higher-order multi-point boundary value problems. Zbl 1278.34019
Henderson, Johnny; Luca, Rodica
2013
Positive solutions for a system of second-order nonlinear multi-point eigenvalue problems. Zbl 1333.34032
Henderson, Johnny; Luca, Rodica
2013
A third order boundary value problem with jumping nonlinearities. Zbl 1266.34033
Benmezaï, Abdelhamid; Henderson, Johnny; Meziani, Mohamed
2013
Anti-periodic solutions for a gradient system with resonance via a variational approach. Zbl 1283.34016
Tian, Yu; Henderson, Johnny
2013
Existence of positive solutions for singular second order boundary value problems under eigenvalue criteria. Zbl 1285.34015
Benmezai, Abdelhamid; Esserhane, Wassila; Henderson, Johnny
2013
Positive solutions for a $$p(t)$$-Laplacian three point boundary value problem. Zbl 1291.34048
Benkaci-Ali, Nadir; Benmezaï, Abdelhamid; Henderson, Johnny
2013
Preface: Fractional differential equations and their applications. Zbl 1298.00097
Benchohra, Mouffak; Cabada, Alberto; Henderson, Johnny
2013
Positive solutions for a system of second-order multi-point boundary value problems. Zbl 1251.34039
Henderson, Johnny; Luca, Rodica
2012
Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory. Zbl 1254.34026
Tian, Yu; Henderson, Johnny
2012
Existence and multiplicity for positive solutions of a multi-point boundary value problem. Zbl 1348.34055
Henderson, Johnny; Luca, Rodica
2012
A Filippov’s theorem, some existence results and the compactness of solution sets of impulsive fractional order differential inclusions. Zbl 1263.34010
Henderson, Johnny; Ouahab, Abdelghani
2012
Global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order. Zbl 1269.26003
Abbas, Saïd; Benchohra, Mouffak; Henderson, Johnny
2012
Positive solutions for a system of second-order multi-point discrete boundary value problems. Zbl 1259.39001
Henderson, Johnny; Luca, Rodica
2012
Existence of positive solutions for a system of higher-order multi-point boundary value problems. Zbl 1311.34051
Henderson, Johnny; Luca, Rodica
2012
Infinitely many solutions for nonlocal elliptic systems of ($$p_1,\dots ,p_n$$)-Kirchhoff type. Zbl 1254.35006
Heidarkhani, Shapour; Henderson, Johnny
2012
Weak solutions for hyperbolic partial fractional differential inclusions in Banach spaces. Zbl 1268.35122
Benchohra, Mouffak; Henderson, Johnny; Mostefai, Fatima-Zohra
2012
Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type. Zbl 1251.45003
Darwish, Mohamed Abdalla; Henderson, Johnny
2012
Multiple solutions for a nonlocal perturbed elliptic problem of $$p$$-Kirchhoff type. Zbl 1267.35095
Heidarkhani, Shapour; Henderson, Johnny
2012
Boundary value problems for fractional differential inclusions in Banach spaces. Zbl 1415.34103
Benchohra, Mouffak; Henderson, Johnny; Seba, Djamila
2012
Uniqueness implies existence and uniqueness conditions for a class of $$(k+j)$$-point boundary value problems for $$n$$-th order differential equations. Zbl 1254.34029
Eloe, Paul W.; Henderson, Johnny; Khan, Rahmat Ali
2012
On a system of second-order multi-point boundary value problems. Zbl 1260.34044
Henderson, Johnny; Luca, Rodica
2012
...and 273 more Documents
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#### Cited by 1,924 Authors
155 Henderson, Johnny Lee 110 Agarwal, Ravi P. 106 Benchohra, Mouffak 79 Ge, Weigao 77 O’Regan, Donal 73 Ahmad, Bashir 69 Ntouyas, Sotiris K. 55 Wang, Jinrong 53 Wong, Patricia J. Y. 51 Nieto Roig, Juan Jose 49 Graef, John R. 47 Luca, Rodica 44 Abbas, Said 44 Zhou, Yong 43 Kong, Lingju 38 Ma, Ruyun 38 Ouahab, Abdelghani 32 Eloe, Paul W. 31 Liu, Lishan 30 Feng, Meiqiang 30 Heidarkhani, Shapour 29 Li, Wan-Tong 27 Al-saedi, Ahmed Eid Salem 27 Liu, Yuji 24 Sun, Jianping 24 Zhang, Xuemei 23 Li, Yongxiang 23 Tariboon, Jessada 22 Anderson, Douglas Robert 22 Thompson, Harold Bevan 22 Tisdell, Christopher C. 22 Yang, Bo 20 Tian, Yu 20 Wu, Yonghong 20 Yaslan Karaca, Ilkay 19 Chang, Yong-Kui 18 Stamova, Ivanka Milkova 18 Sun, Hongrui 17 Avery, Richard I. 17 Hernández, Eduardo M. 17 Pandey, Dwijendra Narain 17 Pei, Minghe 17 Prasad, Kapula Rajendra 17 Shen, Jianhua 17 Xu, Jiafa 17 Yan, Zuomao 17 Zhang, Jihui 16 Afrouzi, Ghasem Alizadeh 16 Bai, Zhanbing 16 Gou, Haide 16 Kong, Qingkai 15 Chen, Haibo 15 Darwish, Mohamed Abdalla 15 Goodrich, Christopher S. 15 N’Guérékata, Gaston Mandata 15 Rachůnková, Irena 15 Stamov, Gani Trendafilov 15 Yang, Zhilin 14 Cui, Yujun 14 Fečkan, Michal 14 Guo, Yanping 13 Băleanu, Dumitru I. 13 Cabada, Alberto 13 Caristi, Giuseppe 13 Liang, Sihua 13 Sun, Yongping 13 Wei, Wei 13 Wong, Fu-Hsiang 12 Chyan, Chuan Jen 12 Davis, John M. 12 Gao, Chenghua 12 He, Zhimin 12 Henríquez, Hernán R. 12 Li, Baolin 12 Li, Jianli 12 Liu, Xia 12 Rao, Sabbavarapu Nageswara 12 Tokmak Fen, Fatma 12 Zhai, Chengbo 12 Zhang, Yuanbiao 11 Arjunan, Mani Mallika 11 Dogan, Abdulkadir 11 Ji, Dehong 11 Kosmatov, Nickolai 11 Liang, Jin 11 Minhós, Feliz Manuel 11 Shi, Haiping 11 Wang, Dabin 11 Yao, Qingliu 11 Zhao, Yulin 11 Zhou, Zhan 10 Bai, Chuanzhi 10 Chadha, Alka 10 Lee, Yong-Hoon Lee 10 Lv, Zhiwei 10 Moradi, Shahin 10 Pang, Huihui 10 Su, Youhui 9 Bohner, Martin J. 9 Du, Zengji ...and 1,824 more Authors
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#### Cited in 252 Serials
199 Advances in Difference Equations 184 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 162 Journal of Mathematical Analysis and Applications 148 Computers & Mathematics with Applications 129 Applied Mathematics and Computation 124 Boundary Value Problems 114 Applied Mathematics Letters 88 Abstract and Applied Analysis 72 Journal of Difference Equations and Applications 69 Journal of Applied Mathematics and Computing 54 Mathematical and Computer Modelling 53 Journal of Computational and Applied Mathematics 45 Mediterranean Journal of Mathematics 42 Fractional Calculus & Applied Analysis 35 Discrete Dynamics in Nature and Society 33 Journal of Function Spaces 32 Applicable Analysis 29 Rocky Mountain Journal of Mathematics 26 Communications in Nonlinear Science and Numerical Simulation 22 Journal of Inequalities and Applications 22 Nonlinear Analysis. Real World Applications 21 Journal of Differential Equations 20 Mathematical Problems in Engineering 20 International Journal of Differential Equations 20 Nonlinear Analysis. Theory, Methods & Applications 19 Differential Equations and Dynamical Systems 19 Nonlinear Analysis. Modelling and Control 19 Journal of Applied Mathematics 16 Journal of Optimization Theory and Applications 16 Journal of Fixed Point Theory and Applications 16 Journal of Nonlinear Science and Applications 15 Nonlinear Dynamics 15 Nonlinear Analysis. Hybrid Systems 15 Differential Equations and Applications 14 Acta Applicandae Mathematicae 14 Acta Mathematicae Applicatae Sinica. English Series 14 Fractional Differential Calculus 13 Journal of the Franklin Institute 13 Mathematical Methods in the Applied Sciences 13 Proceedings of the American Mathematical Society 13 Results in Mathematics 13 International Journal of Nonlinear Sciences and Numerical Simulation 12 Zeitschrift für Analysis und ihre Anwendungen 12 Positivity 12 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 12 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 11 Chaos, Solitons and Fractals 10 Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 10 Acta Mathematica Sinica. English Series 10 Open Mathematics 9 Ukrainian Mathematical Journal 9 Journal of Dynamical and Control Systems 9 Fixed Point Theory and Applications 8 Demonstratio Mathematica 8 Mathematische Nachrichten 8 Journal of Integral Equations and Applications 8 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 8 Asian-European Journal of Mathematics 7 Czechoslovak Mathematical Journal 7 Bulletin of the Iranian Mathematical Society 7 Applied Mathematics. Series B (English Edition) 7 Turkish Journal of Mathematics 7 Journal of Applied Analysis 7 Afrika Matematika 6 Lithuanian Mathematical Journal 6 Applied Mathematics and Mechanics. (English Edition) 6 Opuscula Mathematica 6 Complexity 6 Qualitative Theory of Dynamical Systems 6 Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 6 Discrete and Continuous Dynamical Systems. Series B 6 Cubo 6 Mathematical Sciences 5 Journal of Mathematical Physics 5 Collectanea Mathematica 5 International Journal of Mathematics and Mathematical Sciences 5 Numerical Functional Analysis and Optimization 5 Proceedings of the Edinburgh Mathematical Society. Series II 5 Neural Networks 5 Applications of Mathematics 5 Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences 5 Indagationes Mathematicae. New Series 5 Journal of Mathematical Sciences (New York) 5 Georgian Mathematical Journal 5 Journal of Function Spaces and Applications 5 Symmetry 5 Arabian Journal of Mathematics 5 Journal of Applied Analysis and Computation 5 Journal of Mathematics 4 Bulletin of the Australian Mathematical Society 4 ZAMP. Zeitschrift für angewandte Mathematik und Physik 4 Linear Algebra and its Applications 4 Electronic Journal of Differential Equations (EJDE) 4 Random Operators and Stochastic Equations 4 The ANZIAM Journal 4 Boletim da Sociedade Paranaense de Matemática. Terceira Série 4 Communications in Mathematical Analysis 4 Advances in Mathematical Physics 4 ISRN Mathematical Analysis 4 Mathematics ...and 152 more Serials
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#### Cited in 38 Fields
2,182 Ordinary differential equations (34-XX) 787 Operator theory (47-XX) 447 Difference and functional equations (39-XX) 243 Real functions (26-XX) 179 Partial differential equations (35-XX) 175 Integral equations (45-XX) 105 Systems theory; control (93-XX) 90 Numerical analysis (65-XX) 78 Global analysis, analysis on manifolds (58-XX) 52 Biology and other natural sciences (92-XX) 50 Calculus of variations and optimal control; optimization (49-XX) 34 Probability theory and stochastic processes (60-XX) 25 Dynamical systems and ergodic theory (37-XX) 18 Mechanics of deformable solids (74-XX) 14 Functional analysis (46-XX) 10 General topology (54-XX) 9 Special functions (33-XX) 7 Operations research, mathematical programming (90-XX) 5 Functions of a complex variable (30-XX) 5 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 4 Approximations and expansions (41-XX) 4 Integral transforms, operational calculus (44-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 3 Abstract harmonic analysis (43-XX) 2 General and overarching topics; collections (00-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Statistics (62-XX) 2 Computer science (68-XX) 2 Mechanics of particles and systems (70-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Quantum theory (81-XX) 2 Information and communication theory, circuits (94-XX) 1 Combinatorics (05-XX) 1 Number theory (11-XX) 1 Potential theory (31-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Fluid mechanics (76-XX) 1 Statistical mechanics, structure of matter (82-XX) | 2021-10-19 08:24:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5948140621185303, "perplexity": 5676.452240955918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585246.50/warc/CC-MAIN-20211019074128-20211019104128-00578.warc.gz"} |
https://math.stackexchange.com/questions/2202308/how-to-find-constant-term-in-two-quadratic-equations | # How to find constant term in two quadratic equations
Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - x + p=0$ and let $\gamma$ and $\delta$ be the roots of the equation $x^2 -4x+q=0$. If $\alpha , \beta , \gamma , \delta$ are in Geometric progression then what is the value of $p$ and $q$?
My approach:
From the two equations, $$\alpha + \beta = 1$$, $$\alpha \beta = p$$, $$\gamma + \delta = 4$$, and, $$\gamma \delta = q$$. Since $\alpha , \beta , \gamma , \delta$ are in G. P., let $\alpha = \frac{a}{r^3}$, $\beta = \frac{a}{r^1}$, $\gamma = ar$, $\delta = ar^3$. $$\therefore \alpha \beta \gamma \delta = a^4 = pq$$ Now, $$\frac{\alpha + \beta}{\gamma + \delta} = \frac{1}{r^4}$$ $$\frac{1}{4} = \frac{1}{r^4}$$ $$\therefore r = \sqrt(2)$$
From here I don't know how to proceed. Am I unnecessarily complicating the problem??
Use your $r$ and your definitions for $\alpha(a,r)$ and $\beta(a,r)$ in your first equation, and solve for $a$. Once you have $a$ and $r$, you've determined $\alpha, \beta, \gamma, \delta$, and so you can use those values to determine $p,q$.
• Note that you should get the same value for $a$ if you use $\gamma(a,r)$ and $\delta(a,r)$ in the third equation. – AlexanderJ93 Mar 25 '17 at 10:44
let $$\beta=\alpha y$$,$$\gamma=\alpha y^2$$,$$\delta=\alpha y^3$$ then we get from the first equation $$\alpha^2-\alpha=\alpha^2y^2-\alpha y$$ from here we get $$\alpha=\frac{1}{1+y}$$ analogously we get from the second equation:$$\alpha=\frac{4}{y^2(1+y)}$$ combining these equations we have $$y^2=4$$ can you finish now?
$\alpha + \beta = 1$
Also,
$\alpha + \beta = \frac{a}{r^3} + \frac{a}{r^1}$
So we have,
$\frac{a}{r^3} + \frac{a}{r^1} = 1$
Put value of $r$,
$\frac{a}{2\sqrt2} + \frac{a}{\sqrt 2}= 1$
$a = \frac{2\sqrt2}{3}$
Now you can find roots. | 2019-10-14 15:20:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9691728353500366, "perplexity": 73.6642021153182}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986653876.31/warc/CC-MAIN-20191014150930-20191014174430-00463.warc.gz"} |
http://tex.stackexchange.com/questions/553/what-packages-do-people-load-by-default-in-latex/97045 | # What packages do people load by default in LaTeX?
I'm getting the impression from reading the answers written by some of the real experts here that there are quite a few little packages that just tweak LaTeX2e's default behaviour a little to make it more sensible here and there.
Rather than try to pick these up one by one as I read answers to questions (and thus risk missing them), I thought I'd ask up front what LaTeX2e packages people load by default in (almost) every document.
As this is a "big list" question, I'm making it CW. I don't know if there are standard rules across all SE/SO sites for such questions, but on MathOverflow the rule is generally: one thing (in this case, package) per answer. I guess that if a couple of packages really do go together then it would be fine to group them.
This is perhaps a little subjective and a little close to the line, so I'll not be offended if it gets closed or voted down! (But please explain why in the comments.)
Also see our community poll question: “I have used the following packages / classes”
-
Personally, I'd find a single list, separated by headings (Ex. Format, Math, Bib,Images, Other for this question), with a list of everyone's packages and how they're different from other packages in the section much more readable and useful. That amsmath is the highest voted just says that the MO community is here in full force. The less-known, but equally relevant formatting packages linked by Vivi, Joseph, and András are invisible without a lot of scrolling and reading. – Kevin Vermeer Jul 29 '10 at 22:37
I think the list of one package per answer is a good idea, as we can vote on individual packages... – Amir Rachum Jul 30 '10 at 11:30
It can be good to have a single answer that is just an index of all the other answers, and accept that, so that it floats to the top. – naught101 Aug 30 '12 at 3:44
I save my documents in an SVN repository. The svn package helps to extract some informations out of the version control system. The document has somewhere a hint what revision number and what date it is. For this you have to set svn keywords and declare in your LaTeX document what you need:
\SVN $LastChangedRevision$
\SVN $LastChangedDate$
Wihtin the document you can refer to that information with \SVNLastChangedRevision and \SVNDate.
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This has been mentioned in some of the “big answers”, but thought it deserved special attention. Probably most documents should include:
\usepackage[T1]{fontenc}
This is to resolve some deficiencies and inconsistencies of the default OT1 font encoding; while improving the support of special characters (e.g. the ability to copy&paste from the generated pdf document).
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A nice commenting environment is provided by the package:
\usepackage{verbatim}
For debugging purposes I find this package indispensable. Before I found this package I would have to enter % before each line I wished to comment. The environment works as follows:
\begin{comment}
Text in this environment will be ignored by LaTeX.
\end{comment}
The packages
\usepackage{comment}
\usepackage{xcomment}
provide even greater commenting capabilities (i.e. the ability to selectively typeset certain environments) though I personally haven't had much use for these extended features.
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In any decent editor, you can easily comment out/in several lines at once. Due to that, I find the usefulness of the comment environment greatly reduced – i.e. I don’t use it at all. – Konrad Rudolph Sep 11 '10 at 8:14
I simply use \newcommand{\comment}[1]{}. Put \comment{ before the block and } after to comment out any part of the file. – András Salamon Sep 11 '10 at 10:45
Usually I write German texts. We have new and old rules for spelling. The package hyphsubst provides some new hyphenation pattern. That's why I load it in addition to babel:
\RequirePackage[ngerman=ngerman-x-latest]{hyphsubst}
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This question assumes you are making a LaTeX document for personal use. If you are planning to submit the document to a journal, it's safer to avoid using too many unusual classes, because they may be incompatible with the journal's LaTeX classes or may be incompatible with the style that the journal will impose on your paper. Very common packages like amsthm are usually safe. (I would leave this as a comment, but I don't have enough reputation yet.)
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Yes and no. Given that I rarely know what paper it is intended for when I start writing a paper, and given how useful some of these packages are, I include them all and try to get away with it! Sometimes I'm successful, sometimes I need to include the package .sty file along with my submission. – Loop Space Aug 4 '10 at 7:03
Edited by doncherry: Removed packages mentioned in separate answers.
I use TeX for a variety of documents: research papers, lectures/tutorials, presentations, miscellaneous documents (some in Japanese). Each of these different uses, requires different packages.
Depending on my mood, I like to use different fonts. A particular nice combination for mathematics papers is
\usepackage[T1]{fontenc} % better treatment of accented words
\usepackage{eulervm} % Zapf's Euler fonts
\usepackage{tgpagella} % TeXGyre Pagella fonts
For references,...
\usepackage[notref,notcite]{showkeys} % useful when writing the paper
\usepackage[noadjust]{cite} % [1,2,3,4,5] --> [1-5] useful in hep-th!
For lecture notes (again mathematical) I often like to section the document into "lectures" instead of sections and to add some colours to the titles,.... To do this it's useful to use
\usepackage{fancyhdr} % fancy headers
\usepackage{titlesec} % to change how sections are displayed
\usepackage{color} % to be able to do this in colour
and I also like to decorate using some silly glyphs, for which these fonts are useful:
\usepackage{wasysym,marvosym,pifont}
and also box equations and other things
\usepackage{fancybox,shadow}
\usepackage[rflt]{floatflt}
\usepackage{graphicx,subfigure,epic,eepic}
You may want to hide the answers to tutorial exercises, problems,... and this can be achieved with
\usepackage{version,ifthen} % ifthen allows controlling exclusions
I use XeLaTeX for documents containing Japanese, which works better with
\usepackage{fontspec} % makes it very easy to select fonts in XeLaTeX
\usepackage{xunicode} % accents
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As the question suggested, could you write an answer per package/topic and explain what these packages do or why do you need them? – Juan A. Navarro Jul 29 '10 at 10:51
can you please add comments like \ usepackage{foo} % to get following features within your code? – Dima Jul 29 '10 at 11:06
To avoid breaking them up all the way, you could try grouping them a little (say, if there's one package that you wouldn't consider using without another one then put them together). – Loop Space Jul 29 '10 at 13:04
\usepackage{fancyvrb}
I use it for highly customisable verbatim. The abstract of the package documentation reads:
This package provides very sophisticated facilities for reading and writing verbatim TeX code. Users can perform common tasks like changing font family and size, numbering lines, framing code examples, colouring text and conditionally processing text.
Here's an example using the SaveVerbatim environment in combination with the \fcolorbox command:
-
I also find package lipsum fun to use. It lets you generate several versions of lorem ipsum placeholder text to see what your document would look like.
-
blindtext is much more mighty, it has several languages and can use some example math. – MaxNoe Jan 17 '15 at 14:01
I'm not just feigning surprise when I say I'm shocked that such an incredibly useful package set as xparse/expl3 (the latter is loaded by the former) hasn't been mentioned yet. I invariably find myself typing:
\documentclass{article}
\usepackage{xparse}
to begin a document.
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So, what does it do? – fifaltra Dec 24 '13 at 0:32
with xparse, one can define commands and environments with multiple optional arguments before, between, and after mandatory arguments. Several new type of arguments can be defined, starred commands, and much more. – Michael P May 7 '14 at 10:17
\usepackage[scaled=0.8]{luximono}
which is a fixed-width font which supports boldface. This is useful when typesetting source code.
-
\usepackage{docmute}
I use this in my syllabus preparation as I can make each of the subordinate documents fully standalone, yet do a complete compile of everything at once to verify I have all the corrections made.
It does require that all of the preambles are identical.
This then allows me to work only on one course syllabus or schedule or homework assignments with very fast compiles. Also during the semester I can do corrections on individual documents.
My main document preamble is
\documentclass[10pt,letterpaper]{article}
\input{commonpreamble}
\usepackage{docmute}
\begin{document}
And the subordinate documents have this preamble
\documentclass[10pt,letterpaper]{article}
\input{commonpreamble}
\begin{document}
Notice: Only one master document and the \usepackage{docmute} is only in that file.
Also all subordinate document must be only loaded with \input or \include from the main document. Only one level down is allowed.
I keep one copy of the preamble as commonpreamble. And all files are kept in one folder. This system works very well with Texmaker or TexStudio as the structure of the document is always displayed regardless of choosing a "Master Document".
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As long as this list is, minted is missing. For code syntax highlighting it works really well and includes the long list of languages of pygments. The pieces of code end up looking like this:
\begin{minted}{language}
code
\end{minted}
In Beamer it requires frames to be marked as [fragile], and it takes some skill to set it up on Windows. But the results are well worth the effort.
-
@Christian: the main difference is that you can tap directly into pygments, which is a (very) well maintained source for syntax colouring for many languages and is used in many places other than LaTeX. There is a full discussion on the differences between lstlisting and minted here: tex.stackexchange.com/questions/102596/…, – FvD Jun 28 '13 at 13:17
\usepackage{etex}
to be able to include e.g. TikZ without strange errors.
-
– Ben Feb 10 '12 at 11:01
For the natural scientists among us, the package mhchem makes it very easy to typeset chemical symbols and chemical equations.
-
I usually use relsize package. It's easy to use it. It changes the font size of part of your text. Just type \relsize{x} where x is the number of steps you want to move through the hierarchy of font sizes.
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When I'm writing package documentation using ltxdoc it likes using three columns for the index. I'd prefer two. I fix it with the idxlayout package:
\usepackage[columns=2]{idxlayout}
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I always use
\usepackage[retainorgcmds]{IEEEtrantools} % sophisticated equation arrays
It offers a sophisticated environment for formatting equation arrays,IEEEeqnarray and also offers a few other constructions. I don't use the traditional eqnarrays any more. I usually set the option [retainorgcmds] because it prevents the package from overwriting the itemize, enumerate and description definitions.
Check out How to Typeset Equations in LaTeX. The author gives some good examples of how and why to use this package instead of the traditional ones. The Not So Short Introduction to LaTeX 2ε also mentions the package in section 3.5.2. This section actually seems to be a copy of the first link ;)
-
I always load the package xy to produce diagrams.
Also tikz to draw figures.
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I use tikz-cd to get commutative diagrams drawn with tikz with a syntax highly reminiscent of the xy syntax. – Charles Staats Dec 6 '12 at 3:22
\usepackage{mciteplus}
Allows you to combine multiple references: \cite{refa, *refc, *refc, refd} will produce one references with refa, refb, and refc combined (if they are not used independently elsewhere).
-
BTW: natbib supports this feature too. See p.19 in the documentation mirrors.ctan.org/macros/latex/contrib/natbib/natbib.pdf – amorua May 2 '12 at 1:24
The following command before the \documentclass command permits Computer Modern fonts at arbitrary sizes: \RequirePackage{fix-cm}.
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Very often a requirement for the documents I write is that the font should be Times (or Times New Roman), so the package I use to set the main roman font to Times and acceptable math is mathptmx.
Recently, I have experimented with newtxtext and newtxmath but, personally, I do not like the design of some symbols and there are a few cases where the spacing between characters is too tight.
For personal use I set the font to New Century Schoolbook and Fourier (for math) with the fouriernc package.
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pageslts: for being able to refer to the last page of a document
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No one mention tabulary.
Sometimes I make tables with multiline cells in several columns, where the total width must be just \textwidth. Use tabular with p{} columns here is a pain since one must take into account \tabcolsep.
For this, the sibling tabularx (cited in another answer) could make a good work ( X columns take all the available space), but often I need columns weighted according to the amount of text rather and with different alignments, but X columns of tabularx share equally that space.
Instead, tabulary allow the use L, C, R and J columns o automatic variable width. Not always a column layouts as LLCRL produce the desired result but since it is possible mix L,C,R columns with basic types (l,r,c,p{}, m{}...) find the best fit (i.e., some like Lcp{5em}RL) is a child play.
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I just discovered the xparse package. It lets you define more flexible macros with more than one optional argument. I used it to make a very general partial derivative function.
\usepackage{xparse}
\DeclareDocumentCommand{\pder}{ O{} O{} m }{\frac{\partial^{#2}#1}{\partial#3^{#2}}}
Example
\pder{x} will give you
\pder[f]{x} will give you
\pder[f][3]{x} will give you
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I have a whole slew of commands that that provide a nice short hand for standard idioms of mine. (and which if I ever share tex source would make someone grumpy if i made it a package)
So the meta habit is: whatever personal short hands you think would be nice, have them defined at the top of your template file!
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I include: \usepackage{outlines} in my preamble. outlines is a quick and easy way to generate hierarchically embedded lists. Especially useful when I'm drafting up a paper (I like to outline it) or if I'm quickly typing up notes, e.g., at a conference.
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I always end up loading the same packages, some of which were suggested by some answers to this question, such as hyperref, amsmath, nag, etoolbox, xparse, and others.
I created a style file latexdev.sty that I use in almost all my notes and publications, which loads all these standard packages:
https://github.com/olivierverdier/latexdev
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When using class book, I always load package emptypage.
It needs no particular skill since it doesn't introduce any new command to use, it removes headers and footers from empty pages at the end of chapters just by adding \usepackage{emptypage} in your preamble.
The default option is odd.
- | 2016-05-29 19:15:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7963257431983948, "perplexity": 2173.304490012806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049281876.4/warc/CC-MAIN-20160524002121-00115-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/23129-midpoint-rule.html | # Thread: The Midpoint Rule
1. ## The Midpoint Rule
Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.
2. umm sorry for the waste of time, I cracked it...
3. Originally Posted by winterwyrm
umm sorry for the waste of time, I cracked it...
Congratulations on getting the answer! If you feel like it, enlighten us. Your post might help someone else.
-Dan | 2017-02-28 09:53:43 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8608941435813904, "perplexity": 2052.266448753712}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501174157.36/warc/CC-MAIN-20170219104614-00514-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/light-cones-tipping-over.140217/ | # Light cones tipping over
1. Oct 27, 2006
### MeJennifer
"Light cones tipping over"
A common phrase used to show alleged time travel solutions in GR.
Even a person like Kip Thorne uses it.
But my question is, is that an accurate representation of GR in strong gravitational fields?
The Schwarzschild metric expressed using the Eddington-Finkelstein coordinates show those "light cones tipping over", and eventually the radial and time coordinates reverse.
But this reversal, and even the tipping over seems to me a peculiarity of the choice of coordinates. It seems to me that it is assumed that there is a particular relationship between the radial and time coordinate.
That seems to be a rather liberal interpretation of the metric.
Afteral nothing in the Minkowskian metric of ds2 = dt2 - dr2 implies that if dt2 becomes negative it must be considered space and if dr2 becomes negative it must be considered time.
Any comments or explanations?
Last edited: Oct 27, 2006
2. Oct 27, 2006
### Aether
Minkowski space-time, at least according to one of my textbooks, is specifically defined as a 4-D real linear space. Neither dt2 nor dr2 can ever become negative while dr and dt are real numbers. That wouldn't apply to a complex linear space though.
3. Oct 27, 2006
### robphy
http://casa.colorado.edu/~ajsh/schwp.html may have useful diagrams for this question. Physically, the tipping of the light cones [which are traced out by null geodesics (a coordinate invariant idea)] is an indication that timelike observers travelling across the horizon cannot escape back out beyond the horizon.
4. Oct 28, 2006
### Thrice
Yeah the tipping just means the future part of the light cones point into a point in space (the black hole).
5. Oct 29, 2006
### cesiumfrog
Yep, keep in mind that "r" and "t" are just coordinates, they don't necessarilly have any global meaning. In particular, t certainly isn't purely timelike for all observers. The original motivation for my current work is all the existing EFE solutions which don't yet have much physical interpretation.
6. Oct 29, 2006
### MeJennifer
But even in coordinate space, what is the compelling argument for calling this a change from one to the other? Would that not imply that t and r are both related to each others imaginary plane? So that if t2 becomes negative it becomes space and that if r2 becomes negative it becomes time?
Would that not be a rather liberal interpretation of the Minkowski metric?
In plain Schwarzschild the light cones get narrower the closer one gets to the event horizon and eventually becomming zero. I would think it much more logical to see them get an imaginary size past the event horizon than to assume that, magically, the time and space coordinates get switched.
You say to keep in mind that coordinate space is not observer space is of course true. But, people use the tipping of light cones, which by the way is only happening when you use the Eddington-Finkelstein coordinates, to "explain" closed time curves. That again is a rather premature conclusion IMHO.
But on the other hand we see that people like Kip Thorne and more surprisingly Roger Penrose use it as well. Penrose who at one point proposed the use of a complex compactified Minkowski spacetime. Obviously this spacetime does not satisfy the condition that t and r are both related to each others imaginary plane.
So go figure, what's the real story here?
Last edited: Oct 29, 2006
7. Oct 29, 2006
### cesiumfrog
Normally you would leave r as simply a real valued coordinate; it's the metric components (eg. $g_{rr}$ vs. $g_{tt}$) that may change in sign.
I don't know that it makes any sense to say that (by what additional metric are you measuring the cones?)..
In the plain old (ie. Boyer-Lindquist) coordinates, the light cones "tip" as you approach the event horizon, which means light rays begin to traverse greater intervals of r coordinate (note we're implicity comparing to a euclidean metric over the same chart). From points on the horizon (neglecting that the chart is actually invalid just there), the cone would be rotated such that the future cone is centered towards the negative r direction, and the past cone in the direction of increasing r coordinate.
The tipping happens in general, unless you specifically choose a coordinate system to avoid it. The relation to causality violation can occur for example if null cones in a coordinate space tip such that the $\phi$ coordinate becomes everywhere timelike AND if there is also some kind of closed geometry such that having zero $\phi$ coordinate be identified with having $\phi$ coordinate equal to $2\pi$.
8. Oct 29, 2006
### MeJennifer
Using standard Schwarzschild coordinates light slows down for decreasing values of r. That is why the cones get narrower. But remember that r is nothing but a coordinate, it does not represent the physical distance from the center of gravity.
Using standard Schwarzschild coordinates light cones do not tip over.
They also do not tip using Kruskal coordinates. However they do tip over using the Eddington-Finkelstein coordinates.
Anyway we can do a lot of juggling with coordinates and then demonstrating all kinds of effects that are completely non physical but only due to the usage of certain coordinate systems.
More important is the critical review of the suggestion that time can become like space and space can become like time. I know for some that is a done deal and I simply do not "understand" it but to me it is not a done deal at all.
Last edited: Oct 29, 2006
9. Oct 29, 2006
### cesiumfrog
By saying "standard", you are referring to "Boyer-Lindquist", right?
In any case, say you draw a t vs. r plot (standard coords) depicting the null cones for observers at different positions.. would you not describe them as "tipping"?
Hmm. I would really avoid saying "time can become like space" or vice-versa, because I don't think it means anything. Physically, to an observer journeying across the event horizon, nothing ever happens of local significance. It's a confusing coincidence that the original Schwarzschild coordinates have undesirable features exactly where the event horizon is... Just that when we try to draw some set of coordinate axis lines (initially chosen somewhat arbitrarily), we sometimes find one of those lines is timelike in one region, and spacelike in another.
10. Oct 29, 2006
### MeJennifer
Boyer-Lindquist is used for a rotating mass not for a non rotating mass.
Using Schwarzschild coordinates there is only one r coordinate. Eddington-Finkelstein introduces an additional r* called the tortoise coordinate. With this in place we get tilting cones.
And if one additional coordinate is not enough we can always use the Krusal coordinates that introduces two coordinates that relate to t and r. In Krusal coordinates the cones are always straight up and also 45 degrees (so no narrowing).
A very good question!
For starters it would not be applicable for Schwarzschild coordinates since this is a view of the situation from the perspective of a distant observer far removed from the gravitational field. So the point is that in Schwarzschild coordinates the observer is fixed.
But with regards to a observer in free fall who is approaching the event horizon and continuing towards the singularity, the speed of light will remain c, at least locally, so that implies that the angle of the cone would remain constant. However we can hardly speak of a cone from the observer's perspective, it would seem clear that since the curvature for this observer is so strong that a construction of an orthonormal coordinate system would show anything but a cone for incoming and outgoing light rays except for a very small local region. Past the event horizon the observer would still measure the speed of light at c, again only locally, but because the curvature is getting so strong here that the closure between the observer and the center of mass is faster than the width angle of the observer's light cone would allow. Now some seem to interpret this as time becoming space and vice versa, which is the whole point of this topic.
To me that is not a coincidence at all. Remember that the Schwarzschild model is a view from the perspective of an observer far removed from the gravitational influence.
Last edited: Oct 29, 2006
11. Oct 29, 2006
### cesiumfrog
So your answer is no?
If so I disagree. Regardless of whether you use Schwarzschild or Ingoing (Eddington-Finkelstein) coordinates, the tilting is what you see when you draw light "cones" on a t vs. r (or t* vs. r*) plot. It's a coordinate effect.
Since coordinates are nothing physical, and you can't ever speak of shape or orientation of a light cone from the observer's perspective, why make a big deal of the tilting at all?
As for time travel, that occurs when the tilting is such that one's future light cone intersects one's past light cone. It isn't an issue in a black hole (the future cone is directed at the singularity, closed timelike loops don't become possible), it only means that from our perspective the gravity of a black hole sucks things away too strongly for their rockets or even their radio messages to approach us.
12. Oct 30, 2006
### MeJennifer
Well I am quite sure that you would not accept from me that using Schwarzschild coordinates light cones do not tip, and that instead you need Eddington-Finkelstein coordinates for that.
Perhaps from someone else you will.
Last edited: Oct 30, 2006
13. Oct 30, 2006
### cesiumfrog
I accept I may be wrong (you haven't yet motivated me to draw the plot myself) but my reasoning is that in Schwarzschild coordinates: far from the black hole, light tends to travel in the time direction (the cone spreading equally in +/- radial directions); inside the even horizon light must always travel radially inward (presumably also spreading in the t coordinate directions); if anything Ingoing coordinates would demonstrate less tilt (limited to 45 degrees).
Regardless, if you believe the tilting appears only in one of two equally valid coordinate systems, do you agree that "space becoming time" is not a physical effect (in black holes)?
14. Oct 31, 2006
### MeJennifer
The whole issue IMHO with cones is that they work well with SR but miserably fail with GR.
In flat spacetime we have cones, but not in curved spacetime. Yes of course if we take a very small region that we can consider flat then we have a mini cone but in curved spacetime we cannot speak about a cone at all. Sure we can insist on cones by picking the proper coordinate system, but it does not mean anything, for all intents and purposes we might as well create a coordinate system that shows cones as passa doblé steps.
I agree that such coordinate representations are more or less meaningless from a physical perspective.
But the problem is that tipping of cones is shown as theoretical evidence for things like closed time loops. Even people like Kip Thorne, Roger Penrose and Stephen Hawking seem run away with it and write books that "clearly shows" what is going on. In my understanding at least it seem that they should know better, but clearly I just don't seem to understand why it is obvious that time and space can flip inside the event horizon. Any takers on a simple explanation?
The main issue that I wanted to bring up in this topic is that suggestion, the suggestion that time becomes like space and vice versa. I don't see any indication for that, except for when we make a particular interpretation of the time and space relationship in the Minkowski metric.
Last edited: Oct 31, 2006
15. Oct 31, 2006
### Haelfix
The interpretation is natural though since asymptotically thats the preffered coordinate system.
The choice of tipping is indeed a coordinate artifact, but it makes good sense from a physical standpoint to compare it with what we know best. Eg a rockets journey starts out with the obvious Minkowski coordinates system, and as you get close and pass through the blackholes event horizon it serves as an illustration how either you have to abandon your choice of coordinates, or you have to admit that ones notion of time/space are going to warp and switch places.
And theres nothing wrong with writing down lightcones locally. Indeed we specifically choose not to pick coordinates where they are torus's or something like that, b/c no one has any intuition whatseover about that and indeed most calculational strategies vanish with such a stupid choice of local coordinates.
16. Oct 31, 2006
### MeJennifer
So demonstrate to me how they switch place!
How do two separate dimensions get intertwined on a Riemann surface?
Last edited: Oct 31, 2006
17. Oct 31, 2006
### robphy
In Minkowski spacetime M, we have two ways to think of the "light cone of an event p"...
It is the "set of events in M" that can only reach or be reached by the vertex event p by a light ray. It also forms the boundary between events that can be causally connected to p from those that cannot.
It is also a "set of directions [set of vectors] in TpM (the tangent space at p)" that are tangent to lightlike paths through event p (i.e., "set of lightlike tangent vectors at p"). It also forms the boundary between the spacelike and non-spacelike (i.e. causal) tangent vectors. TpM is a vector space with a Minkowski metric.
In a general curved spacetime, the notion of "light cone of an event p" usually means the "set of lightlike tangent vectors at p".
Nothing I have said above depends on any choice of coordinates.
To draw these light cones in the spacetime diagram of a spacetime, curved or otherwise, one must identify all of the lightlike geodesics (a coordinate invariant notion). At a particular event, its light cone is determined by the tangents to these geodesics.
Depending on your choice of coordinates, the image of these geodesics in your coordinate chart may trace out all sorts of crazy looking paths (akin to the distortions one gets from various map projections of the earth). In some cases, the image of these light cones may look tipped or distorted relative to the images of other light cones in your coordinate chart.
Regardless of appearances in your chart, the physics is determined by the lightlike geodesics, essentially telling you which events are in the causal future [and causal past] of events in spacetime (i.e. the causal connectivity of events). The worldlines of observers are bounded by these lightlike geodesics.
It may turn out that the null geodesics tell you that you might have closed causal curves. Or it may turn out that certain sets of events have causal futures that don't extend to spatial or null infinity. Or something other feature that one doesn't see in Minkowski spacetime.
Depending on your choice of chart (and thus the images of the light cones), it may be easier or harder to tell the story of what is going on physically. Depending on the particular aspect of the story you want to tell, some charts are better suited than others.
More correctly, these show that closed time loops are mathematically possible, given the constraints imposed in the situation. In other words, saying that one has a 4-manifold with a Lorentzian-signature metric places some restrictions on what "physics" is possible. However, by themselves, they don't restrict the possibility of closed time loops or other pathologies. Even imposing the field equations might still allow pathologies. That is why one is led to the notion of "causality conditions" and the study of "causal structure", which were developed using "global methods" (i.e. geometric, coordinate-free methods). One may also impose other conditions like "energy conditions", "asymptotic conditions", etc...
When I find the time, I'll try to address your concern about "switching".
Last edited: Oct 31, 2006
18. Oct 31, 2006
### MeJennifer
Sure I follow what you say here.
Sure in certain coordinates.
Completely agree!
Well at one point the time part of the geodesic has to connect to another time part of the same geodesic while the spatial parts are irrelevant. Apart from a closed spacetime or a wormhole I do not see how that can be the case. Can you?
Please do, to me it makes absolutely no sense.
19. Nov 1, 2006
### Thrice
Ok you guys are much better than I am at this, but I thought I had a handle on it. Lets look at this in the Schwarzschild coordinates. Is it sufficient to show t becomes imaginary & that a decrease in r becomes as inevitable as going forward in time?
Last edited: Nov 1, 2006
20. Nov 3, 2006
### coalquay404
In case anyone's still interested in this, the clearest pedagogical explanation of this is (unsurprisingly) in MTW. Check out pp. 823-826 for a discussion that uses the Schwarzschild geometry and the surface at $$r=2M$$ as an example.
21. Nov 4, 2006
### MeJennifer
See this posting in the https://www.physicsforums.com/showpost.php?p=1146536&postcount=21" topic for a comment that I think is applicable to this topic as well.
Last edited by a moderator: Apr 22, 2017
22. Nov 19, 2006
### Chris Hillman
Appearance of light cones in curved spacetimes
Hi again, Jennifer,
It is when Kip Thorne uses it! :-/ I know that because he can provide a correct figure which conforms to this informal description, as I can verify using my own computations.
I prefer to be more specific about this "tipping". I guess you are talking about light cones in the Boyer-Lindquist chart for the Kerr vacuum solution in gtr, which does feature closed timelike curves in the interior region, or light cones in the Goedel lambdadust solution, which also features closed timelike curves (see for example the beautiful figures in Hawking and Ellis, Large Scale Structure of Space-Time, for both of these examples).
But let's study an even simpler example:
Specifically, consider the advanced (infalling) Eddington chart, in which the line element takes the form
$ds^2 = -(1-2 m/r) \, du^2 + 2 \, du \, dr + r^2 \, \left( d\theta^2 + \sin(\theta) \, d\phi^2 \right),$
$-\infty < u < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi$
We can write down a "frame field" consisting of four orthonormal vector fields, a timelike unit vector
$\vec{e}_1 = \partial_u - m/r \, \partial_r$
plus three spacelike unit vectors
$\vec{e}_2 = \partial_u - (1-m/r) \, \partial_r$
$\vec{e}_3 = 1/r \, \partial_\theta$
$\vec{e}_4 = 1/r/\sin(\theta) \, \partial_\phi$
You can use these to draw the light cones. If you do it right, they will all be tangent to the null vector field $$\partial_r$$ and as r decreases, they lean inwards, until at $$r=2 m$$ they are also tangent to $$\partial_u$$.
Many people, even some who ought to know better, do talk that way, and invariably they wind up confusing everyone, including themselves. What they should really say is that the vectors $$\partial_u$$ are timelike outside the horizon, null at the horizon, and spacelike inside the horizon. Nothing "reverses"; in particular, the frame vectors given above are unambiguously timelike throughout (for the first) or spacelike throughout (for the remaining three).
George Jones is completely correct: of course "time and space" do not "swap roles" inside the horizon, that would be nonsense!
To elaborate on one point he alluded to, the coordinate basis vector field $$\partial_u$$ happens to be a Killing vector field; that is, the Schwarzschild vacuum is invariant under time translation. Similarly, the coordinate basis vector $$\partial_\phi$$ is a spacelike Killing vector whose integral curves are circles; that is, the Schwarzschild vacuum is invariant under rotation about the axis r=0.
The fact that in the exterior we have an irrotational timelike Killing vector and a spacelike Killing vector (whose integral curves are circles) means that the exterior region is static and axisymmetric. (This is also true of the Kerr vacuum solution.) Inside, we have two spacelike Killing vectors, but no timelike Killing vector; the solution is NOT static inside the horizon. Of course not, since otherwise an observer could use his rocket engine to hover at some Schwarzschild radius $$0 < r < 2m$$.
Not sure I understand that, but it sounds like you did correctly recognize that the coordinate basis vector $$\partial_u$$ changes character at the horizon.
Hope this helps,
Chris Hillman
Last edited: Nov 19, 2006
23. Nov 19, 2006
### cesiumfrog
Could you elaborate a little on exactly where those closed timelike curves are in the Kerr solution?
24. Nov 21, 2006
### Chris Hillman
CTC's in the Kerr vacuum
Hi, Cesium,
The system still seems (as least to me) to be quite unstable (but then I've only been here for a few days), so I daren't try to write very much (having lost quite a bit of work here in the past few days), but briefly, the CTCs in the Kerr vacuum are all located in the "deep interior" blocks (referring to the usual Carter-Penrose block or conformal diagram). The no-hair theorems do not imply that the -interior- geometry prefers to be Kerr-like, and there are various considerations which suggest that it should not be, quite apart from our natural desire to avoid predicting CTCs even in places where, even if a physicist should experience such weirdness, he'd be unable to report this to his colleages in the exterior.
The Taub-NUT vacuum (Misner's "counterexample to everything") and Goedel lambdadust also exhibit some startling causal structure. In fact, the best example to become familiar with CTC's is probably the Goedel lambdadust solution. As it happens, I just came across a spanking new arXiv eprint which offers an extensive and well illustrated discussion; see http://www.arxiv.org/abs/gr-qc/0611093
Chris Hillman
25. Nov 21, 2006
### Los Bobos
Regarding the "tipping of lightcones" can someone point a coordinate system for Schwarzschild black hole, where this does not happen? At least with the usual suspects this seems to happen ("the nature of dx -> dt" in the usual way or the relationship of the light cones and the tangent of fwo-path in Kruskal coordinates).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook | 2018-06-22 04:24:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7551605701446533, "perplexity": 859.2850912792763}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864343.37/warc/CC-MAIN-20180622030142-20180622050142-00590.warc.gz"} |
http://www.inf.u-szeged.hu/actacybernetica/edb/vol05n1/Imreh_1980_ActaCybernetica.xml | . . .
About Us Education Research PhD Acta Cybernetica Conferences Sponsors Departments: - Image Processing and Computer Graphics - Technical Informatics - Foundations of Computer Science - Computer Algorithms and Artificial Intelligence - Computational Optimization - Software Engineering - Research Group on Artificial Intelligence [University of Szeged]
Institute of Informatics>>> Acta Cybernetica>>> Past Issues>>> Volume 5, Number 1, 1980>>> Magyarul
# On isomorphic representations of commutative automata with respect to $\alpha_i$-products
### Abstract (in LaTeX format)
Abstract is not available.
### Full text
Available electronic editions: PDF.
Note that full text is available only for papers that are at least 3 years old. For more recent papers only the first page of the paper is provided.
### BibTeX entry
@article{Imreh:1980:ActaCybernetica,
author = {Bal{\'a}zs Imreh},
title = {On isomorphic representations of commutative automata with respect to $\alpha_i$-products},
journal = {Acta Cybernetica},
year = {1980},
volume = {5},
pages = {21--32},
number = {1}
}
Webmaster:webmaster@inf.u-szeged.hu | 2013-05-22 18:01:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23077069222927094, "perplexity": 6054.736827645197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368702185502/warc/CC-MAIN-20130516110305-00056-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://arxivsorter.org/ | ### 54 articles on Wednesday, October 23
arXiv:1910.09541v1 [pdf, other]
Comparison of the Scaling Properties of EUV Intensity Fluctuations in Coronal Hole and Quiet-Sun Regions
Using detrended fluctuation analysis (DFA) and rescaled range (R/S) analysis, we investigate the scaling properties of EUV intensity fluctuations of low-latitude coronal holes (CHs) and neighboring quiet-Sun (QS) regions in signals obtained with the Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA) instrument. Contemporaneous line-of-sight SDO/Helioseismic and Magnetic Imager (HMI) magnetic fields provide a context for the physical environment. We find that the intensity fluctuations in the time series of EUV images present at each spatial point a scaling symmetry over the range $\sim 20$ min to $\sim$ 1 hour. Thus we are able to calculate a generalized Hurst exponent and produce image maps, not of physical quantities like intensity or temperature, but of a single dynamical parameter that sums up the statistical nature of the intensity fluctuations at each pixel. In quiet-Sun (QS) regions and in coronal holes (CHs) with magnetic bipoles, the scaling exponent ($1.0 < \alpha \leq 1.5$) corresponds to anti-correlated turbulent-like processes. In coronal holes, and in quiet-Sun regions primarily associated with (open) magnetic field of dominant polarity, the generalized exponent (0.5 $< \alpha <$ 1) corresponds to positively-correlated (persistent) processes. We identify a tendency for $\alpha$ $\sim$ $1$ near coronal hole boundaries and in other regions in which open and closed magnetic fields are in proximity. This is a signature of an underlying $1/f$ type process that is characteristic for self-organized criticality and shot-noise models.
arXiv:1910.09552v1 [pdf, other]
Empirical constraints on the formation of early-type galaxies
Comments: 20 pages, 17 figures, 1 table, submitted to MNRAS
We present constraints on the formation and evolution of early-type galaxies (ETGs) with the empirical model EMERGE. The parameters of this model are adjusted so that it reproduces the evolution of stellar mass functions, specific star formation rates, and cosmic star formation rates since $z\approx10$ as well as 'quenched' galaxy fractions and correlation functions. We find that at fixed halo mass present-day ETGs are more massive than late-type galaxies, whereas at fixed stellar mass ETGs populate more massive halos in agreement with lensing results. This effect naturally results from the shape and scatter of the stellar-to-halo mass relation and the galaxy formation histories. The ETG stellar mass assembly is dominated by 'in-situ' star formation below a stellar mass of $3\times10^{11}\mathrm{M}_\odot$ and by merging and accretion of 'ex-situ' formed stars at higher mass. The mass dependence is in tension with current cosmological simulations. Lower mass ETGs show extended star formation towards low redshift in agreement with recent estimates from IFU surveys. All ETGs have main progenitors on the 'main sequence of star formation' with the 'red sequence' appearing at $z \approx 2$. Above this redshift, over 95 per cent of the ETG progenitors are star-forming. More than 90 per cent of $z \approx 2$ 'main sequence' galaxies with $m_* > 10^{10}\mathrm{M}_\odot$ evolve into present-day ETGs. Above redshift 6, more than 80 per cent of the observed stellar mass functions above $10^{9}\mathrm{M}_\odot$ can be accounted for by ETG progenitors with $m_* > 10^{10}\mathrm{M}_\odot$. This implies that current and future high redshift observations mainly probe the birth of present-day ETGs. The source code and documentation of EMERGE are available at github.com/bmoster/emerge.
arXiv:1910.09554v1 [pdf, other]
Simple Yet Powerful: Hot Galactic Outflows Driven by Supernovae
Comments: 9 pages, 4 figures, submitted to ApJL
Supernovae (SNe) drive multiphase galactic outflows, impacting galaxy formation; however, cosmological simulations mostly use \textit{ad hoc} feedback models for outflows, making outflow-related predictions from first principles problematic. Recent small-box simulations resolve individual SNe remnants in the interstellar medium (ISM), naturally driving outflows and permitting a determination of the wind loading factors of energy \etaE, mass \etam, and metals \etaZ. In this Letter, we compile small-box results, and find consensus that the hot outflows are much more powerful than the cool outflows: (i) their energy flux is 2-20 times greater, and (ii) their specific energy $e_{s,h}$ is 10-1000 times higher. Moreover, the properties of hot outflows are remarkably simple: $e_{s,h} \propto \eta_{E,h}/\eta_{m,h}$ is almost invariant over four orders of magnitude of star formation surface density. Also, we find tentatively that $\eta_{E,h}/\eta_{Z,h} \sim$ 0.5. If corroborated by more simulation data, these correlations reduce the three hot phase loading factors into one. Finally, this one parameter is closely related to whether the ISM has a "breakout" condition. The narrow range of \esh\ indicates that hot outflows cannot escape dark matter halos with log $M_{\rm{halo}}\ [M_\odot] \gtrsim 12$. This mass is also where the galaxy mass-metallicity relation reaches its plateau, implying a deep connection between \textit{hot} outflows and galaxy formation. We argue that hot outflows should be included explicitly in cosmological simulations and (semi-)analytic modeling of galaxy formation.
arXiv:1910.09558v1 [pdf, other]
Double-White-Dwarf Merger Products among High-Mass White Dwarfs
Double-white-dwarf (double-WD) binaries may merge within a Hubble time and produce high-mass WDs. Compared to other high-mass WDs, the double-WD merger products have higher velocity dispersion because they are older. With the power of Gaia data, we show strong evidence for double-WD merger products among high-mass WDs by analyzing the transverse-velocity distribution of more than a thousand high-mass WDs (0.8-1.3 $M_\odot$). We estimate that the fraction of double-WD merger products in our sample is about 20 %. We also calculate the double-WD merger rate and its mass dependence. Our results agree with binary population synthesis results and support the idea that double-WD mergers can contribute to a large fraction of type-Ia supernovae.
arXiv:1910.09561v1 [pdf, other]
The large-separation expansion of peak clustering in Gaussian random fields
In the peaks approach, the formation sites of observable structures in the Universe are identified as peaks in the matter density field. The statistical properties of the clustering of peaks are particularly important in this respect. In this paper, we investigate the large-separation expansion of the correlation function of peaks in Gaussian random fields. The analytic formula up to third order is derived, and the resultant expression can be evaluated by a combination of one-dimensional fast Fourier transforms, which are evaluated very fast. The analytic formula obtained perturbatively in the large-separation limit is compared with a method of Monte-Carlo integrations, and a complementarity between the two methods is demonstrated.
arXiv:1910.09565v1 [pdf, other]
Baryonic effects on CMB lensing and neutrino mass constraints
Comments: 9 pages, 2 figures, 2 tables
Measurements of gravitational lensing of the cosmic microwave background (CMB) hold the promise of yielding unique insights into cosmology at high redshift. Uncertainties due to baryonic effects associated with galaxy formation and evolution, including gas cooling, star formation, and feedback from active galactic nuclei (AGN) and supernovae, have typically been neglected when forecasting the sensitivity of future CMB surveys. In this paper, we determine the impact of these effects using four suites of hydrodynamical simulations which incorporate various prescriptions for baryonic processes, namely OWLS, BAHAMAS, Horizon, and IllustrisTNG. Our analysis shows characteristic power suppressions of several percent in CMB lensing due to baryonic effects, compared to dark-matter only simulations, at experimentally observable angular scales. We investigate the associated bias in the inferred neutrino mass for experiments like the upcoming Simons Observatory and CMB-S4. Depending on the experimental precision and the strength of the baryonic feedback within the simulations, biases in the neutrino mass sum show significant dispersion, ranging from very small to an over-estimation by 1.1$\sigma$. We conclude that baryonic effects will likely be non-negligible for a detection of neutrino mass using CMB lensing.
arXiv:1910.09566v1 [pdf, other]
Galactic outflow rates in the EAGLE simulations
Comments: 23 pages, submitted to MNRAS
We present measurements of galactic outflow rates from the EAGLE suite of cosmological simulations. We find that gas is removed from the interstellar medium (ISM) of central galaxies with a dimensionless mass loading factor that scales approximately with circular velocity as $V_{\mathrm{c}}^{-3/2}$ in the low-mass regime where stellar feedback dominates. Feedback from active galactic nuclei causes an upturn in the mass loading for halo masses $> 10^{12} \, \mathrm{M_\odot}$. We find that more gas outflows through the halo virial radius than is removed from the ISM of galaxies, particularly at low redshift, implying substantial mass loading within the circum-galactic medium. Outflow velocities span a wide range at a given halo mass/redshift, and on average increase positively with redshift and halo mass up to $M_{200} \sim 10^{12} \, \mathrm{M_\odot}$. We present a number of like-for-like comparisons to outflow rates from other recent cosmological hydrodynamical simulations, and show that comparing the propagation of galactic winds as a function of radius reveals substantial discrepancies between different models. Relative to some other simulations, EAGLE favours a scenario for stellar feedback where agreement with the galaxy stellar mass function is achieved by removing smaller amounts of gas from the ISM, but with galactic winds that then propagate and entrain ambient gas out to larger radii.
arXiv:1910.09572v1 [pdf, other]
Super-Massive Neutron Stars and Compact Binary Millisecond Pulsars
Comments: Invited review article, submitted for publication in the proceedings of the XIII Multifrequency Behaviour of High Energy Cosmic Sources Workshop (June 3-8, 2019, Palermo, Italy); 12 pages, 1 table, 4 figures
The maximum mass of a neutron star has important implications across multiple research fields, including astrophysics, nuclear physics and gravitational wave astronomy. Compact binary millisecond pulsars (with orbital periods shorter than about a day) are a rapidly-growing pulsar population, and provide a good opportunity to search for the most massive neutron stars. Applying a new method to measure the velocity of both sides of the companion star, we previously found that the compact binary millisecond pulsar PSR J2215+5135 hosts one of the most massive neutron stars known to date, with a mass of 2.27$\pm$0.16 M$_\odot$ (Linares, Shahbaz & Casares, 2018). We reexamine the properties of the 0.33 M$_\odot$ companion star, heated by the pulsar, and argue that irradiation in this "redback" binary is extreme yet stable, symmetric and not necessarily produced by an extended source. We also review the neutron star mass distribution in light of this and more recent discoveries. We compile a list of all (nine) systems with published evidence for super-massive neutron stars, with masses above 2 M$_\odot$. We find that four of them are compact binary millisecond pulsars (one black widow, two redbacks and one redback candidate). This shows that compact binary millisecond pulsars are key to constraining the maximum mass of a neutron star.
arXiv:1910.09575v1 [pdf, other]
Vertical position of the Sun with $γ$-rays
Comments: 5 pages, 5 figures, accepted by A&A
We illustrate a method for estimating the vertical position of the Sun above the Galactic plane by $\gamma$-ray observations. Photons of $\gamma$-ray wavelengths are particularly well suited for geometrical and kinematic studies of the Milky Way because they are not subject to extinction by interstellar gas or dust. Here, we use the radioactive decay line of $\mathrm{^{26}Al}$ at $1.809\,\mathrm{MeV}$ to perform maximum likelihood fits to data from the spectrometer SPI on board the INTEGRAL satellite as a proof-of-concept study. Our simple analytic 3D emissivity models are line-of-sight integrated, and varied as a function of the Sun's vertical position, given a known distance to the Galactic centre. We find a vertical position of the Sun of $z_0 = 15 \pm 17\,\mathrm{pc}$ above the Galactic plane, consistent with previous studies, finding $z_0$ in a range between $5$ and $29\,\mathrm{pc}$. Even though the sensitivity of current MeV instruments is several orders of magnitude below that of telescopes for other wavelengths, this result reveals once more the disregarded capability of soft $\gamma$-ray telescopes. We further investigate possible biases in estimating the vertical extent of $\gamma$-ray emission if the Sun's position is set incorrectly, and find that the larger the true extent, the less is it affected by the observer position. In the case of $\mathrm{^{26}Al}$ with an exponential scale height of $150,\mathrm{pc}$ ($700\,\mathrm{pc}$) in the inner (full) Galaxy, this may lead to misestimates of up to $25\,\%$.
arXiv:1910.09582v1 [pdf, other]
The matter beyond the ring: the recent evolution of SN 1987A observed by the Hubble Space Telescope
Comments: Accepted for publication in ApJ
The nearby SN 1987A offers a spatially resolved view of the evolution of a young supernova remnant. Here we precent recent Hubble Space Telescope imaging observations of SN 1987A, which we use to study the evolution of the ejecta, the circumstellar equatorial ring (ER) and the increasing emission from material outside the ER. We find that the inner ejecta have been brightening at a gradually slower rate and that the western side has been brighter than the eastern side since ~7000 days. This is expected given that the X-rays from the ER are most likely powering the ejecta emission. At the same time the optical emission from the ER continues to fade linearly with time. The ER is expanding at 680\pm 50 km/s, which reflects the typical velocity of transmitted shocks in the dense hotspots. A dozen spots and a rim of diffuse H-alpha emission have appeared outside the ER since 9500 days. The new spots are more than an order of magnitude fainter than the spots in the ER and also fade faster. We show that the spots and diffuse emission outside the ER may be explained by fast ejecta interacting with high-latitude material that extends from the ER toward the outer rings. Further observations of this emission will make it possible to determine the detailed geometry of the high-latitude material and provide insight into the formation of the rings and the mass-loss history of the progenitor.
arXiv:1910.09613v1 [pdf, other]
Overview to the Hard X-ray Modulation Telescope (Insight-HXMT) Satellite
Comments: 29 pages, 40 figures, 6 tables, to appear in Sci. China-Phys. Mech. Astron. arXiv admin note: text overlap with arXiv:1910.04434
As China's first X-ray astronomical satellite, the Hard X-ray Modulation Telescope (HXMT), which was dubbed as Insight-HXMT after the launch on June 15, 2017, is a wide-band (1-250 keV) slat-collimator-based X-ray astronomy satellite with the capability of all-sky monitoring in 0.2-3 MeV. It was designed to perform pointing, scanning and gamma-ray burst (GRB) observations and, based on the Direct Demodulation Method (DDM), the image of the scanned sky region can be reconstructed. Here we give an overview of the mission and its progresses, including payload, core sciences, ground calibration/facility, ground segment, data archive, software, in-orbit performance, calibration, background model, observations and some preliminary results.
arXiv:1910.09619v1 [pdf, other]
Two-index model for characterizing site-specific night sky brightness patterns
Comments: Author formatted text of the accepted version of the paper published in Monthly Notices of the Royal Astronomical Society (2019)
Determining the all-sky radiance distribution produced by artificial light sources is a computationally demanding task that generally requires an intensive calculation load. We develop in this work an analytic formulation that provides the all-sky radiance distribution produced by an artificial light source as an explicit and analytic function of the observation direction, depending on two single parameters that characterize the overall effects of the atmosphere. One of these parameters is related to the effective attenuation of the light beams, whereas the other accounts for the overall asymmetry of the combined scattering processes in molecules and aerosols. By means of this formulation a wide range of all-sky radiance distributions can be efficiently and accurately calculated in a short time. This substantial reduction in the number of required parameters, in comparison with other currently used approaches, is expected to facilitate the development of new applications in the field of light pollution research.
arXiv:1910.09634v1 [pdf, other]
Magnetic fields and cosmic rays in M 31. I. Spectral indices, scale lengths, Faraday rotation & magnetic field pattern
Comments: Accepted for publication in Astronomy 6 Astrophysics
Three deep radio continuum surveys of the Andromeda galaxy, M 31, were performed at 11.3, 6.2, and 3.6 cm wavelength with the Effelsberg 100-m telescope. At all wavelengths, the total and polarized emission is concentrated in a ring-like structure between about 7 kpc and 13 kpc radius from the center. Propagation of cosmic rays away from star-forming regions is evident: The ring of synchrotron emission is wider than the ring of the thermal radio emission and the radial scale length of synchrotron emission is larger than that of thermal emission. The polarized intensity from the ring varies double-periodically with azimuthal angle, indicating that the ordered magnetic field is almost oriented along the ring, with a pitch angle of -14{\deg} $\pm$ 2{\deg}. Faraday rotation measures (RM) show a large-scale sinusoidal variation with azimuthal angle, signature of an axisymmetric spiral (ASS) regular magnetic field, plus a superimposed double-periodic variation of a bisymmetric spiral (BSS) field with about 6x smaller amplitude. The dominating ASS field of M 31 is the most compelling case so far of a field generated by the action of a mean-field dynamo. The RM amplitude between 6.2 cm and 3.6 cm is about 50% larger than between 11.3 cm and 6.2 cm, indicating that Faraday depolarization at 11.3 cm is stronger than at 6.2 cm and 3.6 cm. The phase of the sinusoidal RM variation of -7{\deg} $\pm$ 1{\deg} is interpreted as the average spiral pitch angle of the regular field. The average pitch angle of the ordered field, as derived from the intrinsic orientation of the polarized emission (corrected for Faraday rotation), is significantly smaller: -26{\deg} $\pm$ 3{\deg}. The difference in pitch angle of the regular and the ordered fields indicates that the ordered field contains a significant fraction of an anisotropic turbulent field that has a different pattern than the regular (ASS + BSS) field.
arXiv:1910.09637v1 [pdf, other]
Probing Primordial Stochastic Gravitational Wave Background with Multi-band Astrophysical Foreground Cleaning
The primordial stochastic gravitational wave background (SGWB) carries first-hand messages of early-universe physics, possibly including effects from inflation, preheating, cosmic strings, electroweak symmetry breaking, and etc. However, the astrophysical foreground from compact binaries may mask the SGWB, introducing difficulties in detecting the signal and measuring it accurately. In this Letter, we propose a foreground cleaning method taking advantage of gravitational wave observations in other frequency bands. We apply this method to probing the SGWB with space-borne gravitational wave detectors, such as the Laser Interferometer Space Antenna (LISA). We find that the spectral density of the LISA-band astrophysical foreground can be predicted with percent-level accuracy assuming $10$-years' observations of third-generation GW detectors, e.g., Cosmic Explorer. After the foreground cleaning, LISA's sensitivity to the primordial SGWB will be substantially improved.
arXiv:1910.09668v1 [pdf, other]
The LOFAR Tied-Array All-Sky Survey (LOTAAS): Characterization of 20 pulsar discoveries and their single-pulse behavior
We are using the LOw-Frequency ARray (LOFAR) to perform the LOFAR Tied-Array All-Sky (LOTAAS) survey for pulsars and fast transients. Here we present the astrometric and rotational parameters of 20 pulsars discovered as part of LOTAAS. These pulsars have regularly been observed with LOFAR at 149 MHz and the Lovell telescope at 1532 MHz, supplemented by some observations with the Lovell telescope at 334 MHz and the Nancay Radio Telescope at 1484 MHz. Timing models are calculated for the 20 pulsars, some of which are among the slowest-spinning pulsars known. PSR J1236-0159 rotates with a period P ~ 3.6 s, while 5 additional pulsars show P > 2 s. Also, the spin-down rates Pdot are, on average, low, with PSR J0815+4611 showing Pdot ~ 4E-18. Some of the pulse profiles, generically single-peaked, present complex shapes evolving with frequency. Multi-frequency flux measurements show that these pulsars have generically relatively steep spectra but exceptions are present, with values ranging between ~ -4 and -1. Among the pulsar sample, a large fraction shows large single-pulse variability, with 4 pulsars being undetectable more than 15% of the time and one tentatively classified as a Rotating Radio Transient. Two single-peaked pulsars show drifting sub-pulses.
arXiv:1910.09680v1 [pdf, other]
The population of Galactic planetary nebulae: a study of distance scales and central stars based on the second GAIA release
Comments: The Astrophysical Journal, in press
We matched the astrometry of central stars (CSs) of spectroscopically-confirmed Galactic planetary nebulae (PNe) with DR2 Gaia parallaxes ($p$), finding 430 targets in common with $p>0$ and $|\sigma_{\rm p}/p|<1$. A catalog of PNe whose CSs have DR2 Gaia parallaxes is presented in Table 1. We compared DR2 parallaxes with those in the literature, finding a good correlation between the two samples. We used PNe parallaxes to calibrate the Galactic PN distance scale. Restricting the sample to objects with 20$\%$ parallax accuracy, we derive the distance scale ${\rm log}(R_{\rm pc})=-(0.226\pm0.0155)\times{\rm log}(S_{\rm H\beta})-(3.920\pm0.215)$, which represents a notable improvement with respect to previous ones. We found that the ionized mass vs. optical thickness distance scale for Galactic PNe is not as well constrained by the Gaia calibrators, but gives important insight on the nature of the PNe, and is essential to define the domain for our distance scale application. We placed the CSs whose distance has been determined directly by parallax on the HR diagram, and found that their location on the post-AGB H-burning evolutionary tracks is typical for post-AGB stars.
arXiv:1910.09683v1 [pdf, other]
Multiple populations in globular clusters and their parent galaxies
Comments: 18 pages, 13 figures, accepted for publication in MNRAS
The 'chromosome map' diagram (ChM) proved a successful tool to identify and characterize multiple populations (MPs) in 59 Galactic Globular Clusters (GCs). Here, we construct ChMs for 11 GCs of both Magellanic Clouds (MCs) and with different ages to compare MPs in Galactic and extra-Galactic environments, and explore whether this phenomenon is universal through 'place' and 'time'. MPs are detected in five clusters. The fractions of 1G stars, ranging from about 50% to more than 80%, are significantly higher than those observed in Galactic GCs with similar present-day masses. By considering both Galactic and MC clusters, the fraction of 1G stars exhibits: (i) a strong anti-correlation with the present-day mass, and (ii) with the present-day mass of 2G stars; (iii) a mild anti-correlation with 1G present-day mass. All Galactic clusters without MPs have initial masses smaller than ~1.5 10^5 solar masses but a mass threshold governing the occurrence of MPs seems challenged by massive simple-population MC GCs; (iv) Milky Way clusters with large perigalactic distances typically host larger fractions of 1G stars, but the difference disappears when we use initial cluster masses. These facts are consistent with a scenario where the stars lost by GCs mostly belong to the 1G. By exploiting recent work based on Gaia, half of the known Type II GCs appear clustered in a distinct region of the integral of motions space, thus suggesting a common progenitor galaxy. Except for these Type II GCs,we do not find any significant difference in the MPs between clusters associated with different progenitors.
arXiv:1910.09693v1 [pdf, other]
Toward the Detection of Relativistic Image Doubling in Imaging Atmospheric Cerenkov Telescopes
Cosmic gamma-ray photons incident on the upper atmosphere create air showers that move to the Earth's surface with superluminal speed, relative to the air. Even though many of these air showers remain superluminal all along their trajectories, the shower's velocity component toward a single Imaging Atmospheric Cherenkov Telescope (IACT) may drop from superluminal to subluminal. When this happens, an IACT that is able to resolve the air shower both in time and angle should be able to document an unusual optical effect known as relativistic image doubling (RID). The logic of RID is that the shower appears to precede its own Cherenkov radiation when its speed component toward the IACT is superluminal, but appears to trail its own Cherenkov radiation when its speed component toward the IACT is subluminal. The result is that the IACT will see the shower start not at the top of the atmosphere but in the middle -- at the point along the shower's path where its radial velocity component drops to subluminal. Images of the shower would then be seen by the IACT to go both up and down simultaneously. A simple simulation demonstrating this effect is presented. Clear identification of RID would confirm in the atmosphere a novel optical imaging effect caused not by lenses but solely by relativistic kinematics, and may aid in the accuracy of path and speed reconstructions of the relativistic air shower.
arXiv:1910.09709v1 [pdf, other]
Searching for Planets Orbiting Alpha Centauri A with the James Webb Space Telescope
Alpha Centauri A is the closest solar-type star to the Sun and offers an excellent opportunity to detect the thermal emission of a mature planet heated by its host star. The MIRI coronagraph on JWST can search the 1-3 AU (1"-2") region around alpha Cen A which is predicted to be stable within the alpha Cen AB system. We demonstrate that with reasonable performance of the telescope and instrument, a 20 hr program combining on-target and reference star observations at 15.5 um could detect thermal emission from planets as small as ~5 RE. Multiple visits every 3-6 months would increase the geometrical completeness, provide astrometric confirmation of detected sources, and push the radius limit down to ~3 RE. An exozodiacal cloud only a few times brighter than our own should also be detectable, although a sufficiently bright cloud might obscure any planet present in the system. While current precision radial velocity (PRV) observations set a limit of 50-100 ME at 1-3 AU for planets orbiting alpha Cen A, there is a broad range of exoplanet radii up to 10 RE consistent with these mass limits. A carefully planned observing sequence along with state-of-the-art post-processing analysis could reject the light from alpha Cen A at the level of ~10^-5 at 1"-2" and minimize the influence of alpha Cen B located 7-8" away in the 2022-2023 timeframe. These space-based observations would complement on-going imaging experiments at shorter wavelengths as well as PRV and astrometric experiments to detect planets dynamically. Planetary demographics suggest that the likelihood of directly imaging a planet whose mass and orbit are consistent with present PRV limits is small, ~5%, and possibly lower if the presence of a binary companion further reduces occurrence rates. However, at a distance of just 1.34 pc, alpha Cen A is our closest sibling star and certainly merits close scrutiny.
arXiv:1910.09712v1 [pdf, other]
LRP2020: The cosmic origin and evolution of the elements
Comments: White paper submitted to the Canadian Long Range Plan 2020. Minor formatting changes relative to submitted version
The origin of many elements of the periodic table remains an unsolved problem. While many nucleosynthetic channels are broadly understood, significant uncertainties remain regarding certain groups of elements such as the intermediate and rapid neutron-capture processes, the p-process, or the origin of odd-Z elements in the most metal-poor stars. Canada has a long tradition of leadership in nuclear astrophysics, dating back to the work of Alastair Cameron in the 1950s. Recent faculty hires have further boosted activity in the field, including transient observation and theory, survey science on galactic nucleosynthesis, and nuclear experiments. This white paper contains a brief overview of recent activity in the community, highlighting strengths in each sub-field, and provides recommendations to improve interdisciplinary collaboration. Sustaining Canadian leadership in the next decade will require, on the observational side, access to transient and non-transient surveys like LSST, SKA, or MSE, support for target-of-opportunity observing in current and future Canadian telescopes, and participation in next-generation X-ray telescopes such as ATHENA. State-of-the-art theoretical predictions will require an ambitious succession plan for the Niagara supercomputer to support large parallel jobs. We propose a funding instrument for postdoctoral training that reflects the interdisciplinary nature of nuclear astrophysics research, and the creation of a national collaborative funding program that allows for joint projects and workshop organization.
arXiv:1910.09730v1 [pdf, other]
Understanding Galaxy Evolution through Emission Lines
We review the use of emission-lines for understanding galaxy evolution, focusing on excitation source, metallicity, ionization parameter, ISM pressure and electron density. We show that the UV, optical and infrared contain complementary diagnostics that can probe the conditions within different nebular ionization zones. In anticipation of upcoming telescope facilities, we provide new self-consistent emission-line diagnostic calibrations for complete spectral coverage from the UV to the infrared. These diagnostics can be used in concert to understand how fundamental galaxy properties have changed across cosmic time. We describe new 2D and 3D emission-line diagnostics to separate the contributions from star formation, AGN and shocks using integral field spectroscopy. We discuss the physics, benefits, and caveats of emission-line diagnostics, including the effect of theoretical model uncertainties, diffuse ionized gas, and sample selection bias. Accounting for complex density gradients and temperature profiles is critical for reliably estimating the fundamental properties of H ii regions and galaxies. Diffuse ionized gas can raise metallicity estimates, flatten metallicity gradients, and introduce scatter in ionization parameter measurements. We summarize with a discussion of the challenges and major opportunities for emission-line diagnostics in the coming years.
arXiv:1910.09735v1 [pdf, other]
The Structure of Solar Coronal Mass Ejections in the Extreme-Ultraviolet Passbands
Comments: 15 pages, 5 figures, accepted by ApJ
So far most studies on the structure of coronal mass ejections (CMEs) are conducted through white-light coronagraphs, which demonstrate about one third of CMEs exhibit the typical three-part structure in the high corona (e.g., beyond 2 Rs), i.e., the bright front, the dark cavity and the bright core. In this paper, we address the CME structure in the low corona (e.g., below 1.3 Rs) through extreme-ultraviolet (EUV) passbands and find that the three-part CMEs in the white-light images can possess a similar three-part appearance in the EUV images, i.e., a leading edge, a low-density zone, and a filament or hot channel. The analyses identify that the leading edge and the filament or hot channel in the EUV passbands evolve into the front and the core later within several solar radii in the white-light passbands, respectively. What's more, we find that the CMEs without obvious cavity in the white-light images can also exhibit the clear three-part appearance in the EUV images, which means that the low-density zone in the EUV images (observed as the cavity in white-light images) can be compressed and/or transformed gradually by the expansion of the bright core and/or the reconnection of magnetic field surrounding the core during the CME propagation outward. Our study suggests that more CMEs can possess the clear three-part structure in their early eruption stage. The nature of the low-density zone between the leading edge and the filament or hot channel is discussed.
arXiv:1910.09737v1 [pdf, other]
A New Type of Jets in a Polar Limb of Solar Coronal Hole
Comments: 21 pages, 33 figures, published in ApJ Letters
A new type of chromospheric jets in a polar limb of a coronal hole is discovered in the Ca II filtergram of the Solar Optical Telescope on board the \textit{Hinode}. We identify 30 jets in the Ca II movie of duration of 53 min. The average speed at their maximum heights is found to be 132$\pm$44 km s$^{-1}$ ranging from 57 km s$^{-1}$ to 264 km s$^{-1}$ along the propagation direction. The average lifetime is 20$\pm$6 ranging from 11 seconds to 36 seconds. The speed and lifetime of the jets are located at end-tails of those parameters determined for type II spicules, hence implying a new type of jets. To confirm whether these jets are different from conventional spicules, we construct a time-height image averaged over horizontal region of 1$\arcsec$, and calculate lagged cross-correlations of intensity profiles at each height with the intensity at 2 Mm. From this, we obtain a cross-correlation map as a function of lag and height. We find that the correlation curve as a function of lag time is well fitted into three different Gaussian functions whose standard deviations of the lag time are 193 seconds, 42 seconds, and 17 seconds. The corresponding propagation speeds are calculated to be 9 km s$^{-1}$, 67 km s$^{-1}$, and 121 km s$^{-1}$, respectively. The kinematic properties of the former two components seem to correspond to the 3 minutes oscillations and type II spicules, while the latter component to the jets addressed in this study.
arXiv:1910.09740v1 [pdf, other]
Nonparametric Inference of Neutron Star Composition, Equation of State, and Maximum Mass with GW170817
Comments: 23 pages, 7 figures, 10 tables
The detection of GW170817 in gravitational waves provides unprecedented constraints on the equation of state (EOS) of the ultra-dense matter within the cores of neutron stars (NSs). We extend the nonparametric analysis first introduced in Landry & Essick (2019), and confirm that GW170817 favors soft EOSs. We infer macroscopic observables for a canonical 1.4 $M_{\odot}$ NS, including the tidal deformability $\Lambda_{1.4} = 211^{+312}_{-137}$ ($491^{+216}_{-181}$) and radius $R_{1.4}= 10.86^{+2.04}_{-1.42}$ ($12.51^{+1.00}_{-0.88}$) km, as well as the maximum mass for nonrotating NSs, $M_{max} = 2.064^{+0.260}_{-1.34}$ ($2.017^{0.238}_{-0.087}$) $M_\odot$, with nonparametric priors loosely (tightly) constrained to resemble candidate EOSs from the literature. Furthermore, we find weak evidence that GW170817 involved at least one NS based on gravitational-wave data alone ($B^{NS}_{BBH}= 3.3 \pm 1.4$), consistent with the observation of electromagnetic counterparts. We also investigate GW170817's implications for the maximum spin frequency of millisecond pulsars, and find that the fastest known pulsar is spinning at more than 50% of its breakup frequency at 90% confidence. We additionally find modest evidence in favor of quark matter within NSs, and GW170817 favors the presence of at least one disconnected hybrid star branch in the mass--radius relation over a single stable branch by a factor of 2. Assuming there are multiple stable branches, we find a suggestive posterior preference for a sharp softening around nuclear density followed by stiffening around twice nuclear density, consistent with a strong first-order phase transition. While the statistical evidence in favor of new physics within NS cores remains tenuous with GW170817 alone, these tantalizing hints reemphasize the promise of gravitational waves for constraining the supranuclear EOS.
arXiv:1910.09743v1 [pdf, other]
GRB 180620A: Evidence for late-time energy injection
Comments: 13 pages, 2 figures, 2 tables, submitted to ApJ
The early optical emission of gamma-ray bursts gives an opportunity to understand the central engine and first stages of these events. About 30\% of GRBs present flares whose origin is still a subject of discussion. We present optical photometry of GRB 180620A with the COATLI telescope and RATIR instrument. COATLI started to observe from the end of prompt emission at $T+39.3$~s and RATIR from $T+121.4$~s. We supplement the optical data with the X-ray light curve from \emph{Swift}/XRT. %The optical and X-ray light curves show very unusual behavior with features clearly beyond the standard fireball model. We observe an optical flare from $T+110$ to $T+550$~s, with a temporal index decay $\alpha_\mathrm{O,decay}=1.32\pm 0.01$, and a $\Delta t/t=1.63$, which we interpret as the signature of a reverse shock component. After the initial normal decay the light curves show a long plateau from $T+500$ to $T+7800$~s both in X-rays and the optical before decaying again after an achromatic jet break at $T+7800$~s. Fluctuations are seen during the plateau phase in the optical. Adding to the complexity of GRB afterglows, the plateau phase (typically associated with the coasting phase of the jet) is seen in this object after the ''normal'' decay phase (emitted during the deceleration phase of the jet) and the jet break phase occurs directly after the plateau. We suggest that this sequence of events can be explained by a rapid deceleration of the jet with $t_d\lesssim 40$ s due to the high density of the environment ($\approx 100$ cm$^{-3}$) followed by reactivation of the central engine which causes the flare and powers the plateau phase.
arXiv:1910.09794v1 [pdf, other]
Model-independent determination of cosmic curvature based on Padé approximation
Comments: 8 pages, 2 figures, submitted to ApJ
Given observations of the standard candles and the cosmic chronometers, we apply Pad\'{e} parameterization to the comoving distance and the Hubble paramter to find how stringent the constraint is set to the curvature parameter by the data. A weak informative prior is introduced in the modeling process to keep the inference away from the singularities. Bayesian evidence for different order of Pad\'{e} parameterizations is evaluated during the inference to select the most suitable parameterization in light of the data. The data we used prefer a parameterization form of comoving distance as $D_{01}(z)=\frac{a_0 z}{1+b_1 z}$ as well as a competitive form $D_{02}(z)=\frac{a_0 z}{1+b_1 z + b_2 z^2}$. Similar constraints on the spatial curvature parameter are established by those models and given the Hubble constant as a byproduct: $\Omega_k = 0.25^{+0.14}_{-0.13}$ (68\% confidence level [C.L.]), $H_0 = 67.7 \pm 2.0$ km/s/Mpc (68\% C.L.) for $D_{01}$, and $\Omega_k = -0.01 \pm 0.13$ (68\% C.L.), $H_0 = 68.8 \pm 2.0$ km/s/Mpc (68\% C.L.) for $D_{02}$. The evidence of different models demonstrates the qualitative analysis of the Pad\'{e} parameterizations for the comoving distance.
arXiv:1910.09795v1 [pdf, other]
Experimental characterization of modal noise in multimode fibers for astronomical spectrometers
Comments: 7 pages, 6 figures, accepted by Astronomy and Astrophysics
Starting from our puzzling on-sky experience with the GIANO-TNG spectrometer we set up an infrared high resolution spectrometer in our laboratory and used this instrument to characterize the modal noise generated in fibers of different types (circular and octagonal) and sizes. Our experiment includes two conventional scrambling systems for fibers: a mechanical agitator and an optical double scrambler. We find that the strength of the modal noise primarily depends on how the fiber is illuminated. It dramatically increases when the fiber is under-illuminated, either in the near field or in the far field. The modal noise is similar in circular and octagonal fibers. The Fourier spectrum of the noise decreases exponentially with frequency; i.e., the modal noise is not white but favors broad spectral features. Using the optical double scrambler has no effect on modal noise. The mechanical agitator has effects that vary between different types of fibers and input illuminations. In some cases this agitator has virtually no effect. In other cases, it mitigates the modal noise, but flattens the noise spectrum in Fourier space; i.e., the mechanical agitator preferentially filters the broad spectral features. Our results show that modal noise is frustratingly insensitive to the use of octagonal fibers and optical double scramblers; i.e., the conventional systems used to improve the performances of spectrographs fed via unevenly illuminated fibers. Fiber agitation may help in some cases, but its effect has to be verified on a case-by-case basis. More generally, our results indicate that the design of the fiber link feeding a spectrograph should be coupled with laboratory measurements that reproduce, as closely as possible, the conditions expected at the telescope
arXiv:1910.09809v1 [pdf, other]
Investigating the nature of the extended structure around the Herbig star RCrA using integral field and high-resolution spectroscopy
We present a detailed analysis of the extended structure detected around the young and close-by Herbig Ae/Be star RCrA. This is a young triple system with an intermediate mass central binary whose separation is of the order of a few tens of the radii of the individual components, and an M-star companion at about 30 au. Our aim is to understand the nature of the extended structure by means of combining integral-field and high-resolution spectroscopy. We conducted the analysis based on FEROS archival optical spectroscopy data and adaptive optics images and integral-field spectra obtained with SINFONI and SPHERE at the VLT. The observations reveal a complex extended structure that is composed of at least two components: a non-uniform wide cavity whose walls are detected in continuum emission up to 400~au, and a collimated wiggling-jet detected in the emission lines of Helium and Hydrogen. Moreover, the presence of [FeII] emission projected close to the cavity walls suggests the presence of a slower moving wind, most likely a disk wind. The multiple components of the optical forbidden lines also indicate the presence of a high-velocity jet co-existing with a slow wind. We constructed a geometrical model of the collimated jet flowing within the cavity using intensity and velocity maps, finding that its wiggling is consistent with the orbital period of the central binary. The cavity and the jet do not share the same position angle, suggesting that the jet is itself experiencing a precession motion possibly due to the wide M-dwarf companion. We propose a scenario that closely agrees with the general expectation of a magneto-centrifugal-launched jet. These results build upon the extensive studies already conducted on RCrA.
arXiv:1910.09811v1 [pdf, other]
A data-driven model of nucleosynthesis with chemical tagging in a lower-dimensional latent space
Chemical tagging seeks to identify unique star formation sites from present-day stellar abundances. Previous techniques have treated each abundance dimension as being statistically independent, despite theoretical expectations that many elements can be produced by more than one nucleosynthetic process. In this work we introduce a data-driven model of nucleosynthesis where a set of latent factors (e.g., nucleosynthetic yields) contribute to all stars with different scores, and clustering (e.g., chemical tagging) is modelled by a mixture of multivariate Gaussians in a lower-dimensional latent space. We use an exact method to simultaneously estimate the factor scores for each star, the partial assignment of each star to each cluster, and the latent factors common to all stars, even in the presence of missing data entries. We use an information-theoretic Bayesian principle to estimate the number of latent factors and clusters. Using the second Galah data release we find that six latent factors are preferred to explain N = 2,566 stars with 17 chemical abundances. We identify the rapid- and slow-neutron capture processes, as well as latent factors consistent with Fe-peak and \alpha-element production, and another where K and Zn dominate. When we consider N ~ 160,000 stars with missing abundances we find another 7 factors, as well as 16 components in latent space. Despite these components showing separation in chemistry that is explained through different yield contributions, none show significant structure in their positions or motions. We argue that more data, and joint priors on cluster membership that are constrained by dynamical models, are necessary to realise chemical tagging at a galactic-scale. We release software that allows for model parameters to be optimised in seconds given a fixed number of latent factors, components, and $10^7$ abundance measurements.
arXiv:1910.09815v1 [pdf, other]
Mapping electron temperature variations across a spiral arm in NGC 1672
Comments: 9 pages, 5 figure, accepted to ApJ Letter
We report one of the first extragalactic observations of electron temperature variations across a spiral arm. Using MUSE mosaic observations of the nearby galaxy NGC 1672, we measure the [N II]5755 auroral line in a sample of 80 HII regions in the eastern spiral arm of NGC1672. We discover systematic temperature variations as a function of distance perpendicular to the spiral arm. The electron temperature is lowest on the spiral arm itself and highest on the downstream side. Photoionization models of different metallicity, pressure, and age of the ionizing source are explored to understand what properties of the interstellar medium drive the observed temperature variations. An azimuthally varying metallicity appears to be the most likely cause of the temperature variations. The electron temperature measurements solidify recent discoveries of azimuthal variations of oxygen abundance based on strong lines, and rule out the possibility that the abundance variations are artefacts of the strong-line calibrations.
arXiv:1910.09849v1 [pdf, other]
Galaxies through cosmic time illuminated by gamma-ray bursts and quasars
Comments: PhD thesis, October 2019, University of Iceland, 83 pages. Abridged version from original 276 pp
In the early Universe, most of the cold neutral gas that will later form into individual stars and galaxies is practically invisible to us. These neutral gas reservoirs can, however, be illuminated by bright cosmic lightsources such as gamma-ray bursts (GRBs) and quasars. The aim of this thesis is to use these luminous objects as tools to study the environments of intervening or host galaxy absorption systems through cosmic time. Part I is dedicated to examining the gas, dust and metals in the immediate region surrounding GRBs. Part II presents a search for and the study of cold and molecular gas in high-z GRB host galaxy absorption systems. Part III focuses on using quasars to examine gas-rich intervening galaxies in the line of sight, with specific focus on absorption systems rich in dust and metals. This thesis demonstrates the importance of observing large samples of GRB afterglows to 1) allow for statistical studies of the GRB phenomena itself and the associated host galaxy environments and 2) to obtain spectra of peculiar or unusual GRB afterglows, that is only observed rarely. In addition, it highlights that defining a complete and unbiased sample of quasars is vital to fully exploit the potential of quasars as probes of cosmic chemical evolution. In this version of the thesis only the title, author list and abstract for each published paper, presented as individual chapters, are provided. Principal supervisor: Prof. P\'all Jakobsson.
arXiv:1910.09853v1 [pdf, other]
Non-minimal dark sector physics and cosmological tensions
We explore whether non-standard dark sector physics might be required to solve the existing cosmological tensions. The properties we consider in combination are: \textit{(a)} an interaction between the dark matter and dark energy components, and \textit{(b)} a dark energy equation of state $w$ different from that of the canonical cosmological constant $w=-1$. In principle, these two parameters are independent. In practice, to avoid early-time, superhorizon instabilities, their allowed parameter spaces are correlated. Moreover, a clear degeneracy exists between these two parameters in the case of CMB data. We analyze three classes of extended interacting dark energy models in light of the 2019 \textit{Planck} CMB results and Cepheid-calibrated local distance ladder $H_0$ measurements of Riess et al. (R19), as well as recent BAO and SNeIa distance data. We find that in \textit{quintessence} coupled dark energy models, where $w > -1$, the evidence for a non-zero coupling between the two dark sectors can surpass the $5\sigma$ significance. Moreover, for both Planck+BAO or Planck+SNeIa, we found a preference for $w>-1$ at about three standard deviations. Quintessence models are, therefore, in excellent agreement with current data when an interaction is considered. On the other hand, in \textit{phantom} coupled dark energy models, there is no such preference for a non-zero dark sector coupling. All the models we consider significantly raise the value of the Hubble constant easing the $H_0$ tension. In the interacting scenario, the disagreement between Planck+BAO and R19 is considerably reduced from $4.3\sigma$ in the case of $\Lambda$CDM to about $2.5\sigma$. The addition of low-redshift BAO and SNeIa measurements leaves, therefore, some residual tension with R19 but at a level that could be justified by a statistical fluctuation. (abridged)
arXiv:1910.09860v1 [pdf, other]
Horizontal shear instabilities in rotating stellar radiation zones: I. Inflectional and inertial instabilities and the effects of thermal diffusion
The so-called rotational mixing, which transports angular momentum and chemical elements in stellar radiative zones, is one of the key processes for modern stellar evolution. In the two last decades, the stress has been put on the turbulent transport induced by the vertical shear instability. However, the instabilities of horizontal shears and the strength of the anisotropic turbulent transport they may trigger are still largely unknown. In this paper, we investigate the combined effects of stable stratification, rotation, and thermal diffusion on the instabilities of horizontal shears in the context of stellar radiative zones. The eigenvalue problem describing the instabilities of a flow with a hyperbolic-tangent horizontal shear profile is solved numerically and asymptotically by means of the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) analysis to provide explicit asymptotic dispersion relations in non-diffusive and highly diffusive limits. Two types of instabilities are identified: the inflectional and the inertial instabilities. The inflectional instability is most unstable at finite streamwise wavenumber and zero vertical wavenumber, independently of the stratification, rotation, and thermal diffusion. It is favored by stable stratification but stabilized by thermal diffusion. The inertial instability is driven by rotation and the WKBJ analysis reveals that the growth rate reaches its maximum in the inviscid limit: $\sqrt{f(1-f)}$ (where $f$ is the dimensionless Coriolis parameter). The inertial instability for finite vertical wavenumber is stabilized as the stratification increases for non-diffusive fluids, while it becomes independent of the stratification and stronger for fluids with high thermal diffusivity. Furthermore, we found a self-similarity of the instabilities based on the rescaled parameter $PeN^{2}$ with the P\'eclet number $Pe$ and the Brunt-V\"ais\"al\"a frequency $N$.
arXiv:1910.09864v1 [pdf, other]
First imaging spectroscopy observations of solar drift pair bursts
Comments: Accepted for publication in Astronomy & Astrophysics Letters
Drift pairs are an unusual type of fine structure sometimes observed in dynamic spectra of solar radio emission. They appear as two identical short narrowband drifting stripes separated in time; both positive and negative frequency drifts are observed. Using the Low Frequency Array (LOFAR), we report unique observations of a cluster of drift pair bursts in the frequency range of 30-70 MHz made on 12 July 2017. Spectral imaging capabilities of the instrument have allowed us for the first time to resolve the temporal and frequency evolution of the source locations and sizes at a fixed frequency and along the drifting pair components. Sources of two components of a drift pair have been imaged and found to propagate in the same direction along nearly the same trajectories. Motion of the second component source is delayed in time with respect to that of the first one. The source trajectories can be complicated and non-radial; positive and negative frequency drifts correspond to opposite propagation directions. The drift pair bursts with positive and negative frequency drifts, as well as the associated broadband type-III-like bursts, are produced in the same regions. The visible source velocities are variable from zero to a few $10^4$ (up to ${\sim 10^5}$) km/s, which often exceeds the velocities inferred from the drift rate ($\sim 10^4$ km/s). The visible source sizes are of about $10'-18'$; they are more compact than typical type III sources at the same frequencies. The existing models of drift pair bursts cannot adequately explain the observed features. We discuss the key issues that need to be addressed, and in particular the anisotropic scattering of the radio waves. The broadband bursts observed simultaneously with the drift pairs differ in some aspects from common type III bursts and may represent a separate type of emission.
arXiv:1910.09871v1 [pdf, other]
Stellar Proton Event-induced surface radiation dose as a constraint on the habitability of terrestrial exoplanets
The discovery of terrestrial exoplanets orbiting in habitable zones around nearby stars has been one of the significant developments in modern astronomy. More than a dozen such planets, like Proxima Centauri b and TRAPPIST-1 e, are in close-in configurations and their proximity to the host star makes them highly sensitive to stellar activity. Episodic events such as flares have the potential to cause severe damage to close-in planets, adversely impacting their habitability. Flares on fast rotating young M stars occur up to 100 times more frequently than on G-type stars which makes their planets even more susceptible to stellar activity. Stellar Energetic Particles (SEPs) emanating from Stellar Proton Events (SPEs) cause atmospheric damage (erosion and photochemical changes), and produce secondary particles, which in turn results in enhanced radiation dosage on planetary surfaces. We explore the role of SPEs and planetary factors in determining planetary surface radiation doses. These factors include SPE fluence and spectra, and planetary column density and magnetic field strength. Taking particle spectra from 70 major solar events (observed between 1956 and 2012) as proxy, we use the GEANT4 Monte Carlo model to simulate SPE interactions with exoplanetary atmospheres, and we compute surface radiation dose. We demonstrate that in addition to fluence, SPE spectrum is also a crucial factor in determining the surface radiation dose. We discuss the implications of these findings in constraining the habitability of terrestrial exoplanets.
arXiv:1910.09877v1 [pdf, other]
The Kepler-11 system: evolution of the stellar high-energy emission and {initial planetary} atmospheric mass fractions
The atmospheres of close-in planets are strongly influenced by mass loss driven by the high-energy (X-ray and extreme ultraviolet, EUV) irradiation of the host star, particularly during the early stages of evolution. We recently developed a framework to exploit this connection and enable us to recover the past evolution of the stellar high-energy emission from the present-day properties of its planets, if the latter retains some remnants of their primordial hydrogen-dominated atmospheres. Furthermore, the framework can also provide constraints on planetary initial atmospheric mass fractions. The constraints on the output parameters improve when more planets can be simultaneously analysed. This makes the Kepler-11 system, which hosts six planets with bulk densities between 0.66 and 2.45g cm^{-3}, an ideal target. Our results indicate that the star has likely evolved as a slow rotator (slower than 85\% of the stars with similar masses), corresponding to a high-energy emission at 150 Myr of between 1-10 times that of the current Sun. We also constrain the initial atmospheric mass fractions for the planets, obtaining a lower limit of 4.1% for planet c, a range of 3.7-5.3% for planet d, a range of 11.1-14% for planet e, a range of 1-15.6% for planet f, and a range of 4.7-8.7% for planet g assuming a disc dispersal time of 1 Myr. For planet b, the range remains poorly constrained. Our framework also suggests slightly higher masses for planets b, c, and f than have been suggested based on transit timing variation measurements. We coupled our results with published planet atmosphere accretion models to obtain a temperature (at 0.25 AU, the location of planet f) and dispersal time of the protoplanetary disc of 550 K and 1 Myr, although these results may be affected by inconsistencies in the adopted system parameters.
arXiv:1910.09881v1 [pdf, other]
Measuring the Hubble constant from the cooling of the CMB monopole
The cosmic microwave background (CMB) monopole temperature evolves with the inverse of the cosmological scale factor, independent of many cosmological assumptions. With sufficient sensitivity, real-time cosmological observations could thus be used to measure the local expansion rate of the Universe using the cooling of the CMB. We forecast how well a CMB spectrometer could determine the Hubble constant via this method. The primary challenge of such a mission lies in the separation of Galactic and extra-Galactic foreground signals from the CMB at extremely high precision. However, overcoming these obstacles could potentially provide an independent, highly robust method to shed light on the current low-/high-$z$ Hubble tension. We find that a 3\% measurement of the Hubble constant requires an effective sensitivity to the CMB monopole temperature of approximately $60~\mathrm{pK \sqrt{yr}}$ throughout a 10-year mission. This sensitivity would also enable high-precision measurements of the expected $\Lambda$CDM spectral distortions, but remains futuristic at this stage.
arXiv:1910.09893v1 [pdf, other]
A MUSE study of the inner bulge globular cluster Terzan 9: a fossil record in the Galaxy
Context. Moderately metal-poor inner bulge globular clusters are relics of a generation of long-lived stars that formed in the early Galaxy. Terzan 9, projected at 4d 12 from the Galactic center, is among the most central globular clusters in the Milky Way, showing an orbit which remains confined to the inner 1 kpc. Aims. Our aim is the derivation of the cluster's metallicity, together with an accurate measurement of the mean radial velocity. In the literature, metallicities in the range between have been estimated for Terzan 9 based on color-magnitude diagrams and CaII triplet (CaT) lines. Aims. Our aim is the derivation of the cluster's metallicity, together with an accurate measurement of the mean radial velocity. In the literature, metallicities in the range between -2.0 and -1.0 have been estimated for Terzan 9 based on color-magnitude diagrams and CaII triplet (CaT) lines. Methods. Given its compactness, Terzan 9 was observed using the Multi Unit Spectroscopic Explorer (MUSE) at the Very Large Telescope. The extraction of spectra from several hundreds of individual stars allowed us to derive their radial velocities, metallicities, and [Mg/Fe]. The spectra obtained with MUSE were analysed through full spectrum fitting using the ETOILE code. Results. We obtained a mean metallicity of [Fe/H] -1.10 0.15, a heliocentric radial velocity of vhr = 58.1 1.1 km/s , and a magnesium-to-iron [Mg/Fe] = 0.27 0.03. The metallicity-derived character of Terzan 9 sets it among the family of the moderately metal-poor Blue Horizontal Branch clusters HP 1, NGC 6558, and NGC 6522.
arXiv:1910.09906v1 [pdf, other]
Two-dimensional spectral simulations of neutron star spreading layers
Comments: submitted to A&A; 20 pages, 17 figures; abstract shortened with respect to the submitted version
When the accretion disc around a weakly magnerized neutron star (NS) meets the surface of the star, it should brake down to match the rotation of the NS, forming a boundary layer. As the mechanisms potentially responsible for this braking are apparently inefficient, it is reasonable to consider this layer as a spreading layer (SL) with negligible radial extent and structure. We perform spectral 2D simulations of an SL, considering the disc as a source of matter and angular momentum. Interaction of new, rapidly rotating matter with the pre-existing, relatively slow material co-rotating with the star leads to shear and thermal instabilities capable of transferring angular momentum and creating variability on dynamical time scales. We compute artificial light curves of an SL viewed at different inclination angles. Most of the simulated light curves show oscillations at frequencies close to 1kHz. Oscillations observed in our simulations are most likely inertial modes excited by shear instabilities near the boundary of the SL. Their frequencies, dependence on flux, and amplitude variations closely resemble those of the high-frequency pair quasi-periodic oscillations (QPOs) observed in many low-mass X-ray binaries.
arXiv:1910.09924v1 [pdf, other]
The First Billion Years Project: Finding Infant Globular Clusters at z=6
Comments: 15 pages, 8 figures, submitted to MNRAS
We explore a suite of high-resolution cosmological simulations from the First Billion Years (FiBY) project at $z \geq 6$ to identify low-mass stellar systems, with a particular focus on globular clusters (GCs). Within the demographics of substructures found in the simulations, two distinct groups of objects emerge. We associate the first group, which appear to have a high baryon fraction ($f_{\rm{b}} \geq 0.95$), with infant GC candidates. The second group exhibit a high stellar fraction ($f_* \geq 0.95$) and show a resemblance to ultra-faint dwarf galaxies. The infant GC candidates are characterised by a stellar content similar to the one observed in present-day GCs, but they still contain a high gas fraction ($f_{\rm{gas}} \sim 0.95$) and a relatively low amount of dark matter. They are very compact systems, with high stellar mass densities and sizes which are consistent with recent estimates based on the first observations of possible proto-GCs at high redshifts. Such infant GCs appear to be more massive and more abundant in massive host galaxies, indicating that the assembly of galaxies via mergers may play an important role in shaping up several GC-host scaling relations. Specifically, we express the relation between the mass of the most massive infant GC and its host stellar mass as $\log (M_{\rm cl}) = (0.31\pm0.15)\log (M_{\rm *,gal}) + (4.17\pm1.06)$. Finally, we assess that the present-day GC mass -- halo mass relation offers a satisfactory description of the behaviour of our infant GC candidates at high redshift, suggesting that such a relation may be set at formation.
arXiv:1910.09925v1 [pdf, other]
Neutron stars: new constraints on asymmetric dark matter
We study an impact of asymmetric dark matter on properties of the neutron stars and their ability to reach the two solar masses limit, which allows us to present a new upper constraint on the mass of dark matter particle. Our analysis is based on the observational fact of existence of three pulsars reaching this limit and on the theoretically predicted reduction of the neutron star maximal mass caused by accumulation of dark matter in its interior. Using modern data on spatial distribution of baryon and dark matter in the Milky Way we argue that particles of dark matter can not be heavier than 5 GeV. We also demonstrate that light dark matter particles with masses below 0.2 GeV can create an extended halo around the neutron star leading not to decrease, but to increase of its visible gravitational mass. Furthermore, we predict that high precision measurements of the neutron stars maximal mass near the Galactic center, will put a stringent constraint on the mass of the dark matter particle. This last result is particularly important to prepare ongoing, and future radio and X-ray surveys.
arXiv:1910.09940v1 [pdf, other]
Low-degree mixed modes in red giant stars with moderate core magnetic fields
Comments: 18 pages, 4 figures, accepted for publication in MNRAS
Observations of pressure-gravity mixed modes, combined with a theoretical framework for understanding mode formation, can yield a wealth of information about deep stellar interiors. In this paper, we seek to develop a formalism for treating the effects of deeply buried core magnetic fields on mixed modes in evolved stars, where the fields are moderate, i.e. not strong enough to disrupt wave propagation, but where they may be too strong for non-degenerate first-order perturbation theory to be applied. The magnetic field is incorporated in a way that avoids having to use this. Inclusion of the Lorentz force term is shown to yield a system of differential equations that allows for the magnetically-affected eigenfunctions to be computed from scratch, rather than following the approach of first-order perturbation theory. For sufficiently weak fields, coupling between different spherical harmonics can be neglected, allowing for reduction to a second-order system of ordinary differential equations akin to the usual oscillation equations that can be solved analogously. We derive expressions for (i) the mixed-mode quantisation condition in the presence of a field and (ii) the frequency shift associated with the magnetic field. In addition, for modes of low degree we uncover an extra offset term in the quantisation condition that is sensitive to properties of the evanescent zone. These expressions may be inverted to extract information about the stellar structure and magnetic field from observational data.
arXiv:1910.09975v1 [pdf, other]
Connecting early and late epochs by f(z)CDM cosmography
The cosmographic approach is gaining considerable interest as a model-independent technique able to describe the late expansion of the universe. Indeed, given only the observational assumption of the cosmological principle, it allows to study the today observed accelerated evolution of the Hubble flow without assuming specific cosmological models. In general, cosmography is used to reconstruct the Hubble parameter as a function of the redshift, assuming an arbitrary fiducial value for the current matter density, $\Omega_m$, and analysing low redshift cosmological data. Here we propose a different strategy, linking together the parametric cosmographic behavior of the late universe expansion with the small scale universe. In this way, we do not need to assume any "a priori" values for the cosmological parameters, since these are constrained at early epochs using both the Cosmic Microwave Background Radiation (CMBR) and Baryonic Acoustic Oscillation (BAO) data. In order to test this strategy, we describe the late expansion of the universe using the Pad\'e polynomials. This approach is discussed in the light of the recent $H(z)$ values indicators, combined with Supernovae Pantheon sample, galaxy clustering and early universe data, as CMBR and BAO. We found an interesting dependence of the current matter density value with cosmographic parameters, proving the inaccuracy of setting the value of $\Omega_m$ in cosmographic analyses, and a non-negligible effect of the cosmographic parameters on the CMBR temperature anisotropy power spectrum. Finally, we found that the cosmographic series, truncated at third order, shows a better $\chi^2$ best fit value then the vanilla $\Lambda$CDM model. This can be interpreted as the requirement that higher order corrections have to be considered to correctly describe low redshift data and remove the degeneration of the models.
arXiv:1910.09987v1 [pdf, other]
Constraining the cosmic ray spectrum in the vicinity of the supernova remnant W28: from sub-GeV to multi-TeV energies
Comments: 8 pages, 6 figures, submitted
Supernova remnants interacting with molecular clouds are ideal laboratories to study the acceleration of particles at shock waves and their transport and interactions in the surrounding interstellar medium. In this paper, we focus on the supernova remnant W28, which over the years has been observed in all energy domains from radio waves to very-high-energy gamma rays. The bright gamma-ray emission detected from molecular clouds located in its vicinity revealed the presence of accelerated GeV and TeV particles in the region. An enhanced ionization rate has also been measured by means of millimetre observations, but such observations alone cannot tell us whether the enhancement is due to low energy (MeV) cosmic rays (either protons or electrons) or the X-ray photons emitted by the shocked gas. The goal of this study is to determine the origin of the enhanced ionization rate and to infer from multiwavelength observations the spectrum of cosmic rays accelerated at the supernova remnant shock in the unprecedented range spanning from MeV to multi-TeV particle energies. We developed a model to describe the transport of X-ray photons into the molecular cloud, and we fitted the radio, millimeter, and gamma-ray data to derive the spectrum of the radiating particles. The contribution from X-ray photons to the enhanced ionization rate is negligible, and therefore the ionization must be due to cosmic rays. Even though we cannot exclude a contribution to the ionization rate coming from cosmic ray electrons, we show that a scenario where cosmic ray protons explain both the gamma-ray flux and the enhanced ionization rate provides the most natural fit to multiwavelength data. This strongly suggests that the intensity of CR protons is enhanced in the region for particle energies in a very broad range covering almost 6 orders of magnitude: from $\lesssim 100$ MeV up to several tens of TeV.
arXiv:1910.09988v1 [pdf, other]
New constraint on the atmosphere of (50000) Quaoar from a stellar occultation
Comments: 12 pages, 3 figures, accepted for publication in the Astronomical Journal
We report observations of a stellar occultation by the classical Kuiper belt object (50000) Quaoar occurred on 28 June 2019. A single-chord high-cadence (2 Hz) photometry dataset was obtained with the Tomo-e Gozen CMOS camera mounted on the 1.05 m Schmidt telescope at Kiso Observatory. The obtained ingress and egress data do not show any indication of atmospheric refraction and allow to set new $1\sigma$ and $3\sigma$ upper limits of 6 and 16 nbar, respectively, for the surface pressure of a pure methane atmosphere. These upper limits are lower than the saturation vapor pressure of methane at Quaoar's expected mean surface temperature ($T \sim 44$ K) and imply the absence of a $\sim$10 nbar-level global atmosphere formed by methane ice on Quaoar's surface.
arXiv:1910.09994v1 [pdf, other]
Amateur telescopes discover a kilometre-sized Kuiper belt object from stellar occultation
Comments: 23 pages, 10 figures, author final submission version, published in Nature Astronomy on 28 January 2019 (free version: https://rdcu.be/biPGW)
Kuiper belt objects (KBOs) are thought to be the remnant of the early solar system, and their size distribution provides an opportunity to explore the formation and evolution of the outer solar system. In particular, the size distribution of kilometre-sized (radius = 1-10 km) KBO represents a signature of initial planetesimal sizes when planets form. These kilometre-sized KBOs are extremely faint, and it is impossible to detect them directly. Instead, monitoring of stellar occultation events is one possible way to discover these small KBOs. Hitherto, however, there has been no observational evidence for the occultation events by KBOs with radii of 1-10 km. Here we report the first detection of a single occultation event candidate by a KBO with a radius of $\sim$1.3 km, which is simultaneously provided by two low-cost small telescopes coupled with commercial CMOS cameras. From this detection, we conclude that a surface number density of KBOs with radii exceeding $\sim 1.2$ km is $\sim 6 \times 10^5 \ {\rm deg^{-2}}$. This surface number density favours a theoretical size distribution model with an excess signature at a radius of 1-2 km. If this is a true detection, this implies that planetesimals before their runaway growth phase grow into kilometre-sized objects in the primordial outer solar system and remain as a major population of the present-day Kuiper belt.
arXiv:1910.10010v1 [pdf, other]
Random fragmentation of turbulent molecular clouds lying in the central region of giant galaxies
A stochastic model of fragmentation of molecular clouds has been developed for studying the resulting Initial Mass Function (IMF) where the number of fragments, inter-occurrence time of fragmentation, masses and velocities of the fragments are random variables. Here two turbulent patterns of the velocities of the fragments have been considered, namely, Gaussian and Gamma distributions. It is found that for Gaussian distribution of the turbulent velocity, the IMFs are shallower in general compared to Salpeter mass function. On the contrary, a skewed distribution for turbulent velocity leads to an IMF which is much closer to Salpeter mass function. The above result might be due to the fact that strong driving mechanisms e.g. shocks, arising out of a big explosion occurring at the centre of the galaxy or due to big number of supernova explosions occurring simultaneously in massive parent clouds during the evolution of star clusters embedded into them are responsible for stripping out most of the gas from the clouds. This inhibits formation of massive stars in large numbers making the mass function a steeper one.
arXiv:1910.10023v1 [pdf, other]
Metal-enriched Galaxies in the First ~1 Billion Years: Evidence of a Smooth Metallicity Evolution at z ~ 5
Comments: 19 pages, 21 figures, accepted for publication in MNRAS
We present seven new abundance measurements of the elements O, C and Si at z > 4.5, doubling the existing sample of weakly depleted elements in gas-rich galaxies, in order to constrain the first ~1 billion years of cosmic metal evolution. These measurements are based on quasar spectra of damped Lyman-alpha absorbers (DLAs) and sub-DLAs obtained with the Magellan Inamori Kyocera Echelle (MIKE) and Magellan Echellette (MagE) spectrographs on Magellan-South, and the X-Shooter spectrograph on the Very Large Telescope. We combine these new measurements with those drawn from the literature to estimate the NHI-weighted binned mean metallicity of -1.51 +\- 0.18 at z = 4.8. This metallicity value is in excellent agreement with the prediction from lower redshift DLAs, supporting the interpretation that the metallicity evolution is smooth at z ~ 5, rather than showing a sudden decline at z > 4.7. Furthermore, the metallicity evolution trends for the DLAs and sub-DLAs are similar within our uncertainties. We also find that the [C/O] ratios for z ~ 5 DLAs are consistent with those of the very metal-poor DLAs. Additionally, using [C/O] and [Si/O] to constrain the nucleosynthesis models, we estimate that the probability distributions of the progenitor star masses for three relatively metal-poor DLAs are centered around 12 M_{\odot} to 17 M_{\odot}. Finally, the z ~ 5 absorbers show a different metallicity-velocity dispersion relation than lower redshift DLAs, suggesting that they may be tracing a different population of galaxies.
arXiv:1910.10030v1 [pdf, other]
Diffusive acceleration in relativistic shocks: particle feedback
The spectral index $s$ of particles diffusively accelerated in a relativistic shock depends on the unknown angular diffusion function $\mathcal{D}$, which itself depends on the particle distribution function $f$ if acceleration is efficient. We develop a relaxation code to compute $s$ and $f$ for an arbitrary functional $\mathcal{D}$ that depends on $f$. A local $\mathcal{D}(f)$ dependence is motivated and shown, when rising (falling) upstream, to soften (harden) $s$ with respect to the isotropic case, shift the angular distribution towards upstream (downstream) directions, and strengthen (weaken) the particle confinement to the shock; an opposite effect on $s$ is found downstream. However, variations in $s$ remain modest even when $\mathcal{D}$ is a strong function of $f$, so the standard, isotropic-diffusion results remain approximately applicable unless $\mathcal{D}$ is both highly anisotropic and not a local function of $f$. A mild, $\sim 0.1$ softening of $s$, in both 2D and 3D, when $\mathcal{D}(f)$ rises sufficiently fast, may be indicated by ab-initio simulations.
arXiv:1910.10038v1 [pdf, other]
Identification of a Group III CEMP-no Star in the Dwarf Spheroidal Galaxy Canes Venatici I
Comments: 18 pages, 3 tables, 9 figures, submitted to the Astrophysical Journal
CEMP-no stars, a subclass of carbon-enhanced metal-poor (CEMP) stars, are one of the most significant stellar populations in Galactic Archaeology, because they dominate the low end of the metallicity distribution function, providing information on the early star-formation and chemical-evolution history of the Milky Way and its satellite galaxies. Here we present an analysis of medium-resolution ($R \sim 1,800$) optical spectroscopy for a CEMP giant, SDSS J132755.56+333521.7, observed with the Large Binocular Telescope (LBT), one of the brightest ($g \sim 20.5$) members of the classical dwarf spheroidal galaxy, Canes Venatici I (CVn I). Many CEMP stars discovered to date have very cool effective temperatures ($T_{\mathrm{eff}}< 4500$ K), resulting in strong veiling by molecular carbon bands over their optical spectra at low/medium spectral resolution. We introduce a technique to mitigate the carbon-veiling problem to obtain reliable stellar parameters, and validate this method with the LBT medium-resolution optical spectra of the ultra metal-poor ([Fe/H] = $-4.0$) CEMP-no dwarf, G 77-61, and seven additional very cool CEMP stars, which have published high-resolution spectroscopic parameters. We apply this technique to the LBT spectrum of SDSS J132755.56+333521.7. We find that this star is well-described with parameters $T_{\mathrm{eff}}=4530$ K, log $g=$ 0.7, [Fe/H] $= -3.38$, and absolute carbon abundance $A$(C) = 7.23, indicating that it is likely the first Group III CEMP-no star identified in CVn I. The Group III identification of this star suggests that it is a member of the extremely metal-poor population in CVn I, which may have been accreted into its halo.
arXiv:1910.10047v1 [pdf, other]
Complex organic molecules in comets from remote-sensing observations at millimeter wavelengths
Remote observations of comets, especially using high spectral resolution millimeter spectroscopy, have enabled the detection of over 25 molecules in comets for the last twenty years. Among the molecules identified at radio wavelengths, complex organic molecules (COMs) such as acetaldehyde, ethylene-glycol, formamide, methyl-formate or ethanol have been observed in several comets and their abundances relative to water and methanol precisely determined. Significant upper limits on the abundance of several other COMs have been determined and put constraints on the dominant isomer for three of them. The abundances measured in comets are generally of comparable order of magnitude as those measured in star-forming regions, suggesting that comets contain preserved material from the presolar cloud from which the solar system was born.
arXiv:1910.10058v1 [pdf, other]
Effect of the isotropic collisions with neutral hydrogen on the polarization of the CN solar molecule
Comments: Accepted for publication in MNRAS (Accepted October 22, 2019). 23 pages, 3 tables, 13 figures
Our work is concerned with the case of the solar molecule CN which presents conspicuous profiles of scattering polarization. We start by calculating accurate PES for the singlet and triplet electronic ground states in order to characterize the collisions between the CN molecule in its $X \; ^2\Sigma$ state and the hydrogen in its ground state $^2S$. The PES are included in the Schr\"oodinger equation to obtain the scattering matrix and the probabilities of collisions. Depolarizing collisional rate coefficients are computed in the framework of the infinite order sudden approximation for temperatures ranging from $T= 2000$ K to $T= 15000$ K. Interpretation of the results and comparison between singlet and triplet collisional rate coefficients are detailed. We show that, for typical photospheric hydrogen density ($n_{H} = 10^{15}-10^{16}$ cm$^{-3}$), the $X \; ^2\Sigma$ state of CN is partially or completely depolarized by isotropic collisions.
arXiv:1910.10092v1 [pdf, other]
ESA Voyage 2050 white paper: A Polarized View of the Hot and Violent Universe
Comments: White Paper submitted in response to the ESA Voyage 2050 call, 20 pages + title page + references + list of team members, 10 figures
Since the birth of X-ray Astronomy, spectacular advances have been seen in the imaging, spectroscopic and timing studies of the hot and violent X-ray Universe, and further leaps forward are expected in the future. On the other hand, polarimetry is very much lagging behind: after the measurements of the Crab Nebula and Scorpius X-1, obtained by OSO-8 in the 70s, no more observations have been performed in the classical X-ray band, even if some interesting results have been obtained in hard X-rays and in soft gamma-rays. The NASA/ASI mission IXPE, scheduled for the launch in 2021, is going to provide for the first time imaging X-ray polarimetry in the 2-8 keV band thanks to its photoelectric polarimeter, coupled with ~25'' angular resolution X-ray mirrors. Its orders of magnitude improvement in sensitivity with respect to the OSO-8 Bragg polarimeter implies scientifically meaningful polarimetric measurements for at least the brightest specimens of most classes of X-ray sources. In 2027, the Chinese-led mission eXTP should also be launched. In addition to timing and spectroscopic instruments, eXTP will have on board photoelectric polarimeters very similar to those of IXPE, but with a total effective area 2-3 times larger. Building on IXPE results, eXTP will increase the number of sources for which significant polarimetric measurements could be obtained. However, further progresses, such as exploring a broader energy range, considering a larger effective area, improving the angular resolution, and performing wide-field polarization measurements, are needed to reach a mature phase for X-ray polarimetry. In the first part of this White Paper we will discuss a few scientific cases in which a next generation X-ray Polarimetry mission can provide significant advances. In the second part, a possible concept for a medium-class Next Generation X-ray Polarimetry (NGXP) mission will be sketched.
arXiv:1910.10134v1 [pdf, other]
Stellar Density Profiles of Dwarf Spheroidal Galaxies
Comments: 22 pages, 10 figures. Submitted to ApJ
We apply a flexible parametric model, a combination of generalized Plummer profiles, to infer the shapes of the stellar density profiles of the Milky Way's satellite dwarf spheroidal galaxies (dSphs). We apply this model to 38 dSphs using star counts from the Sloan Digital Sky Survey, PanStarrs-1 Survey, and Dark Energy Survey. Using mock data, we examine systematic errors associated with modelling assumptions and identify conditions under which our model can identify "non-standard" stellar density profiles that have central cusps and/or steepened outer slopes. Applying our model to real dwarf spheroidals, we do not find evidence for centrally cusped density profiles among the ~10 Milky Way satellites for which our tests with mock data indicate there would be sufficient detectability. We do detect steepened (with respect to a standard Plummer model) outer profiles in several dSphs - Fornax, Leo I, Leo II, Reticulum II - which may point to distinct evolutionary pathways for these objects. However, the outer slope of the stellar density profile does not yet obviously correlate with other observed galaxy properties. | 2019-10-23 05:36:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6337766647338867, "perplexity": 1969.545621200718}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987829458.93/warc/CC-MAIN-20191023043257-20191023070757-00442.warc.gz"} |
https://www.physicsforums.com/threads/how-can-i-convert-a-rate-of-behavioral-responses-into-a-probability.682153/ | # How can I convert a rate of behavioral responses into a probability?
1. Mar 31, 2013
### DaveJersey
B.F. Skinner's law of effect said when a response is followed by a reinforcing stimulus, the rate of response increases. So when a response produces a reinforcing stimulus, responses per unit of time (rate) increases. He said the response strengthens. He said when the rate increases, the probability of response increases. So how can I convert, for example, a rate of five responses per minute into a probability number between 0 and 1? Is that possible?
I anticipate your replies.
Dave in New Jersey
2. Apr 1, 2013
### Stephen Tashi
You'll get better advice if you describe a specific experimental situation.
"The probability of a response" and "the rate of a response" are not precise descriptions. In particlar, the "rate" of a physical process is often a deterministic measurement ( for example, a flow in gallons per minute). A probability "of a response" only meaningful if the "response" is a precisely defined event. For example, is the "response" something that does or does-not happen or is there a strength or degree of the response?
3. Apr 1, 2013
### DaveJersey
A rate of response can be precisely described. A rat pressing a lever is defined by the electric switch the lever activates when the rat presses the lever. A person switching on a light can be defined the same way. Response rates are therefore in this scenario the number of light switches or the number of lever-presses per unit of time. Dimensions such as the force or magnitude of a response can also be measured objectively, as in the force by which a pigeon presses a button, or the decibel level of a classroom of "noisy" children. In an experiment, when an underweight (hungry) rat presses a lever the response results in a mechanical delivery of a pellet of grain. The rate of lever presses is shown to increase on a cumulative recorder that draws a line corresponding with total responses. The slope of the line indicates the rate of response. Behavior strength is said to increase with the increased rate of response after reinforcement. Skinner said the probability of response increased, which was shown by the increased rate of response. He did not, however, convert the rate of response into a number between 0 and 1 to indicate the probability. My question is, can the rate of response be converted into a probability value between 0 and 1. Let's say the maximum number of discrete responses possible in one minute is sixty and the rate of response is one per second, can rate be converted to probability?
Last edited: Apr 1, 2013
4. Apr 1, 2013
### Stephen Tashi
The question is still too general to have a definite mathematical answer. To have a well posed mathematical problem, you have to add more detail, which usually means making some specific assumptions.
To speak of probability, you need a probability model for the responses. A simple model would be that after one response is given, there is a random draw to determine the time of the next response. So the random variable is "time to next response". A commonly used distribution for "waiting times" is the exponential distribution. This distribution has a parameter whose size determines the shape of the distribution and hence the statistics of the distribution, such as the mean time to the response. The effect of a reward could be modeled as changing the value of the parameter. You wouldn't be representing the effect of a reward as "a probability", you would represent it as a change in a parameter, which affects the probabilities for the all the events in the distribution (e.g. the probability of responding in l1 second or less, the probability of responding in l.5 seconds or less, etc.)
The fact that the above model is simple and often used doesn't prove it applies to your experiment. It's just the first thing that comes to mind.
To model how a reward strengthens a response, you would still need to define quantitatively how the reward changes the parameter, i.e. give some formula such as $\lambda[n+1] = \lambda[n] + c$ where $\lambda[n+1]$ is the value of the parameter after $n+1$ rewards, $\lambda[n]$ is its value after $n$ rewards and $c$ is some constant. If you try various formulas then there are mathematical ways to compare them to experimental data. If you assume formulas with certain unknown constants then there are ways of estimating the constants from experimental data. | 2018-02-23 12:45:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8464202284812927, "perplexity": 659.557031745216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814700.55/warc/CC-MAIN-20180223115053-20180223135053-00204.warc.gz"} |
https://nrich.maths.org/public/topic.php?code=71&cl=4&cldcmpid=257 | # Resources tagged with: Mathematical reasoning & proof
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### There are 172 results
Broad Topics > Thinking Mathematically > Mathematical reasoning & proof
### Exhaustion
##### Age 16 to 18 Challenge Level:
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
### Dalmatians
##### Age 14 to 18 Challenge Level:
Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.
### Unit Interval
##### Age 14 to 18 Challenge Level:
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
### Breaking the Equation ' Empirical Argument = Proof '
##### Age 7 to 18
This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.
### Polite Numbers
##### Age 16 to 18 Challenge Level:
A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?
### Binomial
##### Age 16 to 18 Challenge Level:
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
### Diverging
##### Age 16 to 18 Challenge Level:
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
### Basic Rhythms
##### Age 16 to 18 Challenge Level:
Explore a number pattern which has the same symmetries in different bases.
##### Age 16 to 18 Challenge Level:
Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
### Euclid's Algorithm II
##### Age 16 to 18
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
### Proofs with Pictures
##### Age 14 to 18
Some diagrammatic 'proofs' of algebraic identities and inequalities.
### Rational Roots
##### Age 16 to 18 Challenge Level:
Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.
### Telescoping Functions
##### Age 16 to 18
Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.
### Polynomial Relations
##### Age 16 to 18 Challenge Level:
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
### Proof: A Brief Historical Survey
##### Age 14 to 18
If you think that mathematical proof is really clearcut and universal then you should read this article.
### And So on - and on -and On
##### Age 16 to 18 Challenge Level:
Can you find the value of this function involving algebraic fractions for x=2000?
### Euler's Formula and Topology
##### Age 16 to 18
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
### The Clue Is in the Question
##### Age 16 to 18 Challenge Level:
Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?
### Leonardo's Problem
##### Age 14 to 18 Challenge Level:
A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
### A Computer Program to Find Magic Squares
##### Age 16 to 18
This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.
### Continued Fractions II
##### Age 16 to 18
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
### Our Ages
##### Age 14 to 16 Challenge Level:
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
### Modulus Arithmetic and a Solution to Dirisibly Yours
##### Age 16 to 18
Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.
### Notty Logic
##### Age 16 to 18 Challenge Level:
Have a go at being mathematically negative, by negating these statements.
### More Sums of Squares
##### Age 16 to 18
Tom writes about expressing numbers as the sums of three squares.
### Some Circuits in Graph or Network Theory
##### Age 14 to 18
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
### Three Ways
##### Age 16 to 18 Challenge Level:
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
### Mechanical Integration
##### Age 16 to 18 Challenge Level:
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
### Little and Large
##### Age 16 to 18 Challenge Level:
A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?
### Where Do We Get Our Feet Wet?
##### Age 16 to 18
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
### Yih or Luk Tsut K'i or Three Men's Morris
##### Age 11 to 18 Challenge Level:
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .
### Modular Fractions
##### Age 16 to 18 Challenge Level:
We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.
##### Age 16 to 18 Challenge Level:
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
### Magic Squares II
##### Age 14 to 18
An article which gives an account of some properties of magic squares.
### Magic W Wrap Up
##### Age 16 to 18 Challenge Level:
Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.
### Always Perfect
##### Age 14 to 16 Challenge Level:
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
### Transitivity
##### Age 16 to 18
Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.
### Modulus Arithmetic and a Solution to Differences
##### Age 16 to 18
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
### Pair Squares
##### Age 16 to 18 Challenge Level:
The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
### Target Six
##### Age 16 to 18 Challenge Level:
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
### Prime AP
##### Age 16 to 18 Challenge Level:
What can you say about the common difference of an AP where every term is prime?
### Big, Bigger, Biggest
##### Age 16 to 18 Challenge Level:
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
### Sums of Squares and Sums of Cubes
##### Age 16 to 18
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.
### Sixational
##### Age 14 to 18 Challenge Level:
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .
### Contrary Logic
##### Age 16 to 18 Challenge Level:
Can you invert the logic to prove these statements?
##### Age 14 to 16 Challenge Level:
Kyle and his teacher disagree about his test score - who is right?
### Perfectly Square
##### Age 14 to 16 Challenge Level:
The sums of the squares of three related numbers is also a perfect square - can you explain why?
### Never Prime
##### Age 14 to 16 Challenge Level:
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
### An Introduction to Number Theory
##### Age 16 to 18
An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions) | 2020-07-12 20:38:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48725828528404236, "perplexity": 1367.7973325664975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657139167.74/warc/CC-MAIN-20200712175843-20200712205843-00101.warc.gz"} |
http://mathoverflow.net/questions/17115/restriction-of-a-complex-polynomial-to-the-unit-circle | # Restriction of a complex polynomial to the unit circle
I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).
Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there is the following dichotomy. Either we can write $f(z)=g(z^k)$ for some other polynomial $g$ and some integer $k>1$, or the restriction of $f(z)$ to the unit circle is a loop with only finitely many self-intersections. (Which means, more concretely, that there are only finitely many pairs $(z,w)$ such that $|z|=1=|w|$, $z\neq w$ and $f(z)=f(w)$.)
EDIT. Here are a couple reasons why I believe the statement is correct.
1) The statement is equivalent to the following assertion. Consider the set of all ratios $z/w$, where $|z|=1=|w|$ and $f(z)=f(w)$ (here we allow $z=w$). If $f$ is a nonconstant polynomial, then this set is finite.
[[ Here is a proof that the latter assertion implies the original statement. Suppose that there are infinitely many pairs $(z,w)$ such that $|z|=1=|w|$, $z\neq w$ and $f(z)=f(w)$. Then some number $c\neq 1$ must occur infinitely often as the corresponding ratio $z/w$. However, this would imply that $f(cz)=f(z)$ (as polynomials). It is easy to check that this forces $c$ to be a root of unity, and if $k$ is the order of $c$, then $f(z)=g(z^k)$ for some polynomial $g(z)$. ]]
Going back to the latter assertion, note that the set of all such ratios is a compact subset of the unit circle, and it is not hard to see that 1 must be an isolated point of this set. So it is plausible that the whole set is discrete (which would mean that it is finite).
2) If I am not mistaken, experiments with polynomials that involve a small number of nonzero monomials (such as 2 or 3) also confirm the original conjecture.
-
What makes you so sure of the truth of the statement? – José Figueroa-O'Farrill Mar 4 '10 at 17:51
is there any relation to Mikio Sato hyperfunctions? – kakaz Mar 4 '10 at 18:24
I don't know whether there is a relation with hyperfunctions – senti_today Mar 4 '10 at 20:35
It looks like $x^n + (1-\epsilon)x$ has $(n-1)^2$ self-intersections. – Douglas Zare Mar 5 '10 at 6:57
I previously posted an answer that was wrong with no redeeming qualities. Then I did some Googling. Douglas Zare found an example which reaches the upper bound proved in Quine's paper. – Jonas Meyer Mar 5 '10 at 7:04
You're right. Quine proved in "On the self-intersections of the image of the unit circle under a polynomial mapping" that if the degree is $n$ and $f(z)\neq g(z^k)$ with $k>1$, then the number of points with at least 2 distinct preimage points is at most $(n-1)^2$. An example shows that this is sharp. Here's the review in MR.
After the proof, there is a remark:
As a simple consequence of this theorem we note that a polynomial $p$ cannot map $|z| < 1$ conformally onto a domain with a slit, for in this case $p(e^{i\phi})$ would have an infinite number of vertices.
-
Thank you so much! (My own attempts at Googling this have failed earlier.) – senti_today Mar 5 '10 at 16:00
The image of the unit circle is a real-algebraic curve, so the number of self-intersections should be finite.
Addendum: I'm not sure how to complete the argument, but here's a heuristic (following Speyer's suggestion). The curve $x^2+y^2=1$ is a rational curve (i.e., birational to $\mathbb{CP}^1$, the Riemann sphere), so it's image is also a rational curve (here, I'm thinking of the map $p:\mathbb{C}\to \mathbb{C}$ as a polynomial map $\mathbb{R}^2 \to \mathbb{R}^2$, and its extension to $\mathbb{C}^2\to \mathbb{C}^2$ and $\mathbb{CP}^2\to \mathbb{CP}^2$). Complex conjugation gives an antiholomorphic involution of $x^2+y^2=1$, fixing the circle (on the Riemann sphere, this must be conjugate to complex conjugation). The image (in $\mathbb{CP}^2$) is a singular sphere, and again complex conjugation should be an anti-holomorphic involution fixing the image of the unit circle. So the map should be a composition of a polynomial map with a map of the Riemann sphere sending the circle to the circle. All such maps of the Riemann sphere are products of Mobius transformations of the form $\frac{z-\varphi}{1-\overline{\varphi}z}$ (this is an exercise in Ahlfors, making use of the Schwarz lemma). If the composition is to be a polynomial map, then $\varphi$ must $=0$ in each factor, and the map is of the form $z^k$. My algebraic geometry is quite weak, so I'm not sure if this argument can be made rigorous (and I probably shouldn't have posted an answer in the first place!).
-
The problem is that f(z)=z^2 has the image going over itself, so this abstract topological approach doesn't seem to work. – Kevin Buzzard Mar 5 '10 at 7:27
This does prove that, if the number of intersections is not finite, then the unit circle must be a branched cover of the image curve. I've been thinking about how to turn that into a proof and making some progress, but Jonas apparently found a relevant reference first. – David Speyer Mar 5 '10 at 11:59
David, It would still be interesting to see a new proof if you're interested in completing it. – Jonas Meyer Mar 5 '10 at 18:32 | 2014-11-28 05:51:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9356734156608582, "perplexity": 128.92422101474293}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931009777.87/warc/CC-MAIN-20141125155649-00225-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.hepdata.net/record/ins1277076 | • Browse all
Updated measurements of exclusive $J/\psi$ and $\psi$(2S) production cross-sections in pp collisions at $\sqrt{s}=7$ TeV
The collaboration
J.Phys. G41 (2014) 055002, 2014.
Abstract (data abstract)
CERN-LHC. The differential cross-section as a function of rapidity has been measured for the exclusive production of $J/\psi$ and $\psi(2S)$ mesons in proton-proton collisions at $\sqrt{s}=7$ TeV, using data collected by the LHCb experiment, corresponding to an integrated luminosity of 930 pb$^{-1}$. The cross-sections times branching fractions to two muons having pseudorapidities between 2.0 and 4.5 are measured to be $$\begin{array}{rl} \sigma_{pp\rightarrow J/\psi\rightarrow{\mu^+}{\mu^-}}(2.0<\eta_{\mu^\pm }<4.5)=&291\pm 7\pm19 {\rm \ pb},\\ \sigma_{pp\rightarrow\psi(2S)\rightarrow{\mu^+}{\mu^-}}(2.0<\eta_{\mu^\pm}<4.5)=&6.5\pm 0.9\pm 0.4 {\rm \ pb},\end{array}$$ where the first uncertainty is statistical and the second is systematic. The measurements agree with next-to-leading order QCD predictions as well as with models that include saturation effects.
• #### Table 1
Data from T 4
10.17182/hepdata.66883.v1/t1
Cross section times branching ratio to two muons with pseudorapidities between 2.0 and 4.5.
• #### Table 2
Data from T 2
10.17182/hepdata.66883.v1/t2
Cross-section measurements for $J/\psi$ and $\psi(2s)$ decaying into two muons (pb) as a function of rapidity. The tabulated errors are... | 2020-08-08 20:30:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9497319459915161, "perplexity": 2824.9087222808}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738351.71/warc/CC-MAIN-20200808194923-20200808224923-00404.warc.gz"} |
https://computergraphics.stackexchange.com/questions/6120/generating-and-combining-spectral-colors | # Generating and Combining Spectral Colors
I have two somewhat related questions to ask:
1. What is the most accurate way to get the colors of a spectrum, without having to go deep into physics and simulating the universe? ;) Right now, I'm using these color matching functions, and simply reading off the XYZ color of a particular wavelength and converting it to sRGB. I know sRGB can't represent most spectral colors, but let's ignore that.
2. If I'm trying to make a color, say white, out of the spectral colors, how should I combine them? Should I add the color in XYZ space, or in RGB space, or is neither sufficient?
My plan is to experiment with spectral rendering, so it should be accurate enough to avoid biased renders. It'll most likely be on the GPU, and I need accuracy over speed.
• What's best depends a lot on your application and what operations you want to do on these spectra. Could you edit your question to add some more about that? – Dan Hulme Jan 15 '18 at 22:17
• Just edited my question to add a bit of clarification. – Daniel Kareh Jan 15 '18 at 22:40
## 1 Answer
Your way of calculating XYZ functions is probably the most efficient way to go about calculating accurate colors from a spectrum. It is standard practice afaik, for examples the books Physically Based Rendering (3rd) and Real-Time Rendering (3rd) both use this method.
You can add the colors in RGB space, but only if you convert from sRGB to linear RGB first. Otherwise you need to take into account, that sRGB sums will not lead to correct colors. The blogpost Adventures with Gamma-Correct Rendering by Naty Hoffman is a good read regarding this, topc:
Computing shading in sRGB space is like doing math in a world where 1+1=3.
This is the problem you have when adding colors in sRGB.
As to whether $XYZ$ can be summed, I think so. If we look at the definition of the coordinates (for $S$ is the spectral function and $x, y, z$ are the CIE functions):
$X = \frac{1}{\int_\lambda y(\lambda) d\lambda} \int_\lambda S(\lambda) x(\lambda) d\lambda$
Now two $X$ coordinates $X_1, X_2$ would have the same definition, only the spectrals are different ($S_1, S_2$).
Thus you have
$X_1 + X_2 = \frac{1}{\int_\lambda y(\lambda) d\lambda} \int_\lambda S_1(\lambda) x(\lambda) d\lambda + \frac{1}{\int_\lambda y(\lambda) d\lambda} \int_\lambda S_2(\lambda) x(\lambda) d\lambda\\ = \frac{1}{\int_\lambda y(\lambda) d\lambda} \left(\int_\lambda S_1(\lambda) x(\lambda) d\lambda + \int_\lambda S_2(\lambda) x(\lambda) d\lambda\right)\\ = \frac{1}{\int_\lambda y(\lambda) d\lambda}\int_\lambda S_1(\lambda) x(\lambda) + S_2(\lambda) x(\lambda) d\lambda \\ = \frac{1}{\int_\lambda y(\lambda) d\lambda}\int_\lambda (S_1(\lambda) + S_2(\lambda)) x(\lambda) d\lambda\\ =X$
The same holds for other coordinates.
Therefore, your best option is probably to sum your XYZ coordinates and convert them to sRGB space in the end.
• Comforting to know that I'm doing this well. I also like your little quote ;) Well said, your answer is accepted. – Daniel Kareh Jan 16 '18 at 21:43 | 2019-06-17 21:27:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5014183521270752, "perplexity": 839.8921336523983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998580.10/warc/CC-MAIN-20190617203228-20190617225228-00434.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-x-3-1 | # How do you solve -x-3=1?
Jan 26, 2016
$x = - 4 , - x = 4$
#### Explanation:
$- x - 3 = 1$
$\rightarrow - x - 3 = 1$
$\rightarrow - x = 1 + 3 = 4$
$S o , - x = 4 , x = - 4$
Feb 3, 2016
A very slightly different way of saying the same thing!
$x = - 4$
#### Explanation:
Given: $- x - 3 = 1$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Objective}}$
$\textcolor{b r o w n}{\text{To end up with ( +x ) on one side of the equals sign and everything}}$$\textcolor{b r o w n}{\text{else on the other side.}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Multiply everything by (-1)}}$
$+ x + 3 = - 1$
$\textcolor{b l u e}{\text{Subtract 3 from both sides}}$
$x + 3 - 3 = - 1 - 3$
$x + 0 = - 4$
$x = - 4$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{p u r p \le}{\text{Shortcut method}}$
Given: $- x - 3 = 1$
Using the rule that for + or - swapping sides of the equals sign changes the sign of the value.
$x = - 3 - 1$
$x = - 4$ | 2021-12-01 17:44:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 20, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7942830324172974, "perplexity": 3127.8151172637117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964360881.12/warc/CC-MAIN-20211201173718-20211201203718-00436.warc.gz"} |
https://formulasearchengine.com/index.php?title=Negative_predictive_value&oldid=238882 | # Negative predictive value
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
{{ safesubst:#invoke:Unsubst||$N=Merge |date=__DATE__ |$B= Template:MboxTemplate:DMCTemplate:Merge partner }}
In statistics and diagnostic testing, the negative predictive value (NPV) is a summary statistic used to describe the performance of a diagnostic testing procedure. It is defined as the proportion of subjects with a negative test result who are correctly diagnosed. A high NPV for a given test means that when the test yields a negative result, it is most likely correct in its assessment. In the familiar context of medical testing, a high NPV means that the test only rarely misclassifies a sick person as being healthy. Note that this says nothing about the tendency of the test to mistakenly classify a healthy person as being sick.
## Definition
The Negative Predictive Value is defined as:
${\displaystyle {\rm {NPV}}={\frac {\rm {number\ of\ True\ Negatives}}{{\rm {number\ of\ True\ Negatives}}+{\rm {number\ of\ False\ Negatives}}}}={\frac {\rm {number\ of\ True\ Negatives}}{\rm {number\ of\ Negative\ calls}}}}$
where a "true negative" is the event that the test makes a negative prediction, and the subject has a negative result under the gold standard, and a "false negative" is the event that the test makes a negative prediction, and the subject has a positive result under the gold standard.
The following diagram illustrates how the positive predictive value, negative predictive value, sensitivity, and specificity are related.
Note that the positive and negative predictive values can only be estimated using data from a cross-sectional study or other population-based study in which valid prevalence estimates may be obtained. In contrast, the sensitivity and specificity can be estimated from case-control studies.
If the prevalence, sensitivity, and specificity are known, the negative predictive value can be obtained from the following identity:
${\displaystyle {\rm {NPV}}={\frac {({\rm {specificity}})({\rm {1-prevalence}})}{({\rm {specificity}})({\rm {1-prevalence}})+(1-{\rm {sensitivity}})({\rm {prevalence}})}}.}$
## Worked example
Suppose that a fecal occult blood (FOB) screen test is used in 2030 people to detect bowel cancer:
In this setting, with NPV = 99.5%, a negative test result may provide some reassurance that the subject is unlikely to have cancer. This high NPV value would be particularly notable if the cancer were relatively common. For example, if 5% of people in the population had bowel cancer, then a NPV of 99.5% would indicate that a person with a negative test result has much lower than the average population risk for bowel cancer. However if the prevalence of bowel cancer were 0.5%, a negative test result in this setting would be uninformative.
## Relation to negative post-test probability
Although sometimes used synonymously, a negative predictive value generally refers to what is established by control groups, while a negative post-test probability rather refers to a probability for an individual. Still, if the individual's pre-test probability of the target condition is the same as the prevalence in the control group used to establish the negative predictive value, then the two are numerically equal. | 2020-11-28 10:58:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6509397029876709, "perplexity": 1189.873049605577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141195417.37/warc/CC-MAIN-20201128095617-20201128125617-00437.warc.gz"} |
https://www.vedantu.com/iit-jee/gravitation | Gravitation
View Notes
Newton’s law of Universal Gravitation
Newton in 1686 stated that each particle of matter gets attracted to every other particle in the universe. This universal attractive force is called gravitation. Newton gave the following law about gravitation.
The force of attraction between any two material particles is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them. It acts along the line joining the two particles.
Consider two particles of masses m1 and m2 are positioned at a distance ‘r’ apart. If the force of attraction acting between them is F, then according to Newton’s law of gravitation, we have
F = G$\frac{m_1m_2}{r^2}$
The proportionality constant G is called ‘gravitational constant’. Its value is the same for all pairs of particles in the universe. Hence G is a ‘universal constant’.
Although Newton's law of gravitation applies strictly to particles (point-masses), it can be applied to extended objects as well, provided the sizes of the object are small compared to the distance between them. For example, the moon and the earth are located far enough from each other so that, to a good approximation, we can treat both the bodies as point masses and apply the law of gravitation. Spherically symmetric bodies tend to attract external particles as if their entire mass were concentrated at their centres of mass. Hence the law of gravitation can be applied to them, irrespective of whatever their size be.
Law of Gravitation in Vector Form
We are considering two-point masses m1 and m2 at a distance ‘r’ apart. Let r12 be a unit vector pointing from mass m1 to mass m2, and r21 a unit vector pointing from m2 to m1. Then the gravitational force F12 exerted on m1 by m2, is given in magnitude and direction by the vector relation:
F12 = -G$\frac{m_2m_1}{r^2}r^{21}$
The minus sign indicates that F12 points in a direction opposite to r21, that is gravitational force is attractive, m1 experiencing a force directed towards m2.
The force exerted on m2 by m1 is similarly
F21 = -G$\frac{m_2m_1}{r^2}r^{12}$
Because r21 = - r12, equation 1 and 2 show that
F12 = -F21
Gravitational Constant G
Newton’s law of gravitation is
F = G$\frac{m_2m_1}{r^2}$
If we put m1 = m2 = 1 and r = 1, then G = F
Thus, the gravitational constant G is numerically equal to the force with which two particles, each of the unit mass and placed a unit distance apart, attract each other.
The SI units of force F, distance r and mass m1 (or m2) are Newton (N), meter (m) and kilogram (kg) respectively. Therefore, the SI unit of G is Nm2 kg-2 or m3 kg-1 s-1 (because N = kg m s-1).
Kepler’s Laws of Planetary Motion
Our solar system consists of a sun that is stationary at the centre of the universe and nine planets that revolve around the sun in separate orbits. Also, there are celestial bodies that move around the planets. These are called satellites. For example, the moon revolves around the earth, hence the moon is a satellite of the earth. Similarly, Mars has two satellites, Jupiter has sixteen satellites, Saturn has nineteen and so on.
Kepler found important regularities in the motion of the planets. These regularities are known as ‘Kepler’s three laws of planetary motion’.
1. Law of Orbits: All the planets move around the sun in elliptical orbits having the sun at one focus of the orbit.
2. Law of Areas: The line joining any planet to the sun sweeps out equal areas in equal times, that is, the areal velocity of the planet remains constant.
3. Law of Periods: The square of the period of revolution of any planet around the sun is directly proportional to the cube of its mean distance from the sun.
Solved Example
Example 1) The acceleration due to gravity at the moon’s surface is 1.67m s-2. If the radius of the moon is 1.74 x 106m, calculate its mass. Take G = 6.67 x 10-11N m2 kg-2
Solution 1) Let Mm be the mass, Rm the radius of the moon, and g be the acceleration due to gravity on its surface. Then,
Mm = $\frac{gR_m^2}{G}$ = $\frac{1.67Nkg^{-1} × (1.74×10^6m)^2}{6.67 × 10^{-11}N m^2kg^-2}$ = 7.58 × 1022kg
Example 2) A body of mass 100 kg falls on Earth from infinity. What will be its velocity on reaching the earth? Calculate its Energy. The radius of the Earth is 6400 km and g = 9.8 m s-2.Air friction is negligible.
Solution 2) A body projected up with the escape velocity ve will go to infinity. Therefore, the velocity of the body falling on the earth from infinity will be ve. Now, the escape velocity of the earth is
ve = $\sqrt{2gR_e}$= $\sqrt{2 × (9.8 m s^{-2}) × (6400 × 10^3m)}$
=1.12 × 104ms-1 = 11.2 kms-1.
The kinetic energy required by the body is
K = ½ m ve2 = ½ × 100 kg × (11.2 × 103m s-1)2 = 6.27 × 109J
Q1. What is the evidence that supports Newton’s Law of Gravitation?
Ans: Various pieces of evidence that supports Newton’s Law of Gravitation are:
1. It is a universal law. It explains the motion of heavenly bodies, particularly that of the planets, the moon, and the sun.
1. The prediction of the solar and the lunar eclipses based on the law come out in perfect agreement with the actual observation.
2. Tides are formed in the oceans due to gravitational attraction between the moon and ocean water.
3. The predictions about the orbits and time period of artificial satellites are made based on this law are found to be correct.
Q2. What are the characteristics of Gravitational Force?
Ans: The vector nature of the law of gravitation and experimental observations reveal the following characteristics of the gravitational force between two bodies:
1. These are always forces of attraction.
2. They form an action-reaction pair, that is, the forces exerted by two bodies on each other are equal in magnitude but oppositely directed.
3. These are central forces, that is they act along the line joining the centres of two bodies.
4. These are completely independent of the presence of other bodies and the properties of the intervening medium.
Q3. What is a Gravitational Field?
Ans: Every particle in the universe attracts every other particle with a force called the ‘Gravitational Force’. The space around the attracting particle in which the gravitational force of that particle can be experienced is called the ‘Gravitational Field’ of that particle. The force experienced by a unit mass placed at a point in a gravitational field is called the ‘Gravitational Field Strength’ or ‘Intensity of Gravitational Field’ at that point. This definition is given on the assumption that the unit mass itself does not make any changes in the original gravitational field. | 2021-02-28 15:09:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7559748888015747, "perplexity": 491.34822269375513}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178361510.12/warc/CC-MAIN-20210228145113-20210228175113-00300.warc.gz"} |
https://sheishe.xyz/post/the-little-schemer-note-3-3/ | # The Little Schemer speedy referring note (3/3)
Posted by Katherine's Blog on Monday, January 6, 2020
## TOC
This is a quick reference note that I pull from the book The Little Schemer. The code in this note is from here with edition:
the-little-schemer/02-do-it-again.ss at master · pkrumins/the-little-schemer
The former chapters can be easily understood from reading the code without counting parenthesis. However from this chapter, it is highly recommended to download Drracket and use a stepper to run all the recurions. For example, the stepper make the answer of soegaard very straightforward:
scheme - Y combinator discussion in “The Little Schemer” - Stack Overflow
This chapter introduces the idea of Y combinator based on recursion. We’ve seen that recursion is a function calling itself during defining itself, but when the function is just an lambda expression without name, what do we do?
The Y combinator provides a solution by designing an high order function, which is a function that takes a function as an argument and returns a function. Taking factorial as an example, we deduce a function G where G(factorial)=factorial. Let’s learn how to deduce step by step.
## Chapter 9 and Again and Again and Again
The key of writing recursion is making sure there is termination condition. That’s the basic requirement for function in both computation and mathmatics area: a function is a mapping procedure which takes in an argument and produces an output accordingly. We need to aviod any algorithm leading to infinite loop.
Here is an example of failed design: the (keep-looking) calls (pick) to see if a is equal to the random atom in lat (assuming the numbers in lat is random).
;pick return n-th element in lat:
(define pick
(lambda (n lat)
(cond
((zero? (sub1 n)) (car lat))
(else
(pick (sub1 n) (cdr lat))))))
(define keep-looking
(lambda (a sorn lat)
(cond
((number? sorn)
(keep-looking a (pick sorn lat) lat))
(else (eq? sorn a )))))
(define looking
(lambda (a lat)
(keep-looking a (pick 1 lat) lat)))
; Example of looking
(looking 'caviar '(6 2 4 caviar 5 7 3)) ; #t
(looking 'caviar '(6 2 grits caviar 5 7 3)) ; #f
In the first test case we can find that, the 'caviar is the 4th element in the first example, and the list contains 4. So running it is very likely to hit the termination condition. But in second example the 'caviar is the 4th element whereas no 4 is contained in the list, so the recursion will run forever, meaning the function won’t always return a value for an input. This illed function is called partial function as opposed to total function defined previously.
Let’s see another example. We’ve defined pair is a list containing two s-expressions (s-expression: a binary tree). The shift takes a pair whose first component is a pair and builds a pair by shifting the second part of the first component into the second component.
(define first
(lambda (p)
(car p)))
(define second
(lambda (p)
(car (cdr p))))
(define build
(lambda (s1 s2)
(cons s1 (cons s2 '()))))
;(shift '((a b) c)) -> '(a (b c))
;(shift '((a b) (c d))) -> '(a (b (c d)))
(define shift
(lambda (pair)
(build (first (first pair))
(build (second (first pair))
(second pair)))))
(define a-pair?
(lambda (x)
(cond
((atom? x) #f)
((null? x) #f)
((null? (cdr x)) #f)
((null? (cdr (cdr x))) #t)
(else #f))))
(define align
(lambda (pora)
(cond
((atom? pora) pora)
((a-pair? (first pora))
(align (shift pora))) ;******alarming
(else (build (first pora)
(align (second pora)))))))
Based on (shift) we further creat (align). Don’t rush to run the function. Remember the seventh commandment emphasizes “recursion should happen on the subparts that are of the same nature: either on the sublists of a list; or on the subexpressions of an arithmetic expression”. We notice something alarming in the starred line in (align): the (align) as well as (keep-looking) both creat new argument in recursion that is not part of the original argument. It’s an indicator of ill, but (align) will still generate output for every input, so it’s not partial function.
We will continue and define a very similar function (shuffle) below, which is partial. It won’t produce value for some cases, since the a-pair predicate will always swap the items of pair, which makes any input with form of ((a b) (c d)) trapped in infinite item swapping loop.
;(revpair '((a b) (c d))) -> ((c d) (a b))
(define revpair
(lambda (p)
(build (second p) (first p))))
(define shuffle
(lambda (pora)
(cond
((atom? pora) pora)
((a-pair? (first pora))
(shuffle (revpair pora)))
(else
(build (first pora)
(shuffle (second pora)))))))
(shuffle '(a (b c))) ; '(a (b c))
(shuffle '(a b)) ; '(a b)
(shuffle '((a b) (c d))) ; infinite swap pora Ctrl + c to break and input q to exit
We just define two different ways to measure the mass of the first component of (align). The (length*) measures every atom with same weight, whereas the (weight*) puts twice as much weight to the first component.
(define length*
(lambda (pora)
(cond
((atom? pora) 1)
(else
(+ (length* (first pora))
(length* (second pora)))))))
;(length* '((a b) c)) -> 3
;(length* '(a (b c)) -> 3
(define weight*
(lambda (pora)
(cond
((atom? pora) 1)
(else
(+ (* (weight* (first pora)) 2)
(weight* (second pora)))))))
;(weight* '((a b) c)) -> 7
;(weight* '(a (b c)) -> 5
From (align) and (shuffle), we realize that whether the arguments will decrease in recursion is not the key to infer whether a function is total. We start to think if possible to develop a diagnose function to detect the partial function. Let’s imaging making up a function (will-stop?) without getting into detail. We want it to return #t if the function would eventually terminate with returning value, and return #f it does not stop. And itself has to be a total function, in which case (will-stop? will-stop>) has to return #t.
What would happen if the input are (length) and (eternity) like these? Sounds cool: the (length) stops when the input is '(), so the (will-stop?) returns #t, great! Meanwhile the (eternity) is partial and won’t stop for any input, which makes the (will-stop?) returns #f whatsoever.
(define eternity
(lambda (x)
(eternity x)))
(define will-stop?
(lambda (x)
(eternity x)))
(define length
(lambda (lat)
(cond
((null? lat) 0)
(define will-stop?
(lambda (x)
(length x)))
But you might also sense logical flaws already: if there is a function can make (will-stop?) return #t but won’t stop, the partial function detection function may eventually NOT exist. If you cannot, the author has carefully come up with an example to show why.
Let’s see this example:
(define last-try
(lambda (x)
(and (will-stop? last-try)
(eternity x))))
In order to test if ths is a right function, we input () and that requires evaluate (will-stop? last-try). Provided it returns #f (aka the last-try will not stop with null input), then the whole function will stop and return #f. Clearly this mother function (last-try (quote())) stops for null input, which goes against the aka part in parenthesis. So we try the opposite hypothesis: it returns #t (aka the last-try will stop with null input), and then we get to evaluate (eternity (quote())), which never stops. And this time, it logically goes against the hypothesis again. This makes the (will-stop?) a function we can describe but can not define. The example delicately demonstrates this by constructing a function based on (will-stop?) which (will-stop?) cannot judge.
You might ask what if we just ban the logical contradiction part, such as “preventing tested function from directly or indirectly calling (will-stop?)”. In that case, will (will-stop?) exist? I am not sure. But that “using functions without recursively calling itself” is exactly what we want to do with the Y combinator.
A function calling itself directly or indirectly, is recursion. A nature wonder about recursion is that, if we want to call a function that haven’t been fully defined, how could we do it?
There are many ways to write programs to realize the same procedure. But a specific interpreter has clear rule to implement a function. In this chapter, we can use transformations and derivations to gain fundamental understanding of nested functions, to peel off the syntax sugar so that we can reform them to design elegant form.
We probably can re-write a recursive function to non-recursive one, then we can use it as much as we want. That’s what the Y combinator is for.
Let’s begin with recursive function we have seen, to see what part we can get rid of and reform. We still use (length) as an example. Currently it is a fully defined function recursive function. The input is a list and the output is a value (i.e. the length of a list). It can use recursion inside itself because it has formal function name length.
;you have use to other name in DrRacket, coz length has been a build-in function
(define length
(lambda (l)
(cond
((null? l) 0)
(else
(length '(a b)) ;-> 2
I highlight four key procedures to show how it adapts:
The thing about the recursion (length) is, it may look like the (length) is calling itself. More broadly speaking, it is just calling a function happens to have the same name with itself. The function will be used when the input (cdr l) is not null. It can be any functions, so you can try substituting (length) with any function you like. This equals to having another layer of lambda expression to the function of (length):
(lambda (length)
(lambda (l)
(cond ((null? l) 0)
(else (+ 1 (length (cdr l)))))))
Notice, you can try inputting any function in outer layer but most functions will not work with non-null input list. For example let’s call (eternity) instead of (length): when the list has more than zero atom, the null predicate returns #f and (eternity '()) will be called and the function will trap in infinite loop.
;((lambda (length) ..) eternity)
;length<=0
(lambda (l)
(cond
((null? l) 0)
(else
(add1 (eternity (cdr l))))))
It can still measure null list only in the applicative order interpreting, where the arguments will be instantaneously evaluated the leftmost innermost reducible expression before the function is applied.
But there is still a way to measure list with less than one element. We just have to call length measuring function on top of (length0) one more time.
;length<=1
(lambda (l)
(cond
((null? l) 0)
(else
;length<=1
(lambda (l) ;read more details below if you don't understand here
(cond
((null? l) 0)
(else
((lambda(l)
(cond
((null? l) 0)
(else
(cdr l))))))
Recursively we can develop (length<=2) below. Can you explain why these three functions are identical?
;length<=2
(lambda (l)
(cond
((null? l) 0)
(else
(lambda (l)
(cond
((null? l) 0)
(else
((lambda(l)
(cond
((null? l) 0)
(else
(cdr l))))))
(lambda (l)
(cond
((null? l) 0)
(else
((lambda(l)
(cond
((null? l) 0)
(else
((lambda(l)
(cond
((null? l) 0)
(else
(cdr l))))))
(cdr l))))))
;let's give distinguished names to arguments in every layer
(lambda (l2) ;assume l2 = '(b c)
(cond
((null? l2) 0)
(else
((lambda(l1) ;then l1 <- cdr(l2) = '(c)
(cond
((null? l1) 0)
(else
((lambda(l0) ;then l0 <- cdr(l1) = '( )
(cond
((null? l0) 0) ;so here returns 0, and terminates
(else
(cdr l1))))))
(cdr l2))))))
As you might imagine, the above form is not quite complete, so we were only saying its got a “hidden layer of parameter”. Let’s make it slightly more formal by separating (eternity) and calling it as argument. In the (length<=1) code we just want to use distinctive names for you to see this procedure clearer.
;length<=0
((lambda (length)
(lambda (l)
(cond
((null? l) 0)
eternity)
;length<=1
((lambda (f)
(lambda (l)
(cond
((null? l) 0)
((lambda (g)
(lambda (l)
(cond
((null? l) 0)
eternity))
;length<=2
((lambda (length)
(lambda(l)
(cond)
((null? l) 0)
((lambda (length)
(lambda (l)
(cond
((null? l) 0)
((lambda (length)
(lambda (l)
(cond
((null? l) 0)
eternity)))
It’s absolutely normal to get confused when there are more layers of lambda expressions involved in a function/recursion. It helps to think whether the lambda expressions is being merely defined or being defined and called, i.e. counting the parenthesis very carefully. The difference between defining and calling is that calling a function has arguments involved:
;defining
(lambda (f)
(lambda (g)...)
)
;calling(f) with defining(g)
((lambda (f)
(lambda (g)...))
arguments-for-f)
A more general case of calling with defining in lambda expression is called the omega combinator. It has shape in the below picture and more information can be found at Lambda calculus - Wikipedia
The above functions show repetitive content: the (length) part is being called over and over, working on a shorter and shorter argument. Normally, we would write and save as a named function for calling in the future like (define length). But, if we don’t save it, instead we want to directly address it within other function, or even address itself. How do we do that?
You may have realized the motivation of this question, addressing itself withing itself is exactly the nature of recursion. In this chapter we just want to find a good way to do it for anonymous functions.
If we can define length abstractly, we can call it to simplify the reptitive procedure. This need is particularly necessary when there is going to be many algorithms having similar repetitions as (length<=n).
Ok, we are finally reaching the core reforming step. We are giving the formal calling form as equivalent to (define length):
(define length
(lambda (l)
(cond
((null? l) 0)
(else
(length '( )) ;-> 0
(length '(a b)) ;-> 2
;--------------------------------------------
((lambda (mk-length)
(mk-length eternity))
(lambda (length)
(lambda(l)
(cond
((null? l) 0)
((mk-length eternity) '());-> 0
((mk-length eternity) '(a));-> error
From the above we can see instead of being a name defined by define, length can also work as an parameter/argument.
However there is a major difference: (define length) has only one lambda expression, the input must be a list and the output is value. But the anonymous definition is adding another layer of lambda expression, the input and output for the outer lambda expression (i.e. the whole function) will have to be functions. It’s the output “function (λ l)” that will return value like the defined length function.
Another very important note is that, the (define length) function can evalue list with any length. the anonymous function can only evaluate input of null list, why? Because the input for length must be function ((cdr l) sure won’t fit), so if we don’t want get error message, it will have to terminate at (null? l) stage. Like we said, we will have to use call function more times if we want to measure longer lists:
;length<=1
((lambda (mk-length)
(mk-length
(mk-length eternity)))
(lambda (length)
(lambda(l)
(cond
((null? l) 0)
;length<=2
((lambda (mk-length)
(mk-length
(mk-length
(mk-length eternity))))
(lambda (length)
(lambda(l)
(cond
((null? l) 0)
(else (add1 (length (cdr l))))))))
In the (length<=0) function the only working part is ((null? l) 0) and the (else) predicate would never got triggered. So in that predicate it doesn’t really matter whether we call function (eternity), or itself (mk-length). We just change (eternity) to (mk-length):
; length<=0
((lambda (mk-length)
(mk-length mk-length)) ;<- we change here
(lambda (length)
(lambda (l)
(cond
((null? l) 0)
(else
(add1 (length (cdr l))))))))
The above code is still ONLY able to measure null list, because for other input, it will have to expand (add1 (length (cdr l))) where the input of argument has to be a function, defined by the (lambda (mk-length)(mk-length mk-length). But the input (cdr l) is a list. See how it fails in stepper:
As you might guess, we can just pass a random/any function to make it successfully measure the length one list, since it will stop in (null? list) in the second loop. For example any of these three functions could work:
((lambda (mk-length)
(mk-length mk-length))
(lambda (length)
(lambda (l)
(cond
((null? l) 0)
(else
((lambda (mk-length)
(mk-length mk-length))
(lambda (length)
(lambda (l)
(cond
((null? l) 0)
(else
;This one is fundamentally different from the above two, why?
((lambda (mk-length)
(mk-length mk-length))
(lambda (length)
(lambda (l)
(cond
((null? l) 0)
(else
(add1 ((length length) (cdr l))))))))
The former two functions can only help you measure length one list. For length two input it would fail in the function expansion, resulting in non-legitimate functions (add1 add1) (eternity eternity) in the second loop. However, the third one won’t fail because (length length) is a legitimate function, it can help us measure list with any length. So functionally speaking, these two are finally consistent:
(define length
(lambda (l)
(cond
((null? l) 0)
(else
((lambda (mk-length)
(mk-length mk-length))
(lambda (length)
(lambda (l)
(cond
((null? l) 0)
(else
(add1 ((length length) (cdr l))))))))
All left job is just to make things extremely pithy. Let’s adding some syntax sugar here: since it doesn’t matter what we name the inner arguments, because it’s just an pseudo name inside a function. So we can name it anything as long as we keep naming and calling consistent. The last function above can be written as:
((lambda (mk-length)
(mk-length mk-length))
(lambda (mk-length)
(lambda (l)
(cond
((null? l) 0)
(else
(add1 ((mk-length mk-length) (cdr l))))))))
The exercise in page 166 will help you on how it works. The instruction can be found in the answer of soegaard : scheme - Y combinator discussion in “The Little Schemer” - Stack Overflow
When running it with stepper in DrRacket, there are 27 steps for a case (l is (' a b c)), I only demonstrate 4 steps here (press ctrl and + to see the enlarged image)
You can try to play with longer list, such as this:
(((lambda (mk-length)
(mk-length mk-length))
(lambda (mk-length)
(lambda (l)
(cond
((null? l) 0 )
((mk-length mk-length)
(cdr l))))))))
'(a b c)) ;<- it works with lists in any length, try it
You would realize that this is just more recurrences of that “bear in mind” picture, aka calling (mk-length mk-length) one more time before applying a shorter candidate list, until the (cdr l) runs out of atom. In the end, the null list becomes the termination condition, without triggering it, we will go stack overflow by calling (mk-length mk-length) infinite times.
If you find it confusing, read this preview of omega combinator in the first answer of this post:scheme - Y combinator discussion in “The Little Schemer” - Stack Overflow
Let’s further abstract the function with the legitimate though interrupting (mk-length mk-length) part. Simple, just add another layer, I add some syntax salt (mk-length-two) to distinguish with the original (mk-length i.e. mk-length-one) so you can see it clearer:
((lambda (mk-length)
(mk-length mk-length))
(lambda (mk-length-two)
((lambda (mk-length-one)
(lambda (l)
(cond
((null? l) 0)
(lambda (l)
((mk-length-two mk-length-two) l)))))
Then we add one last more layer, (I swear it’s the last layer) to switch the order a bit, moving the actual length measuring part to make it look nicer:
((lambda (mk-length-one)
((lambda (mk-length)
(mk-length mk-length))
(lambda (mk-length-two)
(mk-length-one (lambda (l)
((mk-length-two mk-length-two) l))))))
(lambda (mk-length-one)
(lambda (l)
(cond
((null? l) 0)
(else (add1 (mk-length-one (cdr l))))))))
That’s very complex, let’s simplify it
((lambda (f)
((lambda (x) (x x))
(lambda (x)
(f (lambda (l)
((x x) l))))))
(lambda (mk-length-one)
(lambda (l)
(cond
((null? l) 0)
(else (add1 (mk-length-one (cdr l))))))))
Define and name the first part as Y combinator
(define Y
(lambda (f)
((lambda (x) (x x))
(lambda (x)
(f (lambda (l)
((x x) l)))))))
;and call Y with any function you want
((Y
(lambda (len)
(lambda (l)
(cond
((null? l) 0)
(else (+ 1 (len (cdr l)))))))) '(a b c d))
Done. Happy hacking!
## Chapter 10 What is the value of all of this?
Firstly this chapter defines entry and table/enviroment. An entry consists of a pair of lists of equal length (length is the number of first level atoms in a list). A list of entries is table/environment.
The entries can be built by (cons) lists.
; Let's build entries with build from chapter 7 (07-friends-and-relations.ss)
(define build
(lambda (s1 s2)
(cons s1 (cons s2 '()))))
(define new-entry build)
; Test it out and build the example entries above
(build '(appetizer entree bevarage)
'(pate boeuf vin))
We are using entries and tables to write an interpreter in this chapter, which refers to find names with matching values. So in our interested entry, the first list is usually a set of names, and the second list is a set of values corresponding to every names.
Given a function (lookup-in-entry), we would be able to find a value for every name.
;(lookup-in-entry name entry)
;food --
'((appetizer entree bevarage)
(pate boeuf vin))
(lookup-in-entry entree food)
; -> boeuf
Let’s try writing it. The (lookup-in-entry) works this way: the (eq?) checks input with every element in name lists, and return the corresponding value in the second list. (Remember we always need to return ('()) first if the input is null.)
;(define second
; (lambda (p)
; (car (cdr p))))
;the entry-f take null λ function to make it not break
(define lookup-in-entry
(lambda (name entry entry-f)
(lookup-in-entry-help
name
(first entry)
(second entry)
entry-f)))
; lookup-in-entry uses lookup-in-entry-help helper function
(define lookup-in-entry-help
(lambda (name names values entry-f)
(cond
((null? names) (entry-f name))
((eq? (car names) name) (car values))
(else
(lookup-in-entry-help
name
(cdr names)
(cdr values)
entry-f)))))
; Let's try out lookup-in-entry
(lookup-in-entry
'entree
'((appetizer entree bevarage) (pate boeuf vin))
(lambda (n) '()))
;-> boeuf
; The null function make sure the function doesn't break with incorrect input.
(lookup-in-entry
'no-such-item
'((appetizer entree bevarage) (pate boeuf vin))
(lambda (n) '()))
;-> '()
Putting the above code to DrRacket and run with stepper, you can see how things are achieved.
The table/environment can be extended by adding more new pairs (aka entries) on top of the old table/entries.
(define extend-table cons)
We can write another (lookup-in-entry) working as above But notice the take the (car (cdr table)) as input in every recursions, which means the function will immediately cease and return value when the first name matches the input.
; lookup-in-table finds an entry in a table
(define lookup-in-table
(lambda (name table table-f)
(cond
((null? table) (table-f name))
(else
(lookup-in-entry
name
(car table)
(lambda (name)
(lookup-in-table
name
(cdr table)
table-f)))))))
; Let's try lookup-in-table
(lookup-in-table
'beverage
'(((entree dessert) (spaghetti spumoni))
((appetizer entree beverage) (food tastes good)))
(lambda (n) '()))
; -> 'good
Then let’s look at value and type. When asking what’s the value of an S-expression, in most of the cases, it returns the nature of itself. For example, see the value of these S-expression:
(value e) where e is (car (quote (a b c)))
; is a, coz the first element of '(a b c) is an atom
(value e) where e is (quote (car (quote (a b c))))
; is (car (quote (a b c)))
; coz (quote()) make the whole argument literal as a list
; the inner car won't be called.
(value e) where e is
((lambda (nothing)
(cond
(nothing (quote something))
(else (quote nothing))))
#t)
;is something
;coz the e is evaluated first, returned 'something
;'something is an atom
What is type? Based on the working essence, all the functions we have seen in this book can be described in six types: (const, quote, identifier, lambda, cond, application). In interpreting program, implementing a type classification function in the first step is an efficiency booster. Because functions in the same type work in similar way. If we can write an universal function for each type, we can call and optimize each function based on their distinctive characteristics.
The similar part of interpretation is, recognizing the type and arguments of function, finding the corresponding value for each type and arguments (use (lookup-in-table)), then implementing the value to the type and arguments.
So firstly let’s how the type classification works. The first layer of classifier is (expression-to-action) and followed by other more detailed ones. This is going to be the most important function in this chapter: it classifies any input into the above six categories based on the “nature of action” of the input. Then each categories has its own running/interpreting rules with starred names (e.g. *const, *identifier). With these starred functions, we can roughly find values for anything, with a proper table.
Since everything can be classified with the above function, we can use (meaning e table) to find value for both functions and arguments, and merge the type and value together in a table to further writing implementing procedure. The above tree code blocks can be linked by two functions as:
; The value function takes an expression and evaulates it
(define value
(lambda (e)
(meaning e '())))
; The meaning function translates an expression to its meaning
(define meaning
(lambda (e table)
((expression-to-action e) e table)))
Most of the running rules for the six categories are just writing down details of its working nature. And the examples on book are easy to follow. We will see some difficult ones together: *lambda, *cond, *application.
(1) lambda
We can see that e as an lambda expression, when passed to (meaning e table) it got classified as *lambda. The following rule of *lambda action defines e as non-primitive and attaches the table to its remainder. Later we will see in (apply-closure), for a function defined as non-primitive, we will further deconstruct the content of itself to run.
(2) cond
The rule for (*cond) category is similar to lambda. As we know an eternal condition of (cond) is (else) so it’s the first one got translated into meaning. The other predicates conditions of cond (cond-lines-of e) do not have much varieties either, no much more than (atom? eq? null? zero? o< o>) with combinations of (and or). These primitive ones got translated in the second predicate with meaning of (*const) from the (atom-to-action). The remaining non-usual ones which either goes to atom/identifier or got translated into (*application).
If you wonder, put the book example in DrRacket:
(*cond (cond (coffee klatsch)(else party))
(((coffee)(#t))
((klatsch party)(5 (6)))))
And my answer: The coffee example roughly shows an idea of searching function for meaning/action in the table when it’s not primitive like (eq?).
(3) application
This category processes all other complex function expressions. For example, a S-expression applying value to a lambda function.
Starting from a simple example, implementing a lambda expression, is equal to find the meaning of an e with a table like this:
It equals evaluating =(*application (cons x y) table) too. We are not sure what kind of tables will always work, but We would like to have an universal form to do this, maybe with a closure that got functions, arguments and corresponding values stored in a structural form. Evaluating any non-primitive function would be like evaluating the meaning/*application of a standard object.
In fact yes, it is achievable. As is mention in *lambda part, applying a non-primitive function - a closure - to a list of values is the same as finding the meaning of the closure’s body with its table extended by an entry of the form (formals values). In this entry, formals is the formals of the closure and values is the results of evlis.
There may be a thousand possible closures, but let’s start with a simple one matching a function with known form.
This is how it evolves: | 2020-07-08 11:04:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6600525975227356, "perplexity": 4860.020945572803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655896932.38/warc/CC-MAIN-20200708093606-20200708123606-00344.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcdss.2011.4.565 | # American Institute of Mathematical Sciences
June 2011, 4(3): 565-579. doi: 10.3934/dcdss.2011.4.565
## Isotropic-nematic phase transitions in liquid crystals
1 Department of Mathematics, Piazza di Porta S. Donato 5, 40127-Bologna 2 Dipartimento di Matematica, Via Valotti 9, 25133 Brescia, Italy 3 DIBE, Via Opera Pia 11a, 16145 Genova, Italy
Received April 2009 Revised August 2009 Published November 2010
The paper derives the evolution equations for a nematic liquid crystal, under the action of an electromagnetic field, and characterizes the transition between the isotropic and the nematic state. The non-simple character of the continuum is described by means of the director, of the degree of orientation and their space and time derivatives. Both the degree of orientation and the director are regarded as internal variables and their evolution is established by requiring compatibility with the second law of thermodynamics. As a result, admissible forms of the evolution equations are found in terms of appropriate terms arising from a free-enthalpy potential. For definiteness a free-enthalpy is then considered which provides directly the dielectric and magnetic anisotropies. A characterization is given of thermally-induced transitions with the degree of orientation as a phase parameter.
Citation: Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565
##### References:
show all references
##### References:
[1] Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163 [2] Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997 [3] Valeria Berti, Mauro Fabrizio, Diego Grandi. A phase field model for liquid-vapour phase transitions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 317-330. doi: 10.3934/dcdss.2013.6.317 [4] Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591 [5] Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1 [6] Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks & Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257 [7] Jiayan Yang, Dongpei Zhang. Superfluidity phase transitions for liquid $^{4}$He system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5107-5120. doi: 10.3934/dcdsb.2019045 [8] Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623 [9] Marita Thomas, Sven Tornquist. Discrete approximation of dynamic phase-field fracture in visco-elastic materials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (11) : 3865-3924. doi: 10.3934/dcdss.2021067 [10] Claude Vallée, Camelia Lerintiu, Danielle Fortuné, Kossi Atchonouglo, Jamal Chaoufi. Modelling of implicit standard materials. Application to linear coaxial non-associated constitutive laws. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1641-1649. doi: 10.3934/dcdss.2013.6.1641 [11] Honghu Liu. Phase transitions of a phase field model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883 [12] Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681 [13] Carlos J. García-Cervera, Sookyung Joo. Reorientation of smectic a liquid crystals by magnetic fields. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1983-2000. doi: 10.3934/dcdsb.2015.20.1983 [14] Jinhae Park, Feng Chen, Jie Shen. Modeling and simulation of switchings in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1419-1440. doi: 10.3934/dcds.2010.26.1419 [15] Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure & Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545 [16] Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139 [17] Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106 [18] Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211 [19] Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445 [20] Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757
2020 Impact Factor: 2.425 | 2021-10-16 08:01:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48570239543914795, "perplexity": 5762.460864440018}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323584554.98/warc/CC-MAIN-20211016074500-20211016104500-00402.warc.gz"} |
http://www.gradesaver.com/textbooks/math/trigonometry/trigonometry-10th-edition/chapter-1-trigonometric-functions-section-1-1-angles-1-1-exercises-page-9/107 | ## Trigonometry (10th Edition)
1. 0-90 degrees in the first quadrant. Therefore 0$^{\circ}$$\leq75^{\circ}$$\leq$90$^{\circ}$ 2. Coterminal angles always in the same quadrant with original angle, because of full resolutions (2k$\pi$) 75° + 360° = 435° 75° - 360° = -285° | 2017-04-28 22:08:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.878325343132019, "perplexity": 8995.931690260863}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123097.48/warc/CC-MAIN-20170423031203-00638-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://experts.mcmaster.ca/display/publication188528 | # THE PHYSICAL CHARACTERISTICS OF THE GAS IN THE DISK OF CENTAURUS A USING THEHERSCHEL SPACE OBSERVATORY Academic Article
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### abstract
• We search for variations in the disk of Centaurus A of the emission from atomic fine structure lines using Herschel PACS and SPIRE spectroscopy. In particular we observe the [C II](158 $\mu$m), [N II](122 and 205 $\mu$m), [O I](63 and 145 $\mu$m) and [O III](88 $\mu$m) lines, which all play an important role in cooling the gas in photo-ionized and photodissociation regions. We determine that the ([C II]+[O I]$_{63}$)/$F_{TIR}$ line ratio, a proxy for the heating efficiency of the gas, shows no significant radial trend across the observed region, in contrast to observations of other nearby galaxies. We determine that 10 - 20% of the observed [C II] emission originates in ionized gas. Comparison between our observations and a PDR model shows that the strength of the far-ultraviolet radiation field, $G_0$, varies between $10^{1.75}$ and $10^{2.75}$ and the hydrogen nucleus density varies between $10^{2.75}$ and $10^{3.75}$ cm$^{-3}$, with no significant radial trend in either property. In the context of the emission line properties of the grand-design spiral galaxy M51 and the elliptical galaxy NGC 4125, the gas in Cen A appears more characteristic of that in typical disk galaxies rather than elliptical galaxies.
### authors
• Parkin, TJ
• Wilson, Christine D
• Schirm, MRP
• Baes, M
• Boquien, M
• Boselli, A
• Cormier, D
• Galametz, M
• Karczewski, OŁ
• Lebouteiller, V
• De Looze, I | 2019-05-24 11:44:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5382432341575623, "perplexity": 7024.641711520606}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257605.76/warc/CC-MAIN-20190524104501-20190524130501-00457.warc.gz"} |
https://swmath.org/?term=node%20classification | • # HIN2Vec
• Referenced in 6 articles [sw37750]
• them as features for multi-label node classification and link prediction applications on those networks ... micro$-$f_1\$ in multi-label node classification...
• # RolX
• Referenced in 8 articles [sw32343]
• making, searching for similar nodes, and node classification. This paper addresses the question: Given ... automatically discover roles for nodes? We propose RolX (Role eXtraction), a scalable (linear...
• # DropEdge
• Referenced in 3 articles [sw37753]
• Towards Deep Graph Convolutional Networks on Node Classification. Over-fitting and over-smoothing ... deep Graph Convolutional Networks (GCNs) for node classification. In particular, over-fitting weakens the generalization...
• # node2vec
• Referenced in 83 articles [sw27202]
• define a flexible notion of a node’s network neighborhood and design a biased random ... techniques on multi-label classification and link prediction in several real-world networks from diverse...
• # nodeHarvest
• Referenced in 2 articles [sw33154]
• nodeHarvest: Node Harvest for Regression and Classification. Node harvest is a simple interpretable tree-like ... high-dimensional regression and classification. A few nodes are selected from an initially large ensemble...
• # NetKit
• Referenced in 27 articles [sw22730]
• present NetKit, a modular toolkit for classification in net- worked data, and a case-study ... learning research. NetKit is based on a node-centric framework in which classifiers comprise...
• # CogDL
• Referenced in 2 articles [sw37740]
• tasks in the graph domain, including node classification, link prediction, graph classification, and other graph ... Most of the graph embedding methods learn node-level or graph-level representations...
• # HEPTHools
• Referenced in 3 articles [sw30615]
• each other via field redefinitions of the nodes. We extend this to non-valise adinkras ... Python code, providing a complete eigenvalue classification of “node-lifting” for all 36,864 valise...
• # struc2vec
• Referenced in 14 articles [sw36495]
• state-of-the-art techniques for learning node representations fail in capturing stronger notions ... experiments indicate that struc2vec improves performance on classification tasks that depend more on structural identity...
• # Sub2vec
• Referenced in 1 article [sw41562]
• learning algorithms for mining tasks like node classification and edge prediction. However, most ... work focuses on distributed representations of nodes that are inherently ill-suited to tasks such ... mining tasks, like community detection and graph classification. We show that Sub2Vec gets significant gains...
• # Devign
• Referenced in 3 articles [sw40145]
• neural network based model for graph-level classification through learning on a rich ... learned rich node representations for graph-level classification. The model is trained over manually labeled...
• # RANKS
• Referenced in 1 article [sw39968]
• RANKS: a flexible tool for node label ranking and classification in biological networks. Summary:RANKS ... bioinformatics task formalizable as ranking of nodes with respect to a property given...
• # Cheops
• Referenced in 1 article [sw29858]
• tend to break down. Hierarchies above 5000 nodes usually require special modifications such as clustering ... Decimal Classification system, which can contain between a million and a billion nodes. The Cheops ... deep classification hierarchy, which if fully populated would contain over 19 million nodes... | 2022-08-18 07:45:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24549315869808197, "perplexity": 7655.116825861109}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573172.64/warc/CC-MAIN-20220818063910-20220818093910-00056.warc.gz"} |
https://whatshap.readthedocs.io/en/latest/faq.html | # Questions and Answers¶
• Can WhatsHap use a reference panel? Reference panels are used by population-based phasers (like Beagle or ShapeIt). Although we are considering integrating this, WhatsHap cannot take advantage of reference panels right now. In case you have population data, we suggest to produce a population-based phasing (using Beagle, ShapeIt, etc.) and a read-based phasing using WhatsHap separately and then compare/integrate results in a postprocessing step.
• Will Illumina data lead to a good read-based phasing? Illumina paired-end data is not ideal for read-based phasing, since most pairs of heterozygous SNPs will not be bridged by a read pair. However, WhatsHap will attempt to produce as long haplotype blocks as possible. Running WhatsHap will hence tell you how phase-informative your input data is. Just take a look at the number (and sizes) of produced haplotype blocks.
• How large can/should a pedigree be for pedigree-aware read-based phasing (i.e. using option --ped )? The pedigree mode in WhatsHap is intended for intermediate-size pedigrees. The runtime of the core phasing step will be linear in $$2^{2t}$$, where $$t$$ is the number of trio relationships (= number of children) in your pedigree. We do not recommend to use pedigrees with $$t>5$$. For such pedigrees, read-data is unnecessary in most cases anyway and a very high-quality phasing can be obtained by genetic haplotyping methods (like MERLIN). | 2020-02-26 17:13:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6522209644317627, "perplexity": 6249.560826936326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146414.42/warc/CC-MAIN-20200226150200-20200226180200-00448.warc.gz"} |
https://www.oyohyee.com/post/hihocoder/1383/ | 639
# 题目
## 描述
The history of Peking University Library is as long as the history of Peking University. It was build in 1898. At the end of year 2015, it had about 11,000 thousand volumes of books, among which 8,000 thousand volumes were paper books and the others were digital ones. Chairman Mao Zedong worked in Peking University Library for a few months as an assistant during 1918 to 1919. He earned 8 Dayang per month there, while the salary of top professors in Peking University is about 280 Dayang per month.
Now Han Meimei just takes the position which Chairman Mao used to be in Peking University Library. Her first job is to rearrange a list of books. Every entry in the list is in the format shown below:
CATEGORY 1/CATEGORY 2/..../CATEGORY n/BOOKNAME
It means that the book BOOKNAME belongs to CATEGORY n, and CATEGORY n belongs to CATEGORY n-1, and CATEGORY n-1 belongs to CATEGORY n-2...... Each book belongs to some categories. Let's call CATEGORY1 "first class category", and CATEGORY 2 "second class category", ...ect. This is an example:
MATH/GRAPH THEORY
ART/HISTORY/JAPANESE HISTORY/JAPANESE ACIENT HISTORY
ART/HISTORY/CHINESE HISTORY/THREE KINDOM/RESEARCHES ON LIUBEI
ART/HISTORY/CHINESE HISTORY/CHINESE MORDEN HISTORY
ART/HISTORY/CHINESE HISTORY/THREE KINDOM/RESEARCHES ON CAOCAO
Han Meimei needs to make a new list on which the relationship between books and the categories is shown by indents. The rules are:
1) The n-th class category has an indent of 4×(n-1) spaces before it.
2) The book directly belongs to the n-th class category has an indent of 4×n spaces before it.
3) The categories and books which directly belong to a category X should be list below X in dictionary order. But all categories go before all books.
4) All first class categories are also list by dictionary order.
For example, the book list above should be changed into the new list shown below:
ART
HISTORY
CHINESE HISTORY
THREE KINDOM
RESEARCHES ON CAOCAO
RESEARCHES ON LIUBEI
CHINESE MORDEN HISTORY
JAPANESE HISTORY
JAPANESE ACIENT HISTORY
MATH
GRAPH THEORY
>>
## 输入
There are no more than 10 test cases.
Each case is a list of no more than 30 books, ending by a line of "0".
The
## Description of a book contains only uppercase letters, digits, '/' and spaces, and it's no more than 100 characters.
Please note that, a same book may be listed more than once in the original list, but in the new list, each book only can be listed once. If two books have the same name but belong to different categories, they are different books.
## 输出
For each test case, print "Case n:" first(n starts from 1), then print the new list as required.
B/A
B/A
B/B
0
A1/B1/B32/B7
A1/B/B2/B4/C5
A1/B1/B2/B6/C5
A1/B1/B2/B5
A1/B1/B2/B1
A1/B3/B2
A3/B1
A0/A1
0
Case 1:
B
A
B
Case 2:
A0
A1
A1
B
B2
B4
C5
B1
B2
B6
C5
B1
B5
B32
B7
B3
B2
A3
B1
# 代码
/*
By:OhYee
Github:OhYee
Blog:http://www.oyohyee.com/
Email:oyohyee@oyohyee.com
かしこいかわいい？
エリーチカ！
*/
#include <cstdio>
#include <algorithm>
#include <cmath>
#include <cstring>
#include <iomanip>
#include <iostream>
#include <map>
#include <set>
#include <list>
#include <queue>
#include <stack>
#include <string>
#include <vector>
#include <bitset>
#include <functional>
using namespace std;
typedef long long LL;
const int INF = 0x7FFFFFFF;
const double eps = 1e-10;
int kase = 1;
struct CATEGORY {
list<CATEGORY> *child;
list<CATEGORY> *book;
string s;
CATEGORY(string b) {
list<CATEGORY> *p = new list<CATEGORY>;
p->clear();
child = p;
p = new list<CATEGORY>;
p->clear();
book = p;
s = b;
}
bool operator < (string rhs) {
return s < rhs;
}
};
list<CATEGORY> L;
list<CATEGORY>::iterator pushsk(list<CATEGORY>::iterator t,string s) {
list<CATEGORY>::iterator it = t->child->begin();
it = lower_bound(t->child->begin(),t->child->end(),s);
if(it == t->child->end() || it->s != s) {
t->child->insert(it,CATEGORY(s));
it--;
}
return it;
}
void pushs(list<CATEGORY>::iterator t,string s) {
list<CATEGORY>::iterator it = t->book->begin();
it = lower_bound(t->book->begin(),t->book->end(),s);
if(it == t->book->end() || it->s != s) {
t->book->insert(it,CATEGORY(s));
it--;
}
}
void DFS(list<CATEGORY> *L,int n) {
if(L == NULL)
return;
list<CATEGORY>::iterator it = L->begin();
while(it != L->end()) {
for(int i = 0;i < n;i++)
cout << " ";
if(n != -1)
cout << it->s << endl;
DFS(it->child,n + 1);
DFS(it->book,n + 1);
delete it->child;
delete it->book;//处理内存泄露问题
++it;
}
}
bool Do() {
L.clear();
L.push_back(CATEGORY("MAIN"));//主书库
string s;
while(1) {
if(!(getline(cin,s)))
return false;
if(s == "0")
break;
size_t len = s.size();
string t = "";
list<CATEGORY>::iterator it = L.begin();
for(size_t i = 0;i < len;i++) {
if(s[i] == '/') {
it = pushsk(it,t);
t = "";
} else {
t += s[i];
}
}
pushs(it,t);
}
cout << "Case " << kase++ << ":" << endl;
DFS(&L,-1);
return true;
}
int main() {
cin.tie(0);
cin.sync_with_stdio(false);
while(Do());
return 0;
}
• 点击查看/关闭被识别为广告的评论 | 2019-12-11 22:23:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17262327671051025, "perplexity": 10919.09174579373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540533401.22/warc/CC-MAIN-20191211212657-20191212000657-00494.warc.gz"} |
https://quant.stackexchange.com/questions/21874/understanding-capm-cml-and-efficient-portfolios | # Understanding CAPM, CML, and efficient portfolios
I'm trying to understand the CAPM model and how we can use it to understand efficient portfolios. Specfically, I'm trying to use the CML line (mapping expected returns and standard deviations of portfolios) to value proposed portfolios.
In this scenario: risk free rate = 2%. Expected excess return on market portfolio is 8% (so, I'm assuming, the expected return on the the market portfolio is 10%). The last given value is that the standard deviation of the market portfolio is 20.
I have to analyze 3 portfolios:
A: E(r) = 8%, SD = 10% B: E(r) = 12% SD = 25% C: E(r) = 13% SD = 30%
Based on the Sharpe Ratio (ie: the slope of the CML), I deduced that portfolio A is unfeasible and C is inefficient, whereas B falls on the CML and must therefore be efficient for the level of risk.
The next question I'm posed with is "How can the expected return of the wining portfolio be achieved? Specify the amount invested in each asset/portfolio of assets?"
It is given that I have some number X to invest, but I'm not quite sure how to approach this problem. The question does not seem too clear to me.
You just need: $\alpha * .08 + (1-\alpha) * .02 = .12$. Solve for alpha and then check the standard deviation that should be .25. | 2019-07-19 06:29:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7004299759864807, "perplexity": 1193.5136102545177}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526064.11/warc/CC-MAIN-20190719053856-20190719075856-00298.warc.gz"} |
https://www.iith.ac.in/~subruk/publication/countingtm/ | # Exact computation of the number of accepting paths of an NTM
### Abstract
We look at the problem of counting the exact number of accepting computation paths of a given nondeterministic Turing machine (NTM). We give a deterministic algorithm that runs in time $\widetilde{O}(\sqrt{S})$, where $S$ is the size (number of vertices) of the configuration graph of the NTM, and prove its correctness. Our result implies a deterministic simulation of probabilistic time classes like $\mathsf{PP}$, $\mathsf{BPP}$, and $\mathsf{BQP}$ in the same running time. This is an improvement over the currently best known simulatio by van Melkebeek and Santhanam [SIAM J. Comput., 35(1), 2006], which uses time $\widetilde{O}(S^{1 - \delta})$. It also implies a faster deterministic simulation of the complexity classes $\mathsf{\oplus P}$ and $\mathsf{Mod_k P}$.
Type
Publication
Proceedings of the 4th International Conference on Algorithms and Discrete Applied Mathematics - CALDAM 2018, Guwahati, India
Date | 2019-08-25 09:31:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5276089310646057, "perplexity": 393.55484181842667}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027323246.35/warc/CC-MAIN-20190825084751-20190825110751-00294.warc.gz"} |
https://www.neetprep.com/question/25902-/126-Physics--Mechanical-Properties-Fluids/685-Mechanical-Properties-Fluids | The flow rate from a tap of diameter 1.25 cm is 3 lit/min. The coefficient of viscosity of water is ${10}^{-3}$ Pas. The nature of flow is :
1. Turbulent
2. Laminar
3. Neither laminar nor turbulent
Concept Questions :-
Types of flow
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
Water flowing from a hose pipe fills a 15-liter container in one minute. The speed of water from the free opening of radius 1 cm is (in ms${}^{-1}$) :
1. 2.5
2. $\frac{\mathrm{\pi }}{2.5}$
3. $\frac{2.5}{\mathrm{\pi }}$
4. 5 $\mathrm{\pi }$
Concept Questions :-
Equation of continuity
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
Work of 6.0 x 10${}^{-4}$ Joule is required to be done in increasing the size of a soap film from 10cm x 6cm to 10cm x 11cm. The surface tension of the film is :
1. 5 x 10${}^{-2}$ N/m
2. 6 x 10${}^{-2}$ N/m
3. 1.5 x 10${}^{-2}$ N/m
4. 1.2 x 10${}^{-1}$ N/m
Concept Questions :-
Surface tension
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
When a cylindrical tube is dipped vertically into a liquid the angle of contact is 140o. When the tube is dipped with an inclination of 40o, the angle of contact is-
1.
2.
3.
4. $60°$
Concept Questions :-
Surface tension
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
Two liquid drops have their diameters as 1 mm and 2 mm. The ratio of excess pressure in them is :
1. 1:2
2. 2:1
3. 4:1
4. 1:4
Concept Questions :-
Surface tension
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
If a soap bubble of radius 3 cm coalesce with another soap bubble of radius 4 cm under isothermal conditions, the radius of the resultant bubble formed is in cm-
1. 7
2. 1
3. 5
4. 12
Concept Questions :-
Surface tension
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
1. 5:3:7
2. 7:3:5
3. 21:35:15
4. 1:1:1
Concept Questions :-
Equation of continuity
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
W is the work done in forming a bubble of radius r, the work is done in forming a bubble of radius 2r will be:-
1. 4W
2. 3W
3. 2W
4. W
Concept Questions :-
Surface tension
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
Several spherical drops of a liquid each of radius r coalesce to form a single drop of radius R. If T is the surface tension, then the energy liberated will be -
Concept Questions :-
Surface tension
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
A sample of metal weighs 210 g in air, 140 g in water and 120 g in an unknown liquid. Then -
1. The density of the metal is
2. The density of the metal is
3. The density of the metal is 4 times the density of unknown liquid
4. The metal still float in water
Concept Questions :-
Archimedes principle | 2020-07-04 17:57:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6926355361938477, "perplexity": 5085.124422313602}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886516.43/warc/CC-MAIN-20200704170556-20200704200556-00317.warc.gz"} |
https://de.zxc.wiki/wiki/GMR_%28Signaturverfahren%29 | # GMR (signature process)
GMR is a digital signature process that is named after its inventors Shafi Goldwasser , Silvio Micali and Ronald L. Rivest .
Like RSA , GMR is based on the factoring assumption that there are bijective functions that can be calculated quickly, but for which the calculation of the inverse function is very time-consuming.
In contrast to RSA, however, it can be proven for GMR that even in the case of an adaptive active attack it is not possible to forge even a new signature.
## The procedure in detail
You need a collision-resistant pair of permutations with a secret with the domain of definition . The owner of the secret can calculate the inverse functions and easily. It's difficult for everyone else. ${\ displaystyle f_ {0} ^ {}, f_ {1} ^ {}}$${\ displaystyle D}$${\ displaystyle f_ {0} ^ {- 1}}$${\ displaystyle f_ {1} ^ {- 1}}$
In order to sign a single message, the sender must choose a reference at random and publish it authentically. In order to sign an n-bit message , it calculates the signature . The receiver can calculate the inverse function of this and compare the result with the reference. ${\ displaystyle D}$${\ displaystyle m_ {1}, m_ {2}, ..., m_ {n}}$${\ displaystyle f_ {m_ {1}} ^ {- 1} (f_ {m_ {2}} ^ {- 1} (.. f_ {m_ {n}} ^ {- 1} (R) ..)) }$
Obviously the problem is to publish a new reference for each message. This is done with reference trees.
## Individual evidence
1. Shafi Goldwasser, Silvio Micali and Ronald L. Rivest: A Digital Signature Scheme Secure Against Adaptive Chosen-Message Attacks . ( psu.edu ). | 2023-04-01 19:44:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.714566707611084, "perplexity": 4125.87327001215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950247.65/warc/CC-MAIN-20230401191131-20230401221131-00122.warc.gz"} |
https://math.stackexchange.com/questions/1811434/relation-between-the-frattini-property-and-pronormal-subgroups-of-a-solvable-gro | # Relation between the Frattini Property and Pronormal subgroups of a solvable group
A subgroup $H$ of $G$ is said to satisfy the Frattini Property if for any subgroup $K$ and $L$ such that $H\leq K \unlhd L$ implies that $L \leq N_L(H)K$
A subgroup is $H$ is pronormal in $G$ if for each $g \in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x = H^g$
A theorem characterising pronormal subgroups of soluble groups was proved by T. Peng which stated that: if $G$ is soluble group, $H$ is pronormal in $G$ $\iff$ H satisfies the Frattini Property
The $\Rightarrow$ direction is true in general since for any $g\in G$, $\langle H, H^g \rangle \leq H^{\langle g \rangle}$ and using my previous question Frattini Property of a subgroup
For the $\Leftarrow$ direction, solvability of the group will be needed. Would induction on $|G|$ be the way of solving this implication?
• I think you already asked this question before, or at least a very similar one. That Frattini property seems to be a rather involved, very specific property and it seems to be not many around here can handle all the info. Try to look for some help perhaps in MathOverflow, – DonAntonio Jun 3 '16 at 19:14
• You can only use induction on $|G|$ for a finite group $G$, and you do not appear to be assuming that $G$ is finite. – Derek Holt Jun 3 '16 at 19:23
• In any case, if it is a published result then why don't you try and read the proof there? – Derek Holt Jun 3 '16 at 19:29
• I don't have access to the paper which is in an Oxford Journal. My university which is the University of Kwa-Zulu Natal in South Africa also does not have library access to this paper. And yes, I missed out the crucial info of $G$ being finite – R Maharaj Jun 3 '16 at 19:47
• @Joanpemo Thank you. I did not know that another Math forum apart from this one existed. – R Maharaj Jun 3 '16 at 19:52 | 2019-06-24 13:28:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6122288703918457, "perplexity": 300.6227405870529}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999539.60/warc/CC-MAIN-20190624130856-20190624152856-00415.warc.gz"} |