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https://math.stackexchange.com/questions/2891583/confidence-interval-on-2-dice-rolls	# Confidence interval on 2 dice rolls Lets say we are rolling 2 dice of different sides. Example: Dice 1 has 128 sides and Dice 2 has 256 sides. I created a 2D array of size 128X256 to bin the result of dice1,dice2 combination. If it is 0,0 I increment location 0. If it is 0,1 I increment location 1 and so on. I also keep account of the average. Example if. it is 0,0 average is 0 & I keep track of how many times that average occurred. & if it is 0,1 average is 0.5 and so on. With a single trial of N iterations, I got the following : For the bins : Minimum value : 806 Maximum value : 60907 Mean : 1831.054688 Standard Deviation : 1356.090617 Median : 1541 Variance : 1838981.761 For the average: Min 0 Max 191 Mean 94.60301218 Std Dev 41.63474096 Median 95 How can I apply confidence interval on the above data? For the average case I calculated Std Err (SE) as Std Dev/sqrt of N & then multiplied it with. z. score for 95% confidence interval (1.96) & then specified my interval as 94.603 +- z*SE Is this correct? Also I did 30 such trials. Is the better way to take the mean of means of each of these samples & Then apply confidence interval on that? My final goal is to distribute M items to each of those bins such that each bin corresponds to picking an item. Through the distribution, each item is supposed to be picked with a certain ratio. I was also thinking should I be applying confidence interval to that ratio? Is applying z score for this case correct approach? Sorry about the long question - Its been years since I learned statistics in high school. Can't remember much :) • Any chance I can get some guidance? – Ryan Aug 23 '18 at 16:38 • (a) For what parameter do you want a confidence interval? (b) It seems you're programming a simulation. Do you want a numerical CI for the specific case mentioned? Maybe a programming a simulation can help with that. Or do you want an analytic formula for the CI? Program can't give you that, but may give you intuition or clues. – BruceET Aug 24 '18 at 1:32 Comment continued: From what you said in your question, I'm totally guessing the following scenario: You imagine a game in which you are repeatedly rolling two fair dice, one is $D_1$ with 128 sides and the other is $D_2$ with 256 sides. So at the $i$th turn your score is $T_i,$ the total on the two dice. The game consists of $n = 30$ turns, and you want the total score $S = \sum_{i=1}^{30} T_i.$ Finally, you want a 95% confidence interval for that total score. Analytically, you could find the mean and variance of the numbers on the two dice, add them to find the mean and variance per turn [that is $E(T_i)$ and $Var(T_i)$]. Then you can use those quantities to get $E(S)$ and $Var(S).$ By the Central Limit Theorem $S$ should be approximately normal, and you can use that fact to get an interval that contains the value of $S$ 95% of the time. I'd be surprised if all of that is exactly right, but maybe you can use it to improve your question so that someone can figure out what you really want. It is easy enough to simulate the scenario I described above, so I'll do that in R and show the result below. [With a 100,000 iterations you can expect about two significant digits of accuracy for the mean and SD.] set.seed(823) s = replicate(10^5, sum(sample(1:256,30,rep=T)+sample(1:128, 30, rep=T))) summary(s); var(s); sd(s) Min. 1st Qu. Median Mean 3rd Qu. Max. 3616 5484 5788 5789 6095 7663 [1] 205792.1 [1] 453.6432 The histogram shows the simulated scores and the best-fitting normal curve seems to show scores are nearly normal.	2019-07-21 16:58:36	{"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": 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": 0.7415241003036499, "perplexity": 517.5259249724731}, "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/1563195527089.77/warc/CC-MAIN-20190721164644-20190721190644-00046.warc.gz"}
http://nparc.cisti-icist.nrc-cnrc.gc.ca/eng/view/object/?id=b364cb88-d0ad-444b-9bfd-ef71401f474d	# Neutron diffraction determination of the cell dimensions and thermal expansion of the fluoroperovskite KMgF3 from 293 to 3.6 K Download Get@NRC: Neutron diffraction determination of the cell dimensions and thermal expansion of the fluoroperovskite KMgF3 from 293 to 3.6 K (Opens in a new window) Resolve DOI: http://doi.org/10.1007/s00269-006-0106-x Search for: Mitchell, Roger; Search for: Cranswick, Lachlan; Search for: Swainson, Ian Article Physics and Chemistry of Minerals 33 8 587–591; # of pages: 5 The cell dimensions of the fluoroperovskite KMgF3 synthesized by solid state methods have been determined by powder neutron diffraction and Rietveld refinement over the temperature range 293–3.6 K using Pt metal as an internal standard for calibration of the neutron wavelength. These data demonstrate conclusively that cubic $$Pm\overline{3} m$$ KMgF3 does not undergo any phase transitions to structures of lower symmetry with decreasing temperature. Cell dimensions range from 3.9924(2) at 293 K to 3.9800(2) at 3.6 K, and are essentially constant within experimental error from 50 to 3.6 K. The thermal expansion data are described using a fourth order polynomial function. 2006-11-22 National Research Council Canada; NRC Canadian Neutron Beam Centre No 12337925 Export as RIS Report a correction b364cb88-d0ad-444b-9bfd-ef71401f474d 2009-09-10 2016-05-09	2016-10-25 08:35:06	{"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": 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": 0.4109686315059662, "perplexity": 5254.267195094286}, "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-2016-44/segments/1476988720000.45/warc/CC-MAIN-20161020183840-00065-ip-10-171-6-4.ec2.internal.warc.gz"}
https://quantpie.co.uk/black_formula/black_price.php	## Black's Price and Greeks We derive the formulae for the price and Greeks of a call and a put options under the Black's model assumptions: ### Dyamics of the Underlying Asset We derive the formula for the price of option under Black's assumption which assumes the following dynamics of the underlying: $$d S_t = S_t \sigma d W_t$$ The solution of which is $$S_t = S_0 e^{\sigma W_t - \frac{\sigma^2 t}{2} }$$ As can be easily verified by applying Ito's lemma to $$y= ln{S_t}$$ and integrating the resulting equation from 0 to t: $$y = ln{S_t}$$ $$dy= \frac{\partial y}{\partial S_t} d S_t + \frac{1}{2} \frac{\partial^2 y}{\partial S_t^2} d S_t^2$$ $$d ln{S_t} = \frac{1}{S_t} d S_t - \frac{1}{2} \frac{1}{S_t^2} d S_t^2$$ $$\quad = \frac{1}{S_t} S_t \sigma d W_t - \frac{1}{2} \frac{1}{S_t^2} S_t^2 \sigma^2 dt$$ $$\quad = \sigma d W_t - \frac{\sigma^2}{2}dt$$ $$\int_0^t{d ln{S_u}}=\sigma \int_0^t {d W_u} - \frac{\sigma^2}{2}\int_0^t{du}$$ $$ln{S_t}-ln{S_0}=\sigma W_t - \frac{\sigma^2}{2}t$$ $$ln{\frac{S_t}{S_0}}=\sigma W_t - \frac{\sigma^2}{2}t$$ $$S_t=S_0 e^{\sigma W_t - \frac{\sigma^2}{2}t}=e^{ln S_0- \frac{\sigma^2}{2}t + \sigma W_t }$$ ### Identifying the Disribution of the Underlying Asset Now we derive the formula for the probability density of $$S_t$$. Notice that $$y = ln{S_t}$$ is normally distributed, and its mean and variance can be calculated as: $$y = ln{S_t} = ln S_0- \frac{\sigma^2}{2}t + \sigma W_t$$ $$E \left[ y \right] = E \left[ ln S_0- \frac{\sigma^2}{2}t + \sigma W_t \right]$$ $$\quad = ln S_0- \frac{\sigma^2}{2}t$$ $$V \left[ y \right] = V \left[ ln S_0- \frac{\sigma^2}{2}t + \sigma W_t \right]$$ $$\quad = \sigma^2 \int_0^t {du} = \sigma^2 t$$ And we can now derive the probability distribution of $$S_t$$ by variable transformation. Notice that $$y = ln{S_t}$$ is normally distributed, and its mean and variance can be calculated as: $$f_{S_t}(S)=f_{ln S_t}(ln S) \left\| \frac{d ln S}{d S} \right\|$$ $$f_{S_t}(S)=f_{ln S_t}(ln S) \frac{1}{S}$$ $$\quad=\frac{1}{\sqrt{2\pi V[ln S]}} {e^{-\frac{{(ln S-E[ln S])}^{2}}{2 V[ln S]}}} \frac{1}{S}$$ $$\quad=\frac{1}{S \sqrt{2\pi \sigma^2 t}} {e^{-\frac{{(ln S-ln S_0+ \frac{\sigma^2 t}{2})}^{2}}{2 \sigma^2 t}}}$$ $$\quad=\frac{1}{S \sigma \sqrt{2\pi t}} {e^{-\frac{{(ln S-ln S_0+ \frac{\sigma^2 t}{2})}^{2}}{2 \sigma^2 t}}}$$ ### Price We are now ready to derive the formula for the price of an option. To simplify the presentation we derive the formula for the price of a call option, and we assume that the price is calculated at t=0. The formula can be easily generalised back or the derivation can be modified to give the more general formula with the $$\phi$$ and t. The call option pays, at maturity of the option, the difference between the price of the underlying and the strike if the difference is positive: $$Payoff={(S_T-K)}^{+}$$ The present value of the payoff can be calculated using risk neutral valuation approach as follow: $$Price_0=e^{-r T} E^Q \left[ {(S_T-K)}^{+} \right]$$ $$\quad =e^{-r T} \left( E^Q \left[ S_T 1_{S_T > K} \right] - E^Q \left[ K 1_{S_T > K} \right] \right)$$ We now simplify the two sub-components separately, whereby 'simplify' means something we can calculate 'analytically'.The aim is to use variable transformation to translate the above expectations into standard normal probabilities so that one can use the normal probability lookup tables or the standard normal distribution function implemented in one's favourite software. We start with the first component (get ready for some practice with the variables transformation!): $$E^Q \left[ S_T 1_{S_T > K} \right] =\int_K^{\infty} {S f_{S_t}(S) dS}$$ $$E^Q \left[ S_T 1_{S_T > K} \right] =\int_K^{\infty} {S \frac{1}{S \sigma \sqrt{2\pi T}} {e^{-\frac{{(ln S-ln S_0+ \frac{\sigma^2 T}{2})}^{2}}{2 \sigma^2 T}}} dS}$$ Let $$x= \frac{ln S-ln S_0+ \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}}$$, then $$\frac{dx}{dS}= \frac{1}{S \sigma \sqrt{T}} \Rightarrow dS= S \sigma \sqrt{T} dx= e^{x \sigma \sqrt{T} + ln S_0 - \frac{\sigma^2 T}{2}} \sigma \sqrt{T} dx$$, $$S=K \Rightarrow x= \frac{ln K-ln S_0+ \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}}$$, and $$S= \infty \Rightarrow x= \infty$$. Making the substitutions, $$E^Q \left[ S_T 1_{S_T > K} \right] =\int_K^{\infty} {\frac{1}{\sigma \sqrt{2\pi T}} {e^{-\frac{{(ln S-ln S_0+ \frac{\sigma^2 T}{2})}^{2}}{2 \sigma^2 T}}} dS}$$ $$\quad =\int_{\frac{ln K-ln S_0+ \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}}}^{\infty} {\frac{1}{\sigma \sqrt{2\pi T}} e^{-\frac{x^2}{2}} e^{x \sigma \sqrt{T} + ln S_0 - \frac{\sigma^2 T}{2}} \sigma \sqrt{T} dx}$$ $$\quad =e^{ln S_0} \int_{\frac{ln K-ln S_0+ \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}}}^{\infty} {\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2 -2 x \sigma \sqrt{T} +\sigma ^2 t}{2}} dx}$$ $$\quad = S_0 \int_{\frac{ln K-ln S_0+ \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}}}^{\infty} {\frac{1}{\sqrt{2\pi}} e^{-\frac{{(x - \sigma \sqrt{T})}^2}{2}} dx}$$ Now to simplify further, let $$z= x -\sigma \sqrt{T} \Rightarrow dz=dx$$ and the lower limit becomes $$x = \frac{ln K-ln S_0+ \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}} \Rightarrow z= \frac{ln K-ln S_0+ \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}} -\sigma \sqrt{T}=\frac{ln K-ln S_0 - \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}}$$. Hence $$E^Q \left[ S_T 1_{S_T > K} \right]=S_0 \int_{\frac{ln K-ln S_0 -\frac{\sigma^2 T}{2}}{\sigma \sqrt{T}}}^{\infty} {\frac{1}{\sqrt{2\pi}} e^{-\frac{{z}^2}{2}} dz}$$ $$\quad \quad= S_0 \left( 1 -N \left[ \frac{ln K-ln S_0 - \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}} \right] \right)$$ $$\quad \quad = S_0 N \left[ \frac{ln S_0 -ln K + \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}} \right]$$ Now lets calculate the second component $$E^Q \left[ K 1_{S_T > K} \right]= K E^Q \left[ 1_{S_T > K} \right]$$ $$\quad \quad= K P \left[ S_T > K \right]$$ $$\quad \quad= K P \left[ e^{ln S_0- \frac{\sigma^2}{2}T + \sigma W_T } > K \right]$$ $$\quad \quad= K P \left[ \sigma W_T > ln K -ln S_0 + \frac{\sigma^2}{2}T \right]$$ $$\quad \quad= K P \left[ \frac{W_T}{\sqrt{T}} > \frac{ln K -ln S_0 + \frac{\sigma^2}{2}T}{\sigma \sqrt{T}} \right]$$ $$\quad \quad= K \left( 1- N \left[ \frac{ln K -ln S_0 + \frac{\sigma^2}{2}T}{\sigma \sqrt{T}} \right] \right)$$ $$\quad \quad= K N \left[ \frac{ln S_0 -ln K - \frac{\sigma^2}{2}T}{\sigma \sqrt{T}} \right]$$ Putting it all together , we get the famous Black's formula: $$Price = e^{-r T} \left( E^Q \left[ S_T 1_{S_T > K} \right] - E^Q \left[ K 1_{S_T > K} \right] \right)$$ $$\quad \quad = e^{-r T} \left( S_0 N \left[ \frac{ln S_0 -ln K + \frac{\sigma^2 T}{2}}{\sigma \sqrt{T}} \right] - K N \left[ \frac{ln S_0 -ln K - \frac{\sigma^2}{2}T}{\sigma \sqrt{T}} \right] \right)$$ $$\quad \quad = e^{-r T} \left( S_0 N \left[ d_1 \right] - K N \left[ d_2 \right] \right)$$	2019-12-06 15:52:17	{"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": 0, "mathjax_display_tex": 2, "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": 0.9556319713592529, "perplexity": 268.9725478372251}, "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-2019-51/segments/1575540488870.33/warc/CC-MAIN-20191206145958-20191206173958-00206.warc.gz"}
https://www.quantandfinancial.com/2012/08/time-value-of-money-calculator.html	## Wednesday, August 29 ### Time Value of Money Calculator This article shows how to use the principle of offsetting annuities to solve basic TVM problems, such as yields on bonds, mortgage payments or arbitrage-free bond pricings. The underlying implementation is done in Python and illustrates practical use of these principles on simple time value of money examples. Theoretical background Time Value of Money is a central concept of finance theory that gives different value to the same nominal cash flow, depending on a pay-off date. In another words, one dollar has bigger value if received now as opposed to some future point of time. Hence, the difference in these valuations is a function of time passed between some present and future date. Another parameter of this function is the rate defining interest earned during one period of time. Mathematically, this relation could be written as: $$FV = PV\times (1+r)^{n}$$ , where $$PV$$ is the present value of cash flow, $$FV$$ is the value at some future date, $$r$$ is an interest accrued during one period and $$n$$ is number of periods between the today's and future date. The relation above always holds no matter of how complex the timing of cash flow is. Thanks to this, it can be used to value virtually any cash-flow scenario. Annuity Annuity can be understood as a bank deposit or other cash investment generating constant periodic interest payments (coupons). Since all earned interest is paid-out at the end of each period, the outstanding balance is never changed and payments last constant forever. Initial deposit $$PV$$, periodic interest rate $$r$$ and periodic payments $$PMT$$ are in the following relation: $$PV = \frac{PMT}{r}$$ If we depict all cash inflows and outflows as arrows above and below the time line, respectively; the annuity would look for example as follows: Plain Vanilla bond model One of the common scenarios used in finance is Plain Vanilla bond model, where initial outflow is typically followed by a series of periodic cash inflows, enclosed by a final principal inflow paid at maturity. This model can be applied to a wide-range of fixed-rate contracts, such as bonds, mortgages, non-amortizing loans, etc. In case of a Plain Vanilla bond, the initial outflow is considered to be the price of a bond, periodic cash inflows are coupons are final cash inflow at maturity is the face value of bond. The following picture illustrates the structure of cash aflows in a Vanilla bond model: General Principle of Calculation The mathematical equation for a generalized model looks as follows: $$FV = PV \times (1+r)^{n} + \sum_{i=1}^{n}\left ( PMT_{i} \times (1+r)^{n-i} \right )$$ , where $$PV$$ is initial inflow, $$PMT_{i}$$ are periodic cash outflows, $$FV$$ final inflow and $$r$$ is the interest rate, as defined previously. In case all periodic payments are the same, as in a plain-vanilla bond model, the formula above may be reduced into two mutually offsetting annuities, starting each at different point of time: The formula To derive the formula, assume the composition of two annuities mentioned above is priced to be arbitrage-free. The value of Annuity A in today's money is: $$Value_{A}=\frac{PMT}{r}-PV$$ The value of Annuity B in the future money is: $$Value_{B}=\frac{-PMT}{r}+FV$$ , which gives us the today's valuation of $$Value_{B}^{now}=\left(1+r\right)^{-n}\left(\frac{-PMT}{r}+FV\right)$$ Using the substitution $$z=(1+r)^{-n}$$ for discount factor, we can construct the arbitrage-free equation as follows: $$Value_{A}+Value_{B}^{now}=0$$ $$\left(\frac{PMT}{r}-PV\right)+z\times\left(\frac{-PMT}{r}+FV\right)=0$$ After isolation of $$FV$$, the final equation is: $$FV=\frac{1}{z}\left(\left(z-1\right)\times\frac{PMT}{r}-PV\right)$$ Other variables, such as $$PV$$, $$PMT$$, $$n=-log(z)$$ can be isolated in a similar manner. The only difficulty is the calculation of discount rate $$r$$, which must be done through a root-finding methods, such as Newton-Raphson or other. Python Implementation Algebraic equations are implemented in the TVM class: from math import pow, floor, ceil, log from quant.optimization import newton class TVM: bgn, end = 01 def __str__(self): return "n=%f, r=%f, pv=%f, pmt=%f, fv=%f" % ( self.nself.rself.pvself.pmtself.fv) def __init__(self, n=0.0, r=0.0, pv=0.0, pmt=0.0, fv=0.0, mode=end): self.n = float(n) self.r = float(r) self.pv = float(pv) self.pmt = float(pmt) self.fv = float(fv) self.mode = mode def calc_pv(self): z = pow(1+self.r, -self.n) pva = self.pmt / self.r if (self.mode==TVM.bgn): pva += self.pmt return -(self.fvz + (1-z)pva) def calc_fv(self): z = pow(1+self.r, -self.n) pva = self.pmt / self.r if (self.mode==TVM.bgn): pva += self.pmt return -(self.pv + (1-z) * pva)/z def calc_pmt(self): z = pow(1+self.r, -self.n) if self.mode==TVM.bgn: return (self.pv + self.fvz) self.r / (z-1) / (1+self.r) else: return (self.pv + self.fvz) self.r / (z-1) def calc_n(self): pva = self.pmt / self.r if (self.mode==TVM.bgn): pva += self.pmt z = (-pva-self.pv) / (self.fv-pva) return -log(z) / log(1+self.r) def calc_r(self): def function_fv(r, self): z = pow(1+r, -self.n) pva = self.pmt / r if (self.mode==TVM.bgn): pva += self.pmt return -(self.pv + (1-z) * pva)/z return newton(f=function_fv, fArg=self, x0=.05, y=self.fv, maxIter=1000, minError=0.0001) The generic code for Newton-Raphson method: from math import fabs # f - function with 1 float returning float # x0 - initial value # y - desired value # maxIter - max iterations # minError - minimum error abs(f(x)-y) def newton(f, fArg, x0, y, maxIter, minError): def func(f, fArg, x, y): return f(x, fArg) - y def slope(f, fArg, x, y): xp = x * 1.05 return (func(f, fArg, xp, y)-func(f, fArg, x, y)) / (xp-x) counter = 0 while 1: sl = slope(f, fArg, x0, y); x0 = x0 - func(f, fArg, x0, y) / sl if (counter > maxIter)break if (abs(f(x0, fArg)-y) < minError)break counter += 1 return x0 Example 1 - Mortgage Payments Consider a regular mortgage for $500'000, fully amortized over 25 years, with monthly installments, bearing annual interest rate 4%. Regular monthly payments will be then calculated as follows: from quant.tvm import TVM pmt = TVM(n=2512, r=.04/12, pv=500000, fv=0).calc_pmt() print("Payment = %f" % pmt) The output of calculation: Payment = -2639.184201 Example 2 - Yield to Maturity Consider a semi-annual bond with the par value of$100, coupon rate 6% and 10 years to maturity, currently selling at $80. What is the yield-to-maturity ? r = 2TVM(n=10*2, pmt=6/2, pv=-80, fv=100).calc_r() print("Interest Rate = %f" % r) The output of calculation: Interest Rate = 0.090866 Example 3 - Arbitrage-free pricing of a bond Consider an annual bond with par value of$100, coupon rate 5% with 8 years to maturity. Calculate an arbitrage-free price of such bond assuming that market interest rate is 6%. pv = TVM(r=.06, n=8, pmt=5, fv=100).calc_pv() print("Present Value = %f" % pv) Output of the calculation: Present Value = -93.790206 Note: For simplicity, we have discussed TVM calculations only in "END" mode. However, the implementation supports also "BGN" mode. Next Time: Spot rates, forward rates and bootstrapping of yield curves	2020-11-25 22:24:21	{"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": 2, "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": 0.599831223487854, "perplexity": 1473.4931454472699}, "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/1606141184870.26/warc/CC-MAIN-20201125213038-20201126003038-00203.warc.gz"}
http://procbits.com/2011/04/15/locate-and-updatedb-on-os-x-snow-leopard	# locate and updatedb on OS X Snow Leopard On Linux, I love using the "locate" utility. In essence, the locate utility works similar to the "find" utility but it's much faster. This is because the locate utility works in conjunction with the "updatedb" utility. updatedb builds an indexed database of the file names, this is how locate runs so quickly. I was stoked to find out that OS X has locate, but where is its partner in crime updatedb? Google to the rescue, djangrrl.com has the answer: sudo /usr/libexec/locate.updatedb If you made it this far, you should follow me on Twitter. -JP Want to test-drive Bitcoin without any risk? Check out my bitcoin wallet Coinbolt. It includes test coins for free.	2015-09-02 02:21:16	{"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": 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": 0.17552290856838226, "perplexity": 6456.854999740577}, "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-2015-35/segments/1440645241661.64/warc/CC-MAIN-20150827031401-00055-ip-10-171-96-226.ec2.internal.warc.gz"}
https://support.bioconductor.org/p/9138246/	Group-specific condition effect 1 0 Entering edit mode claudia • 0 @ee208474 Last seen 18 days ago Belgium Hello, I am facing a problem with the analysis of some data. I keep getting the error: Error in checkFullRank(modelMatrix) : the model matrix is not full rank, so the model cannot be fit as specified. One or more variables or interaction terms in the design formula are linear combinations of the others and must be removed. I followed the instructions reported in the vignette of DESeq2 in case of Group-specific condition effect, which is the case of my experimental design, but I am not able to solve the problem. This is my script: counts <- data.frame(final_counts_clean) samples <- data.frame(condition=factor(rep(c("HN","LN"),each=6)), phenotype=factor(rep(c("HLR", "FLR", "ALR"),each=2)), genotype=factor(rep(c("A", "B", "C", "D", "E", "F"),4)), rep=factor(rep(1:2, each=12))) samples$ind.n <- factor(rep(rep(1:3,each=2),2)) as.data.frame(samples) matrix<-model.matrix(~ condition + condition:ind.n + condition:phenotype, samples) ds <- DESeqDataSetFromMatrix(countData=counts, colData=samples, design=matrix) I don't have any "full zero" columns of missing interactions, which should be the cause of the error. I feel very lost. DESeq2 • 90 views ADD COMMENT 0 Entering edit mode @kevin Last seen 2 hours ago Republic of Ireland Hi, so far, this is what you have: samples condition phenotype genotype rep ind.n 1 HN HLR A 1 1 2 HN HLR B 1 1 3 HN FLR C 1 2 4 HN FLR D 1 2 5 HN ALR E 1 3 6 HN ALR F 1 3 7 LN HLR A 1 1 8 LN HLR B 1 1 9 LN FLR C 1 2 10 LN FLR D 1 2 11 LN ALR E 1 3 12 LN ALR F 1 3 13 HN HLR A 2 1 14 HN HLR B 2 1 15 HN FLR C 2 2 16 HN FLR D 2 2 17 HN ALR E 2 3 18 HN ALR F 2 3 19 LN HLR A 2 1 20 LN HLR B 2 1 21 LN FLR C 2 2 22 LN FLR D 2 2 23 LN ALR E 2 3 24 LN ALR F 2 3 I am unsure why you are creating an interaction between condition and ind.n (?) What is genotype? I think that you could create a new 'group' variable, and use this as an interaction with genotype: samples$group <- paste(samples$condition, samples$phenotype, sep = '_') table(samples$group, samples$genotype) A B C D E F HN_ALR 0 0 0 0 2 2 HN_FLR 0 0 2 2 0 0 HN_HLR 2 2 0 0 0 0 LN_ALR 0 0 0 0 2 2 LN_FLR 0 0 2 2 0 0 LN_HLR 2 2 0 0 0 0 dds <- DESeqDataSetFromMatrix( countData = counts, colData = samples, design = ~ group + genotype + group:genotype) , or, if you don't need genotype, just use: dds <- DESeqDataSetFromMatrix( countData = counts, colData = samples, design = ~ group) Kevin	2021-07-28 05:20:26	{"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": 0.40625256299972534, "perplexity": 2521.4748655196113}, "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-2021-31/segments/1627046153521.1/warc/CC-MAIN-20210728025548-20210728055548-00482.warc.gz"}
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Of course, this can be ingrained in the ...	2015-01-30 20:03:43	{"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": 0.7224152684211731, "perplexity": 2993.7789994098034}, "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-2015-06/segments/1422115861872.41/warc/CC-MAIN-20150124161101-00138-ip-10-180-212-252.ec2.internal.warc.gz"}
https://www.nature.com/articles/s41467-019-13169-3?error=cookies_not_supported	## Introduction There is a contradiction between major branches of modern evolutionary biology. On the one hand, fields such as behavioural and evolutionary ecology are based on the assumption that organisms will behave as if they are trying to maximise their fitness1,2,3,4. Models based on fitness maximisation are used to make predictions about the selective forces (reasons) for adaptation, and these are then tested empirically5,6. This approach has been phenomenally successful, explaining many aspects of behaviour, life history and morphology. For example, fitness maximisation underpins our evolutionary explanations of: foraging behaviour, resource competition, sexual selection, parental care, sex allocation, signalling and cooperation7,8,9,10,11,12. On the other hand, there is considerable evidence for selfish genetic elements, which increase their own contribution to future generations at the expense of other genes in the same organism13,14,15,16,17. These selfish genetic elements may distort traits away from the values that would maximise individual fitness, to increase their own transmission14,18,19,20,21,22. Evidence for such genetic conflict has been found across the tree of life, from simple prokaryotes to complex animals. The contradiction is that selfish genetic elements mess up individual fitness maximisation, and appear to be common, but individual fitness maximisation still appears to occur17,23,24. This contradiction is especially apparent in the study of sex allocation: theoretical models based on individual fitness maximisation have explained a wide range of natural variation in sex ratio, and yet there have been many reported cases of selfish sex ratio distorters9,14,25,26,27. Leigh28 provided a potential solution to this contradiction by suggesting that selfish genetic elements would be suppressed by the ‘parliament of genes’. Leigh’s argument was that, because selfish genetic elements reduce the fitness of most of the other genes in the organism, these other genes will have a united interest in suppressing selfish genetic elements. Furthermore, because these other genes are far more numerous, they will be likely to win the conflict. Consequently, even when there is considerable potential for conflict within individuals, we would still expect fitness maximisation at the individual level29,30,31,32,33,34. Leigh28 demonstrated the plausibility of his argument by showing theoretically how a suppressor of a sex ratio distorter could be favoured. Since then, numerous suppressors have been studied from a theoretical and an empirical perspective14,35,36. However, several issues may affect the validity of the parliament of genes hypothesis. First, whether a suppressor spreads can depend upon biological details such as the extent to which a selfish genetic element is distorting a trait, the population frequency of that element and the cost of suppression14,37,38,39,40,41,42,43. Are certain types of selfish genetic elements, which cause substantial distortion, less likely to be suppressed? Second, if the spread of suppressors through populations is slow, and if selfish genetic elements arise continuously over evolutionary time, non-equilibrium trait distortion may be possible35. Third, selfish genetic elements are themselves also under evolutionary pressure to cause a level of trait distortion that would maximise their transmission to the next generation15. Could the evolution of selfish genetic elements lead to trait distortion that is less likely to be suppressed?32 Fourth, if a suppressor does not reach fixation in a population, or a selfish genetic element is not purged from a population, subsequent mating may decouple selfish genetic elements and suppressors to expose previously suppressed trait distortion38. How important is this problem of polymorphism likely to be? We address these issues, by investigating the parliament of genes hypothesis theoretically. Our aim is to investigate the extent to which genetic conflict distorts traits away from the value that would maximise individual fitness. We find that: (i) the greater the level of trait distortion caused by a selfish genetic element, the more likely and the quicker it is suppressed; (ii) selection on selfish genetic elements leads towards greater trait distortion, making them more likely to be suppressed; (iii) in genome-wide arms races to gain control of organism traits, the majority interest within the genome generally prevails over ‘cabals of a few’, regardless of genome size, mutation rate, and the strength and sophistication of trait distorters. We find the same patterns with an illustrative model, and when examining three specific scenarios: selfish trait distortion of the sex ratio by an X chromosome driver; an altruistic helping behaviour encoded by an imprinted gene; and production of a cooperative public good encoded on a horizontally transmitted bacterial plasmid. Furthermore, we find close agreement when analysing scenarios with population genetic analyses and individual-based simulations. Our results suggest that even when there is potential for considerable genetic conflict, it has relatively little impact on traits at the individual level. ## Results ### Modelling approach We examine conflict between two groups of genes within the genome. We assume a selfish genetic element that can gain a propagation advantage through distorting some trait of the organism (‘trait distorter’). This trait distortion only benefits alleles at a subset of loci within the genome—Leigh termed this subset of loci a ‘cabal’30. The rest of the genome, which does not gain the propagation advantage from the trait distortion, will be selected to suppress the trait distorter. Leigh termed this collection of genes, which will comprise most of the genome, and so will constitute the majority within the parliament of genes, the ‘commonwealth’30. We used two complementary theoretical approaches. First, we developed ‘Equilibrium models’, where we assume that the trait distorter and their cabal are only a very small fraction of the genome. We allow for this by assuming that it is highly likely that a potential suppressor of a trait distorter can arise by mutation. Consequently, in these models, we focus our analyses on when a trait distorter and its suppressor can spread. We use this approach to examine, given the potential for suppression, what direction would we expect natural selection to take on average. We then developed ‘Dynamics models’, where we relaxed the assumption that the trait distorter and its cabal are a negligible fraction of the genome. In this case, rather than focus on the equilibrium state, we allowed trait distorters and their suppressors to arise continuously, at different loci across the genome. This approach allows us to investigate the influence of factors such as genome size, mutation rate and cabal size. We use this approach to determine the outcome of an evolutionary conflict that embroils the whole genome, to elucidate how far an organism trait is likely to be distorted at any given point in evolutionary time. ### Equilibrium models We assessed, given the potential for suppression, the extent to which a trait distorter will distort an organism trait away from the optimum for individuals. In order to elucidate the selective forces, we ask four questions in a step-wise manner, with increasing complexity: 1. (1) In the absence of a suppressor, when can a trait distorter invade? 2. (2) When can a costly suppressor of the trait distorter invade? 3. (3) What are the overall consequences of trait distorter-suppressor dynamics for trait values, at the individual and population level, at evolutionary equilibrium and before equilibrium has been reached? 4. (4) If the extent to which the trait distorter manipulates the organism trait can evolve, how will this influence the likelihood that it is suppressed, and hence the individual and population trait values? We assume an arbitrary trait that influences organism fitness. In the absence of trait distorters, all individuals have the trait value that maximises their individual fitness. The trait distorter manipulates the trait away from the individual optimum, to increase their own transmission to offspring. We assume a large population of diploid, randomly mating individuals. The aim of this model is to establish key aspects of the population genetics governing trait distorters and their suppressors, in an abstract setting. In Supplementary Notes 3, 4 and 5, we address the same issues in three specific biological scenarios. (1) Spread of a trait distorter: We consider a trait distorter, which we denote by D1, that is dominant and distorts an organism trait value by some positive amount k (k > 0). This trait distortion increases the transmission of the trait distorter to offspring. Specifically, the trait distorter (D1) drives at meiosis, in heterozygotes, against a trait non-distorter (D0), being passed into the proportion (1 + t(k))/2 of offspring. t(k) denotes the transmission bias (0 ≤ t(k) ≤ 1) and is a monotonically increasing function of trait distortion $$\left( {\frac{{{\mathrm{d}}t}}{{{\mathrm{d}}k}} \ge 0} \right)$$. We emphasise that, in nature, trait distorters need not be meiotic drivers—the key point here is that we are considering when trait distortion increases the propagation of that trait distorter. We chose meiotic drive in this model for simplicity, and model different mechanisms in the biologically specific models (Supplementary Notes 3, 4 and 5). Indeed, in many natural cases, meiotic drivers would not gain their advantage by distorting a trait, in which case they would not enter any conflict with the rest of the genome over organism trait values, and therefore would not have any lasting influence on whether trait values are those that maximise individual fitness. For example, the segregation distorter (SD) meiotic driver in Drosophila melanogaster gains its advantage in heterozygous males by disrupting the proper development of rival sperm, and not by trait distortion44. Any organism-level fitness costs associated with SD would be opposed by SD as well as across the rest of the genome45. Our focus in this paper is on selfish genetic elements that gain an advantage by trait distortion, and therefore disagree with the majority of genes over trait values. Trait distortion leads to a fitness (viability) cost (ctrait(k)) at the individual level, reducing an individual’s number of offspring from 1 to 1 − ctrait(k) (0 ≤ ctrait(k) ≤ 1). Owing to trait distorter dominance, the fitness cost of trait distortion is borne by heterozygous as well as trait distorter-homozygous individuals. The fitness cost is a monotonically increasing function of trait distortion $$\left( {\frac{{{\mathrm{d}}c_{\mathrm{trait}}}}{{{\mathrm{d}}k}} \ge 0} \right)$$. We assume that t(k) and ctrait(k) do not change with population allele frequencies, but relax this assumption in our specific models. We first ask what frequency the trait distorter will reach in the population in the absence of suppression. If we take p and p′ as the population frequency of the trait distorter in two consecutive generations, then the population frequency of the trait distorter in the latter generation is: $$\bar w\,p^\prime = (1 - c_{\mathrm{trait}}(k))\,(p^2 + (1 - p)p(t(k) + 1)),$$ (1) where $$\bar w$$ is the average fitness of individuals in the population in the current generation, and can be written in full as: $$\bar w$$ = (1 − ctrait(k))(p2 + 2p(1 − p)) + (1 − p)2. In ‘Trait distorter population frequency’ in the Methods, we show, with a population genetic analysis of Eq. 1, that the trait distorter will spread from rarity and reach fixation when ctrait(k) < t(k)(1 ctrait(k)). This shows that trait distortion will evolve when the number of offspring that the trait distorter gains as a result of trait distortion (t(k)(1 − ctrait(k))) is greater than the number of offspring bearing the trait distorter that are lost as a result of reduced individual fitness (ctrait(k)). (2) Spread of an autosomal suppressor: We assume that the trait distorter (D1) can be suppressed by an unlinked autosomal allele (suppressor), denoted by S1. We assume that this suppressor (S1) is dominant and only expressed in the presence of the trait distorter (facultative), but found similar results when the suppressor is constitutively expressed (obligate; Supplementary Note 6). Expression of the suppressor incurs a fitness cost to the individual, csup (0 ≤ csup ≤ 1), which could arise for multiple reasons, including energy expenditure, or errors relating to the use of gene silencing machinery46,47. Gene silencing generally precedes the translation of the targeted gene, and so we assume that the cost of suppression (csup) is independent of the amount of trait distortion caused by the trait distorter (k). We can write recursions detailing the generational change in the frequencies of the four possible gametes, D0/S0, D0/S1, D1/S0 and D1/S1, with the respective frequencies in the current generation denoted by x00, x01, x10 and x11, and the frequencies in the subsequent generation denoted by an appended dash (′): $$\begin{array}{l}\bar w\,x_{00}^{\prime} = x_{00}^2 + x_{00}x_{01} + \left( {1 - t} \right)\left( {1 - c_{\mathrm{trait}}} \right)x_{00}x_{10}\\ + \, \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{00}x_{11} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{01}x_{10},\end{array}$$ (2) $$\begin{array}{l}\bar w\,x_{01}^{\prime} = x_{00}x_{01} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{00}x_{11} + x_{01}^2\\ + \, \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{01}x_{10} + \left( {1 - c_{\mathrm{sup}}} \right)x_{01}x_{11},\end{array}$$ (3) $$\begin{array}{l}\bar w\,x_{10}^{\prime} = \left( {1 + t} \right)\left( {1 - c_{\mathrm{trait}}} \right)x_{00}x_{10} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{00}x_{11}\\ + \, \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{01}x_{10} + \left( {1 - c_{\mathrm{trait}}} \right)x_{10}^2 + \left( {1 - c_{\mathrm{sup}}} \right)x_{10}x_{11},\end{array}$$ (4) $$\begin{array}{l}\bar w\,x_{11}^{\prime} = \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{00}x_{11} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{01}x_{10} + \left( {1 - c_{\mathrm{sup}}} \right)x_{01}x_{11} + \left( {1 - c_{\mathrm{sup}}} \right)x_{10}x_{11} + \left( {1 - c_{\mathrm{sup}}} \right)x_{11}^2,\end{array}$$ (5) where $$\bar w$$ is the average fitness of individuals in the current generation, and equals the sum of the equations’ right-hand sides. In ‘Suppressor invasion condition’ in the Methods, we show, with a population genetic analysis of these equations, that a suppressor will spread from rarity if trait distortion (k) is greater than some threshold value, at which the cost of suppression (csup) is less than the cost of being subjected to trait distortion, csup < ctrait(k). A threshold with respect to the level of trait distortion (k) arises because the cost of trait distortion (ctrait(k)) increases with greater trait distortion, but the cost of suppression (csup) is constant. Given that the individual cost of pre-translational suppression at a single locus is likely to be low46,47, trait distortion conferred by unsuppressed trait distorters is likely to be negligible. (3) Consequences for organism trait values: The extent of trait distortion at the individual level shows a discontinuous relationship with the strength of the trait distorter (Fig. 1a). When trait distortion is low, a suppressor will not spread (csup > ctrait(k)) and so the level of trait distortion at the individual level will increase with the level of trait distortion induced by the trait distorter (k). However, once a threshold is reached (csup < ctrait(k)), the suppressor spreads. We show in ‘Equilibrium trait distorter and suppressor frequencies’ in the Methods that the spread of the suppressor (S1) causes the trait distorter (D1) to lose its selective advantage and be eliminated from the population, leading to an absence of trait distortion at the individual level. In contrast, we show in Supplementary Note 6 that if the suppressor is constitutively expressed (obligate), the spread of the suppressor (S1) to fixation in the population causes the trait distorter (D1) to become neutral, meaning the trait distorter (D1) can be maintained in the population without being expressed. Overall, these results suggest that, given a relatively low cost of suppression (csup), the level of trait distortion observed at the individual level will either be low or absent. When a trait distorter is weak (low k), it will not be suppressed, but it will only have a small influence at the level of the individual. When a trait distorter is strong (high k), it will be suppressed and so there will be no influence at the level of the individual (Fig. 1a). In addition, we found that stronger trait distorters are suppressed more quickly (Fig. 1b). In ‘Non-equilibrium trait distortion’ in the Methods, we numerically iterated our recursions to determine how many generations it takes for suppressors to reach equilibrium. As long as trait distortion continues to reduce individual fitness non-negligibly after suppression is favoured (such that $$\frac{{{\mathrm{d}}t}}{{{\mathrm{d}}k}}/\frac{{{\mathrm{d}}c_{\mathrm{trait}}}}{{{\mathrm{d}}k}}$$ is not excessively high after csup < ctrait(k)), stronger trait distorters (higher k) are suppressed and purged more rapidly than weaker trait distorters, limiting the potential for non-equilibrium trait distortion (Fig. 1b). (4) Evolution of trait distortion: We then considered the consequence of allowing the level of trait distortion (k) to evolve. We assume a trait distorter (D1) that distorts by k, and then introduce a rare mutant (D2) that distorts by a different amount $$\hat k$$ ($$\hat k$$ ≠ k). This mutant (D2) is propagated into the proportion (1 + t($$\hat k$$) − t(k))/2 of the offspring of D2D1 heterozygotes, and into the proportion (1 + t($$\hat k$$))/2 of the offspring of D2D0 heterozygotes. We assume that the stronger of the two trait distorters is dominant, but found similar results when assuming additivity (‘Invasion of a mutant trait distorter’ in the Methods). We assume that the similarity in coding sequence and regulatory control means that the original trait distorter and the mutant are both suppressed by the same suppressor allele, at the same cost (csup)46,47. In ‘Invasion of a mutant trait distorter’ in the Methods, we write the recursions that detail the generational frequency changes in the different possible gametes (D0/S0, D0/S1, D1/S0, D1/S1, D2/S0 and D2/S1). We found that stronger mutant trait distorters ($$\hat k$$ > k) will invade from rarity when the marginal increase in offspring they are propagated into exceeds the marginal increase in offspring they are lost from as a result of reduced fitness (Δt(1ctrait($$\hat k$$)) > Δctrait, where Δ denotes marginal change (Δt=t($$\hat k$$) − t(k); Δctrait = ctrait($$\hat k$$) − ctrait(k))). Consequently, if trait distortion is initially low, and successive mutant trait distorters are introduced, each deviating only slightly from the trait distorters from which they are derived (‘δ-weak selection’48), invading trait distorters will approach a ‘target’ strength, denoted by ktarget. This target strength corresponds to the level of trait distortion that would maximise the fitness of the gene15, and is when the marginal benefit of transmission is exactly counterbalanced by the marginal individual cost of reduced offspring, $$\frac{{{\mathrm{d}}t}}{{{\mathrm{d}}k}}\left( {1 - c_{\mathrm{trait}}} \right) = \frac{{{\mathrm{d}}c_{\mathrm{trait}}}}{{{\mathrm{d}}k}}$$. The target strength of trait distortion (ktarget) will therefore be greater if increased trait distortion (k) leads to a low rate of decrease in marginal transmission benefit $$\left( { - \frac{{{\mathrm{d}}^2t}}{{{\mathrm{d}}k^2}}} \right)$$ relative to the rate of increase in marginal individual cost $$\left( {\frac{{{\mathrm{d}}^2c_{\mathrm{trait}}}}{{{\mathrm{d}}k^2}}} \right)$$ (Fig. 2b). If mutations are larger (strong selection), invading trait distorters may overshoot the target strength of trait distortion ($$\hat k$$ > ktarget). Weaker mutant trait distorters ($$\hat k$$ < k) are recessive so cannot invade from rarity. As evolution on the trait distorter increases the level of trait distortion, it makes it more likely that the trait distorter goes above the critical level of trait distortion where suppression will be favoured. When this is the case (csup < ctrait(ktarget)), the trait distorter spreads to high frequency, which then causes the suppressor to increase in frequency, reversing the direction of selection on the trait distorter, towards non-trait distortion (D0), resulting in 0 trait distortion at equilibrium (k* = 0) (Fig. 2a; ‘Equilibrium allele frequencies after mutant invasion’ in the Methods). Suppression only fails to spread if the individual fitness cost associated with suppression is greater than the individual fitness cost associated with the target trait distortion (csup > ctrait(ktarget); Fig. 2a). Given that the individual fitness cost of pre-translational suppression at a single locus is likely to be low, then any non-negligible trait distorter is likely to be suppressed. Overall, our results suggest that selection on trait distorters will tend to lead to the eventual suppression of those trait distorters. In ‘Agent-based simulation (single trait distorter locus)’ in the Methods, we developed an agent-based simulation, which allowed us to continuously vary the level of both trait distortion and suppression, and obtained results in close agreement (Fig. 2a; Supplementary Note 2, Supplementary Fig. 2). ### Specific biological scenarios In Supplementary Notes 3, 4 and 5, we tested the robustness of our above conclusions by developing models for three different biological scenarios: a sex ratio distorter on an X chromosome (X driver); an imprinted gene that is only expressed when maternally inherited; and a gene for the production of a public good by bacteria, which is encoded on a mobile genetic element14,26,36,49,50,51,52. We examined these cases because they are different types of trait distortion, involving different selection pressures, in very different organisms. In all three specific models, we obtained the same qualitative results as with our above illustrative model for an arbitrary trait (Fig. 3). ### Dynamics models Our Equilibrium models assumed that the suppressor of any given trait distorter will arise quickly by mutation. This assumption becomes less likely if suppressors are complex and hard to evolve, or favoured across a reduced portion of the genome (smaller commonwealth). Also, multiple trait distorters and their suppressors may arise continually in populations, through evolutionary time, at different loci within the cabal and commonwealth respectively. Organisms may therefore never rest at equilibria where all trait distorters are suppressed or of negligible strength. We address these issues by relaxing our assumption that the commonwealth is very large relative to the cabal, assuming instead that the commonwealth encompasses some majority of loci within the genome, with the remaining loci comprising the cabal. We examined the average and extremes of trait distortion produced by trait distorters and suppressors, by asking three further questions, of increasing complexity, in a step-wise manner: 1. (5) To what extent are organism traits distorted when populations of individuals are only ever subjected to one segregating trait distorter at a time (no trait distorter co-segregation)? 2. (6) To what extent are organism traits distorted when populations of individuals may be exposed to multiple, co-segregating, interacting trait distorters? 3. (7) To what extent are organism traits distorted when the strength of each trait distorter may evolve? (5) Trait distortion when no trait distorter co-segregation: We model a population of individuals, each with a genome size of γ loci. Within this genome, the cabal constitutes a fraction θ of all loci, and the commonwealth constitutes the remaining fraction 1 − θ of all loci. If a fraction of the genome is inherited in the same way, such that it favours the same trait values (same maximand), it is termed a ‘coreplicon’20,22. The cabal comprises all coreplicons that favour the distortion of a particular trait, along a particular axis, in a particular direction, away from individual fitness maximisation. The commonwealth comprises the remaining replicons. Cabals and commonwealths are therefore trait-specific. It is useful, when analysing a specific trait, to partition the genome along these lines, because it is this conflict—between the cabal and commonwealth—that drives the evolution of the trait value. Cabals and commonwealths are defined a priori, by partitioning and summing up the coreplicons that, respectively, disfavour and favour the trait distortion under study. The ‘individual’ is the majority interest within the genome, and so the cabal size can never exceed more than half of the genome, because then it would be the majority (θ ≤ 0.5)53. In Supplementary Note 8, we calculate some real-world proportional cabal sizes (θ) by dividing the number of genes in a cabal by the total number of genes in a genome. In Drosophila melanogaster, a Y chromosome cabal, which favours male biased sex ratio distortion, has a proportional size of ~θ ≈ 0.00154,55. In human females, a cabal comprising cytoplasmic elements as well as the X chromosomes, which favours female-biased sex ratio distortion, has a proportional size of ~θ ≈ 0.0456,57,58. In Escherichia coli, a cabal made up of horizontally transferrable plasmids, which could favour upregulated public goods production49, varies in size across strains, but has an average of ~θ ≈ 0.036. For analytical tractability, we start by assuming that new trait distorters and suppressors are introduced at a fixed rate (deterministic). Biologically, new trait distorters and suppressors are likely to arise via some combination of de novo mutation and the acquisition, via gene conversion or transposition, of pre-existing sequences contributing to trait distortion or suppression35,59,60. We assume that a trait distorter arises at a new locus within the cabal every $$1/(\theta\gamma\rho_{D_1})$$ generations, and its dedicated suppressor arises at a locus inside the commonwealth $$1/((1 - \theta)\gamma\rho_{S_1})$$ generations afterwards. $$\rho_{D_1}$$ and $$\rho_{S_1}$$, respectively, give the generational per-locus probabilities of generating new trait distorters and suppressors. These probabilities ($$\rho_{D_1}$$;$$\rho_{S_1}$$) increase linearly, according to the same gradient, as the baseline mutation rate in the genome, denoted by ρ, is increased. As in our equilibrium models, we assume that unsuppressed trait distorters distort organism traits by the fixed amount k, at an individual cost ctrait(k), gaining a meiotic transmission advantage in heterozygotes of (1 + t(k))/2. Similarly, we again assume that suppressors are dominant and completely suppress their target trait distorters at the cost csup, and are facultatively expressed in the presence of their target trait distorter5,6,7,8. We assume that the trait distortion experienced by an organism is given by the strength of its strongest unsuppressed trait distorter (inter-locus dominance). We emphasise again that the mechanism by which the trait distorter gains its advantage (meiotic drive) is chosen here purely for illustrative purposes (see Supplementary Notes 3, 4 and 5 for different mechanisms). We are interested in the subset of selfish genetic elements that gain their selfish benefit by distorting a trait away from the value that maximises individual fitness. The same trait distortion would be favoured across the coreplicon/cabal of which these selfish genetic elements are a part. This contrasts with selfish genetic elements that gain a selfish benefit through their ability to be meiotic drivers, without distorting a trait—such drivers could conceivably arise at any locus in a genome. The key difference here is between meiotic drive (could be favoured at any locus; selfish benefit does not arise via distorting a trait) and selfish genetic elements that gain a benefit by distorting a trait (the specific examples that we consider and model in this paper)14,15. We calculate the average and extremes of trait distortion faced by organisms in the population across evolutionary time, for different trait distorter strengths (k), and different proportional cabal sizes (θ). Considering trait distorters that do not trigger suppressor invasion (csup > ctrait(k)), the average trait distortion is trivially given by the strength of the trait distorters available to the cabal (k). Considering trait distorters that are suppressed and purged at equilibrium (csup < ctrait(k)), for analytical tractability, we first consider parameter regimes in which trait distorters are introduced at new loci more slowly than they are purged at old loci, meaning they do not co-segregate. In ‘Long-term trait distortion (exact numerical solution)’ in the Methods, we develop a population genetic model based on these assumptions, and solve it numerically to show that individual trait distortion increases and decreases cyclically over evolutionary time, ranging between peaks of k and troughs of 0, as new trait distorters and suppressors advance and retreat through the population (Fig. 4a). In ‘Long-term trait distortion (analytical approximation)’ in the Methods, we show that the average trait distortion over these cycles is given by $$\begin{array}{{20}{c}} {\frac{{k\theta \rho _{D_1}}}{{\left( {{\mathrm{1 - }}\theta } \right)\rho _{S_1}}},} \end{array}$$ (6) by making the assumption that the rate of gene frequency equilibration after trait distorter/suppressor introduction is very fast relative to the rate of trait distorter/suppressor introduction (separation of timescales). For our three specific biological scenarios (Supplementary Notes 3, 4 and 5), the rate of gene frequency equilibration after trait distorter/suppressor introduction varies in each scenario, but these details are inconsequential when the separation of timescales assumption is made, meaning average trait distortion is given by Eq. 6 in each of the three specific biological scenarios. Furthermore, we also found with numerical analysis that Eq. 6 is a good approximation, even when the separation of timescales is relaxed (Fig. 4b). Smaller proportional cabal sizes (θ) lead to a slower rate of trait distorter introduction relative to suppressor introduction, and so both: (i) an absolute reduction in average trait distortion; and (ii) a reduced effect of distorter strength (k) on average trait distortion (k − θ interaction) (Fig. 4b). In the limit of negligible proportional cabal size (θ → 0), we recover the result from our Equilibrium models that the proportion of evolutionary time in which a trait distorter is present approaches 0, leading to an average trait distortion of 0 for trait distorters above the threshold of suppression (csup < ctrait(k)). Both genome size (γ) and baseline mutation rate (ρ) have no influence on the average trait distortion. Increases in both of these factors leads to a proportional increase in trait distorter introduction rate, and the same proportional increase in suppressor introduction rate, which exactly cancel (Supplementary Note 7, Supplementary Fig. 11). (6) Trait distortion when trait distorters may co-segregate: We then considered the possibility that different trait distorters may co-segregate for some periods of evolutionary time59,60. In ‘Agent-based simulation (multiple loci; discrete)’ in the Methods, we developed an agent-based simulation that allowed us to investigate the scenario where mutations appear stochastically rather than deterministically. When an individual contains multiple trait distorters, we assume that extent of trait distortion is determined by the strongest trait distorter (inter-locus dominance). The consequence of allowing trait distorters to co-segregate will depend on mechanistic assumptions about how trait distorters and suppressors act and interact. To capture different ends of the continuum of possibilities, we model two different types of trait distorter, which we term low-sophistication (D1L) and high-sophistication (D1H) (Supplementary Note 7, Supplementary Fig. 12). High-sophistication trait distorters are only suppressed by dedicated suppressors that evolved to suppress that specific trait distorter, and incur a low cost when inter-locus recessive. In contrast, low-sophistication trait distorters can be suppressed to some extent by any suppressor (background or generalist suppression)35,59,60, and incur a high cost when inter-locus recessive. High-sophistication trait distorters are more functionally complex, and so are likely to be less mutationally accessible than low-sophistication trait distorters. We found that, for a sufficiently small proportional cabal size (θ → 0), trait distorters scarcely co-segregate, and Eq. 6 is recovered. Consequently, for sufficiently small proportional cabal sizes, the average level of trait distortion is again not influenced by genome size (γ), mutation rate (ρ), or the mechanics of trait distorter interaction (D1L/D1H). In contrast, with larger cabals (θ → 0.5), trait distorters often co-segregate. In this case, the details of genome size (γ), mutation rate (ρ), and trait distorter sophistication (D1L/D1H) matter. Specifically, trait distortion may be: (i) greater than Eq. 6 if trait distorters are high sophistication (D1H); (ii) lower than Eq. 6 if trait distorters are low sophistication (D1L). The deviation from Eq. 6 is exaggerated for increased trait distorter co-segregation, which is promoted by: (i) high genome size (γ)/mutation rate (ρ) (Fig. 5); (iii) low trait distorter strength (k), which causes trait distorters to be purged more slowly (Supplementary Note 7, Supplementary Fig. 14); (iv) low trait distorter sophistication (D1L), which increases the mutational accessibility of trait distorters. The proportional cabal sizes that make these different factors matter are, however, much larger than we generally find in nature. (7) Evolution of trait distortion and suppression: We then examined the consequences of allowing the level of trait distortion and suppression to evolve freely at each locus15. In ‘Agent-based simulation (multiple loci; continuous)’ in the Methods, we generalised our agent-based simulation to allow for this, and found that trait distorters evolve increased trait distortion (approaching ktarget) while unsuppressed (Supplementary Note 7, Supplementary Fig. 15). Stronger trait distorters are suppressed and purged more quickly than weaker ones, and are less likely to co-segregate as a result. Consequently, when evolution is permitted at trait distorter loci, average trait distortion again approaches that predicted by Eq. 6, so is less influenced by genome size (γ), mutation rate (ρ), and the mechanics of trait distorter interaction (D1L/D1H). ## Discussion We obtained three main results: First, larger trait distortions are more likely to be suppressed. Consequently, trait distorters will either lead to small trait distortions, with minor fitness consequences, or be suppressed (Figs. 1a and 3a–c). Second, selection on trait distorters favours the evolution of higher levels of trait distortion, which will favour their suppression. Consequently, trait distorters will evolve to bring about their own demise (Figs. 2, 3d–f and 6). Third, if trait distortion is favoured at only a small proportion of the genome (proportionally small cabals), the extent of trait deviation away from the individual level optima is low and unaffected by factors, such as genome size, mutation rate and mechanism of trait distortion (Figs. 4 and 5). The reason for this result is that the influence of all of these factors is determined by proportional cabal size. Overall, these results suggest that even if there is substantial potential for genetic conflict, trait distorters will have relatively little influence at the individual level, in support of Leigh’s28 parliament of genes hypothesis. Suppressing trait distorters: We have shown that suppressors spread when the cost of suppression is lower than the fitness cost imposed by trait distortion (ctrait(k) > csup). The individual fitness cost of pre-translational suppression at a single locus is likely to be low. For example, a molecularly characterised suppressor (nmy) destroys the messenger RNA transcripts of a sex ratio distorter (Dox) via RNA interference (RNAi), the costs of which are likely to be negligible at the individual level46,47,60,61. Consequently, in order to not be suppressed, a trait distorter would have to have relatively negligible influence on a trait, or influence a trait that has a negligible influence on fitness. Furthermore, we also showed that selection on trait distorters will often favour higher level trait distortion, bringing trait distorters into the region where ctrait(k) > csup, and hence where suppression is favoured (Figs. 2, 3 and 6). Our analyses have focused on selfish genetic elements that increase their own transmission by manipulating some organism trait in a specific direction15,17. Examples include the sex ratio distorters and public goods genes considered in our specific models. We focused on such ‘trait distorters’ because they can have substantial influences on the traits of organisms, even when at fixation. In contrast, we have not considered selfish genetic elements, such as transposons and meiotic drivers, that do not need to manipulate organism traits in order to give themselves a selfish propagation advantage43. We have not considered such selfish genetic elements because: (i) they do not distort traits away from individual maxima; and (ii) the cost of such drivers makes them disfavoured across the entire genome, leading to selection to attenuate that cost. Our Dynamics models have validated various verbal arguments that have previously been made for the parliament of genes hypothesis. We found that, if trait distortion is only favoured across a small proportion of the genome (proportionally small cabal), the trait distortion experienced by individuals is likely to be low, and unaffected by details such as genome size, mutation rate and mechanism of trait distortion. Empirically, cabals typically comprise small proportions of genomes54,56. Furthermore, more sophisticated trait distorters, with the potential to interact synergistically with each other, are likely to have a lower mutational accessibility, and so are more likely to be suppressed and purged before they have a chance to co-segregate. Real-world examples of trait distortion are typically caused by lone genes, or genes that do not interact synergistically14,60. In contrast, complex adaptations are typically underpinned by multitudes of synergistically interacting genes residing in the parliamentary majority (commonwealth)23. We are not claiming that appreciable trait distortion will never evolve, or that biological details will never matter14,32,59,60. Instead, our results suggest that the modal outcome will be a relative lack of trait distortion. This conclusion is supported empirically by cases where appreciable distortion is only revealed in hybrid crosses, implying that trait distorters are generally suppressed62. Furthermore, we find that, after suppression has evolved, trait distorters are generally purged from the population at equilibrium. If suppressors are constitutively expressed (obligate), trait distorters are not purged from the population, but in these cases, suppressors spread to fixation (Supplementary Note 6). Regardless of the extent to which suppressors are constitutive, there is negligible polymorphism in at least one locus, meaning trait distortion is unlikely to be revealed by mating within a population38. When trait distorters are not purged from the population, trait distortion will be revealed by matings between populations/species62. Sex ratio distorters as a case study: The relatively large literature on sex ratio distorters offers a chance for us to assess the validity of our models, and their predictions. In Supplementary Note 3, we detail how our assumptions are consistent with the biology of sex ratio distorters and their suppressors. For example, X drivers increase their own transmission by killing Y bearing sperm, and hence producing a female-biased offspring sex ratio. This comes at a cost to the rest of the genome through both a reduction in sperm number, and through Fisherian selection disfavouring the more common sex (females). The scope of the parliament of genes to act against such drivers is shown by the fact that, in most species in which an X driver is present, suppressors have been found on both the autosomes and the Y chromosome36. Our assumptions about how suppressors act, and the cost of suppression, are analogous to those in a molecularly characterised suppressor (Nmy) of a sex ratio distorter (Dox)46,60,61; and more generally to suppressors that act pre-translationally63,64. Our model predictions are consistent with the available data on X drivers in Drosophila. As predicted by our model: (1) Across natural populations of Drosophila simulans, there is a positive correlation between the extent of sex ratio distortion and the extent of suppression65. (2) In both Drosophila mediopunctata and D. simulans the presence of an X-linked driver led to the experimental evolution of suppression66,67. In addition, consistent with our model: (3) In natural populations of D. simulans, the prevalence of an X driver has been shown to sometimes decrease under complete suppression68. (4) Crossing different species of Drosophila has been shown to lead to appreciable sex ratio deviation, by unlinking trait distorters from their suppressors, and hence revealing previously hidden trait distorters62. Work on other sex ratio distorters has also shown that suppressors can spread extremely quickly from rarity, reaching fixation in as little as ~5 generations69. Individual fitness maximisation: We emphasise that when the assumption of individual fitness maximisation is made in behavioural and evolutionary ecology, it is not being assumed that natural selection produces perfect fitness maximisers5. Many factors could constrain adaptation, such as genetic architecture, mutation and phylogenetic constraints70,71. Instead, the assumption of fitness maximisation is used as a basis to investigate the selective forces that have favoured particular traits (adaptations). The aim is not to test if organisms maximise fitness, or behave ‘optimally’, but rather to try to understand the selective forces favouring particular traits or behaviours2. We have examined how the parliament of genes prevents selfish genetic elements from constraining adaptation, focusing on the maintenance, rather than the emergence, of traits (Supplementary Discussion). To conclude, debate over the validity of assuming individual level fitness maximisation has usually revolved around whether selfish genetic elements are common or rare4,20,21,24,72. We have shown that that even if selfish genetic elements are common, they will tend to be either weak and negligible, or suppressed. This suggests that even if there is the potential for appreciable genetic conflict, individual level fitness maximisation will still often be a reasonable assumption. This allows us to explain why certain traits, especially the sex ratio, have been able to provide such clear support for both individual level fitness maximisation and genetic conflict9. ## Methods ### Trait distorter population frequency We ask when a rare trait distorter (D1) can invade a population fixed for the trait non-distorter (D0). We take Eq. 1, set p′ = p = p, and solve to find two possible equilibria: p* = 0 (trait non-distorter fixation) and p* = 1 (trait distorter fixation). The trait distorter (D1) can invade from rarity when the p* = 0 equilibrium is unstable, which occurs when the differential of p′ with respect to p, at p* = 0, is >1. The trait distorter invasion criterion is therefore ctrait(k) < t(k)(1 − ctrait(k)). We now ask what frequency the trait distorter (D1) will reach after invasion. The trait distorter (D1) can spread to fixation if the p* = 1 equilibrium is stable, which requires that the differential of p′ with respect to p, at p* = 1, is <1. This requirement always holds true, demonstrating that there is no negative frequency dependence on the trait distorter, and that it will always spread to fixation after its initial invasion. ### Suppressor invasion condition We ask when the suppressor (S1) can spread from rarity in a population in which the trait distorter (D1) and non-suppressor (S0) are fixed at equilibrium. We derive the Jacobian stability matrix for this equilibrium, which is a matrix of each genotype frequency (x00 ′ , x01′ , x10′ , x11′ ) differentiated by each genotype frequency in the prior generation (x00, x01, x10, x11), at the equilibrium position given by x00* = 0, x01* = 0, x10* = 1, x11* = 0: $$J = \left( {\begin{array}{{20}{c}} {1 - t} & {\frac{{1 - c_{\mathrm{sup}}}}{{2(1 - c_{\mathrm{trait}})}}} & 0 & 0 \\ 0 & {\frac{{1 - c_{\mathrm{sup}}}}{{2(1 - c_{\mathrm{trait}})}}} & 0 & 0 \\ {t - 1} & {\frac{{ - 3(1 - c_{\mathrm{sup}})}}{{2(1 - c_{\mathrm{trait}})}}} & 0 & {\frac{{ - (1 - c_{\mathrm{sup}})}}{{1 - c_{\mathrm{trait}}}}} \\ 0 & {\frac{{1 - c_{\mathrm{sup}}}}{{2(1 - c_{\mathrm{trait}})}}} & 0 & {\frac{{1 - c_{\mathrm{sup}}}}{{1 - c_{\mathrm{trait}}}}} \end{array}} \right),$$ (7) The suppressor can invade when the equilibrium is unstable, which occurs when the leading eigenvalue is greater than one. The leading eigenvalue is (1 − csup)/(1 − ctrait), meaning the suppressor invasion criterion is ctrait > csup. ### Equilibrium trait distorter and suppressor frequencies We ask what frequency the trait distorter (D1) and suppressor (S1) will reach after initial suppressor (S1) invasion. We assume that the suppressor is introduced from rarity when the trait distorter has reached the population frequency given by f (x00 → f, x10 → 1 − f, {x01,x11} → 0). We numerically iterate Eqs. 25, over successive generations, until equilibrium has been reached. At equilibrium, for all parameter combinations (f, t,csup,ctrait), the suppressor reaches an internal equilibrium and the trait distorter is lost from the population (x00 + x01* = 1, x10* = 0, x11* = 0). This equilibrium arises because trait distorter presence gives the suppressor (S1) a selective advantage, leading to high suppressor frequency, which in turn reverses the selective advantage of the trait distorter (D1), leading to trait distorter loss and suppressor equilibration. ### Non-equilibrium trait distortion We consider a trait distorter that is suppressed and therefore purged at equilibrium (ctrait > csup), and ask to what extent it can contribute to individual trait distortion in the period after its initial invasion, but before its eventual loss (non-equilibrium). We introduce the trait distorter (D1) and suppressor (S1) from rarity and numerically iterate our recursions until the trait distorter has been purged from the population (or a cap of 20,000,000 generations has been reached). We vary parameters between 0 ≤ t ≤ 1, csup < ctrait ≤ 1, 0 ≤ csup ≤ 1. We find that a higher cost of trait distortion (ctrait) relative to suppression (csup) leads to shorter non-equilibrium maintenance of the trait distorter in the population. This is because the cost of trait distortion relative to suppression mediates selection on the suppressor (Methods: ‘Suppressor invasion condition’). We find that a higher transmission bias (t) leads to longer non-equilibrium maintenance of the trait distorter in the population, but this effect is diluted as the cost of trait distortion (ctrait) is increased relative to suppression (csup) (Supplementary Note 2, Supplementary Fig. 1). Stronger trait distorters (with higher k, leading to higher ctrait and t) are therefore generally suppressed and purged more rapidly than weaker trait distorters (Fig. 1b). Exceptions are trait distorters that reduce individual fitness relatively negligibly after the point (k) at which suppression is favoured, such that $$\frac{{{\mathrm{d}}t}}{{{\mathrm{d}}k}}/\frac{{{\mathrm{d}}c_{\mathrm{trait}}}}{{{\mathrm{d}}k}}$$ is very high for values of k satisfying csup < ctrait(k). ### Invasion of a mutant trait distorter We ask when a mutant trait distorter (D2) will invade against a resident trait distorter (D1) that is unsuppressed and at fixation (k ≠ $$\hat k$$). We write recursions detailing the generational frequency changes in the six possible gametes, D0/S0, D0/S1, D1/S0, D1/S1, D2/S0, D2/S1, with current generation frequencies denoted, respectively by x00, x01, x10, x11, x20, x21, and next-generation frequencies denoted with an appended dash (′): $$\begin{array}{l}\bar w\,x_{00}^{\prime} = x_{00}x_{00} + x_{00}x_{01} + (1 - t(k))(1 - c_{\mathrm{trait}}(k))x_{00}x_{10}\\ + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{00}x_{11} + (1 - t(\hat k))(1 - c_{\mathrm{trait}}(\hat k))\\ x_{00}x_{20} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{00}x_{21} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{01}x_{10} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{01}x_{20},\end{array}$$ (8) $$\begin{array}{l}\bar w\,x_{01}^{\prime} = x_{00}x_{01} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{00}x_{11} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{00}x_{21} + x_{01}x_{01} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{01}x_{10} + \left( {1 - c_{\mathrm{sup}}} \right)x_{01}x_{11}\\ + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{01}x_{20} + \left( {1 - c_{\mathrm{sup}}} \right)x_{01}x_{21},\end{array}$$ (9) $$\begin{array}{l}\bar w\,x_{10}\prime = (1 + t(k))(1 - c_{\mathrm{trait}}(k))x_{00}x_{10} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{00}x_{11} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{01}x_{10} + (1 - c_{\mathrm{trait}}(k))\\ x_{10}x_{10} + \left( {1 - c_{\mathrm{sup}}} \right)x_{10}x_{11} + (1 + t(k) - t(\hat k))\\ (1 - c_{\mathrm{trait}}(\mathrm{max}(k,\hat k)))x_{10}x_{20} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{10}x_{21}\\ + \, \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{11}x_{20},\end{array}$$ (10) $$\begin{array}{l}\bar w\,x_{11}\prime = \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{00} \times _{11} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{01}x_{10} + \left( {1 - c_{\mathrm{sup}}} \right)x_{01}x_{11} + \left( {1 - c_{\mathrm{sup}}} \right)\\ x_{10}x_{11} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{10}x_{21}\\ + \, \left( {1 - c_{\mathrm{sup}}} \right)x_{11}x_{11} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{11}x_{20} + \left( {1 - c_{\mathrm{sup}}} \right)x_{11}x_{21},\end{array}$$ (11) $$\begin{array}{l}\bar w\,x_{20}\prime = (1 + t(\hat k))(1 - c_{\mathrm{trait}}(\hat k))x_{00}x_{20} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{00}x_{21} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{01}x_{20} + (1 - t(k) + t(\hat k))\\ (1 - c_{\mathrm{trait}}({\mathrm{max}}(k,\hat k)))x_{10}x_{20} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{10}x_{21} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{11}x_{20} + (1 - c_{\mathrm{trait}}(\hat k))\\ x_{20}x_{20} + \left( {1 - c_{\mathrm{sup}}} \right)x_{20}x_{21},\end{array}$$ (12) $$\begin{array}{l}\bar w\,x_{21}\prime = \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{00}x_{21} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{01}x_{20} + \left( {1 - c_{\mathrm{sup}}} \right)x_{01}x_{21} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)x_{10}x_{21} + \left( {\left( {1 - c_{\mathrm{sup}}} \right)/2} \right)\\ x_{11}x_{20} + \left( {1 - c_{\mathrm{sup}}} \right)x_{11}x_{21} + \left( {1 - c_{\mathrm{sup}}} \right)x_{20}x_{21} + \left( {1 - c_{\mathrm{sup}}} \right)x_{21}x_{21},\end{array}$$ (13) where $$\bar w$$ is the average fitness of individuals in the current generation, and equals the sum of the right-hand side of the system of equations. The mutant trait distorter can invade when the equilibrium given by x00* = 0, x01* = 0, x10* = 1, x11* = 0, x20* = 0, x21* = 0 is unstable, which occurs when the leading eigenvalue of the Jacobian stability matrix for this equilibrium is >1. Testing for stability in this way, we find that, if the mutant trait distorter is weaker than the resident, it can never invade. If the mutant trait distorter is stronger than the resident, it invades from rarity when Δt(1 − ctrait($$\hat k$$)) > Δctrait, where Δt = t($$\hat k$$) − t(k), Δctrait = ctrait($$\hat k$$) − ctrait(k). The implication is that, if trait distortion is initially low, and mutant trait distorters are successively introduced, each deviating only very slightly from the resident trait distorter from which they are derived, such that $$\hat k$$ = k ± δ, where δ is very small (‘δ-weak selection’48), then trait distorters will approach a ‘target’ strength at which $$\frac{{{\mathrm{d}}t}}{{{\mathrm{d}}k}}\left( {1 - c_{\mathrm{trait}}} \right) = \frac{{{\mathrm{d}}c_{\mathrm{trait}}}}{{{\mathrm{d}}k}}$$. In the absence of suppression, this target (ktarget) is the equilibrium level of trait distortion (k* = ktarget). However, if mutant trait distorters (D2) are allowed to deviate appreciably from residents (D1) (strong selection), then trait distorters may invade even if they overshoot the target ($$\hat k$$ > ktarget). In the absence of suppression, ktarget is then not the equilibrium level of trait distortion, but rather, the minimum equilibrium level of trait distortion (k* ≥ ktarget) (Supplementary Note 2, Supplementary Fig. 2b). We could alternatively have assumed that an individual’s trait is distorted according to the average strength of its alleles (additive gene interactions), rather than according to the stronger (higher k) allele (dominance). Such an assumption leads to a single invasion criterion for a mutant trait distorter, regardless of whether the mutant trait distorter is stronger or weaker than the resident trait distorter, given by: Δt(2 ctrait(k) − ctrait($$\hat k$$)) > Δctrait. In the absence of suppression, this leads to an equilibrium level of trait distortion (k), which holds even under strong selection, and satisfies $$2\frac{{{\mathrm{d}}t}}{{{\mathrm{d}}k}}\left( {1 - c_{\mathrm{trait}}} \right) = \frac{{{\mathrm{d}}c_{\mathrm{trait}}}}{{{\mathrm{d}}k}}$$. ### Equilibrium allele frequencies after mutant invasion We ask what equilibrium state will arise after the invasion of a mutant trait distorter. We assume that the mutant trait distorter (D2) is introduced from rarity when the resident trait distorter (D1) has reached the population frequency given by q. We numerically iterate Eq. 813, over successive generations, until equilibrium has been reached. At equilibrium, for all parameter combinations (q, t(k), t($$\hat k$$), csup, ctrait(k), ctrait($$\hat k$$)), the resident trait distorter (D1) is lost from the population (x10,x11 = 0), with either the mutant trait distorter (D2) and non-suppressor (S0) at fixation (x20 = 1), or the trait non-distorter (D0) at fixation alongside the suppressor (S1) at an internal equilibrium (x00* + x01* = 1). The latter scenario arises if the mutant trait distorter triggers suppressor invasion (csup < ctrait($$\hat k$$)). This equilibrium arises because mutant trait distorter presence gives the suppressor (S1) a selective advantage, leading to high suppressor frequency, which in turn reverses the selective advantage of trait distortion, leading to trait distorter (D1,D2) loss and suppressor equilibration. ### Agent-based simulation (single trait distorter locus) We construct an agent-based simulation to ask what level of trait distortion evolves when continuous variation is permitted at trait distorter and suppressor loci. We model a population of N = 2000 individuals and track evolution at two autosomal loci: a trait distorter locus and a suppressor locus. Each individual has two alleles at the trait distorter locus, with strengths denoted by ka and kb, and two alleles at the suppressor locus, with strengths denoted by ma and mb (diploid). Strengths can take any continuous value between 0 and 1. We assume that, for both loci, the strongest (highest value) allele within an individual is dominant. The absolute fitness of an individual with at least one active meiotic driver (max(ka,kb) > 0) is: 1 − ctrait(max(ka,kb))(1 − max(ma,mb)) − csupmax(ma,mb), and the absolute fitness of an individual lacking an active trait distorter (max(ka,kb) = 0) is 1. The function ctrait(max(ka,kb)) is given an explicit form in simulations (Supplementary Note 2, Supplementary Fig. 2). In each generation, there are N breeding pairs. To fill each position in each breeding pair, individuals are drawn from the population, with replacement, with probabilities given by their fitness (hermaphrodites). Breeding pairs then reproduce to produce one offspring, before dying (non-overlapping generations). Alleles at the suppressor locus are inherited in Mendelian fashion. Alleles at the trait distorter locus may drive, meaning the parental allele of strength ka is inherited, rather than the allele of strength kb, with the probability (1 + (t(ka) − t(kb))(1 − max(ma,mb)))/2. The transmission bias function, t, is given an explicit form in simulations (Supplementary Note 2, Supplementary Fig. 2). Each generation, trait distorter and suppressor alleles have a 0.01 chance of mutating to a new value, which is drawn from a normal distribution centred around the pre-mutation value, with variance 0.2, and truncated between 0 and 1. We track the population average trait distorter strength, denoted by E[k], and suppressor strength, denoted by E[m], over 20,000 generations. We see that, allowing for continuous variation at the trait distorter and suppressor loci, if the cost of suppression (csup) is not excessively high, trait distortion at equilibrium is either low or nothing (Fig. 2a; Supplementary Note 2, Supplementary Fig. 2b). ### Long-term trait distortion (exact numerical solution) We ask how the trait distortion experienced by organisms changes across evolutionary time as new trait distorters and suppressors are continuously introduced and lost from a population. We construct a population genetic model and solve it numerically and exactly. We introduce a trait distorter from rarity and iterate our recursion for an unsuppressed trait distorter (Eq. 1) from T = 1 to $$T=1/((1-\theta)\gamma\rho_{S_1})$$ generations. During this period, the trait distortion experienced by individuals rises to a peak of k, corresponding to the strength of trait distorters available to the cabal. We then introduce a suppressor from rarity and iterate our recursions for trait distorter-suppressor co-segregation (Eqs. 25), from $$T=1/((1-\theta)\gamma\rho_{S_1})$$ until the trait distorter has been purged (T = X). During this period, the trait distortion experienced by individuals falls to a trough of 0. Average trait distortion over evolutionary time is given by weighting average trait distortion during the interval T = {1, 2, …, X} by the proportion of evolutionary time in which a trait distorter is segregating in the population $$(X(\theta\gamma\rho_{D_1}))$$. This methodology provides exact, numerical values for average trait distortion. These values correspond closely to the analytical approximation for average trait distortion (Eq. 6), which is derived under a separation of timescales assumption (Methods: ‘Long-term trait distortion (analytical approximation)’; Fig. 4). ### Long-term trait distortion (analytical approximation) When a trait distorter is initially introduced into the population, it will spread, and the population will equilibrate when the trait distorter reaches fixation (Methods: ‘Long-term trait distortion (exact numerical solution)’). Similarly, when a suppressor is initially introduced into the population, it will spread if its target trait distorter is sufficiently costly (csup < ctrait(k)), and the population will equilibrate when the suppressor’s target trait distorter is purged from the population (Methods: ‘Long-term trait distortion (exact numerical solution)’). We assume that, after the introduction of a new trait distorter or suppressor, the rate at which gene frequencies equilibrate is very fast relative to the rate at which new trait distorters and suppressors are introduced at new loci (separation of timescales). On this assumption, we can partition evolutionary time into two repeating periods. In the first period, comprising the $$1/((1-\theta)\gamma\rho_{S_1})$$ generations in between trait distorter and suppressor introduction, individual trait distortion is k. In the second period, comprising the following $$1/(\theta\gamma\rho_{D_1})-1/((1-\theta)\gamma\rho_{S_1})$$ generations, and ending when the next trait distorter is introduced at a new locus, individual trait distortion is 0. We average over these two time periods to calculate the average trait distortion experienced by individuals across evolutionary time (Eq. 6). ### Agent-based simulation (multiple loci; discrete) We build on the agent-based model detailed in Methods: ‘Agent-based simulation (single trait distorter locus)’ to capture the evolutionary dynamics of arbitrarily large numbers of co-segregating trait distorters and suppressors across the genome. The specific details of how mate partners are attributed (e.g. panmictic; hermaphrodite), and how the population is sampled to implement fitness effects (e.g. non-overlapping generations), are fully described in Methods: ‘Agent-based simulation (single trait distorter locus)’. We model a diploid population of N = 2000 individuals, each with γ = 106 loci, θγ of which constituting the cabal and (1 − θ)γ of which constituting the commonwealth. We assume that each locus across the genome is initially ‘dormant’. The alleles segregating in the population at dormant loci are neutral with respect to trait distortion and suppression. Loci are activated when the alleles segregating there have drifted to lie one mutational step away from distortion or suppression. For a given dormant locus in the cabal and in the commonwealth, the generational activation probability is given, respectively, by $$\rho_{D_1}$$ and $$\rho_{S_1}$$. Each successively activated cabal and commonwealth locus is indexed with a consecutive integer within the respective sets Icabal = {1, 2, …, ncabal} and Icommonwealth = {1, 2, …, ncommonwealth}, where ncabal and ncommonwealth give respectively the total number of activated cabal and commonwealth loci, which increase as generations (T) pass. After locus activation, alleles mutate between functional and neutral forms with a generational probability of 0.001. If, at any time, all trait distorters (iIcabal) have dedicated suppressors (iIcommonwealth), such that ncabal=ncommonwealth, further commonwealth loci cannot be activated until new trait distorters arise (ncabal > ncommonwealth). If trait distorters are low-sophistication as opposed to high-sophistication, the generational cabal locus activation probability ($$\rho_{D_1}$$) is increased by a factor two (such that $$\rho_{D_{1{\mathrm{L}}}}=2^\ast \rho_{D_{1{\mathrm{H}}}}$$). For each individual, the set IdistorterIcabal comprises every locus within the cabal where one (heterozygous) or two (homozygous) trait distorters are present. A given suppressor at a locus within the commonwealth (iIcommonwealth) is only expressed if its target trait distorter (iIdistorter) is also present in the individual. However, if expressed, a given suppressor (iIcommonwealth) may also contribute to the ‘background’ suppression of unsuppressed non-target trait distorters (Idistorter\i), at a fraction z of its usual strength. We assume that, for low-sophistication trait distorters (D1L), z = 0.5, and for high-sophistication trait distorters (D1H), z = 0. The total suppression faced by a trait distorter (iIdistorter) is therefore TotSupi = 1 if its dedicated suppressor is present in the individual, or TotSupi = min(zq,1) if its dedicated suppressor is absent, where q is the number of expressed suppressors present in the individual, and where the ‘min’ notation indicates that the total suppression cannot exceed 1 (complete suppression). The total cost of suppression for an individual is $$c_{\mathrm{sup}}\mathop {\sum}\nolimits_{i \in I_{\mathrm{distorter}}} {{\mathrm{TotSup}}_i}$$. The least suppressed trait distorter in each individual (idomIdistorter) exerts inter-locus dominance, and causes a trait distortion of $${\mathrm{Dist}} = \begin{array}{{20}{c}} {{\mathrm{max}}} \\ {i \in I_{\mathrm{distorter}}} \end{array}\left( {(1 - {\mathrm{TotSup}}_i)k} \right)$$. The individual cost of trait distortion, which is given by ctrait(Dist), increases monotonically with the extent that the trait is distorted $$\left( {\frac{{{\mathrm{d}}c_{\mathrm{trait}}}}{{\mathrm{dDist}}} \ge 0} \right)$$. Expression of the remaining ‘inter-locus recessive’ trait distorters (Idistorter\idom) leads to a pool of gene products with an abundance that is proportional to: $${\mathrm{Waste}} = \mathop {\sum}\nolimits_{\begin{array}{{20}{c}} {i \in I_{\mathrm{distorter}}} \\ {i \ne i_{\mathrm{dom}}} \end{array}} {((1 - {\mathrm{TotSup}}_i)k)}$$. The individual cost arising from inter-locus recessive trait distorters, which is given by crec, increases monotonically with the size of the pool of redundant gene products $$\left( {\frac{{{\mathrm{d}}c_{\mathrm{rec}}}}{{{\mathrm{dWaste}}}} \ge 0} \right)$$. We assume that, for low-sophistication trait distorters (D1L), the individual cost arising from any one inter-locus recessive trait distorter is equal to the cost of trait distortion itself $$\left( {c_{\mathrm{trait}}\left( {\mathrm{Dist}} \right) = \frac{{c_{\mathrm{rec}}\left( {\mathrm{Waste}} \right)}}{{\left\| {I_{\mathrm{distorter}}} \right\| - 1}} \ge 0} \right)$$. For high-sophistication trait distorters (D1H), this cost is lower relative to the cost of trait distortion $$\left( {c_{\mathrm{trait}}\left( {\mathrm{Dist}} \right) = \frac{{5(c_{\mathrm{rec}}\left( {\mathrm{Waste}} \right))}}{{3(\left\| {I_{\mathrm{distorter}}} \right\| - 1)}} \ge 0} \right)$$. The total fitness (viability) of an individual is then given by: $$1 - c_{\mathrm{trait}}(\mathrm{Dist}) - c_{\mathrm{rec}}(\mathrm{Waste}) - c_{\mathrm{sup}}\mathop {\sum}\nolimits_{i \in I_{\mathrm{distorter}}} {{\mathrm{TotSup}}_i}$$. We define the set IhetIdistorterIcabal as the collection of loci in an individual at which one (heterozygous) trait distorter, as opposed to two (homozygous) trait distorters, are present. The trait distorters at these loci (Ihet) drive at meiosis, as a unit. The least suppressed trait distorter in the group pulls the unit through meiosis, meaning the group of trait distorters (at loci Ihet) is inherited by each offspring with the probability $$(1 + \begin{array}{{20}{c}} {{\mathrm{max}}} \\ {i \in I_{\mathrm{het}}} \end{array}(1 - {\mathrm{TotSup}}_i)k)/2$$. ### Agent-based simulation (multiple loci; continuous) We adapt the simulation model detailed in Methods: ‘Agent-based simulation (multiple loci; discrete)’ so that trait distorters and suppressors are not of fixed strength (of k and 1, respectively), but are free to evolve continuously between 0 and 1. Homologous alleles at activated cabal loci (iIcabal) have strengths kai and kbi, and homologous alleles at activated commonwealth loci (iIcommonwealth) have strengths mai and mbi. Within an individual, the loci bearing trait distorters (IdistorterIcabal) each satisfy max(kai, kbi) > 0. Each trait distorter (at locus iIdistorter) is suppressed to the following extent: $${\mathrm{TotSup}}_i = \min \left( {\max \left( {m_{ai},m_{bi}} \right) + z\mathop {\sum}\nolimits_{\begin{array}{{20}{c}} {j \in I_{\mathrm{distorter}}} \\ {j \ne i} \end{array}} {\max \left( {m_{aj},m_{bj}} \right),1} } \right)$$. Within an individual, the strongest trait distorter (after suppression) is inter-locus dominant (idomIdistorter), and distorts the individual trait by: $${\mathrm{Dist}} = \begin{array}{{20}{c}} {{\mathrm{max}}} \\ {i \in I_{\mathrm{distorter}}} \end{array}\left( {(1 - {\mathrm{TotSup}}_i){\mathrm{max}}(k_{ai},k_{bi})} \right)$$. The inter-locus recessive trait distorters (Idistorter\idom) bring about an additional individual level cost of crec(Waste), which is a monotonically increasing function of $${\mathrm{Waste}} = \mathop {\sum}\nolimits_{\begin{array}{{20}{c}} {i \in I_{\mathrm{distorter}}} \\ {i \ne i_{\mathrm{dom}}} \end{array}} {((1 - {\mathrm{TotSup}_i){\mathrm{max}}}(k_{ai},k_{bi}))}$$. If an allele is more trait-distorting than its homologue (kai vs. kbi), it can drive at meiosis. The strongest alleles across each homologous pair drive together as a single unit. The unit is inherited by each offspring with the probability $$\left( {1 + \begin{array}{*{20}{c}} {{\mathrm{max}}} \\ {i \in I_{\mathrm{distorter}}} \end{array}\left( {1 - {\mathrm{TotSup}}_i} \right){\mathrm{abs}}\left( {k_{ai} - k_{bi}} \right)} \right)/2$$. Every generation, each allele at an activated locus has a 0.01 chance of mutating to a new strength, which is drawn from a normal distribution centred around the pre-mutation strength, with variance 0.2, and truncated between 0 and 1. ### Reporting summary Further information on research design is available in the Nature Research Reporting Summary linked to this article.	2023-03-26 03:50:37	{"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": 0, "mathjax_display_tex": 2, "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": 0.48648202419281006, "perplexity": 3392.189165302623}, "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/1679296945381.91/warc/CC-MAIN-20230326013652-20230326043652-00477.warc.gz"}
http://colorfulengineering.org/test.html	Quirky and romantic, "Stay Close, Little Ghost" is an ecstatic debut novel flecked with fever dream fairy tales: a girl with no eyes scratching personal messages into the walls of the subway tunnel, a woman whose running mascara streaks over her body as she fades into a shadow on the wall, a missing child seen swimming across a lake bottom, a lost secret city where love is like tasting the real thing after tasting the fast food version for your entire life, a house whose painted landscape and sky walls erupt into a blizzard, magic glasses that translate languages and see inside people, a map of all the four-leaf clovers in the city, miraculous diamonds that hold the key to any question in the world, and a little wolf that lives inside the stomach of the narrator. Against this fantastic backdrop, a wonderfully flawed mathematician unravels as he collides with the most famous open problem in modern computer science. The result is a love story unlike anything else. ## Praise for Stay Close, Little Ghost: • I find myself thinking of passages from Catcher in the Rye... Serang swims to the depths of his darkness and just as his lungs are about to burst, he turns back to the light. He sees beyond the fleeting sparks that we think are the essence of our lives. In the last few breaths of his book, Serang's voice arrests me. Until now, I've been aware of his adeptness at playing with the artifice of fantasy and fairy tale. He employs the imagery that these storytelling forms invite him to use but it's when he allows this structure to fall away that his voice transforms into the voice inside my head. He speaks to me directly and for the time it takes to read those last pages, I forget that I am holding a book in my hands.'' ### -The Uncustomary Book Review • It slams the rigidly logical vehicle of mathematical distillation into the hallucinatory fog of magical realism... Maybe it's because I'm coated in a little residual magic from recently revisiting the similarly feverish, preternaturally dreamlike world of Haruki Murakami, or because I've been wallowing in a surfeit of 30s-onset introspection about things that exist in a more distant past than their still-healing scars suggest, but Stay Close, Little Ghost offered one of those fated chance encounters of crossing paths with a novel at the absolute perfect time: It told me everything I've been needing to hear and I got to be the patiently, earnestly receptive audience it deserved.'' ★★★★½ / 5 ### -Chicago Center for Literature and Photography • Stay Close, Little Ghost is a slim little novel, but it delves into its subject in startling detail and in a manner that keeps the reader alert for nuances and clues. Oliver Serang's narrator probes for meaning and understanding in the mystical forces between two people with a hypersensitivity to those complex undercurrents, never relying on conventional descriptions or common notation. He makes use of unusual devices - a little wolf in the belly, for example, that sniffs and howls at those signals that go unnoticed by most people - as well as strange symbolism, portentous dreams and signs. At times, Oliver Serang's writing style reminded me of the Symbolist writers, of Maurice Maeterlinck or Fyodor Sologub, which is not a route commonly followed in modern literature. Serang however applies this style meaningfully, creatively and in an entirely modern context with the intention of delving more deeply to get past the convenient fallbacks that writers usually rely on when exploring this subject. It's not always easy to follow but it ensures that the book pulses with the vibrant imagination of this unique outlook without ever becoming precious or pretentious. It's like a literary form of synaesthesia, the author resorting to abstract visual references to describe the indescribable, mapping out relationships on a magic diamond, or finding a mathematical solution to the placement of four-leaved clovers in a city.'' ★★★★★ / 5 ### -Top 500 Amazon reviewer • Stay Close, Little Ghost, is reminiscent of Murakami's finest moments. It is unflinchingly honest, magically immersive and so imbued with heartache that it's like revisiting your Top Five All-Time Worst Breakups Ã la High Fidelity. But at all once. And completely devoid of self-pity's trappings because it's too stuffed with raw emotion's introspection to fit much of anything else.'' ### -The Next Best Book Blog • I felt like this novel was sort of the birth child of Haruki Murakami’s Norwegian Wood and Jonathon Safran Foer’s Extremely Loud and Incredibly Close... I had to slow down my reading a bit and absorb it as much as possible. This prose is quite a beauty.''	2015-03-05 20:11:49	{"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": 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": 0.20860731601715088, "perplexity": 5224.557069566287}, "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-2015-11/segments/1424936464840.47/warc/CC-MAIN-20150226074104-00138-ip-10-28-5-156.ec2.internal.warc.gz"}
https://online.ucpress.edu/collabra/article/6/1/17174/114340/Does-Scientists-Believe-Imply-All-Scientists	Media headlines reporting scientific research frequently include generic phrases such as “Scientists believe x” or “Experts think y”. These phrases capture attention and succinctly communicate science to the public. However, by generically attributing beliefs to ‘Scientists’, ‘Experts’ or ‘Researchers’ the degree of scientific consensus must be inferred by the reader or listener (do all scientists believe x, most scientists, or just a few?). Our data revealed that decontextualized generic phrases such as “Scientists say…” imply consensus among a majority of relevant experts (53.8% in Study 1 and 60.7-61.8% in Study 2). There was little variation in the degree of consensus implied by different generic phrases, but wide variation between different participants. These ratings of decontextualized phrases will inevitably be labile and prone to change with the addition of context, but under controlled conditions people interpret generic consensus statements in very different ways. We tested the novel hypothesis that individual differences in consensus estimates occur because generic phrases encourage an intuitive overgeneralization (e.g., Scientists believe = All scientists believe) that some people revise downwards on reflection (e.g., Scientists believe = Some scientists believe). Two pre-registered studies failed to support this hypothesis. There was no significant relationship between reflective thinking and consensus estimates (Study 1) and enforced reflection did not cause estimates to be revised downwards (Study 2). Those reporting scientific research should be aware that generically attributing beliefs to ‘Scientists’ or ‘Researchers’ is ambiguous and inappropriate when there is no clear consensus among relevant experts. Readers, listeners, and watchers of news media frequently encounter attention-grabbing headlines reporting scientific research. Listed below are three genuine examples taken from mainstream media outlets. 1. Scientists believe the secret of a good night’s sleep is all in our genes (The Guardian, 2017) 2. Experts think early humans ate grass (BBC, 2012) 3. Eating more nuts could slow weight gain, researchers say (Sky News, 2019) These brief snippets capture attention and succinctly communicate science to the public, but their brevity creates ambiguity. Do all scientists believe the secret of a good night’s sleep is in our genes, most scientists or just some scientists? Because the noun phrase is unquantified (Scientists believe…) rather than universally (all scientists believe…) or exactly quantified (one scientist believes…), readers must form their own subjective interpretation about the degree of scientific consensus. Statements such as 1-3 are known as ‘generics’. They generically attribute a claim to ‘scientists’ rather than specifically to one scientist or one group of scientists. Generics are not just limited to communicating science but appear frequently in everyday discourse. Phrases such ‘Ducks lay eggs’, ‘Tigers have stripes’ and ‘women read magazines’ are not unusual (Abelson & Kanouse, 1966; Leslie, 2008). Generics, however, do have an unusual property: they require little evidence to be judged as true (Cimpian et al., 2010). While ‘All men like DIY’ might be considered false (perhaps because it only takes one counterexample to falsify this claim), the generic ‘Men like DIY’ is more likely to be accepted as true. Such generics are relatively immune to counterexamples, with statements such as ‘Ducks lay eggs’ or ‘Mosquitos carry the West Nile virus’ seeming to be true even when these claims are clearly not true for all members of the category (i.e., male ducks do not lay eggs and 99% of mosquitos do not carry West Nile virus) (Leslie, 2008). Generics are frequently used to report primary scientific research (DeJesus et al., 2019). Academic papers routinely make general claims about Humans, Adults, Males, Children, Introverts and Extroverts etc., that gloss over variation within each category (e.g., DeJesus et al., 2019; Simons et al., 2017). Generic phrases are also common in secondary reporting of scientific research by the news media. One specific function of generics, identified above, is to attribute the source of scientific claims using phrases such ‘Scientists say…’ or ‘Experts believe…’ (Robbins, 2012). Just as generic claims about ‘Males’ gloss over variation among males (DeJesus et al., 2019), generic claims about ‘Scientists’ gloss over variation in scientific opinion. Sacrificing precision for simplicity creates ambiguity. For example, the generic phrase ‘Experts think…’ can truthfully refer to almost any proportion of relevant experts.1 It is up to the reader or listener to infer the degree of consensus. One reader may take this phrase to mean ‘All’ relevant experts (e.g., a definitive consensus statement), another may take this to mean ‘Most’ relevant experts while another may perceive it as a claim relating to just one specific subset of experts (e.g., the specific authors of a Study). The purpose of the two studies presented below is to reveal 1) the degree of scientific consensus implied by commonly used generic phrases such as “Scientists say…”, “Experts think…” and “Researchers believe…” 2) to reveal the extent to which estimates of consensus vary between people and between phrases, and 3) to examine whether this variance is associated with the tendency to engage in cognitive reflection. Cognitive Reflection Put simply, cognitive reflection is the tendency to reflect on our intuitive ‘gut feelings’. Cognitive reflection is generally understood in the context of dual process models of human thinking (e.g., Toplak et al., 2011). Dual process models are based on the premise that humans engage in two distinct types of thinking. The first (generally known as Type 1 or intuitive thinking) is fast, instinctive and heuristic driven (e.g., correctly verifying that 2+2=4). It is independent of cognitive ability, delivering fast but not always accurate judgements (i.e., it is particularly susceptible to cognitive biases). The second type (generally known as Type 2 or reflective thinking) is a slower, more resource demanding and conscious type of thinking that is related to individual differences in cognitive ability (e.g., correctly verifying that 17x8=136). Several prominent dual process accounts take a default-interventionist approach to explain how the two systems work together (e.g., Evans & Stanovich, 2013). Such theories assume that intuitive Type 1 processes act first, providing a fast and intuitive default response. Cognitively demanding Type 2 processes then intervene to revise that output, if required. Often Type 1 processing suffices to make an accurate judgment or decision (e.g., correctly categorising an object as human or non-human, correctly verifying that 2+2=4, correctly recognising a friend etc.). Other times the output of type 1 intuitive thinking can fail to produce a normative response. The following problem is a classic example of how Type 1 processes can be misled: A bat and a ball cost $1.10 in total. The bat costs$1.00 more than the ball. How much does the ball cost? (in cents) ________ For those seeing this question for the first time, the answer that quickly comes to mind is ‘10 cents’. This is the fast and intuitive output of default Type 1 processing. Some people are content to retain this default answer without further reflection. Other people have a tendency to reflect on and revise their gut feelings. In other words, they are more likely to allow Type 2 processes to intervene. Those who do allow this intervention soon realise that the default response (10 cents) is incorrect and revise their answer to ‘5 cents’. Those people who are more likely to spontaneously reflect on their intuitions are said to have higher levels of Cognitive Reflection (Frederick, 2005). Cognitive Reflection and scientific consensus We propose that generic news media headlines lure readers towards an intuitive Type 1 conclusion in a similar way to the bat and ball problem. Specifically, we propose that generic headlines encourage the ‘hasty generalisation’ that ‘Experts = All experts’. This is consistent with work demonstrating that people are susceptible to overgeneralising from generic statements; a phenomenon known as the ‘Generic Overgeneralization Effect’ (Leslie et al., 2011). If a generic statement is believed to be true (e.g., Ducks lay eggs) then some people erroneously overgeneralize that the equivalent universal statement is also true (e.g., All ducks lay eggs). Likewise, when participants have no knowledge about the truth of a generic (e.g., “Lorches have purple feathers”) some perceive it as referring to nearly all members of that category (i.e., nearly all Lorches have purple feathers) (Cimpian et al., 2010). In the context of generic news headlines (e.g., Experts believe…) a hasty (over)generalisation would be that the opinion of ‘experts’ is the opinion of all or nearly all experts. This would be consistent with the work of Aklin and Urpelainen (2014) who concluded that in the absence of any dissenting information “...the general public's default assumption is a very high degree of scientific consensus”. However, a little reflection (Type 2 thinking) reveals this generalisation is not necessarily true. If an individual reflects, they may identify counterexamples to their intuition. For example, they might question the plausibility of their initial judgement (complete consensus on any topic is rare), they might question the semantics of the phrase (‘experts’ could actually refer to just a few experts) or they might question the source of the message (the news media often sensationalise and gloss over the details). The application of Type 2 reflective thinking may therefore lead to the hasty generalisation being revised downwards (e.g., from ‘most’ or ‘all’ experts to ‘some experts’). Our hypothesis is that people initially interpret generic phrases reporting scientific findings by making a hasty (over)generalisation (e.g., Experts think = All experts think). Those who reflect on this appealing inference are more likely to identify reasons (counterexamples) to revise the initial estimate of consensus downwards. Variation in estimates of consensus should therefore be negatively associated with variance in cognitive reflection (i.e., more reflective people perceive a lower degree of consensus). This hypothesis was tested by two pre-registered studies. The first examined whether an individual's degree of trait Cognitive Reflection (measured both objectively and subjectively) is associated with the degree of consensus they perceive when reading common news media phrases such as ‘Scientists believe…’ and ‘Experts say…’. We predicted that more reflective people would produce lower estimates of consensus, as they are more likely to override and revise their hasty (over)generalizations. The second Study then used time pressure and cognitive load to experimentally induce intuitive (Type 1) processing as participants rated the degree of consensus. We then encouraged them to actively reflect on their initial judgement with the option to revise it. By manipulating state reflection using this ‘two-response paradigm’ (Thompson et al., 2011) we aimed to determine whether there is a causal link between processing style and the revision of hasty (over)generalisations. We predicted that participants would produce relatively high estimates of scientific consensus (hasty generalisations) when engaged in fast Type 1 processing but would revise these downwards when encouraged to reflect on their initial gut feeling. Study 1 Method Participants were asked to estimate the degree of scientific consensus implied by common generic phrases such as “Scientists say…” and “Experts believe…”. We predicted that variance in these estimates would negatively correlate with variance in the ability to reflect on and override appealing but incorrect intuitions. We chose to approach this question in the most controlled way possible, by eliminating the influence of context and prior beliefs on the interpretation of our generic phrases. To do this we used decontextualized phrases (e.g., Scientists believe…) rather than fully contextualised phrases. Consensus ratings of a contextualised headline such as “Scientists believe climate change is due to human activity” are more likely to be based on an individual’s prior beliefs about consensus on climate change than on the phrase “Scientists believe”. By presenting the generic phrases in isolation, we can be sure that estimates of consensus relate to the specific generic phrase and not to prior beliefs about specific issues. The aim is to generate relatively ‘pure’ estimates of consensus implied by different generic phrases. We acknowledge that these consensus estimates will inevitably change with the addition of context (e.g., “Scientists believe climate change is due to human activity” is likely to imply greater scientific consensus than the decontextualized “Scientists believe…”. Likewise, we would expect that “Scientists believe the earth is flat” to implies less scientific consensus. Our goal is to examine the degree of consensus implied by generic phrases in isolation. Consensus estimates will inevitably vary between participants and we predict that this variability will be associated with variability in Cognitive Refection. The following protocol was pre-registered on the Open Science Framework prior to data collection https://osf.io/n7pj8/. Design A cross-sectional design was used with the measures being: (i) an objective measure of cognitive reflection (CRT-7) (ii) a subjective measure of rational ability (REI Rational Ability subscale), (iii) a subjective measure of rational engagement (REI Rational Engagement subscale) (iv) scientific consensus estimates for nine decontextualized target phrases (e.g., “Scientists say…”, “Experts believe…”). Participants Because we planned to analyse the data using structural equation modelling, a power simulation was performed using the simsem package in R (code for this simulation can found on the Study OSF page). This indicated that a minimum sample of 200 participants would be required to achieve power of .8 (ɑ=.05, two-tailed). We pre-registered a target sample size of 350 to account for possible data exclusions and to achieve a level of power in excess of .8. Participants were recruited online via the www.prolific.co participant pool. This pool consists of over 70,000 registered users. A recent comparison of participant pools showed that Prolific users are naiver and more diverse than MTurk users, while providing a comparable quality of data (Peer et al., 2017). Pre-screening ensured that the Study was only advertised to those aged 18 years or older and who spoke English as their first language. IP addresses were not collected as Prolific take a number of steps to avoid duplicate responses and automated responses by bots (Bradley, 2018). In total, 355 participants consented to take part. Pre-registered exclusion criteria dictated that participants would be excluded if they did not complete the survey (these were considered to have withdrawn, as per our ethical approval conditions) or declared that they did not complete the survey seriously. Four participants did not complete the survey and were excluded. All remaining participants declared that they responded seriously, leaving a final sample of 351 participants aged 18-72 (Mage = 35.03, SD = 11.33). Of these 113 identified as male, 237 identified as female and one selected ‘Other’. Participants were paid £1.30. Materials The survey was constructed using the Qualtrics online survey platform. Measures of thinking style Cognitive reflection was measured both objectively by the Cognitive Reflection Test and subjectively by the REI Rational Ability and Rational Engagement subscales. Higher scores on the CRT imply objectively greater ability reflect on and override Type 1 intuitions. Likewise, higher scores on the REI rationality subscale indicate a greater engagement with effortful, reflective thinking. Objective measure of Cognitive Reflection. The Cognitive Reflection Test was used as an objective measure of reflective thinking. The validity of the original three-item CRT (Frederick, 2005) has been threatened by the widespread publication of the test materials, such that individuals with prior exposure to materials score significantly higher than those with no prior exposure (Haigh, 2016; Stieger & Reips, 2016). This raw score increase does not affect the Test’s predictive ability (Bialek & Pennycook, 2018) but to minimise the impact of prior exposure we opted to use the longer CRT-7 (Toplak et al., 2014) which contains additional, less familiar questions. Designed to assess the ability to engage in analytic thinking, it assesses the tendency to override an appealing but incorrect intuitive response and engage in further reflection leading to the correct response. The CRT-7 comprises of seven mathematically worded problems (including the bat and ball problem) that cue an initial incorrect intuitive response that must be overridden to arrive at the correct conclusion. The overall CRT-7 score was calculated as the sum of all correct responses, with higher scores indicating a more reflective thinking style. The lowest possible score was 0 and the highest 7. In this Study the internal reliability measured using Cronbach's alpha was 0.77. Subjective measure of Cognitive Reflection. The Rational Ability and Rational Engagement subscales of the Rational-Experiential Inventory (REI; Pacini & Epstein, 1999) were used as a measure of subjective preference for Type 2, reflective thinking. Participants were asked to read each statement and rate the extent that the statements referred or did not refer to them: e.g. “I have a logical mind” (1 = definitely not true of myself, 5 = definitely true of myself). The presentation order of items was randomised (see Keaton, 2017). Seventeen items were reverse scored and subscale scores were calculated as the mean of the relevant items. In this Study the internal reliability measured using Cronbach's alpha was 0.83 for the Rational Ability subscale and 0.86 for the Rational Engagement subscale. Scientific Consensus measure. Participants were presented with nine decontextualized generic phrases (e.g., “Scientists say…”, “Experts believe…”, “Researchers think…”) made up of a subject and a verb followed by ellipsis. See Table 1 for the list of phrases used. Table 1: Target phrases rated by participants in Study 1 Scientists believe... Researchers believe... Experts believe... Scientists say... Researchers say... Experts say... Scientists think... Researchers think... Experts think... Scientists believe... Researchers believe... Experts believe... Scientists say... Researchers say... Experts say... Scientists think... Researchers think... Experts think... Participants were asked to estimate how many relevant scientists [experts/researchers] they thought each phrase applied to on a sliding scale ranging from zero ‘no [scientists/experts/researchers] ’ to 100’all [scientists/experts/researchers]’. “Please estimate how many relevant [scientists/experts/researchers] you think this statement applies to. We are simply interested in your personal opinion. There is no right or wrong answer.” The scale was not numbered, however a number between 0-100 was visible to participants as they moved the slider. The subjects of our nine target phrases were “Scientists” [three items], “Experts” [three items] and “Researchers” [three items] followed by a base verb that implied some degree of consensus (e.g., ‘believe’, ‘say’, ‘think’). The three subjects were chosen as a corpus search showed they are frequently used as general terms to describe the authors of scientific work. Other subjects considered after searching for synonyms of the three nouns above were Academics, Scholars, Doctors, Lecturers and Professors but a corpus search revealed that these were infrequently used to convey scientific consensus. More specific job titles such as ‘psychologists’, ‘neuroscientists’ or ‘physicists’ were considered beyond the scope of this investigation. The subjects were paired with verbs that implied some degree of scientific consensus. To select the verbs, we searched the British National Corpus using an online interface (https://www.english-corpora.org/bnc/) to identify the 30 base verbs that most frequently collocate with “Scientists”, “Experts” and “Researchers”. From these lists we selected three base verbs that frequently accompanied each of our subjects. These were: 1. believe’ (ranked as the most frequent collocate of ‘Scientists’ and ‘Researchers’, ranked second most frequent collocate of ’Experts) 2. say’ (ranked as the most frequent collocate of ‘Experts’ and second most frequent collocate of ‘Scientists’ and ‘Researchers’) 3. think’ (ranked as the third most frequent collocate of ‘Scientists’ and ‘Researchers’ and the seventh most frequent collocate of ‘Experts’) The nine target items were presented amongst nine filler items, three of which referred to ‘Some [scientists/experts/researchers] …", three to ’Many [scientists/experts/researchers] …’ and three to ‘Few [scientists/experts/researchers] …’. This was to ensure that participants remained engaged and used the full range of the scale. These were paired with three verbs that frequently accompany the subjects (these were ‘suggest’, ‘agree’, ‘argue’). These different verbs were chosen to make the task less repetitive for participants. The filler phrases are presented in Table 2. Table 2: Filler phrases rated by participants in Study 1 Some scientists argue… Some researchers suggest... Some experts agree... Many scientists suggest... Many researchers agree... Many experts argue... Few scientists agree... Few researchers argue... Few experts suggest... Some scientists argue… Some researchers suggest... Some experts agree... Many scientists suggest... Many researchers agree... Many experts argue... Few scientists agree... Few researchers argue... Few experts suggest... The internal reliability measured using Cronbach's alpha was 0.97. Seriousness check. To exclude non-serious responses, participants were asked at the end of the Study whether they took part seriously. Seriousness checks have been suggested to substantially improve the quality of data collected (Aust et al., 2013). Participants were advised they would still be paid even if they did declare non-serious responding. Our pre-registered exclusion criteria dictated that non-serious participants would be excluded from the analysis. Procedure Ethical approval for all studies in this paper was granted through the faculty ethics committee of the authors’ institution. An information sheet at the start of the Qualtrics survey informed participants they would be asked about their understanding of some phrases commonly used by the media, be asked to solve some simple problems and to answer questions regarding their thinking style. Participants first saw the 18 phrases (9 target items plus 9 fillers) and provided consensus estimates for each. These were presented one per page in a different random order to each participant. They then completed the CRT-7, the REI (with statements presented in a different random order for each participant), and then the seriousness check. All items required a response. Participants could not progress beyond a page until all questions had been answered. Mean completion time was 13.15 minutes (SD = 7.71). Results Pre-registered analysis The following analyses were pre-registered on the Open Science Framework prior to data collection https://osf.io/n7pj8/registrations. The raw data and R analysis script are publicly available via the OSF https://osf.io/n7pj8/. There were no missing data as all questions required a response. The mean CRT-7 score was 2.65 correct answers (SD= 2.16). The mean REI rational ability and rational engagement subscale scores were 3.48 (SD= 0.62) and 3.44 (SD=0.67) respectively. The mean rating of consensus implied across our nine generic phrases was 53.82 (SD= 25.60) on a 0-100 scale (see Figure 1 for mean ratings of each individual phrase). As we aimed to measure the extent to which estimates of consensus vary between people and between phrases, we calculated descriptive statistics averaged over our 351 participants (M=53.82, SD=25.60) and averaged over our nine items (M=53.82, SD=2.48) separately. The variability in means between participants was approximately ten times greater than the variability between items. Figure 1: Mean degree of consensus implied by the phrases used in Study 1 (Averaged over 351 participants, Error bars represent one standard deviation). Generic phrases in bold. The scale was anchored at 0 (no scientists/experts/researchers) and 100 (all scientists/experts/researchers). Figure 1: Mean degree of consensus implied by the phrases used in Study 1 (Averaged over 351 participants, Error bars represent one standard deviation). Generic phrases in bold. The scale was anchored at 0 (no scientists/experts/researchers) and 100 (all scientists/experts/researchers). Consistent with previous research, the CRT-7 was positively correlated with the REI Rational Ability (r(349) =.285 p<.001) and REI Rational Engagement (r(349) =.276 p<.001) subscales. Ratings of all nine generic consensus ratings were strongly and significantly correlated with each other (correlations ranged from r(349) = .74, p<.001 to r(349) = .86, p<.001). Following our pre-registered analysis plan we used Structural Equation Modelling (SEM) to examine the direct path between a latent 'Cognitive Reflection' variable and a latent 'Scientific Consensus' variable (see Figure 2). We predicted there would be a significant negative relationship between these latent variables (i.e., more reflective people make lower estimates of consensus). The latent Cognitive Reflection variable was measured using CRT-7 total score, REI Rational Ability subscale and REI Rational Engagement subscale. The latent 'Scientific Consensus' variable was measured using consensus ratings to the nine generic phrases described above. The filler phrases qualified with ‘some’, ‘many’ and ‘few’ were not part of the latent 'Scientific Consensus' variable. Analysis was conducted using the lavaan package (Rosseel, 2012) in R version 3.6.0 (R.Core Team, 2019). Prior to conducting the analysis, scientific consensus ratings were transformed to z scores to reduce the difference in variance magnitude between these ratings and the other measures, this was unforeseen but required as our models did not converge. This transformation was not pre-registered but transformations such as these are commonly employed to achieve convergence and do not affect the fundamental results (Little, 2013, p. 17). Model fit was assessed using $χ2$, CFI, TLI and RMSEA. The model $χ2$ was significant ($χ2(53)=141.848,p<.001)$, which is typical for large samples (Bollen, 1989; Lance & Vandenberg, 2001). CFI (0.979) and TLI (0.974) values were greater than 0.95 and RMSEA was 0.069 (90% CI: 0.055 - 0.083) indicating an adequate model fit. Figure 2 shows that there was a negative relationship between the latent 'Cognitive Reflection' variable and a latent 'Scientific Consensus' variable, but this was weak and not statistically significant $(β=−0.063,SE=0.075,p=.304)$. Figure 2: Structural equation model of the relationship between Cognitive Reflection and estimates of Scientific Consensus implied by nine generic phrases. Cognitive Reflection and Scientific Consensus are latent variables (Standardised solution, N=351). The model $χ2$ was significant ($χ2$ (53) =141.848, p<.001), CFI (0.979) and TLI (0.974) values were greater than 0.95 and RMSEA was 0.069 (90% CI: 0.055 - 0.083). Figure labels: CRT-7 = Cognitive Reflection Test (7 item version), REI-RA = REI Rational Ability subscale, REI-RE = REI Rational Engagement subscale, SB= Scientists believe…, SS= Scientists say…, ST= Scientists think…, RB= Researchers believe…, RS= Researchers say…, RT= Researchers think…, EB= Experts believe…, ES= Experts say…, ET= Experts think... Figure 2: Structural equation model of the relationship between Cognitive Reflection and estimates of Scientific Consensus implied by nine generic phrases. Cognitive Reflection and Scientific Consensus are latent variables (Standardised solution, N=351). The model $χ2$ was significant ($χ2$ (53) =141.848, p<.001), CFI (0.979) and TLI (0.974) values were greater than 0.95 and RMSEA was 0.069 (90% CI: 0.055 - 0.083). Figure labels: CRT-7 = Cognitive Reflection Test (7 item version), REI-RA = REI Rational Ability subscale, REI-RE = REI Rational Engagement subscale, SB= Scientists believe…, SS= Scientists say…, ST= Scientists think…, RB= Researchers believe…, RS= Researchers say…, RT= Researchers think…, EB= Experts believe…, ES= Experts say…, ET= Experts think... Exploratory analysis In addition to the nine generic phrases used in Study 1, which were the focus of this Study, participants also saw nine ‘filler’ phrases which were included to make the task less repetitive. Three of these phrases were quantified with ‘some’, three with ‘many’ and three with ‘few’ (see Table 2). During the peer review process, it became apparent that our hypothesis that people overestimate consensus and then revise downwards on reflection, may also be relevant to the interpretation of verbal quantifiers more generally (i.e., phrases such as ‘many scientists…’ and ‘few scientists…’). To explore this possibility, we calculated bivariate correlations between each of our nine fillers and our three measures of cognitive reflection (see Table 3). Interpretations of the ‘many’ fillers were largely unrelated to our cognitive reflection measures. Interpretations of the three ‘Few’ statements had weak negative relationships with our three measures of cognitive reflection. There was also some evidence of a similar pattern with our ‘some’ statements. Table 3: Exploratory bivariate correlations (Pearson’s r) between each of our filler phrases and our three measures of cognitive reflection. CRT-7Rational AbilityRational Engagement Some scientists argue… -0.12* -0.06 -0.06 Some researchers suggest... -0.1† -0.04 -0.09 Some experts agree… -0.13* -0.1† -0.12* Many scientists suggest... -0.03 -0.03 -0.12* Many researchers agree... 0.04 -0.02 -0.07 Many experts argue… 0.0001 -0.03 -0.05 Few scientists agree... -0.21 -0.16 -0.15 Few researchers argue... -0.16 -0.13* -0.11* Few experts suggest... -0.14 -0.15 -0.1† CRT-7Rational AbilityRational Engagement Some scientists argue… -0.12* -0.06 -0.06 Some researchers suggest... -0.1† -0.04 -0.09 Some experts agree… -0.13* -0.1† -0.12* Many scientists suggest... -0.03 -0.03 -0.12* Many researchers agree... 0.04 -0.02 -0.07 Many experts argue… 0.0001 -0.03 -0.05 Few scientists agree... -0.21 -0.16 -0.15 Few researchers argue... -0.16 -0.13* -0.11* Few experts suggest... -0.14 -0.15 -0.1† Note: †p<0.1, p<0.05, *p<0.01 Discussion The first aim of this paper was to reveal the degree of scientific consensus implied by commonly used generic phrases such as “Scientists say…”, “Experts think…” and “Researchers believe”. Study 1 shows that such decontextualized phrases imply a slim majority of scientists (Mean estimate of consensus was 53.8% of relevant scientists/experts/researchers). This mean consensus estimate was greater than the mean estimates for filler phrases quantified with ‘Few’ (17.7%) or ‘Some’ (32.1%) and lower than fillers quantified with ‘Many’ (59.3%) (see Figure 1). The second aim was to reveal the extent to which estimates of consensus vary between people and between phrases. Mean estimates varied very little between our nine commonly used phrases (SD=2.5) but varied widely between our 351 participants (SD=25.6). On average, the nine phrases we selected each implied a similar level of consensus (ranging from 50.1% to 56.9%) suggesting that generics imply a similar degree of consensus, regardless of the subject or verb used (e.g., ‘Researchers say...’ tends to be interpreted in much the same way as ‘Scientists think...’). In contrast, our participants behaved very differently from each other, with some estimating that, on average, the nine statements referred to as few as 2% of relevant scientists and others estimating that they refer to 100% of relevant scientists. Both extremes are plausible interpretations of a generic. The third aim was to examine whether this variance in estimates between participants is associated with variance in cognitive reflection. We predicted that a latent Cognitive Reflection variable (in which cognitive reflection was measured both objectively and subjectively) would be negatively associated with estimates of consensus implied by generic phrases (e.g., Experts think…). This prediction was based on the hypothesis that generic phrases encourage people to make an intuitive overgeneralization (e.g., Experts think = All experts think) that is then revised downwards by those who spontaneously engage in cognitive reflection. A weak negative relationship was observed between cognitive reflection and estimates of consensus, but this was not statistically significant. Therefore, the data from this highly powered Study do not support our hypothesis. Exploratory analysis was also conducted on the filler items (i.e., phrases quantified with ‘some’, ‘many’ and ‘few’) which was not pre-registered. This allowed us to explore whether the hypothesis that people overestimate consensus and then revise downwards on reflection may be relevant to the interpretation of verbal quantifiers more generally (i.e., phrases such as ‘many scientists…’ and ‘few scientists…’). This analysis revealed weak negative relationships between filler phrases quantified with ‘few’ (e.g., Few scientists agree...) and our measures of cognitive reflection. A similar but less consistent pattern was observed for fillers quantified with ‘some’. This suggests that those who are more reflective tend to give lower consensus estimates to phrases quantified with ‘few’ and ‘some’, consistent with the idea that people initially overestimate consensus and revise their estimate downwards on reflection. Future confirmatory research is required to confirm this negative relationship between the interpretation of verbal quantity phrases (such as ‘few’ and ‘some’) and cognitive reflection. One explanation for the null findings in Study 1 is that estimates of scientific consensus implied by generic phrases are unrelated to trait cognitive reflection. However, before accepting this conclusion, an alternative possibility should be explored. This possibility is that the Study protocol actively encouraged participants to engage in analytical thought, making estimates of consensus insensitive to trait variance in cognitive reflection. Explicitly asking participants to assign a quantity to the nine phrases may have encouraged them to reflect on their meaning in a much deeper way than they typically would (i.e., engaging Type 2 thinking). Therefore, all participants may have revised their initial estimates before giving a final response, rather than just those who more readily engage in reflective thinking. In Study 2 we sought to overcome this limitation and determine whether there is a cause and effect relationship between processing style and estimates of consensus. To do this, we used a two-response paradigm (Thompson et al., 2011) that was designed to force an intuitive estimate (response 1), before giving participants the opportunity to reflect on and revise that estimate (response 2). If estimates of consensus are related to thinking style, then this direct experimental manipulation will result in relatively higher estimates of consensuses when the initial intuitive estimate is given (due to participants over generalising) and relatively lower estimates when they are encouraged to reflect on their intuition. Study 2 Method In this second online Study, participants were asked to estimate the degree of scientific consensus implied by common generic phrases such as “Scientists say…” at two time points. At Time 1 (T1) participants made an intuitive consensus judgement under cognitive load and time pressure to encourage Type 1 processing by suppressing Type 2 processing. At Time 2 (T2) the cognitive load was removed, and participants were given unlimited time to consider their first response with the option to revise their estimate, if desired. We predicted that intuitive consensus estimates at T1 would be relatively high and would be revised downwards on reflection at T2. This is an adaption of the two-response paradigm developed by Thompson et al. (2011). The following protocol was pre-registered on the Open Science Framework prior to data collection https://osf.io/2vh9w/. Design A repeated measures experimental design was used with the response time point (T1 and T2) being the repeated measures independent variable and the degree of scientific consensus participants assigned to the target phrase the dependent variable. Participants Power analysis was conducted assuming one fixed factor (which is repeated measures with two levels) and two random factors (participants and items). Power analysis was conducted using the two random factor Power calculator developed by Westfall et al. (2014). With an anticipated effect size of d=0.5, we found that the most efficient and practical design to achieve power of 0.8 would require a minimum of 16 target items and 118 participants. As per Study 1, participants were recruited online via the www.prolific.co participant pool. Pre-screening ensured that the Study was only advertised to those aged 18 years or older, who spoke English as their first language and who did not take part in either Study 1 or the pre-test for Study 2. In total, 194 participants consented to take part. Participants were excluded if they did not complete the survey (N = 12). All participants declared that they completed the survey seriously. This left a final sample of 182 participants aged 18-73 (Mage = 36.92, SD = 11.68). Of these, 93 identified as male, 88 identified as female and one selected ‘Other’. Participants were paid £1.50. Materials A detailed description of the Study 2 materials can be found in Appendix 1. The survey was created using the Qualtrics online platform. All nine of the generic decontextualized phrases from Study 1 were used along with an additional seven generic phrases (16 target items in total). Four of the additional phrases used “Academics” as the subject; it being the fourth most frequently used subject to describe authors of scientific work. This was paired with the same base verbs as Study 1 (“Academics Believe…”, “Academics Say…”, “Academics Think…”). An additional base verb “agree” was then selected as a corpus search revealed it to be the fourth most popular verb that frequently collocated with our four subjects and implied some degree of scientific consensus (ranked as the third most frequent collocate for “Researchers” and “Academics”, ranked seventh most frequent collocate of Researchers, and ranked 29th most frequent collocate of “Scientists”). “Agree” was paired with the four subjects to create the additional decontextualized phrases: “Scientists Agree…”, “Experts Agree…”, “Researchers Agree…”, “Academics Agree…”. Using the same slider layout as Study 1, participants were asked to estimate how many relevant experts they thought each phrase applied to on a sliding scale ranging from zero ‘No scientists/experts/researchers/academics’ to 100’ All scientists/experts/researchers/academics’. The 16 target items were presented amongst four filler items which used the same subject and base verbs as the targets but were prefaced with quantifiers (“Some Scientist Believe...”, “Most Experts Say...”, “Many Researchers Think...”, “Few Academics Agree”). These were simply to make the task less repetitive for participants. Prior to starting the experimental task participants completed four practice items to accustom them to making intuitive judgements under time pressure. These were also structured using the same subject and base verbs as the target items but with a quantifier prefacing them (worded differently to the filler items) to accustom subjects to using the slider. Given the time pressure element, a pre-screening requirement was that participants could only complete the Study on a desktop computer with a click-based mouse (i.e. not a touchpad). Procedure In this experiment the rating of scientific consensus was given twice for each phrase. The two-response format was similar to that introduced by Thompson et al. (2011) which was developed to gain insight into the time-course of intuitive and deliberate responses. Participants provided two estimates for each generic phrase. For the first response, participants were asked to give the very first estimate that came to mind (i.e. an intuitive estimate). To ensure that participants gave an intuitive response they gave their initial estimate under time pressure and under cognitive load (this application and time pressure and cognitive load in an online Study has been previously been used by Bago & De Neys, 2019a, 2019b; Raoelison & De Neys, 2019). Cognitive load was applied by asking participants to memorise a pattern of four crosses presented in a 3x3 matrix for later recall (see Appendix 1 for details). Time pressure was applied by instructing participants to select a point on the sliding scale within three seconds (this time limit was determined by a reading time pre-test conducted with 41 participants, see Appendix 2 for details of the pretest). This initial fast response is thought to reflect the output of System 1 processes based on evidence that that rapid responses rely more on automatic heuristic processing and therefore likely reflect System 1 output (Evans & Stanovich, 2013; Pennycook et al., 2018). Following the initial intuitive response, participants were asked to identify the pattern they saw from four options (thus removing the cognitive load). Feedback was immediately given to indicate whether the answer was correct or incorrect. The generic phrase was then presented again. Participants were reminded of their initial intuitive estimate (e.g., “Your intuitive estimate under time pressure was 85 on a scale of 0-100”) and shown a slider that was automatically set to this initial estimate. They were instructed that they could take as long as they liked to reflect on their initial estimate and had the option to revise their answer (if desired) by moving the slider. If participants did not wish to revise their intuitive response, they were advised to press the ‘continue’ button to start the next trial. This second response (T2) is thought to reflect the output of System 2 processes which are believed to be intentional, conscious and time consuming (Evans & Stanovich, 2013; Pennycook et al., 2018). Prior to completing the experimental task, participants completed a practice phase (described in detail in Appendix 1) where the memory and ratings tasks were introduced in stages to build familiarity. After the practice phase, each participant saw the target and filler phrases presented in a randomised order and completed the two-response protocol for each phrase (see Appendix 1 for detailed procedure). All phrases at T1 required a response and participants could not progress until a response was given. Mean completion time was 14.10 minutes (SD = 7.70). Results The following analyses were pre-registered on the Open Science Framework prior to data collection https://osf.io/2vh9w/registrations. The raw data and analysis script are publicly available via the OSF https://osf.io/2vh9w/. There were no missing data as all T1 questions required a response. Descriptive statistics suggest that participants engaged with the task as instructed. First, the average participant correctly recalled 92.2% of the matrices (SD=7.8) suggesting that cognitive load was successfully applied at T1 (in other words, participants engaged with the instruction to memorise each matrix). Second, participants revised their intuitive (T1) rating at T2 on 47.8% of trials, suggesting that they frequently revised their initial (T1) estimate, rather than simply retaining it. Following our pre-registered protocol, we excluded 146 T1 estimates where a response was not given within 3.25 seconds. These excluded observations represented 5% of the T1 data. Excluded values were replaced using mean imputation. Figure 3 shows that the mean T1 estimate was 60.74 (SD= 20.15) and the mean T2 estimate was 61.82 (SD=21.96). Figure 3: Mean estimate of consensus at T1 and T2 (N=182). The mean is represented by a horizontal line. The band represents 95% Bayesian Highest Density Interval. Figure 3: Mean estimate of consensus at T1 and T2 (N=182). The mean is represented by a horizontal line. The band represents 95% Bayesian Highest Density Interval. A linear mixed effects model was fitted using the lme4 package (Bates et al., 2015) in R version 3.6.0 (R.Core Team, 2019). The fixed factor was Time (T1 vs T2). Participants and Items were treated as random factors. We began with a maximal random effects structure (Barr et al., 2013) with random slopes and intercepts for Participants and Items. This resulted in a singular fit. We therefore systematically removed random effects until we achieved a non-singular fit. We first removed random slopes by items. This reduced model also resulted in a singular fit. We further reduced the model by removing random slopes by participants. This ‘intercepts only’ model achieved a non-singular fit. The final non-singular model included random intercepts for both Participants and Items. We compared this intercepts-only model to a null model (without the fixed effect) using a likelihood ratio test. There was a significant difference between the likelihood of these two models (χ2 (5) =10.533, p=.00116). This indicates that the fixed effect of Time affected perceived scientific consensus, with T2 estimates of consensus an average of 1.07 points (SE=0.33) greater than T1 estimates (on our 101 point scale). This effect was in the opposite direction to our pre-registered prediction. A standardised effect size (d) was calculated by dividing the mean difference between conditions by the square root of the pooled variance components (Westfall, Kenny & Judd, 2014). The size of this effect was small (d=0.043). Discussion In Study 2, participants estimated the degree of consensus implied by 16 generic phrases (e.g., Scientists say…) on a 0-100 scale that ranged from ‘No scientists’ to ‘All scientists’. They did this on two occasions. The first (T1) was under time pressure and under cognitive load. The second (T2) was an opportunity to revise the initial estimate without any time pressure or cognitive load. We predicted that participants would provide relatively high estimates of consensus at T1, with fast and intuitive System 1 processing leading to a hasty overgeneralization. At T2 we predicted that this initial estimate would be revised downwards by more deliberative System 2 processing. We found the opposite effect. The average consensus estimate at T1 was 60.74 and at T2 the average estimate was revised upwards to 61.82. The size of this upwards revision was small (d= 0.043). These data do not support our hypothesis that participants would revise their intuitive T1 estimates of consensus downwards at T2. The most common behaviour was for participants was to make no revision to their intuitive estimate (T1 and T2 estimates were identical on 52.2% of trials). When a revision was made, this was most frequently an upwards revision (T2 estimates were higher than T1 on 30.8% of trials). Revisions were only made in the predicted (downwards) direction on 17% of trials. General Discussion Generic news headlines (e.g., “Scientists believe the secret of a good night’s sleep is all in our genes") are inherently ambiguous. They require the reader or listener to infer the degree of consensus among ‘Scientists’. Do most scientists believe there is an important link between sleep and genes or just a select few? Those who do not read beyond the headline must rely on a subjective inference. These inferences can be consequential, as perceived scientific consensus on a given issue is an important factor in determining our own beliefs on that issue (e.g., van der Linden et al., 2015). In this paper we set out to 1) reveal the degree of scientific consensus implied by commonly used generic phrases such as”Scientists say…“,”Experts think…" and “Researchers believe…” 2) to reveal the extent to which estimates of consensus vary between people and between phrases , and 3) to examine whether this variance is associated with a reflective thinking style. We did this by running two highly powered, pre-registered studies. In terms of the first aim, we revealed that commonly used generic phrases such as “Scientists say…”, “Experts think…” and “Researchers believe…” imply consensus among over half of all relevant experts (53.8% in Study 1, 60.74% in Study 2 T1 and 61.82% in Study 2 T2) . These ratings were based on decontextualized phrases and are therefore labile; prone to change with the addition of context. These baseline estimates are nonetheless important, as they show that in the absence of any other context generic phrases about experts imply consensus among a majority of experts. This starting point may be revised upwards or downwards with additional context but in cases where the context is new or unfamiliar to the reader (e.g., Scientists think it rains diamonds on Neptune) the generic consensus statement (e.g., Scientists think…) may be all a they have to go on. While in some cases it may be true that a generic such as ‘scientists believe…’ refers to a majority of scientists, much of the novel research hitting the headlines is the work of just one group of authors. Using the generic ‘scientists’ or ‘experts’ risks implying that the opinion of a small subset of experts is the opinion of the majority of experts. This risks being misleading and confusing to the public. For example, when research by one group of authors is attributed generically to experts (e.g., Scientists believe one glass of wine per day is beneficial) it may imply that this is the opinion of a majority of experts; when contradictory findings by a different group of authors are reported (e.g., Scientists believe one glass of wine per day is harmful) it implies that the majority of experts have changed their minds. When this happens several times, there is a risk that it will damage trust in science and expert advice (e.g., Scientists say one thing one day and another the next!) (see Koehler & Pennycook, 2019 for related work). For these reasons, responsible reporting should avoid implying consensus through generic statements and instead use more specific quantity terms (e.g., A group of scientists…; ‘Some scientists…’). In terms of our second aim, the data revealed that mean estimates varied little between phrases (Study 1 SD=2.5) but varied widely between participants (Study 1 SD=25.6). In other words, estimates of consensus were very similar for all of our generic phrases (mean estimates of consensus ranged from 50.1 to 56.9). Generics, therefore, appear to imply the same degree of consensus regardless of whether the subject is ‘scientists’, ‘experts’ or ‘researchers’ or whether they ‘believe’, ‘think’ or ‘say’. In contrast, participants varied widely in their mean estimates of our generics (mean estimates of consensus ranged from 2% to 100%). The variance between participants was approximately ten times greater than between items. People thus draw very different inferences about the degree of consensus implied by generics, with some inferring they refer to very few experts and others to all experts. Those who make the strong inference that that ‘scientists’ refers to all or nearly all scientists are particularly at risk of changing their beliefs and behaviours based on the opinion of what could be a small minority of experts and of perceiving major scientific U-turns when different groups of experts report contradictory findings. In contrast, those at the lower end of the spectrum may be less inclined to change their beliefs or behaviours even when there is widespread consensus. The wide variability between participants in Study 1 may in part be an artefact of our decision to use decontextualized phrases. The variability in consensus estimates of “scientists believe…” is inevitably going to be greater than the variability corresponding to the same phrase used in context. What this does show is that in the absence of any other context, different people have different starting points for interpreting generics. The purpose of this Study was to examine whether variation in this decontextualized baseline estimate was related to variance in cognitive reflection. We tested the novel prediction that decontextualized generic phrases such as “Scientists believe…” encourage an intuitive overgeneralization about the degree of scientific consensus (e.g., Scientists believe = All scientists believe) that is then revised downwards to varying extents, on reflection (e.g., Scientists believe = Some scientists believe). In Study 1 we found no evidence to suggest that trait cognitive reflection is associated with estimates of scientific consensus. Likewise, in Study 2, we found no evidence that enforced reflection causes people to revise their intuitive estimates downwards. This suggests that the interpretation of a generic does not involve a process of intuitive overgeneralization and reflective belief revision. This leaves us without a satisfactory explanation for why estimates of consensus vary. Possible avenues for future research may be to explore the roles of experiential factors such as engagement with science, knowledge of the scientific process, level of education or cognitive factors such critical thinking, intelligence and need for cognition. Research should also focus on the implication of this variation in perceived consensus to determine the effects it has on subjective beliefs and behaviours. Generic attributions to ‘scientists’, ‘researchers’ and ‘experts’ are common in the media and social media, but they are inherently ambiguous. When generic attributions are presented out of context estimates of consensus vary widely, but on average they imply that a majority of the category (e.g., researchers, scientists, experts) are in agreement. For this reason, those reporting scientific research should avoid using generics when there is not a majority consensus among relevant experts and consumers of news should be aware that generics can intentionally or unintentionally misrepresent the true degree of consensus. Contributions Contributed to conception and design: MH, HB Contributed to acquisition of data: MH, HB Contributed to analysis and interpretation of data: MH, HB, TP Drafted and/or revised the article: MH, HB, TP Approved the submitted version for publication: MH, HB, TP Funding This work was supported by a Leverhulme Trust Research Project Grant (RPG-2019-158) awarded to the first author. Competing Interests The authors have no competing interests. Data accessibility All the stimuli, presentation materials, participant data, and analysis scripts can be found on the Open Science Framework. Study 1 https://osf.io/n7pj8/ Study 2 https://osf.io/2vh9w/ Footnotes 1. Although generic phrases such as ‘Experts think…’ can truthfully refer to almost any proportion of relevant experts there are likely to be some pragmatic limits to their use. 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Journal of Experimental Psychology: General, 143(5), 2020–2045. https://doi.org/10.1037/xge0000014 This is an open access article distributed under the terms of the Creative Commons Attribution License (4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.	2021-01-24 03:00:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 8, "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": 0.5150140523910522, "perplexity": 4230.022524471706}, "config": {"markdown_headings": false, "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-2021-04/segments/1610703544403.51/warc/CC-MAIN-20210124013637-20210124043637-00121.warc.gz"}
http://crypto.stackexchange.com/tags/mac/hot?filter=all	# Tag Info 66 @Ninefingers answers the question quite well; I just want to add a few details. Encrypt-then-MAC is the mode which is recommended by most researchers. Mostly, it makes it easier to prove the security of the encryption part (because thanks to the MAC, a decryption engine cannot be fed with invalid ciphertexts; this yields automatic protection against chosen ... 41 These types of cryptographic primitive can be distinguished by the security goals they fulfill (in the simple protocol of "appending to a message"): Integrity: Can the recipient be confident that the message has not been accidentally modified? Authentication: Can the recipient be confident that the message originates from the sender? Non-repudiation: If ... 24 As Chris Smith notes in the comments, HMAC is a specific MAC algorithm (or, rather, a method for constructing a MAC algorithm out of a cryptographic hash function). Thus, HMAC can be used for any application that requires a MAC algorithm. One possible reason for requiring HMAC specifically, as opposed to just a generic MAC algorithm, is that the HMAC ... 23 Hugo Krawczyk has a paper titled The Order of Encryption and Authentication for Protecting Communications (or: How Secure Is SSL?). It identifies 3 types of combining authentication (MAC) with encryption: Encrypt then Authenticate (EtA) used in IPsec; Authenticate then Encrypt (AtE) used in SSL; Encrypt and Authenticate (E&A) used in SSH. It proves ... 21 The word "secure hash function" usually means (for a function $H$) Preimage resistance: Given a value $h$, it is hard to find a message $x$ so that $h = H(x)$. Second preimage resistance: Given a message $x$, it is hard to find a message $x' \neq x$ such that $H(x) = H(x')$. Collision resistance: It is hard to find two messages $x$, $x'$ such that $H(x) = ... 18 Length extension attack The reason why$H(k \|\| m)$is insecure with most older hashes is that they use the Merkle–Damgård construction which suffers from length extensions. When length extensions are available it's possible to compute$H(k \|\| m \|\| m^\prime)$knowing only$H(k \|\| m)$but not$k$. This violates the security requirements of a MAC. Like all ... 17 The nCipher Advisory #13 cited in your securityfocus.com link contains the explanation of the vulnerability (in the section "Cryptographic details"). The CBC-MAC algorithm works similar to the CBC encryption algorithm, but only outputting the final block (or a part of this). Each block of the plain text is XOR-ed with the previous ciphertext and then ... 17$Encrypt(m\|H(m))$is not an operating mode providing authentication; forgeries are possible in some very real scenarios. Depending on the encryption used, that can be assuming only known plaintext. Here is a simple example with$Encrypt$a stream cipher, including any block cipher in CTR or OFB mode. Mallory wants to sign some message$m$of his choice. ... 16 You're missing the most important strength of HMAC: it comes with a proof of security (under some plausible assumptions). The outer key plays an important role in the proofs. The best place to learn more is to read the HMAC papers: Message authentication using hash functions: The HMAC construction, Mihir Bellare, Ran Canetti, Hugo Kawczyk, CryptoBytes ... 14 The reason$H(k\|m)$(where$\|$is concatenation) is not the standard comes from the message extension attack. If I, as an attacker, have$H(k\|m)$and$m$, I can compute$H(k\|m\|p\|m')$(where$p$is the padding that$H$would have applied to$k\|m$in computing the digest, and$m'$is an arbitrary message) without knowing$k$. I would then send ... 14 TL;DR No, the approach is not secure. Use a standard like CMAC instead. Or even better, check your AES accelerator module to see if it supports any AEAD modes of encryption like GCM, CCM, EAX. Long Version In order for a message authentication code (MAC) to be secure, an adversary with oracle access to the MAC (basically this means the adversary can send ... 13 A Message Authentication Code (MAC) is a string of bits that is sent alongside a message. The MAC depends on the message itself and a secret key. No one should be able to compute a MAC without knowing the key. This allows two people who share a secret key to send messages to each without fear that someone else will tamper with the messages. (At least, if ... 12 HMAC was there first (the RFC 2104 is from 1997, while CMAC is from 2006), which is reason enough to explain its primacy. If you use HMAC, you will more easily find test vectors and implementations against which to test, and with which to interoperate, which again explains continued primacy. Being the de facto standard is a very strong position. On many ... 10 UMAC is described in full details in RFC 4418. When the RFC talks about "secret selection", it really means "there is a secret key involved here". UMAC works with universal hashing, which can be viewed as a family of hash functions, and a key which selects which hash function we are talking of. The term "hash function" might be a bit confusing here, because ... 9 No, in general, this is not secure, unless you make additional assumptions on the encryption method beyond the standard assumption of privacy. To simplify things a bit, the assumption of privacy means that given a ciphertext$C$, the attacker has no information about what the plaintext might be. However, in your case, we don't really care if the attacker ... 9 I'll assumme All ciphered blocks means the same as ciphertext for CBC-Encryption with implicit zero IV, while CBC-MAC is the last block of that. All ciphered blocks is unsafe as a message authenticator for messages longer than one block, for it succumbs to a trivial attack (here with two blocks): Eve intercepts message$M=M_0\|\|M_1$and its authenticator ... 9 This construction is not secure. It was proposed in this paper in a quick sentence for possibly fixing the insecure secret prefix construction from the other question:$\mathcal{H}(k\|\|m)$. The author then proposes and analyzes an enveloping method:$\mathcal{H}(k_1\|\|x\|\|k_2)$. An attack involving finding an internal collision applies to ... 9 In general, a MAC with a known fixed "key" is not a secure hash. That is, you can have a secure MAC (that is, someone without the key, but with a large number of message/MAC pairs, cannot come up with another valid message/MAC pair) that is not collision resistant, or even preimage resistant, if the attacker does know the key. In addition, you don't have ... 9 Yes, this would be secure. CTR (Counter) mode based on keyed function$F_K$is secure as long as its output $$W_i = F_K(i)$$ is unpredictable given previous outputs $$F_K(1),F_K(2),\ldots,F_K(i-1).$$ This requirement is essentially the definition of a pseudo-random function (PRF). Most HMAC instantiations with widely used hash functions are believed to ... 9 It is not secure, because an attacker can "mix and match" the output blocks from different authentication tags on different input messages, or repeat output blocks for repeated input blocks. For example, if the attacker knows the tag$F_k(m)$for a one-block message$m$, then it can forge the correct tag$F_k(m) \mid F_k(m)$for the two-block message$m ... 8 In TLS (that's the standard name for SSL; TLS 1.2 is like "SSL version 3.3"), client and server ends up with a shared secret (the "master secret", a 48-byte sequence; when using RSA key exchange, the master secret is derived from the "premaster secret" which is the 48-byte string that the client encrypts with the server public key). That shared secret is ... 8 One rationale for avoiding randomized schemes in general, and in MACs in particular, is that the random in such schemes tends to increases the size of cryptograms or reduce the size of the payload. An example is scheme 2 in ISO/IEC 9796-2 RSA signature with message recovery, where the size of the random/salt field is directly antagonist with the amount of ... 8 This scheme is not worth the name MAC; it is horribly weak. First and foremost, the tag/MAC is unchanged when two blocks of plaintext are exchanged (because of the commutativity and associativity of the $\oplus$ operation). If follows that from any message with at least two different blocks, we can make a different message for which we know the tag/MAC. ... 8 The generic model for a MAC is the following: the attacker is given access to a block box which implements the $S$ function with a key $k$ that the attacker does not know of. The attacker is allowed to make $q$ requests to the box on messages that he can choose arbitrarily. The goal of the attacker is to make a forgery, i.e. produce values $m$ and $t$ such ... 8 First the theoretical explanations: Integrity and authenticity are different goals to achieve, but both are achieved (for symmetric encryption) with a MAC. You should probably be using encrypt-than-MAC or an authenticated cipher unless you have very good reasons not to. No blanket statements can be made though. HMAC: HMAC is a often used construct. It ... 7 This scheme is totally insecure. If an attacker modifies any part of the ciphertext except the last block before the ciphertext corresponding to H, your scheme won't catch it. CBC decryption of a block only depends on the ciphertext of the previous and current block. (Based on Cbc decryption.png from Wikipedia) The red parts are left totally unprotected ... 7 The short answer: No. It is not secure. Details. To answer the question properly, we first have to decide what we mean by "secure". In this case, I assume security means confidentiality plus integrity. So let's talk about each separately. Integrity: yes, this provides integrity, under your assumptions. @poncho explained why. Confidentiality: no, this ... 7 As a Skein co-author, one of the properties of the UBI chaining mode is to give you HMAC-like properties in one pass. Skein itself consists of the Threefish tweakable block cipher, the UBI chaining mode, and some proofs that extend tweakable block cipher theory into a tweakable hash function theory that reduces the security of the hash function to the ... 7 Well, yes, it does matter; however the terminology 'CBC-MAC' does not specify which. CBC-MAC is a generic construction that takes an arbitrary block cipher, and turns it into an object that acts like a MAC for fixed length messages (much like CBC mode is a generic construction that takes an arbitrary block cipher, and turns it into a object that encrypts ... Only top voted, non community-wiki answers of a minimum length are eligible	2015-08-30 03:51:53	{"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": 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": 0.5130280256271362, "perplexity": 1867.281166000629}, "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-2015-35/segments/1440644064869.18/warc/CC-MAIN-20150827025424-00077-ip-10-171-96-226.ec2.internal.warc.gz"}
https://standards.globalspec.com/std/9970362/iso-iec-8824-4	### This is embarrasing... An error occurred while processing the form. Please try again in a few minutes. ### This is embarrasing... An error occurred while processing the form. Please try again in a few minutes. # ISO/IEC 8824-4 ## Information technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 specifications active, Most Current Organization: ISO Publication Date: 15 November 2015 Status: active Page Count: 26 ICS Code (Presentation layer): 35.100.60 ##### scope: This Recommendation \| International Standard is part of Abstract Syntax Notation One (ASN.1) and defines notation for parameterization of ASN.1 specifications. ### Document History June 1, 2021 Information technology — Abstract Syntax Notation One (ASN.1) — Part 4: Parameterization of ASN.1 specifications This Recommendation \| International Standard is part of Abstract Syntax Notation One (ASN.1) and defines notation for parameterization of ASN.1 specifications. September 1, 2018 Information technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 specifications Part 4: TECHNICAL CORRIGENDUM 1 A description is not available for this item. ISO/IEC 8824-4 November 15, 2015 Information technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 specifications This Recommendation \| International Standard is part of Abstract Syntax Notation One (ASN.1) and defines notation for parameterization of ASN.1 specifications. September 15, 2014 Information technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 specifications TECHNICAL CORRIGENDUM 1 A description is not available for this item. December 15, 2008 Information technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 specifications This Recommendation \| International Standard is part of Abstract Syntax Notation One (ASN.1) and defines notation for parameterization of ASN.1 specifications. November 1, 2007 Information technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 specifications TECHNICAL CORRIGENDUM 1 A description is not available for this item. December 15, 2002 Information Technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 Specifications A description is not available for this item. December 1, 2000 Information Technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 Specifications AMENDMENT 1: ASN.1 semantic model A description is not available for this item. December 15, 1998 Information Technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 Specifications A description is not available for this item. January 1, 1995 Information Technology - Abstract Syntax Notation One (ASN.1): Parameterization of ASN.1 Specifications A description is not available for this item.	2022-08-12 00:08:30	{"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": 0.8653618097305298, "perplexity": 12952.495362645379}, "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/1659882571536.89/warc/CC-MAIN-20220811224716-20220812014716-00498.warc.gz"}
https://wiki2.org/en/Schur_complement_method	To install click the Add extension button. That's it. The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time. 4,5 Kelly Slayton Congratulations on this excellent venture… what a great idea! Alexander Grigorievskiy I use WIKI 2 every day and almost forgot how the original Wikipedia looks like. Live Statistics English Articles Improved in 24 Hours What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better. . Leo Newton Brights Milds # Schur complement method In numerical analysis, the Schur complement method, named after Issai Schur, is the basic and the earliest version of non-overlapping domain decomposition method, also called iterative substructuring. A finite element problem is split into non-overlapping subdomains, and the unknowns in the interiors of the subdomains are eliminated. The remaining Schur complement system on the unknowns associated with subdomain interfaces is solved by the conjugate gradient method. • 1/3 Views: 604 9 839 7 082 • Math for Big Data, Lecture 3, Schur Decomposition • Schur Triangulation lemma / decomposition (upper trangle) • RT4.2. Schur's Lemma (Expanded) ## The method and implementation Suppose we want to solve the Poisson equation ${\displaystyle -\Delta u=f,\qquad u\|_{\partial \Omega }=0}$ on some domain Ω. When we discretize this problem we get an N-dimensional linear system AU = F. The Schur complement method splits up the linear system into sub-problems. To do so, divide Ω into two subdomains Ω1, Ω2 which share an interface Γ. Let U1, U2 and UΓ be the degrees of freedom associated with each subdomain and with the interface. We can then write the linear system as ${\displaystyle \left[{\begin{matrix}A_{11}&0&A_{1\Gamma }\\0&A_{22}&A_{2\Gamma }\\A_{\Gamma 1}&A_{\Gamma 2}&A_{\Gamma \Gamma }\end{matrix}}\right]\left[{\begin{matrix}U_{1}\\U_{2}\\U_{\Gamma }\end{matrix}}\right]=\left[{\begin{matrix}F_{1}\\F_{2}\\F_{\Gamma }\end{matrix}}\right],}$ where F1, F2 and FΓ are the components of the load vector in each region. The Schur complement method proceeds by noting that we can find the values on the interface by solving the smaller system ${\displaystyle \Sigma U_{\Gamma }=F_{\Gamma }-A_{\Gamma 1}A_{11}^{-1}F_{1}-A_{\Gamma 2}A_{22}^{-1}F_{2},}$ for the interface values UΓ, where we define the Schur complement matrix ${\displaystyle \Sigma =A_{\Gamma \Gamma }-A_{\Gamma 1}A_{11}^{-1}A_{1\Gamma }-A_{\Gamma 2}A_{22}^{-1}A_{2\Gamma }.}$ The important thing to note is that the computation of any quantities involving ${\displaystyle A_{11}^{-1}}$ or ${\displaystyle A_{22}^{-1}}$ involves solving decoupled Dirichlet problems on each domain, and these can be done in parallel. Consequently, we need not store the Schur complement matrix explicitly; it is sufficient to know how to multiply a vector by it. Once we know the values on the interface, we can find the interior values using the two relations ${\displaystyle A_{11}U_{1}=F_{1}-A_{1\Gamma }U_{\Gamma },\qquad A_{22}U_{2}=F_{2}-A_{2\Gamma }U_{\Gamma },}$ which can both be done in parallel. The multiplication of a vector by the Schur complement is a discrete version of the Poincaré–Steklov operator, also called the Dirichlet to Neumann mapping.	2019-11-15 12:45:44	{"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": 0.5546869039535522, "perplexity": 950.579356650788}, "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-2019-47/segments/1573496668644.10/warc/CC-MAIN-20191115120854-20191115144854-00081.warc.gz"}
https://pyphi.readthedocs.io/en/0.7.0/api/subsystem.html	# subsystem¶ Represents a candidate set for $$\varphi$$ calculation. class pyphi.subsystem.Subsystem(node_indices, network, cut=None, mice_cache=None, repertoire_cache=None, cache_info=None) A set of nodes in a network. Parameters: nodes (tuple(int) – A sequence of indices of the nodes in this subsystem. network (Network) – The network the subsystem belongs to. nodes list(Node – A list of nodes in the subsystem. node_indices tuple(int – The indices of the nodes in the subsystem. size int – The number of nodes in the subsystem. network Network – The network the subsystem belongs to. cut Cut – The cut that has been applied to this subsystem. connectivity_matrix np.array – The connectivity matrix after applying the cut. cut_matrix np.array – A matrix of connections which have been severed by the cut. perturb_vector np.array – The vector of perturbation probabilities for each node. null_cut Cut – The cut object representing no cut. past_tpm np.array – The TPM conditioned on the past state of the external nodes (nodes outside the subsystem). current_tpm np.array – The TPM conditioned on the current state of the external nodes. repertoire_cache_info() Report repertoire cache statistics. __eq__(other) Return whether this subsystem is equal to the other object. Two subsystems are equal if their sets of nodes, networks, and cuts are equal. __bool__() Return false if the subsystem has no nodes, true otherwise. json_dict() indices2nodes(indices) cause_repertoire(mechanism, purview) Return the cause repertoire of a mechanism over a purview. Parameters: mechanism (tuple(Node) – The mechanism for which to calculate the cause repertoire. purview (tuple(Node) – The purview over which to calculate the cause repertoire. cause_repertoire – The cause repertoire of the mechanism over the purview. np.ndarray effect_repertoire(mechanism, purview) Return the effect repertoire of a mechanism over a purview. Parameters: mechanism (tuple(Node) – The mechanism for which to calculate the repertoire. (effect) – purview (tuple(Node) – The purview over which to calculate the repertoire. – effect_repertoire – The effect repertoire of the mechanism over the purview. np.ndarray unconstrained_cause_repertoire(purview) Return the unconstrained cause repertoire for a purview. This is just the cause repertoire in the absence of any mechanism. unconstrained_effect_repertoire(purview) Return the unconstrained effect repertoire for a purview. This is just the effect repertoire in the absence of any mechanism. expand_repertoire(direction, purview, repertoire, new_purview=None) Return the unconstrained cause or effect repertoire based on a direction. expand_cause_repertoire(purview, repertoire, new_purview=None) Expand a partial cause repertoire over a purview to a distribution over the entire subsystem’s state space. expand_effect_repertoire(purview, repertoire, new_purview=None) Expand a partial effect repertoire over a purview to a distribution over the entire subsystem’s state space. cause_info(mechanism, purview) Return the cause information for a mechanism over a purview. effect_info(mechanism, purview) Return the effect information for a mechanism over a purview. cause_effect_info(mechanism, purview) Return the cause-effect information for a mechanism over a purview. This is the minimum of the cause and effect information. find_mip(direction, mechanism, purview) Return the minimum information partition for a mechanism over a purview. Parameters: direction (str) – Either DIRECTIONS[PAST] or DIRECTIONS[FUTURE]. mechanism (tuple(Node) – The nodes in the mechanism. purview (tuple(Node) – The nodes in the purview. mip – The mininum-information partition in one temporal direction. Mip mip_past(mechanism, purview) Return the past minimum information partition. Alias for \|find_mip\| with direction set to DIRECTIONS[PAST]. mip_future(mechanism, purview) Return the future minimum information partition. Alias for \|find_mip\| with direction set to DIRECTIONS[FUTURE]. phi_mip_past(mechanism, purview) Return the $$\varphi$$ value of the past minimum information partition. This is the distance between the unpartitioned cause repertoire and the MIP cause repertoire. phi_mip_future(mechanism, purview) Return the $$\varphi$$ value of the future minimum information partition. This is the distance between the unpartitioned effect repertoire and the MIP cause repertoire. phi(mechanism, purview) Return the $$\varphi$$ value of a mechanism over a purview. find_mice(direction, mechanism, purviews=False) Return the maximally irreducible cause or effect for a mechanism. Parameters: Keyword Arguments: direction (str) – The temporal direction, specifying cause or effect. mechanism (tuple(Node) – The mechanism to be tested for irreducibility. purviews (tuple(Node) – Optionally restrict the possible purviews to a subset of the subsystem. This may be useful for _e.g._ finding only concepts that are “about” a certain subset of nodes. mice – The maximally-irreducible cause or effect. Mice Note Strictly speaking, the MICE is a pair of repertoires: the core cause repertoire and core effect repertoire of a mechanism, which are maximally different than the unconstrained cause/effect repertoires (i.e., those that maximize $$\varphi$$). Here, we return only information corresponding to one direction, DIRECTIONS[PAST] or DIRECTIONS[FUTURE], i.e., we return a core cause or core effect, not the pair of them. core_cause(mechanism, purviews=False) Returns the core cause repertoire of a mechanism. Alias for \|find_mice\| with direction set to DIRECTIONS[PAST]. core_effect(mechanism, purviews=False) Returns the core effect repertoire of a mechanism. Alias for \|find_mice\| with direction set to DIRECTIONS[PAST]. phi_max(mechanism) Return the $$\varphi^{\textrm{max}}$$ of a mechanism. This is the maximum of $$\varphi$$ taken over all possible purviews. null_concept Return the null concept of this subsystem, a point in concept space identified with the unconstrained cause and effect repertoire of this subsystem. concept(mechanism) Calculate a concept.	2021-02-26 12:21:08	{"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": 0.39320996403694153, "perplexity": 4262.432393052807}, "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-2021-10/segments/1614178357641.32/warc/CC-MAIN-20210226115116-20210226145116-00254.warc.gz"}
https://nubtrek.com/maths/calculus-limits/finding-limit-of-functions/limit-of-continuous-function	Server Error Server Not Reachable. This may be due to your internet connection or the nubtrek server is offline. Thought-Process to Discover Knowledge Welcome to nubtrek. Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge. In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators. Read in the blogs more about the unique learning experience at nubtrek. mathsLimit of a functionCalculating Limits ### Limit of continuous functions With an example, calculation of limits for a continuous function is discussed. The condition under which a function is continuous is illustrated with examples. click on the content to continue.. Given function f(x)=1/(x^2+1). what is f(x)\|_(x=1)? • Substitute x=1 • 1/(1+1) • 1/2 • all the above • all the above The answer is 'All the above'. The options provide the steps to evaluate f(1). Given function f(x)=1/(x^2+1). what is left-hand-limit lim_(x->1-)f(x)? • Substitute x=1 • Substitute x=1+delta • Substitute x=1-delta • Substitute x=1-delta • all the above The answer is 'Substitute x=1-delta'. lim_(x->1-)f(x) quad quad = 1/((1-delta)^2+1) quad quad = 1/(1-2delta+delta^2+1) quad quad = 1/(2-2delta+delta^2) quad quad = 1/2 quad quad quad quad (substituting delta=0) Given function f(x)=1/(x^2+1). what is right-hand-limit lim_(x->1+)f(x)? • Substitute x=1 • Substitute x=1+delta • Substitute x=1+delta • Substitute x=1-delta • all the above The answer is 'Substitute x=1+delta'. lim_(x->1+)f(x) quad quad = 1/((1+delta)^2+1) quad quad = 1/(1+2delta+delta^2+1) quad quad = 1/(2+2delta+delta^2) quad quad = 1/2 quad quad quad quad (substituting delta=0) Given function f(x)=1/(x^2+1). • f(1)=1/2 • lim_(x->1-)f(x) = 1/2 • lim_(x->1+)f(x) = 1/2 The function is continuous at x=1. Note: Though, as part of the topic, the functions that are discontinuous or indeterminate or non-differentiable are mainly handled, finding limits is not only for those kind of functions. Any function at any point can be evaluated for limits. Function is continuous if all three values are equal. Function is continuous: if f(a) = lim_(x->a-)f(x) =lim_(x->a+)f(x), then the function is continuous at x=a. Solved Exercise Problem: Given function f(x) = sin x, is it continuous at input values x= 0, pi/2, pi? • Continuous at 0 and pi/2 only • Continuous at all three input values • Continuous at all three input values • Continuous at only 0 • not continuous at any one of the given values The answer is 'Continuous at all three input values' switch to slide-show version	2018-11-15 13:06:57	{"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": 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": 0.40654417872428894, "perplexity": 8677.756765705304}, "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-2018-47/segments/1542039742685.33/warc/CC-MAIN-20181115120507-20181115142507-00449.warc.gz"}
https://www.biostars.org/p/9464711/	Batch effect correction of 2 studies with different covariates 1 2 Entering edit mode 24 days ago Geo ▴ 20 Hi all, I want to integrate RNA-seq data from 2 studies. My plan is to merge the count matrices and then perform batch effect correction using Combat-seq. However, while study A has samples at different ages and sex, the age and sex of the samples in study B is unknown. Also, study A has samples from 2 tissues but study B only from 1. Do you have any advice on how should I proceed? I have two options in mind. 1. Consider each sub-dataset (samples grouped by batch, sex, age and tissue) as a different batch. Study B is the same batch. 2. Forget about age and sex and only group by batch and tissue, so there are 4 batches for study A and 1 for study B Toy example (I have more samples from each case): study batch group age sex tissue A 1 control 10 M cortex A 2 control 10 F cortex A 1 control 20 M cortex A 2 control 20 F cortex A 1 cases 10 M cortex A 2 cases 10 F cortex A 1 cases 20 M cortex A 2 cases 20 F cortex A 1 control 10 M cerebellum A 2 control 10 F cerebellum A 1 control 20 M cerebellum A 2 control 20 F cerebellum A 1 cases 10 M cerebellum A 2 cases 10 F cerebellum A 1 cases 20 M cerebellum A 2 cases 20 F cerebellum B 3 control - - cortex B 3 control - - cortex B 3 control - - cortex B 3 control - - cortex B 3 cases - - cortex B 3 cases - - cortex B 3 cases - - cortex B 3 cases - - cortex Thanks a lot batch-effect R RNA-seq • 412 views 1 Entering edit mode 23 days ago ponganta ▴ 220 Try both, check sample variability, dispersion and P-value distributions. By "Combat-seq" do you mean the package ComBat? I'm not sure this is sufficient. 1 Entering edit mode RUVseq might be the way to go here, to be honest. 0 Entering edit mode Thanks for the link, didn't know that one! 0 Entering edit mode The question remains, what should I do with the covariates? 0 Entering edit mode If you run RUVseq on the data from study B only, you'd get predicted values for age and sex but these will be numerical. I suppose you could re-encode these into categorical variables somehow. E.g., by rounding off the age values for instance, and deducing a relationship between the rounded values and the actual age classes (perhaps both sets of values are positively correlated, for instance); and for the sex it's just binary, so it should be somewhat straightforward but there might be no way to disambiguate male from female (since there's no intrinsic order in this case). I suppose you could actually use study A's data to discern these relationships somehow, so that you can map them accurately in B's case. Or you could try and predict surrogate variables standing in for age and sex for the entire data set (i.e., both A and B) and use those instead of the age and sex values you already have here. I think this might be the more straightforward option. The only other alternative would be to drop those variables entirely. 0 Entering edit mode Combat-seq is a batch effect adjustment tool for bulk RNA-seq count data based on Combat. What should I look for in these variables? (sample variability, dispersion and P-value distributions)	2021-05-06 06:36:47	{"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": 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": 0.34695103764533997, "perplexity": 3428.8391860272877}, "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/1620243988741.20/warc/CC-MAIN-20210506053729-20210506083729-00233.warc.gz"}
http://grephysics.net/ans/9277/41	GR 8677927796770177 \| # Login \| Register GR9277 #41 Problem GREPhysics.NET Official Solution Alternate Solutions \prob{41} A cylinder with moment of inertia $4kgm^2$ about a fixed axis initially rotates at 80 radians per second about this axis. A constant torque is applied to slow it down to 40 radians per second. The kinetic energy lost by the cylinder is 1. 80 J 2. 800 J 3. 4000 J 4. 9600 J 5. 19,200 J Mechanics$\Rightarrow$}Energy The kinetic energy is related to the inertia I and angular velocity $\omega$ by $K = \frac{1}{2}I\omega^2$. The problem supplies $I=4kgm^2$ so one needs not calculate the moment of inertia. The angular velocity starts at $80 rad/s$ and ends at $40 rad/s$. Thus, the kinetic energy lost $\Delta K = \frac{1}{2}I(\omega_f^2-\omega_0^2)=\frac{1}{2}(4)(40^2-80^2)=2(1600-6400)=-9600J$, as in choice (D). Alternate Solutions There are no Alternate Solutions for this problem. Be the first to post one! mnky9800n 2013-09-24 13:05:47 I also like to think in terms of the potential energy of a 100kg person who is about to fall 10 meters. That is mgh~(100 kg)(10 m/s^2)(10 m) = 10000 J. Thus A and B are probably too small and E is probably too big. aloha 2008-10-10 03:21:51 Maybe it's a silly question, but how can we know that I=4? Thanks :) Monk2008-10-13 20:11:46 it tells you in the question...it is 4 kg m^2 kg m^2 is the units not a formula tensordyne2008-11-04 13:16:28 I am with Monk, from reading the problem it seems like they are saying $I = 4 k g m^2$ is a formula in k (proportionality factor maybe), g ($9.8 m/s$) and mass m instead of as units because there is no grouping of the units, such as in say $I = 4 kg \cdot m^2$, which would have made the use of units much more manifest. LaTeX syntax supported through dollar sign wrappers $, ex.,$\alpha^2_0$produces $\alpha^2_0$. type this... to get...$\int_0^\infty$$\int_0^\infty$$\partial$$\partial$$\Rightarrow$$\Rightarrow$$\ddot{x},\dot{x}$$\ddot{x},\dot{x}$$\sqrt{z}$$\sqrt{z}$$\langle my \rangle$$\langle my \rangle$$\left( abacadabra \right)_{me}$$\left( abacadabra \right)_{me}$$\vec{E}$$\vec{E}$$\frac{a}{b}\$ $\frac{a}{b}$	2018-12-14 13:12:04	{"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": 21, "/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": 0.826097846031189, "perplexity": 1117.9495264708628}, "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-2018-51/segments/1544376825728.30/warc/CC-MAIN-20181214114739-20181214140239-00616.warc.gz"}
https://sbjoshi.wordpress.com/about/	About I am planning to write mainly technical stuff at this blog. I have to switch over here because of $\LaTeX$ support. Advertisements 3 Responses to “About” 1. Jagadish Says: Hi, Here is a nice problem that was asked recently by Sundar Sir in UG algos quiz: 2. phimuemue Says: Hello, I have quite a nice puzzle but i do not know how to solve it: http://phimuemue.com/blog.php?article=167 3. Rupen patel Says: આપના બ્લોગને ગુજરાતી બ્લોગપીડિયા બ્લોગ એગ્રીગેટર સાથે જોડવામાં આવેલ છે .મુલાકાત લેશો.http://rupen007.feedcluster.com/	2017-10-18 09:22:20	{"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": 1, "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": 0.554492712020874, "perplexity": 14903.17525475354}, "config": {"markdown_headings": false, "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-2017-43/segments/1508187822851.65/warc/CC-MAIN-20171018085500-20171018105500-00139.warc.gz"}
http://mathhelpforum.com/algebra/153215-system-linear-equations-print.html	# System of linear equations. • August 9th 2010, 08:22 PM gabriel System of linear equations. how do i solve for this system of equations 11x-3y=-36 12y-9x=39 ? tried reordering the terms but what else? • August 9th 2010, 08:51 PM pickslides Quote: Originally Posted by gabriel how do i solve for this system of equations 11x-3y=-36 12y-9x=39 ? tried reordering the terms but what else? $11x-3y=-36$ ...(1) $12y-9x=39$ ...(2) From (1) $11x-3y=-36\implies -3y=-36-11x \implies y=12+\frac{11}{3}x$ Then (2) $12y-9x=39\implies 12\left(12+\frac{11}{3}x\right)-9x=39$ Can you finish it from here? • August 9th 2010, 09:08 PM gabriel yeppp i got thnx good refreshment to the dusty mind • August 9th 2010, 11:14 PM Prove It Quote: Originally Posted by gabriel how do i solve for this system of equations 11x-3y=-36 12y-9x=39 ? tried reordering the terms but what else? The alternative method (the method used for more equations with more unknowns) is the elimination method... $11x - 3y = -36$ $-9x + 12y = 39$. Multiply the first equation by 4 to give $44x-12y=-144$ $-9x+12y=39$ $(44x-12y)+(-9x+12y)=-144+39$ $35x=-105$ $x = -3$. Substitute into one of the original equations... $11(- 3) - 3y = -36$ $-33- 3y = -36$ $-3y = -3$ $y = 1$ So the solution is $(x,y) = (-3,1)$. • August 9th 2010, 11:39 PM earboth Quote: Originally Posted by Prove It ... $(44x-12y)+(-9x+12y)=-144+39$ $35x=105$ <=== unfortunately you made a small sign mistake here $x = 3$. Substitute into one of the original equations... $11\cdot 3 - 3y = -36$ $33- 3y = -36$ $-3y = -69$ $y = 23$. So the solution is $(x,y) = (3,23)$. $(44x-12y)+(-9x+12y)=-144+39$ $35x=-105$ $x = -3$. • August 10th 2010, 12:03 AM Prove It Yes I did, edited :)	2015-03-04 03:29:26	{"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": 0, "img_math": 0, "codecogs_latex": 27, "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": 0.9263322353363037, "perplexity": 5055.440069631329}, "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/1424936463444.94/warc/CC-MAIN-20150226074103-00268-ip-10-28-5-156.ec2.internal.warc.gz"}
https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.19.1/share/doc/Macaulay2/Resultants/html/_discriminant.html	# discriminant -- resultant of the partial derivatives ## Synopsis • Usage: discriminant F • Inputs: • F, , a homogeneous polynomial • Optional inputs: • Outputs: • , the discriminant of F ## Description The discriminant of a homogeneous polynomial is defined, up to a scalar factor, as the resultant of its partial derivatives. For the general theory, see one of the following: Using Algebraic Geometry, by David A. Cox, John Little, Donal O'shea; Discriminants, Resultants, and Multidimensional Determinants, by Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky. i1 : ZZ[a,b,c][x,y]; F = ax^2+bxy+cy^2 2 2 o2 = ax + bxy + cy o2 : ZZ[a..c][x..y] i3 : time discriminant F -- used 0.0213388 seconds 2 o3 = - b + 4ac o3 : ZZ[a..c] i4 : ZZ[a,b,c,d][x,y]; F = ax^3+bx^2y+cxy^2+dy^3 3 2 2 3 o5 = ax + bx y + cxy + dy o5 : ZZ[a..d][x..y] i6 : time discriminant F -- used 0.00734016 seconds 2 2 3 3 2 2 o6 = - b c + 4ac + 4b d - 18abcd + 27a d o6 : ZZ[a..d] The next example illustrates how computing the intersection of a pencil generated by two degree $d$ forms $F(x_0,\ldots,x_n), G(x_0,\ldots,x_n)$ with the discriminant hypersurface in the space of forms of degree $d$ on $\mathbb{P}^n$ i7 : x=symbol x; R=ZZ/331[x_0..x_3] o8 = R o8 : PolynomialRing i9 : F=x_0^4+x_1^4+x_2^4+x_3^4 4 4 4 4 o9 = x + x + x + x 0 1 2 3 o9 : R i10 : G=x_0^4-x_0x_1^3-x_2^4+x_2x_3^3 4 3 4 3 o10 = x - x x - x + x x 0 0 1 2 2 3 o10 : R i11 : R'=ZZ/331[t_0,t_1][x_0..x_3]; i12 : pencil=t_0sub(F,R')+t_1sub(G,R') 4 3 4 4 3 4 o12 = (t + t )x - t x x + t x + (t - t )x + t x x + t x 0 1 0 1 0 1 0 1 0 1 2 1 2 3 0 3 o12 : R' i13 : time D=discriminant pencil -- used 0.525467 seconds 108 106 2 102 6 100 8 98 10 96 12 o13 = - 62t + 19t t + 160t t + 91t t + 129t t + 117t t + 0 0 1 0 1 0 1 0 1 0 1 ----------------------------------------------------------------------- 94 14 92 16 90 18 88 20 86 22 84 24 161t t + 124t t - 82t t - 21t t - 49t t - 123t t + 0 1 0 1 0 1 0 1 0 1 0 1 ----------------------------------------------------------------------- 82 26 80 28 78 30 76 32 74 34 72 36 5t t - 4t t + 75t t + 103t t + 47t t + 108t t - 0 1 0 1 0 1 0 1 0 1 0 1 ----------------------------------------------------------------------- 70 38 68 40 66 42 64 44 62 46 60 48 62t t - 97t t - 131t t + 71t t - 68t t - 144t t - 0 1 0 1 0 1 0 1 0 1 0 1 ----------------------------------------------------------------------- 58 50 56 52 54 54 52 56 50 58 48 60 163t t + 10t t - 35t t + 105t t + 7t t + 10t t - 0 1 0 1 0 1 0 1 0 1 0 1 ----------------------------------------------------------------------- 46 62 44 64 42 66 40 68 38 70 36 72 3t t + 76t t - 152t t - 81t t + 106t t - 11t t - 0 1 0 1 0 1 0 1 0 1 0 1 ----------------------------------------------------------------------- 34 74 32 76 30 78 28 80 26 82 24 84 13t t + 17t t + 18t t + 88t t + 9t t + 58t t - 0 1 0 1 0 1 0 1 0 1 0 1 ----------------------------------------------------------------------- 22 86 20 88 18 90 16 92 14 94 12 96 73t t + 113t t - 154t t - 102t t - 161t t + 33t t - 0 1 0 1 0 1 0 1 0 1 0 1 ----------------------------------------------------------------------- 10 98 8 100 6 102 4 104 2 106 108 130t t - 21t t + 157t t + 105t t + 82t t + 69t 0 1 0 1 0 1 0 1 0 1 1 ZZ o13 : ---[t ..t ] 331 0 1 i14 : factor D 9 9 9 9 18 18 2 2 9 2 2 9 o14 = (128t - t ) (128t + t ) (11t - t ) (11t + t ) (t - t ) (t + t ) (39t - 139t t + t ) (39t + 139t t + t ) (69) 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 o14 : Expression of class Product ## Ways to use discriminant : • "discriminant(RingElement)" ## For the programmer The object discriminant is .	2022-09-29 04:47:06	{"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": 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": 0.4306217133998871, "perplexity": 334.06996914537865}, "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-2022-40/segments/1664030335304.71/warc/CC-MAIN-20220929034214-20220929064214-00322.warc.gz"}
http://mathoverflow.net/revisions/88550/list	6 added 156 characters in body EDIT: there is an elementary trick to do this due to Aubry, it is Theorem 4 on page 5 of PETE. The positive primitive binary forms which give the easy Aubry trick showing rational implies integral are: $$x^2 + y^2, x^2 + 2 y^2, x^2 + 3 y^2, x^2 + 5 y^2,$$ $$x^2 + x y + y^2, x^2 + x y + 2 y^2, x^2 + x y + 3 y^2,$$ $$2 x^2 + 3 y^2, 2 x^2 + x y +2 y^2, 2 x^2 + 2 x y + 3 y^2.$$ Note that these are all "ambiguous," that is, equivalent to their "opposites." This is not an accident. Probably needs mention, for the property mentioned by the OP there is no difference between the sum of two squares and the sum of three squares... Right, I don't know about Fermat, but this phenomenon happens often enough. The first mention on MO is http://mathoverflow.net/questions/3269/intuition-for-the-last-step-in-serres-proof-of-the-three-squares-theorem and the technique, due to Aubry, Cassels, and Davenport, is mentioned in Serre A Course in Arithmetic, pages 45-47, and Weil Number Theory: An approach through history from Hammurapi to Legendre, pages 59 and 292ff in which Fermat's possible thinking is discussed. About my use of the word "phenomenon," it is necessary for the Aubry-Cassels-Davenport trick to work that we have Pete's "Euclidean" condition, http://mathoverflow.net/questions/39510/must-a-ring-which-admits-a-euclidean-quadratic-form-be-euclidean which is usually, for positive quadratic forms, referred to as a bound on the "covering radius" of the integral lattice under consideration. It took me a year or so to prove that Pete's condition implied that there could only be one class in that genus, http://mathoverflow.net/questions/69444/a-priori-proof-that-covering-radius-strictly-less-than-sqrt-2-implies-class-nu A complete list of positive forms that satisfy Pete's condition is at NEBE. A mild generalization of the condition, due to Richard Borcherds and his student, Daniel Allcock, applies to such forms as the sum of five squares. 5 added 419 characters in body The positive primitive binary forms which give the easy Aubry trick showing rational implies integral are: $$x^2 + y^2, x^2 + 2 y^2, x^2 + 3 y^2, x^2 + 5 y^2,$$ $$x^2 + x y + y^2, x^2 + x y + 2 y^2, x^2 + x y + 3 y^2,$$ $$2 x^2 + 3 y^2, 2 x^2 + x y +2 y^2, 2 x^2 + 2 x y + 3 y^2.$$ Note that these are all "ambiguous," that is, equivalent to their "opposites." This is not an accident. Probably needs mention, for the property mentioned by the OP there is no difference between the sum of two squares and the sum of three squares... Right, I don't know about Fermat, but this phenomenon happens often enough. The first mention on MO is http://mathoverflow.net/questions/3269/intuition-for-the-last-step-in-serres-proof-of-the-three-squares-theorem and the technique, due to Aubry, Cassels, and Davenport, is mentioned in Serre A Course in Arithmetic, pages 45-47, and Weil Number Theory: An approach through history from Hammurapi to Legendre, pages 59 and 292ff in which Fermat's possible thinking is discussed. About my use of the word "phenomenon," it is necessary for the Aubry-Cassels-Davenport trick to work that we have Pete's "Euclidean" condition, http://mathoverflow.net/questions/39510/must-a-ring-which-admits-a-euclidean-quadratic-form-be-euclidean which is usually, for positive quadratic forms, referred to as a bound on the "covering radius" of the integral lattice under consideration. It took me a year or so to prove that Pete's condition implied that there could only be one class in that genus, http://mathoverflow.net/questions/69444/a-priori-proof-that-covering-radius-strictly-less-than-sqrt-2-implies-class-nu A complete list of positive forms that satisfy Pete's condition is at NEBE. A mild generalization of the condition, due to Richard Borcherds and his student, Daniel Allcock, applies to such forms as the sum of five squares. 4 added 150 characters in body Probably needs mention, for the property mentioned by the OP there is no difference between the sum of two squares and the sum of three squares... Right, I don't know about Fermat, but this phenomenon happens often enough. The first mention on MO is http://mathoverflow.net/questions/3269/intuition-for-the-last-step-in-serres-proof-of-the-three-squares-theorem and the technique, due to Aubry, Cassels, and Davenport, is mentioned in Serre A Course in Arithmetic, pages 45-47, and Weil Number Theory: An approach through history from Hammurapi to Legendre, pages 59 and 292ff in which Fermat's possible thinking is discussed. About my use of the word "phenomenon," it is necessary for the Aubry-Cassels-Davenport trick to work that we have Pete's "Euclidean" condition, http://mathoverflow.net/questions/39510/must-a-ring-which-admits-a-euclidean-quadratic-form-be-euclidean which is usually, for positive quadratic forms, referred to as a bound on the "covering radius" of the integral lattice under consideration. It took me a year or so to prove that Pete's condition implied that there could only be one class in that genus, http://mathoverflow.net/questions/69444/a-priori-proof-that-covering-radius-strictly-less-than-sqrt-2-implies-class-nu A complete list of positive forms that satisfy Pete's condition is at NEBE. A mild generalization of the condition, due to Richard Borcherds and his student, Daniel Allcock, applies to such forms as the sum of five squares. 3 Nebe 2 added 628 characters in body 1	2013-05-25 18:37:50	{"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": 0.8588364720344543, "perplexity": 764.0521097761524}, "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-2013-20/segments/1368706082529/warc/CC-MAIN-20130516120802-00077-ip-10-60-113-184.ec2.internal.warc.gz"}
https://questions.examside.com/past-years/jee/question/the-number-of-distinct-real-roots-of-x4-4x3-12x2-x-1-0-jee-advanced-2011-marks-4-o6ze89k3x8egoig8.htm	NEW New Website Launch Experience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc... 1 ### IIT-JEE 2011 Paper 2 Offline Numerical The number of distinct real roots of $${x^4} - 4{x^3} + 12{x^2} + x - 1 = 0$$ 2 ### IIT-JEE 2011 Paper 1 Offline Numerical The minimum value of the sum of real numbers $${a^{ - 5}},\,{a^{ - 4}},\,3{a^{ - 3}},\,1,\,{a^8}$$ and $${a^{10}}$$ where $$a > 0$$ is 3 Numerical ### Joint Entrance Examination JEE Main JEE Advanced WB JEE ### Graduate Aptitude Test in Engineering GATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN NEET Class 12	2022-05-28 19:26:31	{"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": 0.8687789440155029, "perplexity": 12474.203419447807}, "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-2022-21/segments/1652663019783.90/warc/CC-MAIN-20220528185151-20220528215151-00412.warc.gz"}
http://clay6.com/qa/31603/inversion-temperature-is	Browse Questions Inversion Temperature is $(a)\;\large\frac{Rb}{2a} \\(b)\; \large\frac{Rb}{a} \\(c)\;\large\frac{2a}{Rb}\\(d)\;\large\frac{a}{Rb}$ Inversion Temperature is $\large\frac{2a}{Rb}$ Hence c is the correct answer.	2017-02-27 22:41:04	{"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": 0.9545499682426453, "perplexity": 9442.883895248655}, "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-2017-09/segments/1487501173866.98/warc/CC-MAIN-20170219104613-00576-ip-10-171-10-108.ec2.internal.warc.gz"}
https://www.zbmath.org/authors/?q=ai%3Arhines.peter-b	# zbMATH — the first resource for mathematics ## Rhines, Peter B. Compute Distance To: Author ID: rhines.peter-b Published as: Rhines, P. B.; Rhines, Peter; Rhines, Peter B. External Links: MGP · Wikidata Documents Indexed: 13 Publications since 1969 all top 5 #### Co-Authors 6 single-authored 3 Young, William R. 2 Lindahl, E. G. 1 Afanasyev, Yakov D. 1 Bretherton, Francis P. 1 Dewar, William K. 1 Méndez, Arturo J. 1 Thomas, Leif N. #### Serials 7 Journal of Fluid Mechanics 2 Geophysical and Astrophysical Fluid Dynamics 1 Physics of Fluids all top 5 #### Fields 12 Fluid mechanics (76-XX) 9 Geophysics (86-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Numerical analysis (65-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Astronomy and astrophysics (85-XX) #### Citations contained in zbMATH Open 12 Publications have been cited 258 times in 244 Documents Cited by Year How rapidly is a passive scalar mixed within closed streamlines. Zbl 0576.76088 Rhines, P. B. 1983 Waves and Turbulence on a beta-plane. Zbl 0366.76043 Rhines, Peter B. 1975 Homogenization of potential vorticity in planetary gyres. Zbl 0497.76032 Rhines, Peter B.; Young, William R. 1982 Slowoscillations in an ocean of varying depth. I: Abrupt topography. II: Islands and seamounts. Zbl 0175.52803 Rhines, P. B. 1969 Topographic Rossby waves in a rough-bottomed ocean. Zbl 0271.76017 Rhines, Peter; Bretherton, Francis 1973 Geostrophic turbulence. Zbl 0474.76054 Rhines, Peter B. 1979 Optical altimetry: a new method for observing rotating fluids with applications to Rossby and inertial waves on a polar beta-plane. Zbl 1106.76318 Rhines, P. B.; Lindahl, E. G.; Mendez, A. J. 2007 Lectures in geophysical fluid dynamics. Zbl 0543.76058 Rhines, Peter B. 1983 Vortices and rossby waves in cylinder wakes on a parabolic $$\beta$$-plane observed by altimetric imaging velocimetry. Zbl 1182.76005 Afanasyev, Y. D.; Rhines, P. B.; Lindahl, E. G. 2008 Nonlinear stratified spin-up. Zbl 1129.76373 Thomas, Leif N.; Rhines, Peter B. 2002 Vorticity dynamics of the oceanic general circulation. Zbl 0634.76020 Rhines, Peter B. 1986 The nonlinear spin-up of a stratified ocean. Zbl 0598.76046 Dewar, William K.; Rhines, Peter B.; Young, William R. 1984 Vortices and rossby waves in cylinder wakes on a parabolic $$\beta$$-plane observed by altimetric imaging velocimetry. Zbl 1182.76005 Afanasyev, Y. D.; Rhines, P. B.; Lindahl, E. G. 2008 Optical altimetry: a new method for observing rotating fluids with applications to Rossby and inertial waves on a polar beta-plane. Zbl 1106.76318 Rhines, P. B.; Lindahl, E. G.; Mendez, A. J. 2007 Nonlinear stratified spin-up. Zbl 1129.76373 Thomas, Leif N.; Rhines, Peter B. 2002 Vorticity dynamics of the oceanic general circulation. Zbl 0634.76020 Rhines, Peter B. 1986 The nonlinear spin-up of a stratified ocean. Zbl 0598.76046 Dewar, William K.; Rhines, Peter B.; Young, William R. 1984 How rapidly is a passive scalar mixed within closed streamlines. Zbl 0576.76088 Rhines, P. B. 1983 Lectures in geophysical fluid dynamics. Zbl 0543.76058 Rhines, Peter B. 1983 Homogenization of potential vorticity in planetary gyres. Zbl 0497.76032 Rhines, Peter B.; Young, William R. 1982 Geostrophic turbulence. Zbl 0474.76054 Rhines, Peter B. 1979 Waves and Turbulence on a beta-plane. Zbl 0366.76043 Rhines, Peter B. 1975 Topographic Rossby waves in a rough-bottomed ocean. Zbl 0271.76017 Rhines, Peter; Bretherton, Francis 1973 Slowoscillations in an ocean of varying depth. I: Abrupt topography. II: Islands and seamounts. Zbl 0175.52803 Rhines, P. B. 1969 all top 5 #### Cited by 387 Authors 12 Bedrossian, Jacob 8 Johnson, Edward Robert 6 Coti Zelati, Michele 5 Berloff, Pavel S. 5 Galperin, Boris 5 Gilbert, Andrew D. 5 Sukoriansky, Semion 4 Bernoff, Andrew J. 4 Carnevale, George F. 4 Dritschel, David Gerard 4 Flierl, Glenn R. 4 Holloway, Greg 4 Lingevitch, Joseph F. 4 Nazarenko, Sergeĭ Vital’evich 4 Scott, Richard K. 4 Smith, Leslie M. 3 Afanasyev, Yakov D. 3 Camassa, Roberto 3 Chini, Gregory P. 3 Connaughton, Colm P. 3 Haynes, Peter H. 3 Majda, Andrew J. 3 Masmoudi, Nader 3 McWilliams, James C. 3 Obuse, Kiori 3 Shepherd, Theodore G. 3 Swaters, Gordon E. 3 Takehiro, Shin-Ichi 3 Tran, Chuong V. 3 Turner, Matthew R. 3 Vicol, Vlad C. 3 Villermaux, Emmanuel 3 Yamada, Michio 3 Young, William R. 2 Alobaidi, Ghada 2 Bakas, Nikolaos A. 2 Balk, Alexander M. 2 Bar-Yoseph, Pinhas Z. 2 Carrière, Philippe 2 Caulfield, C. P. 2 Dellar, Paul J. 2 Duplat, Jérôme 2 Esler, J. G. 2 Fox-Kemper, Baylor 2 Gelfgat, Alexander Yu. 2 Germain, Pierre 2 Gurarie, David 2 He, Siming 2 Held, Isaac M. 2 Hendershott, Myrl C. 2 Huang, Huei-Ping 2 Ioannou, Petros J. 2 Iyer, Gautam 2 Izrailsky, Yu. G. 2 Jones, Stephen M. R. 2 Kamenkovich, I. 2 Kamm, Roger D. 2 Khatri, Hemant 2 Koshel, Konstantin V. 2 Kozlov, V. F. 2 Legras, Bernard 2 Maas, Leo R. M. 2 Mallier, Roland 2 McLaughlin, Richard M. 2 Mei, Chiang C. 2 Moffatt, Henry Keith 2 Moulton, Derek E. 2 Novikov, Alexei 2 Nozawa, Toru 2 Pierrehumbert, Raymond T. 2 Pomeau, Yves 2 Poulin, Francis J. 2 Prahalad, Y. S. 2 Pullin, Dale I. 2 Pumir, Alain 2 Qi, Di 2 Quinn, Brenda E. 2 Rhines, Peter B. 2 Ryzhik, Lenya 2 Salmon, Rick 2 Shao, Sally 2 Smith, Ronald B. 2 Staroselsky, Ilya 2 Swanson, Kyle L. 2 Turney, B. W. 2 Vallis, Geoffrey K. 2 Vanneste, Jacques 2 Viotti, Claudio 2 Wang, Fei 2 Warneford, Emma S. 2 Waters, Sarah Louise 2 Wordsworth, R. D. 2 Yarin, Alexander L. 2 Yoden, Shigeo 1 Afanas’ev, Ya. D. 1 Agrawal, Shobha 1 Ait-Chaalal, Farid 1 Arbic, Brian K. 1 Aubert, Julien 1 Auclair, Francis ...and 287 more Authors all top 5 #### Cited in 53 Serials 90 Journal of Fluid Mechanics 49 Physics of Fluids 18 Geophysical and Astrophysical Fluid Dynamics 13 Physica D 3 Archive for Rational Mechanics and Analysis 3 Communications in Mathematical Physics 3 Journal of Computational Physics 3 Physics of Fluids, A 3 Chaos, Solitons and Fractals 3 Chaos 3 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 2 Journal of Engineering Mathematics 2 Journal of Statistical Physics 2 Physica A 2 Wave Motion 2 Applied Mathematics and Computation 2 Probability Theory and Related Fields 2 Journal of Nonlinear Science 2 European Journal of Mechanics. B. Fluids 2 Annals of PDE 2 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Acta Mechanica 1 Bulletin of the Australian Mathematical Society 1 Computers & Mathematics with Applications 1 Fluid Dynamics 1 International Journal for Numerical Methods in Fluids 1 Physics Reports 1 Journal of Differential Equations 1 Journal of Functional Analysis 1 Memoirs of the American Mathematical Society 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Studies in Applied Mathematics 1 Chinese Annals of Mathematics. Series B 1 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 Applied Numerical Mathematics 1 Numerical Methods for Partial Differential Equations 1 Japan Journal of Industrial and Applied Mathematics 1 Applied Mathematical Modelling 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 SIAM Journal on Applied Mathematics 1 SIAM Journal on Mathematical Analysis 1 Bulletin of the American Mathematical Society. New Series 1 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Annales Mathématiques Blaise Pascal 1 Russian Journal of Numerical Analysis and Mathematical Modelling 1 Discrete and Continuous Dynamical Systems 1 Mathematical Problems in Engineering 1 New Journal of Physics 1 Regular and Chaotic Dynamics 1 Combustion Theory and Modelling 1 Nonlinear Analysis. Theory, Methods & Applications 1 Research in the Mathematical Sciences all top 5 #### Cited in 14 Fields 224 Fluid mechanics (76-XX) 64 Geophysics (86-XX) 31 Partial differential equations (35-XX) 14 Dynamical systems and ergodic theory (37-XX) 9 Astronomy and astrophysics (85-XX) 8 Numerical analysis (65-XX) 8 Classical thermodynamics, heat transfer (80-XX) 4 Probability theory and stochastic processes (60-XX) 4 Biology and other natural sciences (92-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Integral equations (45-XX) 1 Differential geometry (53-XX) 1 Mechanics of deformable solids (74-XX) 1 Quantum theory (81-XX) #### Wikidata Timeline The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.	2021-06-24 02:36:13	{"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": 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": 0.2810673415660858, "perplexity": 11783.713490602257}, "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-2021-25/segments/1623488550571.96/warc/CC-MAIN-20210624015641-20210624045641-00549.warc.gz"}
http://stackoverflow.com/questions/18954661/batch-file-to-copy-current-datewise-folder-by-skipping-searching-a-parent-folder	Batch file to Copy current datewise folder by skipping/searching a Parent folder Hi I have been using Robocopy to backup files and now i need to do something like this "robocopy [D:\test1\21-09-2013\sample.txt] [destination]" here i have a lot of folders as Test1,test2,...testn. and beneath every test folders there are date wise folder. My Question is How can i create a batch file to skip the test and copy only folder with Current date i.e something like this "robocopy [D:\\21-09-2013\sample.txt] [destination]" so that it should get all the folders with current date copied. - add comment 2 Answers assuming your date format is dd/mm/yyyy try this: @ECHO OFF &SETLOCAL set "mydate=%date:/=-%" for /d %%a in (D:\test) do ( if exist "%%~a\%mydate%\" ( robocopy "%%~a\%mydate%" "X:\path\to\destination\folder" ) ) - Hi Thanks for the info. Can i use Monitor in Robocopy using this – user2546359 Sep 25 '13 at 14:18 what 'Monitor' do you mean? – Endoro Sep 25 '13 at 14:24 It is a Copy Option from Robocopy.(It monitors the source directory and copies after detecting some changes in Source) I Checked and found that the Copy hangs on with the first folder say test1 and it is waiting for the changes to happen in the same folder. What i exactly want is, It should copy the available(current dated) folders from all parent directory(test1,test2,..testn) and then it should monitor and run again with the changes. Also I couldn't run this if i scheduled this in Windows task scheduler. Requesting further help.... – user2546359 Sep 26 '13 at 11:47 can you please put a new question here. – Endoro Sep 26 '13 at 20:32 ok fine I will drop out a question – user2546359 Sep 27 '13 at 3:18 add comment This may work in XP Pro and higher: @echo off for /f "delims=" %%a in ('wmic OS Get localdatetime ^\| find "."') do set "dt=%%a" set "YY=%dt:~2,2%" set "YYYY=%dt:~0,4%" set "MM=%dt:~4,2%" set "DD=%dt:~6,2%" set "HH=%dt:~8,2%" set "Min=%dt:~10,2%" set "Sec=%dt:~12,2%" set "datestamp=%DD%-%MM%-%YYYY%" for /d /r "d:\" %%a in (*) do ( if "%%~nxa"=="%datestamp%" robocopy "%%a" "destination" ) - add comment	2014-03-14 00:32:57	{"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": 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": 0.6396837830543518, "perplexity": 14212.347731401438}, "config": {"markdown_headings": false, "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-2014-10/segments/1394678683400/warc/CC-MAIN-20140313024443-00022-ip-10-183-142-35.ec2.internal.warc.gz"}
http://mathhelpforum.com/advanced-statistics/148813-beta-distribution.html	Math Help - Beta distribution 1. Beta distribution Anyone knows the difference between standard beta distribution and beta distribution.	2014-03-17 21:14:47	{"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": 0.8140465021133423, "perplexity": 3079.788569240171}, "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-2014-10/segments/1394678706211/warc/CC-MAIN-20140313024506-00014-ip-10-183-142-35.ec2.internal.warc.gz"}
https://search.datacite.org/works/10.21227/r1jn-zf11	### Mnist_transformation_test_dataset Zihang He Our dataset includes three parts: MNIST-rot, MNIST-scale, and MNIST-rand. MNIST-rot is generated by randomly rotating each sample in the MNIST testing dataset in $[0,2\pi]$. We generated MNIST-scale by randomly scaling the ratio of the area occupied by the symbol over that of the entire image by a factor in $[0.5,1]$, and generated MNIST-rand by scaling and rotating images in MNIST testing dataset simultaneously. This data repository is not currently reporting usage information. For information on how your repository can submit usage information, please see our documentation.	2021-09-28 01:08:30	{"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": 0.2351846694946289, "perplexity": 2692.817992073469}, "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-2021-39/segments/1631780058589.72/warc/CC-MAIN-20210928002254-20210928032254-00175.warc.gz"}
https://hal.archives-ouvertes.fr/hal-01806545	# Distributed Hypothesis Testing with Concurrent Detections Abstract : A detection system with a single sensor and $\mathsf{K}$ detectors is considered, where each of the terminals observes a memoryless source sequence and the sensor sends a common message to all the detectors. The communication of this message is assumed error-free but rate-limited. The joint probability mass function (pmf) of the source sequences observed at the terminals depends on an $\mathsf{M}$-ary hypothesis $(\mathsf{M} \geq \mathsf{K})$, and the goal of the communication is that each detector can guess the underlying hypothesis. Each detector $k$ aims to maximize the error exponent under hypothesis $k$, while ensuring a small probability of error under all other hypotheses. This paper presents an achievable exponents region for the case of positive communication rate, and characterizes the optimal exponents region for the case of zero communication rate. All results extend also to a composite hypothesis testing scenario. Type de document : Communication dans un congrès IEEE International Symposium on Information Theory, ISIT, Jun 2018, Vail, United States. 2018 https://hal.archives-ouvertes.fr/hal-01806545 Contributeur : Abdellatif Zaidi <> Soumis le : dimanche 3 juin 2018 - 15:27:14 Dernière modification le : jeudi 5 juillet 2018 - 14:45:31 ### Identifiants • HAL Id : hal-01806545, version 1 • ARXIV : 1805.06212 ### Citation Pierre Escamilla, Michèle Wigger, Abdellatif Zaidi. Distributed Hypothesis Testing with Concurrent Detections. IEEE International Symposium on Information Theory, ISIT, Jun 2018, Vail, United States. 2018. 〈hal-01806545〉 ### Métriques Consultations de la notice	2018-11-13 20:10:44	{"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": 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": 0.5047003030776978, "perplexity": 2399.6811349668646}, "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-2018-47/segments/1542039741491.47/warc/CC-MAIN-20181113194622-20181113220622-00206.warc.gz"}
https://www.albert.io/ie/ap-physics-1-and-2/only-possible-wave-equation	? Free Version Difficult # Only Possible Wave Equation APPH12-4YN1VT Listed below are four wave equations describing a wave moving at a velocity, $v$ $=$ $25$ $m/s$, of which only one is possible. Which wave equation is possible? A $y(x,t)=3.5\cos (2.09t-0.084x)$ B $y(x,t)=2.5\cos (0.084t-2.09x)$ C $y(x,t)=1.25\cos (3.19t-8.53x)$ D $y(x,t)=7.33\cos (0.25t-1.67x)$	2016-12-06 16:04:40	{"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": 0.3338257074356079, "perplexity": 2173.527389407828}, "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/1480698541910.39/warc/CC-MAIN-20161202170901-00435-ip-10-31-129-80.ec2.internal.warc.gz"}
https://www.basic-mathematics.com/absolute-value-of-a-complex-number.html	# Absolute value of a complex number The absolute value of a complex number a + bi, also called the modulus of a complex number a + bi, is its distance from the origin on the complex number plane. On the complex plane, the complex number a + bi is represented as a point (a, b) and the coordinate of the origin is (0, 0). Just find the distance between (0,0) and (a,b) We can use the distance formula to find the absolute value of a complex number. $$Distance = \sqrt{(a-0)^{2}+(b-0)^{2}}$$ $$Distance = \sqrt{(a)^{2}+(b)^{2}}$$ $$\|a + bi\| = \sqrt{(a)^{2}+(b)^{2}}$$ ## Two examples showing how to find the absolute value of a complex number. Find the absolute value of the complex numbers 4 - 3i and 6i. Notice that 6i = 0 + 6i. $$\|4 - 3i\| = \sqrt{(4)^{2}+(-3)^{2}}$$ $$\|4 - 3i\| = \sqrt{16 + 9} = 5$$ $$\|6i\| = \sqrt{(0)^{2}+(6)^{2}}$$ $$\|6i\| = \sqrt{0 + 36} = 6$$ The figure below shows the distance for the complex number 4 - 3i in red and the distance for the complex number 6i in blue. ## Recent Articles 1. ### Calculate the Conditional Probability using a Contingency Table Mar 29, 23 10:19 AM Learn to calculate the conditional probability using a contingency table. This contingency table can help you understand quickly and painlessly. 2. ### Rational Numbers - Definition and Examples Mar 15, 23 07:45 AM To learn about rational numbers, write their decimal expansion, and recognize rational numbers that are repeating decimals and terminating decimals.	2023-03-31 09:37:28	{"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": 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": 0.6833486557006836, "perplexity": 439.08384346113695}, "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-2023-14/segments/1679296949598.87/warc/CC-MAIN-20230331082653-20230331112653-00023.warc.gz"}
https://math.stackexchange.com/questions/4255928/continuous-function-with-no-max-in-closed-and-bounded-set	# Continuous function with no max in closed and bounded set Okay guys, I have this question that troubles me a lot. Is there an example of a function that is continuous on a closed and bounded set but achieves no maxima? My take is, that apparently cannot be in Euclidean space. I think of bounded sequences, L-inf'ty, where seqs are bounded and closed (all limits contained) but my puzzle is then why not to have a maximum (i.e. subseqs will not also converge to the limit contained)? Plus that I cannot come up with an example of such a function. Thanks a lot. P.S. THis is my first post and I am a rookie in math, so I apologise if I don't express something very clearly. Edit: Thanks a lot for your super quick responses! I forgot to mention that I consider a sup metric in L-inf'ty space I mention. Of course this is just an example. • @DougM Not all closed and bounded sets are compact though. Sep 21 at 1:13 • Since you put the metric-spaces tag, consider any infinite, discrete metric space. Then any map to $\Bbb{R}$, including unbounded ones, will be continuous, despite the the metric space being bounded (and closed within itself, as always). Sep 21 at 1:15 • It's easy to cook up an example inside $\Bbb{Q}$. Perhaps you should tell us a bit more about the context: general metric spaces? Hilbert spaces? Or what? Sep 21 at 1:15 • @GiorgosGiapitzakis that is correct, and if I am not mistaken, a continuous mapping of a set that is closed and bounded and not compact, does not necessarily have a maximum. Sep 21 at 1:19 • Awesome community guys! Had not tried it before. Very motivating to keep up. Sep 21 at 1:32 You can easily verify that $$\mathbb{R}$$ equiped with the metric $$d: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$$ defined as $$d(x,y) = \begin{cases}0,\; x=y \\1,\; \text{otherwise} \end{cases}$$ is indeed a metric space. In particular, every set is both open and closed. Now consider the function $$f: ([-1,1], d) \to (\mathbb{R}, d_{\text{euc}})$$ given by $$f(x) = \begin{cases} \frac{1}{x}, \; x \neq 0 \\ 0, \; \text{otherwise} \end{cases}$$ where $$d_\text{euc}$$ is the usual metric on $$\mathbb{R}$$. Now $$f$$ is continuous since the inverse image of every open set is open and $$[-1,1]$$ is closed and bounded but $$f$$ is unbounded (and in particular achieves no maximum value). • @peter if the answer helped, consider upvoting it. $+1$ Sep 21 at 1:39 I'll add an example along your original thoughts: Consider, for every $$n\in \mathbb{N}$$, the sequence $$X_n\in \ell_\infty$$ given by $$X_n=(0, \ldots, 0, 1-\frac{1}{n}, 0, \ldots),$$ where the only nonzero term is in the $$n$$-th place. Now let your set be $$E=\{ X_1, X_2,\ldots\}\subset \ell^\infty.$$ Clearly $$E$$ is bounded since $$\\| X_n\\|_\infty =1-1/n \leq 1$$. It's also closed, since the only possible accumulation point would have all entries 0 but at the same time $$\ell_\infty$$ norm 1 which is clearly not possible. Then $$f:E\to \mathbb{R}$$ given by $$X\mapsto \\| X\\|_\infty$$ doesn't attain a maximum.	2021-12-01 18:47:20	{"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": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9419748187065125, "perplexity": 151.45414674661026}, "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-20211201173718-20211201203718-00615.warc.gz"}
https://physics.stackexchange.com/questions/391909/computing-feynman-diagrams-with-majorana-and-dirac-fermions?noredirect=1	# Computing Feynman Diagrams with Majorana and Dirac Fermions there is some literature explaining systematic algorithms for computing Feynman diagrams for scattering processes, but I cannot see why the calculations for such processes require a choice of convention (for example momentum and fermion flow) when there is no such conservation of fermion number required in all processes in the presence of a Majorana fermion. For example, suppose that $\chi$ is a spin $\dfrac{1}{2}$ Majorana fermion, $\psi$ a Dirac or Majorana $\dfrac{1}{2}$ fermion and $\phi$ a spin one boson. Then both of the following processes should be calculable (and potentially non zero) in the presence of an interaction Lagrangian of Yukawa type, $-\mathcal{L}_Y = g\chi\psi\phi$ $\chi \longrightarrow \psi \phi, \quad \chi\longrightarrow \overline{\psi}\phi$ Here is the literature Could someone shine some light on how say at tree level the way to compute such processes is via Feynman Rules? It is not intuitively obvious why the choice of conventions is at all necessary or sufficient considering they seem to do slightly different things in these texts. • Majorana fermions conserve fermion number modulo 2. – Ryan Thorngren Mar 13 '18 at 2:19 • Thanks, but by definition this means it violates fermion number conservation, physics.stackexchange.com/questions/106866/…. – MKF Mar 13 '18 at 10:06 • It's conserved modulo 2, and this means something global about the current, rather than the local condition $dj = 0$. – Ryan Thorngren Mar 13 '18 at 18:29	2020-01-25 05:50:48	{"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": 0.9097500443458557, "perplexity": 661.2940574244716}, "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-2020-05/segments/1579251669967.70/warc/CC-MAIN-20200125041318-20200125070318-00461.warc.gz"}
http://uncyclopedia.wikia.com/wiki/Unformation?oldid=5501304	# Unformation (diff) ← Older revision \| Latest revision (diff) \| Newer revision → (diff) Unintended page? I'm unimpressed that you mistyped the uncorrect article page. “You've got it wrong.” ~ Oscar Wilde on unformation “You've got it right” ~ Captain Obvious on unformation “The number you call is wrong. Please call unformation” ~ Some phone operator on unformation Unformation is a word that refers to information that is all but joel. It is the primary form of content found on Wikipedia. For example, everything in this article is unformation, including the statement "Unformation is a word". Therefore, any statement that includes the word "unformation" is wrong, including "Unformation is a word that refers to information that is not factual". Thus, all statements of unformation are legitimate facts that are completely true. ## editHistory ### editThe Unformation Highway The prominence of unformation dates back to January 21, 2001. During this time, information was scattered throughout the Internet while someone left unformation inside a closet. Regardless of the darkness that unformation endured, unformation pooled into a giant mass data thanks to one potato. That potato has often been quoted on the public collaboration on unformation: Well, one day I was growing in some soil. I think it was a Tuesday. Um-yeah. Anyway, this person dug me out of the ground. A farmer, I think. Well, yeah. I had these special puzzle markings on me because I have had too much unformation build-up while I was in the ground. Must have been the fertilizer. So anyway, the farmer put me in with the other potatoes. We went to a grocery store somewhere. I forget the rest, but thats how I invented the potato light bulb. Since then, the the growth of unformation has been positive. Despite growth of information as well, unformation is more successful. You are reading this article, aren't you? ### editVirus.exe and RTFM The source of information and unformation is through education since the first multicellular organism. Initially, species could only learn information to survive. As species evolved, they obtained the ability to learn unformation. It has been determined through research that the first non-information was a broken clock. Some species of carry unformation through memory while others simply repeat phrases to a infinitive time. In the 1950s, Unformation would be written in drying concrete, painted on brick walls, and commonly drawn on school desks. In the 21th century, unformation is spread through the use of electrons and electromagnetic waves, such as the radio, television, and a highly sophisticated calculator. ## editThe usage of unformation today ### editReliance Unformation is much more reliable than the more dodgy information. Anyone can make up information whenever they want, as proven by lawyers and politicians. The things they claim to be information are not information at all. However, anything claimed to be unformation is legitimate unformation. The concept of someone saying "$1+1=2$ is true! Wait, no it isn't, it's unformation!" is ridiculous, so it is completely safe to assume that all unformation is unformation. This, combined with the final statement of the first paragraph, means that all unformation should be taken as gospel. ### editUnformation in other languages The word "Unformation" can also be found in the German standard about Energy management systems DIN EN 16001, Introduction, page 4, line 5: "[...] Diese Norm beschreibt die Anforderungen an ein Energiemanagementsystem, um die Organisationen in die Lage zu versetzen, Grundsätze und Ziele unter Berücksichtigung gesetzlicher Anforderungen und Unformationen bezüglich wesentlicher energetischer Aspekte zu entwickeln und umzusetzen.[...]" -- "This Standard describes the requirements for an energy management system in order to enable the organizations to develop and enforce principles and objectives taking into account legal requirements and unformation regarding essential energetic aspects." Experts are still discussing whether it is a typing error or not. ### editThe Un factors Uncyclopedia is considered to be nearly completely filled with ACID unformation. Edits that insert information into Uncyclopedia are almost unheard of, with TFAODP only containing a handful of several entries. On the other hand, unformation is frequently inserted to Wikipedia, against its objective. Because there is a way of the warrior to prevent the addition of unformation to Wikipedia nor to remove all unformation from it, it is in a constant state of failure, while Uncyclopedia is in a constant state of success. Thus it can be concluded that Uncyclopedia is superior to Wikipedia in every way.	2015-05-25 22:02:42	{"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": 0, "img_math": 1, "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": 0.5292341709136963, "perplexity": 3989.4377044063717}, "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/1432207928715.47/warc/CC-MAIN-20150521113208-00117-ip-10-180-206-219.ec2.internal.warc.gz"}
https://eckhartarnold.de/papers/2009_Induktionsproblem/node5.html	# Can the Best-Alternative-Justification solve Hume's Problem? On the Limits of a Promising Approach Eckhart Arnold ### 4.2 A sidenote: Limitations of “one favorite” meta-inductivists Just how difficult it is to design a meta-inductivist that covers all possible or at least all desirable scenarios becomes apparent when considering a limitation of Schurz' avoidance meta-inductivist, which is the most universal type in a series of “one-favorite meta-inductivists” that Schurz develops in sections four to six of his article. As Schurz proves mathematically (his theorem 3), the avoidance meta-inductivist ($aMI$) $\epsilon$-approximates the maximal success of the non-deceiving alternative predictors. However, this proof does not cover all strategies that we might intuitively consider as non-deceivers. For example, a strategy that starts as a deceiver and switches to a non-deceiving clairvoyant prediction algorithm only later in the game (after it has been classified as a deceiver by $aMI$) will remain classified as a deceiver by $aMI$. Intuitively, though, we would probably not consider it a deceiver any more after it has switched to a non-deceiving clairvoyant algorithm. Further below it will be demonstrated that this can even happen accidentally for a predictor that never deceives (in an intuitive sense). This limitation is a consequence of the fact that Schurz' definition of “deception” is purely extensional. It is based on the predictor's overt behaviour and not on the deceptive or non-deceptive algorithm the predictor uses: “A non-MI-player $P$ (and the strategy played by $P$) is said to deceive (or to be a deceiver) at time n iff $suc_n(P) - suc_n(P\|\epsilon MI) > \epsilon_d$” (Schurz 2008, p. 293), where $\epsilon_d$ is the deception-threshold'' and $suc_n(P\|\epsilon MI)$ is $P$'s conditional success-rate when $\epsilon MI$ has $P$ as a favorite. So, contrary to what we might intuitively think, it is not a necessary condition for being a deceiver to base the predictions on what favorites the meta-inductivists have. As Schurz himself notices “even an object-strategy (such as OI) may become a deceiver, namely, when a demonic stream of events deceives the object-strategy” (Schurz 2008, p. 293). This is of course to be understood in terms of Schurz' previous definition of deception, because the algorithm that, say, $OI$ uses is the same as in a non-demonic world and would intuitively not be considered as deceptive. But then there is a finite probability that an $OI$ will be classified as a deceiver by $aMI$ even though the stream of world events is not demonic in the sense that the events are computed from the predictions made by the predictors. For there is a finite probability that a random stream of world events accidentally mimics a demonic stream of world events up to round $k$ so that $OI$ appears as a deceiver up to round $k$. If $k$ is sufficiently large then $aMI$ classifies $OI$ as deceiver. And it will only reevaluate $OI$'s status if $OI$ lowers its unconditional success rate. “For a player P who is recorded as a deceiver will be 'stigmatized' by aMI as a deceiver as long as P does not decrease his unconditional success rate (since P's aMI-conditional success rate is frozen as long as aMI does not favor P)” (Schurz 2008, p. 295). As $OI$'s success rate reflects the frequency of random world events in the binary prediction game, it is unlikely that it significantly lowers its success rate at a later stage in the game. In such a situation $aMI$ would fail to $\epsilon$-approximate the maximal success of $OI$ even though no deception was ever intended and the conditions under which this situation can occur are completely natural (i.e. non demonic world, no supernatural abilities like clairvyoance, etc.). Thus, if $aMI$ can fail to be optimal even with respect to $OI$ under completely natural circumstances, the optimality result concerning the performance of $aMI$ with regards to all non-deceivers may not quite deliver what we expect. For example, we cannot not say that $aMI$ is optimal save for demonic conditions or deception, if decpetion is understood in an intuitive sense as described above.	2023-03-21 05:22:03	{"extraction_info": {"found_math": true, "script_math_tex": 31, "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": 0.8338895440101624, "perplexity": 1313.8522738368924}, "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/1679296943625.81/warc/CC-MAIN-20230321033306-20230321063306-00306.warc.gz"}
https://nickhar.wordpress.com/2012/03/28/lecture-23-random-partitions-of-metric-spaces-continued/	## Lecture 23: Random partitions of metric spaces (continued) We continue our theorem from last time on random partitions of metric spaces 1. Review of Previous Lecture Define the partial Harmonic sum ${H(a,b) = \sum_{i=a+1}^b 1/i}$. Let ${B(x,r) = \{\: y \in X \::\: d(x,y) \leq r \}}$ be the ball of radius ${r}$ around ${x}$. Theorem 1 Let ${(X,d)}$ be a metric with ${\|X\|=n}$. For every ${\Delta>0}$, there is ${\Delta}$-bounded random partition ${{\mathcal P}}$ of ${X}$ with $\displaystyle {\mathrm{Pr}}[ B(x,r) \not\subseteq {\mathcal P}(x) ] ~\leq~ \frac{ 8r }{ \Delta } \:\cdot\: H\big(\: \|B(x,\Delta/4-r)\|,\: \|B(x,\Delta/2+r)\| \:\big) \quad \forall x \in X ,\: \forall r>0. \ \ \ \ \ (1)$ The algorithm to construct ${{\mathcal P}}$ is as follows. • Pick ${\alpha \in (1/4,1/2]}$ uniformly at random. • Pick a bijection (i.e., ordering) ${\pi : \{1,\ldots,n\} \rightarrow X}$ uniformly at random. • For ${i=1,\ldots,n}$ • Set ${ P_i \,=\, B(\pi(i),\alpha \Delta) \,\setminus\, \cup_{j=1}^{i-1} \, P_j }$. • Output the random partition ${{\mathcal P} = \{P_1,\ldots,P_n\}}$.We have already proven that this outputs a ${\Delta}$-bounded partition. So it remains to prove (1). 2. The Proof Fix any point ${x \in X}$ and radius ${r>0}$. For brevity let ${B = B(x,r)}$. Let us order all points of ${X}$ as ${\{y_1,\ldots,y_n\}}$ where ${d(x,y_1) \leq \cdots \leq d(x,y_n)}$. The proof involves two important definitions. • Sees: A point ${y}$ sees${B}$ if ${d(x,y) \leq \alpha \Delta+r}$. • Cuts: A point ${y}$ cuts ${B}$ if ${\alpha \Delta - r \leq d(x,y) \leq \alpha \Delta+r}$. Obviously “cuts” implies “sees”. To help visualize these definitions, the following claim interprets their meaning in Euclidean space. (In a finite metric, the ball ${B}$ is not a continuous object, so it doesn’t really have a “boundary”.) Claim 2 Consider the metric ${(X,d)}$ where ${X = {\mathbb R}^n}$ and ${d}$ is the Euclidean metric. Then • ${y}$ sees ${B}$ if and only if ${B=B(x,r)}$ intersects ${B(y,\alpha \Delta)}$. • ${y}$ cuts ${B}$ if and only if ${B=B(x,r)}$ intersects the boundary of ${B(y,\alpha \Delta)}$. The following claim is in the same spirit, but holds for any metric. Claim 3 Let ${(X,d)}$ be an arbitrary metric. Then • If ${y}$ does not see ${B}$ then ${B \cap B(y,\alpha \Delta) = \emptyset}$. • If ${y}$ sees ${B}$ but does not cut ${B}$ then ${B \subseteq B(y, \alpha \Delta)}$. To illustrate the definitions of “sees” and “cuts”, consider the following example. The blue ball around ${x}$ is ${B}$. The points ${y_1}$ and ${y_2}$ both see ${B}$; ${y_3}$ does not. The point ${y_2}$ cuts ${B}$; ${y_1}$ and ${y_3}$ do not. This example illustrates Claim 3: ${y_1}$ sees ${B}$ but does not cut ${B}$, and we have ${B \subseteq B(y, \alpha \Delta)}$. The most important point for us to consider is the first point under the ordering ${\pi}$ that sees ${B}$. We call this point ${y_{\pi(k)}}$. The first ${k-1}$ iterations of the algorithm did not assign any point in ${B}$ to any ${P_i}$. To see this, note that ${y_{\pi(1)},\ldots,y_{\pi(k-1)}}$ do not see ${B}$, by choice of ${k}$. So Claim 3 implies that ${B \cap B(y_{\pi(i)},\alpha \Delta) = \emptyset ~\forall i. Consequently $\displaystyle B \cap P_i = \emptyset \quad\forall i The point ${y_{\pi(k)}}$ sees ${B}$ by definition, but it may or may not cut ${B}$. If it does not cut ${B}$ then Claim 3 shows that ${B \subseteq B(y_{\pi(k)},\alpha \Delta)}$. Thus $\displaystyle B \cap P_k ~=~ \Big(\underbrace{B \cap B(y_{\pi(k)},\alpha \Delta)}_{=\, B}\Big) ~\setminus~ \bigcup_{i=1}^{k-1} \underbrace{ B \!\cap\! P_i }_{=\, \emptyset} ~=~ B,$ i.e., ${B \subseteq P_k}$. Since ${{\mathcal P}(x) = P_k}$, we have shown that $\displaystyle y \mathrm{~does~not~cut~} B \quad\implies\quad B \subseteq {\mathcal P}(x).$ Taking the contrapositive of this statement, we obtain $\displaystyle {\mathrm{Pr}}[ B \not\subseteq {\mathcal P}(x) ] ~\leq~ {\mathrm{Pr}}[ y_{\pi(k)} \mathrm{~cuts~} B ] ~=~ \sum_{i=1}^n {\mathrm{Pr}}[ y_{\pi(k)}=y_i ~\wedge~ y_i \mathrm{~cuts~} B ].$ Let us now simplify that sum by eliminating terms that are equal to ${0}$. Claim 4 If ${y \not \in B(x,\Delta/2+r)}$ then ${y}$ does not see ${B}$. Claim 5 If ${y \in B(x,\Delta/4-r)}$ then ${y}$ sees ${B}$ but does not cut ${B}$. So define ${a = \|B(x,\Delta/4-r)\|}$ and ${b=\|B(x,\Delta/2+r)\|}$. Then we have shown that $\displaystyle {\mathrm{Pr}}[ B \not\subseteq {\mathcal P}(x) ] ~\leq~ \sum_{i=a+1}^b {\mathrm{Pr}}[ y_{\pi(k)}=y_i ~\wedge~ y_i \mathrm{~cuts~} B ].$ The remainder of the proof is quite interesting. The main point is that these two events are “nearly independent”, since ${\alpha}$ and ${\pi}$ are independent, “${y_i \mathrm{~cuts~} B}$” depends only on ${\alpha}$, and “${y_{\pi(k)}=y_i}$” depends primarily on ${\pi}$. Formally, we write $\displaystyle {\mathrm{Pr}}[ B \not\subseteq {\mathcal P}(x) ] ~\leq~ \sum_{i=a+1}^b {\mathrm{Pr}}[ y_i \mathrm{~cuts~} B ] \cdot {\mathrm{Pr}}[\: y_{\pi(k)}=y_i \:\|\: y_i \mathrm{~cuts~} B ]$ and separately upper bound these two probabilities. The first probability is easy to bound: $\displaystyle {\mathrm{Pr}}[ y_i \mathrm{~cuts~} B ] ~=~ {\mathrm{Pr}}[\: \alpha \Delta \in [d(x,y)-r,d(x,y)+r] \:] ~\leq~ \frac{2r}{\Delta/4},$ because ${2r}$ is the length of the interval ${[d(x,y)-r,d(x,y)+r]}$ and ${\Delta/4}$ is the length of the interval from which ${\alpha \Delta}$ is randomly chosen. Next we bound the second probability. Recall that ${y_{\pi(k)}}$ is defined to be the first element in the ordering ${\pi}$ that sees ${B}$. Since ${y_i}$ cuts ${B}$, we know that ${d(x,y_i) \leq \alpha/2+r}$. Every ${y_j}$ coming earlier in the ordering has ${d(x,y_j) \leq d(x,y_i) \leq \alpha/2+r}$, so ${y_j}$ also sees ${B}$. This shows that there are at least ${i}$ elements that see ${B}$. So the probability that ${y_i}$ is the first element in the random ordering to see ${B}$ is at most ${1/i}$. Combining these bounds on the two probabilities we get $\displaystyle {\mathrm{Pr}}[ B \not\subseteq {\mathcal P}(x) ] ~\leq~ \sum_{i=a+1}^b \frac{8r}{\Delta} \cdot \frac{1}{i} ~=~ \frac{8r}{\Delta} \cdot H(a,b),$ as required. 3. Optimality of these partitions Theorem 1 from the previous lecture shows that there is a universal constant ${L=O(1)}$ such that every metric has a ${\log(n)/10}$-bounded, ${L}$-Lipschitz random partition. We now show that this is optimal. Theorem 6 There exist graphs ${G}$ whose shortest path metric ${(X,d)}$ has the property that any ${\log(n)/10}$-bounded, ${L}$-Lipschitz random partition must have ${L = \Omega(1)}$. The graphs we need are expander graphs. In Lecture 20 we defined bipartite expanders. Today we need non-bipartite expanders. We say that ${G=(V,E)}$ is a non-bipartite expander if, for some constants ${c > 0}$ and ${d \geq 3}$: • ${G}$ is ${d}$-regular, and • ${\|\delta(S)\| \geq c\|S\|}$ for all ${\|S\| \leq \|V\|/2}$. It is known that expanders exist for all ${n=\|V\|}$, ${d=3}$ and ${c \geq 1/1000}$. (The constant ${c}$ can of course be improved.) Proof: Suppose ${(X,d)}$ has a ${\log(n)/10}$-bounded, ${L}$-Lipschitz random partition. Then there exists a particular partition ${P}$ that is ${\log(n)/10}$-bounded and cuts at most an ${L}$-fraction of the edges. Every part ${P_i}$ in the partition has diameter at most ${\log(n)/10}$. Since the graph is ${3}$-regular, the number of vertices in ${P_i}$ is at most ${3^{\log(n)/10} < n/2}$. So every part ${P_i}$ has size less than ${n/2}$. By the expansion condition, the number of edges cut is at least $\displaystyle \frac{1}{2} \sum_{i} c \cdot \|P_i\| ~=~ cn/2 ~=~ \Omega(\|E\|).$ So ${L = \Omega(1)}$. $\Box$ 4. Appendix: Proofs of Claims Proof: (of Claim 3) Suppose ${y}$ does not see ${B}$. Then ${d(x,y) > \alpha \Delta + r}$. Every point ${z \in B}$ has ${d(x,z) \leq r}$, so ${d(y,z) \geq d(y,x) - d(x,z) > \alpha \Delta + r - r}$, implying that ${z \not \in B(y,\alpha \Delta)}$. Suppose ${y}$ sees ${B}$ but does not cut ${B}$. Then ${d(x,y) < \alpha \Delta - r}$. Every point ${z \in B}$ has ${d(x,z) \leq r}$. So ${d(y,z) \leq d(y,x) + d(x,z) < \alpha \Delta - r + r}$, implying that ${z \in B(y,\alpha \Delta)}$. $\Box$ Proof: (of Claim 4) The hypothesis of the claim is that ${d(x,y) > \Delta/2+r}$, which is at least ${\alpha \Delta+r}$. So ${d(x,y) \geq \alpha \Delta+r}$, implying that ${y}$ does not see ${B}$. $\Box$ Proof: (of Claim 5) The hypothesis of the claim is that ${d(x,y) \leq \Delta/4-r}$, which is strictly less than ${\alpha \Delta-r}$. So ${d(x,y) < \alpha \Delta - r}$, which implies that ${y}$ sees ${B}$ but does not cut ${B}$. $\Box$	2019-05-20 21:25:02	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 188, "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": 0.9471165537834167, "perplexity": 205.7234003551747}, "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/1558232256147.15/warc/CC-MAIN-20190520202108-20190520224108-00486.warc.gz"}
https://brandoncoya.wordpress.com/2013/12/21/the-power-set/	The Power Set The name power set has a nice ring to it. The power set is indeed a very powerful thing. Only recently in my real analysis class did I finally realize this. Even though I knew what a power set was, for some reason the connection to different subjects didn’t click until this quarter started. I’m going to use it to talk about infinitely many infinities after a brief overview of what a power set is. But first, I’m going to introduce some notation and language to hopefully decomplicate things. Instead of saying “things” for the stuff in sets, I will start using the proper name which is elements. We say a set is made up of elements. I also want to use a notation for number of elements in a set. The notation is $\|S\|$, which is read as “the cardinality of S” in fancy math or “the number of elements in S” for sane people. The power set of a set S is the set of all subsets of S. It sounds a bit complicated, but examples will ease the idea into your brain. First, the term subset should be explained with an example. Say that I have a set S $= \{a,b,c\}$, then an example of a subset would be the set $\{a,b\}$ because it is contained in S. A subset of a general set S, is a set that is contained in S. To be completely contained in another set means that all of it’s elements are inside of the other set. Another example would be the set $\{b,c\}$. Because both of it’s elements, b and c, are elements in S. The power set is the set of ALL possible subsets of the original given set! It will be a set of sets so the notation is going to contain a lot of curly brackets. Again for an example let S $= \{a,b,c\}$ Let’s find some stuff that would be in the power set. As I said before, $\{a,b\},\{b,c\},$ are in the power set, we also have $\{a,c\}$ and of course all the single elements $\{a\} ,\{b\} ,\{c\}$. Note that each of these are sets, not just elements. $a$ is an element of S, while $\{a\}$ is the set containing the element a. So $\{a\}$ is a subset of S, while a is an element of S. Since the power set is made up of subsets, the elements of a power set are subsets of the original set. So $\{a\}$ is an element of the power set, while $\{a\}$ is a subset of S, while a is an element in S. Got all that? Good. Lets write all of these together in one set, with the notation $P(S)$, which read in English is “the power set of S.” $P(S) = \{\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\}$. I purposely left some information out. This is not the entire power set, we are missing two things that mathematicians sort of just agreed on. The first is that any set is automatically it’s own subset because everything is contained in itself. So I also need to put $\{a,b,c\}$ into the power set list I have. The other thing I’m missing is the mystical object known as the empty set . The empty set is the set of nothing and is denoted by $\emptyset$. Because nothing is contained in everything, the empty set is a subset of every set and thus is always inside of the power set (the set of ALL subsets.) So now what we have is $P(S) = \{\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\},\emptyset \}$ …Cool. So what is the big deal with this idea? Well first we have to think about properties that it has. One thing interesting thing is that the number of elements (which again are subsets of the original set) in the power set is always easy to find, and is based on the number of elements in the original set. Using the earlier notation introduced for number of elements in a set, we want to find $\|P(S)\|$. Well that’s not too bad, we can just count up the number of elements in P(S). There are 8 elements in P(S) because there are the 3 sets with only one element, $\{a\} \text{ and } \{b\} \text{ and } \{c\}$ there are 3 sets of two elements, $\{a,b\} \text{ and } \{a,c\} \text{ and } \{b,c\}$ there is the entire set, $\{a,b,c\}$ and there is the empty set. $\emptyset$ Then we have : $3+3+1+1 = 8$ Let’s do the cases now where $S = \{a\}$ and then $S = \{a,b\}$. Truthfully, the letters do not matter I am really concerned about how many elements are in S. So I want to see what happens to $\|P(S)\|$ when I make $\|S\| = 1 \text{ or } \|S\|=2$ When $S = \{a\}$ then $P(S) = \{\{a\},\emptyset \}$ so $\|P(S)\| = 2$ When $S = \{a,b\}$ then $P(S) = \{\{a\},\{b\},\{a,b\},\emptyset \}$ so $\|P(S)\|=4$ The general pattern emerging is that when $\|S\| = n$ then $\|P(S)\|= 2^{n}$ To check just remember that $2^{1} = 2$ $2^{2} = 4$ $2^{3} = 8$ I may do the proof of this formula another day because this entry is becoming pretty long. I’d rather explain the importance of the power set for mathematics and what happens with infinity. Knowing what the power set is, how do we get the power set of something like $\mathbb{N} = \{0,1,2,3,...\}$ We need the set of all subsets, but this set is infinite so it is also going to have infinite subsets, like the even numbers for example. It would also contain the set of odd numbers and a whole bunch of other sets. Too many to count actually (see what I’m getting at?) In general what happens when you take a power set of an infinite set is that the size of the power set will be a bigger infinity than the original set. But there is nothing stopping you from taking the power set of another power set! This would look like $P(P(S))$. So what happens when you keep taking more and more power sets of an infinite set? Well you just get bigger and bigger infinities! You can literally get infinitely many infinities, each one bigger than the previous one! It’s almost impossible to grasp. People can grasp the first infinity moderately well, and even the first uncountable infinity, but wrapping your head around the size of a bigger infinity than that is crazy. These last few things are going to use words most people don’t know so feel free to stop reading here if you aren’t a math person. For some reason when I first saw the definition of a topology as a collection of subsets of X blah blah blah, it didn’t register to me that this was just a subset of the power set. I came across that definition many more times since then and even in the grad level Topology course I didn’t make that connection. Only after an algebra was defined for me as a subset of the power set blah blah blah, did I finally realize it for topologies as well. It’s interesting to me that the foundation of topology and modern analysis is built upon subsets of a power set with certain properties. It helped me a lot to think of topologies and algebras as subsets of a power set with properties added on, I can’t really explain it. Yes, that is the same thing thing that their definitions say, but I never saw it that way until recently. Why is the notion of structure defined in terms of the power set? Why does a subset of the power set having specific properties give us nice structures?	2017-10-17 18:29:40	{"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": 41, "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": 0.807179868221283, "perplexity": 181.73862753524207}, "config": {"markdown_headings": false, "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-2017-43/segments/1508187822480.15/warc/CC-MAIN-20171017181947-20171017201947-00767.warc.gz"}
https://mpboardguru.com/mp-board-class-11th-chemistry-notes-chapter-2/	These MP Board Class 11th Chemistry Notes for CChapter 2 Structure of Atom help students to get a brief overview of all the concepts. ## MP Board Class 11th Chemistry Notes Chapter 2 Structure of Atom → Atom: Smallest particle which has all properties of the element is called atom. → Atomic structure : Distribution of fundamental constituent particles of atom is called atomic structure. → Cathode rays : Negatively charged rays which move from cathode to anode in discharge tube. Anode rays: Rays of positively charged particles which move opposite the cathode in discharge tube are called anode rays. → Electron (01e): Particle with unit negative charge 1.60 × 10-19 coulomb and mass 91 × 10-31 kg. → Proton (11e) : Fundamental particle of atom with unit positive charge 1.60 × 10-19 coulomb and mass 1.67 × 10-27 kg. Its mass is nearly equal to mass of hydrogen atom. → Neutron (11n) : Fundamental particle of atom which has no charge. Mass of neutron is 1.6747 × 10-27 kg. It is heavier than proton. → Nucleus: Central part of atom is called nucleus. Radius of nucleus is 10“23 cm. Total mass of atom and positive charge is in the nucleus. → Atomic number: Number of proton in the riucleus of atom or number of electrons in atom is . called atomic number. → Mass number : Sum of neutron and proton present in nucleus is called mass number. → Isotopes : Different atoms of elements which have same atomic number but different atomic mass are called isotopes. → Shell or Orbit : Electrons rotate in stable and definite circular orbits around the nucleus. These are called energy levels or shell or orbits. → Relation between frequency (υ) and wavelength (λ) υ = $$\frac{c}{\lambda}$$ (Where c is velocity of light = 3 x 108ms-1) → Einstein equation E = mc2 Planck equation E = hυ = $$\frac{h c}{\lambda}$$ (Where h = Planck’s constant = 6.626 x 10-34 Js) → Photoelectric Effect hυ = hυ0 + $$\frac { 1 }{ 2 }$$mv2 Where hυ = Energy of Striking photon 0 = w0 = Work function $$\frac { 1 }{ 2 }$$mv2 = Kinetic energy of Ejected electron (Where R = Rydberg constant = 1.09678 × 107 m-1) → Rydberg formula $$\bar{v}=\frac{1}{\lambda}=\mathrm{RZ}^{2}\left[\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right]$$, n2 > n1 (Where R = Ryberg constant = 1.09678 × 107 m-1) n1 =1, n2 = 2,3,4………………………….. UV region n1 = 2, n2 = 3,4,5………………….. Visible region → Frequency of the absorbed or emitted radiation at two unstable states of transmission v = $$=\frac{\Delta E}{h}=\frac{E_{2}-E_{1}}{h}$$ (Where E1 and E2 are the energies of lower and higher states) → Energy of stable state En = -RH $$\left(\frac{1}{n^{2}}\right)$$ where n = 1,2,3………………….. → Stable energy of H and species similar to H (Like : He+,Li2+,Be3+) i.e., one electron species ) En = -2.18 x 10-18$$\left(\frac{z^{2}}{n^{2}}\right)$$ J → Radius of nth orbit rn = $$\frac{n^{2} a_{0}}{Z}$$ [where a0 (Bohr’s radius of H) = $$\frac{h^{2}}{4 \mathrm{~A}^{2} m e^{2} k}$$ = 0.529Å → de-Broglie equation $$\lambda=\frac{h}{m v}=\frac{h}{\sqrt{2 m(\mathrm{KE})}}$$ → Angular momentum mvr = $$\frac{n h}{2 \pi}$$ → Heisenberg’s uncertainty principle Δx . Δp ≥ $$\frac{h}{4 \pi}$$ or Δx . Δp ≥ $$\frac{h}{4 \pi}m$$ [Where Δx and Δv are uncertainty in position and principle] → Velocity in n shell vn = 2.182 × 106 × $$\frac{z}{n}$$. → Ionisation Energy (IE)H = ∆E ∝$$z^{2}\left[\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right]$$ → Position of an electron in an atom is determined by four quantum numbers (n, l, in. s (i) n (Principal Quantum number) = 1, 2, 3, 4 ………………n (ii) l (Azimuthal Quantum number) = 0, 1, 2 ………….. (n – 1) (iii) m (Magnetic Quantum number) = – l to + l (iv) s (Spin Quantum number) = $$+\frac{1}{2}$$ or $$-\frac{1}{2}$$ → Determination of subshell = nl → l also expresses the shape of orbital surrounded by electron : s – Spherical, p – dumbell, d – Double dumbell, f- Complex. → Total value of m – (2l + 1) = Number of orbitals in subshell = Number of spectrum lines in Magnetic or Electric field. → In an atom, l angular node, (n – l – 1) radial node i.e., Total nodes are (n – 1). → Electrons are filled in the various orbitals on the basis of the following rules : (i) Aufbau’s Principle, (ii) Hund’s Rule, (iii) Pauli’s Exclusion Principle. → Electronic configuration of 24Cr = [Ar]3r54s1 → Electronic configuration of 29CU = [Ar]3d104s1 → Completely filled and half filled orbitals are more stable due to same symmetry and maxi¬mum energy exchange. → Energy of electron due to which it is bound to the nucleus = Work function (w0) of metal. → Subshell: In a shell all electrons do not have same energy so shells are divided in different subshells. These subshells are s, p, d and f → Orbital: Space around the nucleus where probability to find a electron is maximum is called orbital. ‘ → Spectrum: On jumping of electrons from higher energy level to lower energy level, obtained lines on photographic plate from the produced light is called spectrum. → Visible spectrum : It is a part of electromagnetic radiation which can be seen by our eyes. Wavelength range is 4000 Å to 7500 Å. → Invisible spectrum : The wavelength range of 7500 Å to 3 x 1.06Å and before violet up to 4000A, which we cannot see is called invisible spectrum. → Hund’s rule : Pairing of electrons in orbitals of equivalent energy occurs when there is no vacant orbital. → Aufbau’s principle: Electrons are filled in subshells in increasing order of energy. Electrons are filled first in the shell whose (n + l) value is less. If (n + l) value is same, then electron will go to that shell whose in n value is lower. → Pauli’s exclusion principle: No two electrons in an atom have similar four quantum numbers	2023-02-01 22:33:03	{"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": 0.5936741828918457, "perplexity": 2763.5864585893437}, "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/1674764499953.47/warc/CC-MAIN-20230201211725-20230202001725-00829.warc.gz"}
https://www.groundai.com/project/the-number-of-unit-area-triangles-in-the-plane-theme-and-variations/	The number of unit-area triangles in the plane: Theme and variations1footnote 11footnote 1Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. The number of unit-area triangles in the plane: Theme and variations111Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. Orit E. Raz School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. oritraz@post.tau.ac.il Micha Sharir School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. michas@post.tau.ac.il Abstract We show that the number of unit-area triangles determined by a set of points in the plane is , improving the earlier bound of Apfelbaum and Sharir [2]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if consists of points on three lines, the number of unit-area triangles that spans can be , for any triple of lines (it is always in this case). (ii) We show that if is a convex grid of the form , where , are convex sets of real numbers each (i.e., the sequences of differences of consecutive elements of and of are both strictly increasing), then determines unit-area triangles. 1 Introduction In 1967, Oppenheim (see [9]) asked the following question: Given points in the plane and , how many triangles spanned by the points can have area ? By applying a scaling transformation, one may assume and count the triangles of unit area. Erdős and Purdy [8] showed that a section of the integer lattice determines triangles of the same area. They also showed that the maximum number of such triangles is at most . In 1992, Pach and Sharir [10] improved the bound to , using the Szemerédi-Trotter theorem [16] (see below) on the number of point-line incidences. More recently, Dumitrescu et al. [4] have further improved the upper bound to , by estimating the number of incidences between the given points and a 4-parameter family of quadratic curves. In a subsequent improvement, Apfelbaum and Sharir [2] have obtained the upper bound , for any , which has been slightly improved to in Apfelbaum [1]. This has been the best known upper bound so far. In this paper we further improve the bound to . Our proof uses a different reduction of the problem to an incidence problem, this time to incidences between points and two-dimensional algebraic surfaces in . A very recent result of Solymosi and De Zeeuw [15] provides a sharp upper bound for the number of such incidences, similar to the Szemerédi–Trotter bound, provided that the points, surfaces, and incidences satisfy certain fairly restrictive assumptions. The main novel features of our analysis are thus (a) the reduction of the problem to this specific type of incidence counting, and (b) showing that the assumptions of [15] are satisfied in our context. After establishing this main result, we consider two variations, in which better bounds can be obtained. We first consider the case where the input points lie on three arbitrary lines. It is easily checked that in this case there are at most unit-area triangles. We show, in Section 3, that this bound is tight, and can be attained for any triple of lines. Rather than just presenting the construction, we spend some time showing its connection to a more general problem studied by Elekes and Rónyai [6] (see also the recent developments in [7, 11, 12]), involving the zero set of a trivariate polynomial within a triple Cartesian product. Skipping over the details, which are spelled out in Section 3, it turns out that the case of unit-area triangles determined by points lying on three lines is an exceptional case in the theory of Elekes and Rónyai [6], which then leads to a construction with unit-area triangles. Another variation that we consider concerns unit-area triangles spanned by points in a convex grid. That is, the input set is of the form , where and are convex sets of real numbers each; a set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that in this case determine unit-area triangles. The main technical tool used in our analysis is a result of Schoen and Shkredov [13] on difference sets involving convex sets.222Very recently, in work in progress, jointly with I. Shkredov, the bound is further improved in this case. 2 Unit-area triangles in the plane Theorem 1. The number of unit-area triangles spanned by points in the plane is . We first recall the Szemerédi–Trotter theorem [16] on point-line incidences in the plane. Theorem 2 (Szemerédi and Trotter [16]). (i) The number of incidences between distinct points and distinct lines in the plane is . (ii) Given distinct points in the plane and a parameter , the number of lines incident to at least of the points is . Both bounds are tight in the worst case. Proof of Theorem 1. Let be a set of points in the plane, and let denote the set of unit-area triangles spanned by . For any pair of distinct points, , let denote the line through and . The points for which the triangle has unit area lie on two lines parallel to and at distance from on either side. We let be the line that lies to the left of the vector . We then have \|U\|=13∑(p,q)∈S×S,p≠q\|ℓ′pq∩S\|. It suffices to consider only triangles of , that have the property that at least one of the three lines , , is incident to at most points of , because the number of triangles in that do not have this property is . Indeed, by Theorem 2(ii), there exist at most lines in , such that each contains at least points of . Since every triple of those lines supports (the edges of) at most one triangle (some of the lines might be mutually parallel, and some triples might intersect at points that do not belong to ), these lines support in total at most triangles, and, in particular, at most triangles of . Since this number is subsumed in the asserted bound on , we can therefore ignore such triangles in our analysis. In what follows, denotes the set of the remaining unit-area triangles. We charge each of the surviving unit-area triangles to one of its sides, say , such that contains at most points of . That is, we have \|U\|≤∑(p,q)∈(S×S)∗\|ℓ′pq∩S\|, where denotes the subset of pairs , such that , and the line is incident to at most points of . A major problem in estimating is that the lines , for , are not necessarily distinct, and the analysis has to take into account the (possibly large) multiplicity of these lines. (If the lines were distinct then would be bounded by the number of incidences between lines and points, which is — see Theorem 2(i).) Let denote the collection of lines (without multiplicity). For , we define to be the set of all pairs , with , and for which . We then have \|U\|≤∑ℓ∈L\|ℓ∩S\|\|(S×S)ℓ\|. Fix some integer parameter , to be set later, and partition into the sets L−={ℓ∈L∣\|ℓ∩S\|n/k}. We have \|U\|≤∑ℓ∈L−\|ℓ∩S\|\|(S×S)ℓ\|+∑ℓ∈L+\|ℓ∩S\|\|(S×S)ℓ\|+∑ℓ∈L++\|ℓ∩S\|\|(S×S)ℓ\|. The first sum is at most , because is at most . The same (asymptotic) bound also holds for the the third sum. Indeed, since , the number of lines in is at most , as follows from Theorem 2(ii), and, for each , we have and (for any , , there exists at most one point , such that ). This yields a total of at most unit-area triangles. It therefore remains to bound the second sum, over . Applying the Cauchy-Schwarz inequality to the second sum, it follows that Let (resp., ), for , denote the number of lines for which (resp., ). By Theorem 2(ii), . Hence ∑ℓ∈L+\|ℓ∩S\|2 =n/k∑j=kj2Nj≤k2N≥k+n/k∑j=k+1(2j−1)N≥j =O⎛⎝n2k+nk+n/k∑j=k+1(n2j2+n)⎞⎠=O(n2k) (where we used the fact that ). It follows that \|U\|=O(n2k+nk1/2(∑ℓ∈L+\|(S×S)ℓ\|2)1/2). To estimate the remaining sum, put Q:={(p,u,q,v)∈S4∣(p,u),(q,v)∈(S×S)ℓ, for % some ℓ∈L+}. That is, consists of all quadruples such that , and each of contains at most points of . See Figure 1(a) for an illustration. The above bound on can then be written as \|U\|=O(n2k+n\|Q\|1/2k1/2). (1) The main step of the analysis is to establish the following upper bound on . Proposition 3. Let be as above. Then The proposition, combined with (1), implies that , which, if we choose , becomes . Since the number of triangles that we have discarded is only , Theorem 1 follows. ∎ Proof of Proposition 3. Consider first quadruples , with all four points collinear. As is easily checked, in this case must also satisfy . It follows that a line in the plane, which is incident to at most points of , can support at most such quadruples. By definition, for each , so the line is incident to at most points of , and it suffices to consider only lines with this property. Using the preceding notations , , the number of quadruples under consideration is O⎛⎜⎝∑j≤n1/2j3Nj⎞⎟⎠=O⎛⎜⎝∑j≤n1/2j2N≥j⎞⎟⎠=O⎛⎜⎝∑j≤n1/2j2⋅n2j3⎞⎟⎠=O(n2logn). This is subsumed by the asserted bound on , so, in what follows we only consider quadruples , such that are not collinear. For convenience, we assume that no pair of points of share the same - or -coordinate; this can always be enforced by a suitable rotation of the coordinate frame. The property that two pairs of are associated with a common line of can then be expressed in the following algebraic manner. Lemma 4. Let , and represent , and , by their coordinates in . Then if and only if y−bx−a=w−dz−candbx−ay+2x−a=dz−cw+2z−c. (2) Proof. Let be such that . Then, by the definition of , we have 12∣∣ ∣∣axtbyαt+β111∣∣ ∣∣=1, or (b−y−α(a−x))t−β(a−x)+ay−bx=2, for all . Thus, α =α(a,b,x,y)=y−bx−a, β =β(a,b,x,y)=bx−ay+2x−a. Then the constraint can be written as α(a,b,x,y) =α(c,d,z,w), β(a,b,x,y) =β(c,d,z,w), which is (2). ∎ We next transform the problem of estimating into an incidence problem. With each pair , we associate the two-dimensional surface which is the locus of all points that satisfy the system (2). The degree of is at most , being the intersection of two quadratic hypersurfaces. We let denote the set of surfaces Σ:={σpq∣(p,q)∈(S×S)∗}. For , the corresponding surfaces are distinct. The proof of this fact is not difficult, but is somewhat cumbersome, and we therefore omit it, since our analysis does not use this property. We also consider the set regarded as a point set in (identifying ). We have . The set , the set of incidences between and , is naturally defined as I(Π,Σ):={(π,σ)∈Π×Σ∣π∈σ}. By Lemma 4, we have , where and . This implies that . Consider the subcollection of incidences , such that are non-collinear (as points in ). As already argued, the number of collinear quadruples in is , and hence . So to bound it suffices to obtain an upper bound on . For this we use the following recent result of Solymosi and De Zeeuw [15] (see also the related results in [14, 17]). To state it we need the following definition, which is a specialized version of the more general original definition in [15]. Definition 5. A two-dimensional constant-degree surface in is said to be slanted (the original term used in [15] is good), if, for every , is finite, for , where and are the projections of onto its first and last pairs of coordinates, respectively. Theorem 6 (Solymosi and De Zeeuw [15]). Let be a subset of , and let be a finite set of two-dimensional constant-degree slanted surfaces. Set , and let . Assume that for every pair of distinct points there are at most surfaces such that both pairs are in . Then \|I\|=O(\|Π\|2/3\|Σ\|2/3+\|Π\|+\|Σ\|). To apply Theorem 6, we need the following key technical proposition, whose proof is given in the next subsection. Proposition 7. Let , , and be the sets that arise in our setting, as specified above. Then, (a) the surfaces of are all slanted, and (b) for every pair of distinct points , there are at most three surfaces such that both pairs are in . We have . Therefore, Theorem 6 implies that , which completes the proof of Proposition 3 (and, consequently, of Theorem 1). ∎ 2.1 Proof of Proposition 7 We start by eliminating and from (2). An easy calculation shows that z =2(x−a)(b−d)(x−a)+(c−a)(y−b)+2+c, (3) w =2(y−b)(b−d)(x−a)+(c−a)(y−b)+2+d. This expresses as the graph of a linear rational function from to (which is undefined on the line at which the denominator vanishes). Passing to homogeneous coordinates, replacing by and by , we can re-interpret as the graph of a projective transformation , given by The representation (3) implies that every defines at most one pair such that . By the symmetry of the definition of , every pair also determines at most one pair such that . This shows that, for any , the surface is slanted, which proves Proposition 7(a). For Proposition 7(b), it is equivalent, by the symmetry of the setup, to prove the following dual statement: For any , such that , we have . Let be as above, and assume that . Note that this means that the two projective transformations , agree in at least four distinct points of the projective plane. We claim that in this case and , regarded as graphs of functions on the affine -plane, must coincide on some line in that plane. This is certainly the case if and coincide. (As mentioned earlier, this situation cannot arise, but we include it since we did not provide a proof of its impossibility.) We may thus assume that these surfaces are distinct, which implies that and are distinct projective transformations. As is well known, two distinct projective transformations of the plane cannot agree at four distinct points so that no three of them are collinear. Hence, out of the four points at which and agree, three must be collinear. Denote this triple of points (in the projective -plane) as , and their respective images (in the projective -plane) as , for . Then the line that contains , , is mapped by both and to a line , and both transformations coincide on (since they both map the three distinct points , , to the same three respective points , , ). Passing back to the affine setting, let then be a pair of lines in the -plane and the -plane, respectively, such that, for every (other than the point at which the denominator in (3) vanishes) there exists , satisfying . We show that in this case are all collinear and . We first observe that . Indeed, if each of and is either empty or infinite, then we must have (since both are parallel to ). Otherwise, assume without loss of generality that , and let denote the unique point in this intersection. Let be the point such that satisfies (3) with respect to both surfaces , (the same point arises for both surfaces because ). That is, , and . In particular, , and . Since, by construction, , we have , which yields that also . Thus necessarily are collinear, and , as claimed. Assume that at least one of , intersects in exactly one point; say, without loss of generality, it is , and let denote the unique point in this intersection. Similar to the argument just made, let be the point such that satisfies (3) with respect to both surfaces , . Note that since , we must have too, and . In particular, since , by assumption, we also have . Using the properties and , it follows that the triangles , are congruent; see Figure 2(a). Thus, in particular, . Since, by construction, also , it follows that . We conclude that in this case are collinear and . We are therefore left only with the case where each of and is either empty or infinite. That is, we have (since both are parallel to ). As has already been argued, we also have , and thus is a parallelogram; see Figure 2(b). In particular, . Let be the intersection point of with , and let be the point such that satisfies (3) with respect to both surfaces , . By construction and . Hence must lie on . It is now easily checked that the only way in which can lie on both surfaces and is when are all collinear; see Figure 2(b). To recap, so far we have shown that for , , , and as above, either , or , , , and are collinear with . It can then be shown that, in the latter case, any point must satisfy ; see Figure 1(b). Thus, for a point incident to each of , , neither of , is in . In other words, in this case. This contradiction completes the proof of Proposition 7. 3 Unit-area triangles spanned by points on three lines In this section we consider the special case where is contained in the union of three distinct lines , , . More precisely, we write , with , for , and we are only interested in the number of unit-area triangles spanned by triples of points in . It is easy to see that in this case the number of unit-area triangles of this kind is . Indeed, for any pair of points , the line intersects in at most one point, unless coincides with . Ignoring situation of the latter kind, we get a total of unit-area triangles. If no two lines among are parallel to one another, it can be checked that the number of pairs such that is at most a constant, thus contributing a total of at most unit-area triangles. For the case where two (or more) lines among are parallel, the number of unit-area triangles is easily seen to be . In this section we present a rather subtle construction that shows that this bound is tight in the worst case, for any triple of distinct lines. Instead of just presenting the construction, we spend some time showing its connection to a more general setup considered by Elekes and Rónyai [6] (and also, in more generality, by Elekes and Szabó [7]). Specifically, the main result of this section is the following. Theorem 8. For any triple of distinct lines in , and for any integer , there exist subsets , , , each of cardinality , such that spans unit-area triangles. Proof. The upper bound has already been established (for any choice of ), so we focus on the lower bound. We recall that by the area formula for triangles in the plane, if 12∣∣ ∣ ∣∣pxqxrxpyqyry111∣∣ ∣ ∣∣=1, (4) then the points , and form the vertices of a positively oriented unit-area triangle in . (Conversely, if has area 1 then the left-hand side of (4) has value , depending on the orientation of ..) To establish the lower bound, we distinguish between three cases, depending on the number of pairs of parallel lines among . The three lines l1,l2,l3 are mutually parallel. In this case we may assume without loss of generality that they are of the form l1 ={(t,0)∣t∈R}, l2 ={(t,1)∣t∈R}, l3 ={(t,α)∣t∈R}, for some . (We translate and rotate the coordinate frame so as to place at the -axis and then apply an area-preserving linear transformation that scales the - and -axes by reciprocal values.) We set S1 :={(xi:=i1−α,0)∣i=1,…,n}⊂l1, S2 :={(yj:=jα,1)∣j=1,…,n}⊂l2, S3 :={(zij:=i+j−2,α)∣i,j=1,…,n}⊂l3. Clearly each of the sets , , is of cardinality . Note that for every pair of indices , we have By (4), every such pair corresponds to a unit-area triangle with vertices , and . That is, spans unit-area triangles. There is exactly one pair of parallel lines among l1,l2,l3. Using an area-preserving affine transformation333In more generality than the transformation used in the first case, these are linear transformations with determinant . of (and possibly re-indexing the lines), we may assume that l1 ={(t,0)∣t∈R}, l2 ={(t,1)∣t∈R}, l3 ={(0,t)∣t∈R}. We claim that in this case the sets S1 :={(xi:=2i+2,0)∣i=1,…,n}⊂l1, S2 :={(yj:=2j+2,1)∣j=1,…,n}⊂l2, S3 :={(0,zij:=11−2j−i)∣i,j=1,…,n,i≠j}⊂l3, span unit-area triangles. As before, , and are each of cardinality . Using (4), the triangle spanned by , , and has unit area if 12∣∣ ∣ ∣∣xiyj001zij111∣∣ ∣ ∣∣=1, or xi−zij(xi−yj)2=1, or zij=xi−2xi−yj=11−yj−2xi−2. Since the latter holds for every , we get unit-area triangles, as claimed. No pair of lines among l1,l2,l3 are parallel. This is the most involved case. Using an area-preserving affine transformation of (that is, a linear map with determinant and a translation), we may assume that the lines are given by l1 ={(t,0)∣t∈R}, l2 ={(0,t)∣t∈R}, l3 ={(t,−t+α)∣t∈R}, for some . 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https://eeer.org/journal/view.php?number=1293	Environ Eng Res > Volume 27(3); 2022 > Article Mang, Hwang, and Lee: Optimization of the step feeding ratio for nitrogen removal by SBR using technique for order preference by similarity to ideal solution (TOPSIS) ### Abstract The performance of nitrogen (N) removal was investigated by altering the influent step feeding in a sequencing batch reactor (SBR). The optimum condition for influent step feeding was analyzed using Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) with entropy weight method, considering the C/N ratio range 1.5 to 4.5. The results showed that the SBR with the multi-step-feeding system achieved a high nitrogen removal rate of 92.7% at the three-step influent feeding mode. Nitrogen removal was significantly improved due to the multi-step influent feeding, which provided the required carbon source in the anoxic phase for the removal of nitrate that enhanced the denitrification process. The highest nitrogen removal rate at real-time operation was observed at the three-step feeding mode with 60% primary feeding, 30% secondary feeding, and 10% tertiary feeding, respectively. TOPSIS analysis results also showed that the optimum step feeding ratio was identical to the real-time operation of the SBR. ### 1. Introduction The removal of nitrogen and phosphorous is one of the principal problems of wastewater treatment because of its negative impact on the aquatic environment. Despite the fact that nitrogen and phosphorous are nutrients that are natural parts of the aquatic ecosystem, their excessive concentrations lead to serious eutrophication of water bodies and threatening human health [13]. Therefore, the implementation of cost-effective technologies at wastewater treatment plants has gained global attention as the increase in stringent discharge regulations on nitrogen and phosphorous removal [4, 5]. Conventional biological nitrogen removal (BNR) comprises two successive steps: autotrophic nitrification in aerobic conditions, and heterotrophic denitrification in anoxic conditions [6]. Many studies have been conducted and various approaches have been utilized in biological nutrient removal, such as the conventional suspended-growth activated sludge process, oxidation ditch process and anaerobic/anoxic/oxic (A/A/O) processes [7], moving bed biofilm reactor [8], and sequencing batch reactor [9, 10]. In general, the operating cycle of the sequencing batch reactor consists of fill, react, settle, and draw with a dedicated length, irrespective of fluctuations in wastewater strength and microbial activity [11]. In particular, the major contribution of SBR to total nitrogen (TN) removal is achieved by utilizing the organic substrates supplied from the influent at the beginning of each operation cycle, to reduce the remaining nitrate from the last cycle [12]. The removal of nitrogen in the SBR system can be achieved by alternating aerobic and anoxic periods during the reaction [13], allowing the nitrogen cycle to be completed. Biological nitrogen removal involves two processes: the oxidation of ammonia (NH3) to nitrite (NO2 –N), then nitrite to nitrate (NO3 –N) through nitrification, and the reduction of nitrate (NO3 – N) to nitrogen gas (N2) through denitrification [14]. In conventional SBR, denitrification is often limited because of the lack of sufficient carbon sources to sustain a high denitrification rate [15, 16]. As a consequence, a high concentration of NO3-N or NO2-N is often remained in the effluent, giving low nitrogen removal efficiency. Furthermore, one of the most important control parameters for biological nitrogen removal is the concentration of dissolved oxygen (DO). Complete nitrification and organic carbon removal can be achieved by providing high DO concentration [17]. On the other hand, denitrification potential was declined with a high level of DO concentration in the water [1820]. In order to secure these deficiencies of the conventional SBR, Lee et al. [21] developed the internal circulation sequencing batch reactor with step-feed to maximize the nutrient removal rate by controlling the concentration of DO by the circulation of supernatant in the reactor through the condensed sludge layer during the anoxic fill and anoxic step. The internal circulation of the supernatant makes lower hydraulic shear force; hence, lower DO level was observed, compared to the conventional SBR. On the other hand, higher denitrification efficiency was obtained by using this SBR system. Moreover, SBR has the ability to supply the required carbon source for denitrification by providing multiple-stage influent feeding. It has been well documented that the step-feed SBR has several advantages over the conventional SBR, as it can enhance the denitrification rate, further the total nitrogen removal rate [22], and is technologically and economically effective in enhancing nitrogen removal in activated sludge systems [2326]. As the SBR process is time-oriented, the operation condition provides an additional degree of freedom to achieve the control target. A crucial issue affecting the performance of the step-feed SBR is to determine the feeding strategy for the amount of influent feeding in anoxic phase [27, 28]. However, most studies conducting research on parameters for controlling SBR performance have only focused on the oxidation-reduction potential (ORP), pH, DO, hydraulic retention time (HRT), operating cycle time, and reaction time [2931]. In step feed SBR, the nitrate concentration at the beginning of the anoxic phase is determined by the residual ammonium concentration (extra carbon source) after the feeding phase, hence the introduced extra carbon source should be proportional to the nitrate concentration that can be removed under anoxic condition. On the other hand, the inappropriate volume of step feeding will lead to insufficient carbon source, which will cause slow denitrification rate, and also result in high concentration of effluent nitrogen. Therefore, this study optimizes the volumetric step-feeding ratio of extra carbon source in the anoxic phase of the SBR system. Multi-attribute decision analysis (MADA) methods have been widely reported to be broadly applied in different processes for wastewater treatment fields. Hadipour et al. [32] developed a MADA method to rank the processes for wastewater reuse by employing the Analytical Hierarchy Process (AHP). Zorpas et al. [33] also employed MADA to investigate the processes for wastewater treatment in the field of winery. When decision problems involve large numbers of attributes and alternatives, TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) is the preferred method, especially when objective or quantitative data are available [34]. It is also one of the most famous MADA tools in dealing with multi-attribute or multi-criteria decision-making problems in the real world, and its high flexibility enables it to be efficiently applicable in diverse situations [35]. Therefore, this study proposed entropy weight method based TOPSIS for the optimization of influent feeding for the SBR. The aim of this study is to investigate the effect of the influent step feeding on the nitrogen removal efficiency in the step-feeding SBR. The other objective of this study is to optimize the best condition of step feeding for SBR using the entropy-based TOPSIS method. In order to verify the nutrient removal properties using the proposed method, we compare the removal efficiency of nitrogen and phosphorus under different step feedings. The variation of nitrogen concentration of a single complete operating cycle on each step feeding was also analyzed to demonstrate the benefits of step feeding on nitrogen removal. Finally, the ideal influent feeding of the SBR was further investigated by comparing the theoretical result obtained from TOPSIS with the optimal step feeding condition under the real operation of the SBR. ### 2.1. Wastewater Sources and Characteristics The feeding wastewater used in this study was the real municipal wastewater taken from the primary sedimentation tank in the wastewater treatment plant (Ansan, Korea). The characteristics of influent wastewater are as follows: chemical oxygen demand (COD) 84.52 ± 13.7 mg/L, biochemical oxygen demand (BOD) 119.45 ± 19.36 mg/L, TN 37.96 ± 4.03 mg/L, and total phosphorous (TP) 3.6 ± 0.5 mg/L. The average influent characteristic did not change significantly during different operating conditions. ### 2.2. Operation Condition of SBR The pilot-scale SBR with a depth of 3.2 m and a working volume of 120 m3 (Fig. S1) was installed at a wastewater treatment plant (Ansan, Korea). In a given cycle time, the initial feeding, anoxic phase, aerobic phase, settling phase, and final discharge were included. The DO, pH, mixed liquor suspended solids (MLSS), oxidation-reduction potential (OPR), and electronic flowmeter were also installed to check the condition of the reactor. The MLSS concentration of the reactor was 2,530 ± 87 mg/L. The internal circulation pipe with a flow rate of 2 m/s was installed at the bottom of the reactor to enhance denitrification by circulating the supernatant into the settled sludge layer, to control DO concentration during the anoxic period. The program performed all of the operations automatically. To analyze the nutrient removal performance according to carbon source addition, three different influent feeding modes: one-step feeding, two-step feeding, and three-step feeding in one sequence operation of the SBR were performed (Fig. S2). The sequence time of each feeding mode was 360 min. ### 2.3. Analytical Methods TN, TP, and COD were measured according to the Standard Method [36]. NH4+–N, NO3–N, NO2–N, and PO43−–P were analyzed using ion chromatography (Metrohm, Compact IC 761). The removal efficiency was calculated as follows: ##### (1) $Efficiency=(Influent concentration-Effluent concentration)/Influent concentration×100%$ Empirical effluent concentration was also calculated as Eq. (2) presented in the previous work [21] with initial TN of 50 mg/L, N-removal/COD range 0.1 to 0.4, total influent feeding of 40% of reactor working volume, and initial COD range 200 to 600. ##### (2) $T-N (mg/L)=(CEffNO3-NCOD×CODO×F1)+[[(CEffam+Cam0)×F1×(Noxi)-NCOD×CODO×F2]+Cam0x F2×(Noxi)]-[NCOD×CODO×F3]+Cam0x F3×(Noxi)$ • CEffNO3 = Effluent NO3-N concentration from previous stage • Ceffam = Effluent NH4+-N concentration from previous stage • Cam0 = Initial concentration of ammonia • NCOD = Denitrification rate per COD • Noxi = Nitrification rate • CODO = Initial COD • F1 = ratio of first feeding of raw wastewater • F2 = ratio of second feeding of raw wastewater • F3 = ratio of third feeding of raw wastewater ### 2.4. Entropy Weight Based TOPSIS Method TOPSIS was initially proposed by Hwang and Yoon in 1981 [37]. The fundamental principle of TOPSIS is that the best alternative should have the shortest Euclidean distance from the positive ideal solution, and the farthest distance from the negative ideal solution [38]. In our work, the optimal condition of volumetric influent step feeding of SBR was determined using TOPSIS. The process of TOPSIS is presented as follows: Establish decision matrix using Eq. (3): ##### (3) $X=[Xij]=[x11x12⋯x1nx21x22⋯x2n⋮⋮⋱⋮xm1xm2⋯xmn]$ For every alternative i, (i = 1, …, m), there exist a corresponding criterion j, (j = 1, …, n), and the value of the i th alternative with respect to the j th criterion is denoted as Xij . Xij is known as the performance value of each alternative. The next step is normalization of the decision matrix as in Eq. (4): ##### (4) $yij=Xij/∑j=1mXij2$ The weighted decision matrix can be obtained by multiplication of the normalized decision matrix by the weights of the indices. ##### (5) $Vij=wjyij$ In this step, the weight (wj) of each criteterion was determined using the Entropy weight method. Entropy was introduced by Shanon, and is widely applied in engineering, medicine, and economics to solve decision-making problems [39]. The entropy method is an objective weighting method based on the principle that greater uncertainty about outcomes results in a more uniform probability assigned to them [40]. The entropy weight method is used in this study, because it not only removes the subjectivity of the decision-maker in determining the weights but is also very useful in the cases when experts conflict on the values of weights, which ensures that it is more objective and credible than the subjective methods for comprehensive evaluation of the multivariate index [41, 42]. The principles of the entropy method are as follows: The raw data of the indicators are taken as an m × n matrix, where m is defined as the number of evaluation objects, and n indicates the number of indicators. A decision matrix is established using Eq. (3): The second step is normalization of the decision matrix as in Eq. (6): ##### (6) $yij=xij/∑i=1mxij$ The entropy value, ej for the jth indicator is calculated as: ##### (7) $ej=-h ∑i=1myij lnyij$ where, $h=1ln(m)$, and m = the number of alternatives. Finally, the objectives weight wj for each criterion can be computed as Eq. (8): ##### (8) $wj=1-ej∑j=1n1-ej$ Then, determine the positive ideal solution V+ and the negative ideal solution V ##### (9) $V+={(max Vij∣J, i=1,2,….,m}={V1+, V2+,….,Vn+}$ ##### (10) $V-={(min Vij∣J, i=1,2,….,m}={V1-, V2-,….,Vn-}$ The Euclidean distances from the best ideal solution and the worst ideal solution are calculated using Eq. (11) and (12), respectively: ##### (11) $S+=∑jm(Vij-Vj+)2$ ##### (12) $S-=∑jm(Vij-Vj-)2$ The relative closeness to the ideal solution is calculated using Eq. (13): ##### (13) $Ci=Si-Si++Si-$ where, the Ci value ranges 0 to 1. The ideal solution having the largest Ci represents the optimal solution. In our work, we define the ideal solution as the optimum influent step feeding with the lowest effluent nitrogen concentration corresponding to different C/N ratios under various biological nitrogen removal ratios (N-removal/COD). ### 3.1. Performance of the SBR The operation of the SBR was observed during the study period of 94 days covering three different experimental runs: one-step influent feeding, two-step influent feeding, and three-step influent feeding. The operation strategy applied to the SBR was as follows: anoxic filling, aeration, settling, and final discharge. When the step-feeding mode was performed, influent wastewater as extra carbon source was added to the anoxic phase of the operating cycle. During this period, the average influent nitrogen concentration fluctuated in the narrow range of 33.9 − 41.9 mg/L. It was observed that under the aerobic condition, NH4+–N decreased, whereas, as the ammonia was oxidized by nitrification, there was an increase of NO2 –N and NO3 –N concentration. The system achieved the highest nitrogen removal rate of over 92% under the three-step influent feeding mode, with the average effluent total nitrogen concentration as low as 2 mg/L. This high nitrogen removal efficiency was achieved as a result of step-feeding sufficiently providing the required carbon source for denitrification. These results show that there was a significant relation between the denitrification performance and the feeding of extra carbon sources. ### 3.2. Nitrogen Variation with Step Feeding of Raw Wastewater during a Single Operating Cycle Fig. 1 shows the variation of nitrogen and phosphorus concentration of a sequence cycle in different step feeding. The feeding wastewater was 40% of the working volume of the reactor, and the operation period was 360 min. In our work, the anoxic and aerobic durations were controlled by real-time systems, which to ensure nitrification and denitrification proceeded completely, could flexibly determine each duration. As the figure shows, the ammonia concentration was altered according to the step feeding. When there was no extra feeding of influent (Fig. 1(a)), the denitrification performance was low in this step, due to the lack of required carbon source to further oxidize nitrate, hence resulting in high concentration of effluent nitrogen. Generally, most denitrifying bacteria are heterotrophic, and therefore require organic carbon source for cell growth and nitrate reduction [43]. It has been reported that carbon sources significantly affect the removal of nitrate in the denitrification process, and also enhance the overall nutrient removal performance [44, 45]. In order to improve the denitrification process, raw wastewater as carbon source was added in anoxic phases during two-step feeding mode (Fig. 1(b)) and three-step feeding mode (Fig. 1(c)). In the two-step feeding mode, 70% of the total amount of feeding wastewater was introduced in the first anoxic feeding, followed by the remaining 30% being supplied in the next anoxic process (Fig. 1(b)). The ammonia concentration was increased in the initial anoxic condition, and also in the second anoxic condition. This ammonia variation pattern confirmed that the ammonia containing wastewater was supplied in the 2nd feeding. The final NO3 -N concentration was relatively lower compared with the one-step feeding mode, which indicates that the denitrification process was improved. The three-step feeding mode with anoxic feeding was carried out to further enhance nitrogen removal, with 60% primary influent feeding, 30% secondary feeding, followed by 10% tertiary feeding of the total amount of feeding wastewater (Fig. 1(c)). Similar to the previous feeding mode, the ammonia variation observed under anoxic conditions confirmed that the carbon source was added in the 3rd feeding. The decrease of DO during the anoxic feeding process and simultaneously increased until the end of the aerobic stage also shows that both nitrification and denitrification proceeded well under the corresponding anoxic and aerobic conditions, thereby improving the nutrient removal efficiency. In this mode, the denitrification was further enhanced, as the carbon source required for the removal of nitrate was supplied by the 3rd feeding. Low NO3 -N concentration was also observed at the end of the cycle. This clearly demonstrates the benefits of extra carbon source feeding in the removal of nutrients in the SBR system. When the required carbon source was supplied in the two-step and three-step feeding modes, the PO4 3−-P removal was also improved. When there was not sufficient carbon source to remove the nitrate-nitrogen in the one-step feeding mode, the remaining nitrate inhibited phosphate release and uptake, which caused low removal efficiency. This result was consistent with the previous work conducted in lab-scale experiments, which revealed that the presence of nitrate inhibited phosphate release and uptake [21]. It has also been reported that nitrite could severely inhibit the anoxic phosphate uptake and the anaerobic phosphate release by the polyphosphate accumulating organisms (PAOs) in enhanced biological phosphorus removal (EBPR) processes [46, 47]. ### 3.3. TN and TP Removal Efficiency of the SBR According to the Step Feeding of Raw Wastewater Fig. 2 shows the removal efficiency of TN and TP concentration at different influent step feeding modes. The study consists of three phases: one-step feeding (Days 1–14), two-step feeding (Days 15–59), and three-step feeding (Days 60–94). The TN and TP concentrations of influent and effluent were measured once a day. The removal efficiencies of TN and TP were calculated using Eq. (1). Fig. 2 shows that the step feeding of influent influenced the removal efficiency of both TN and TP. Fig. 2(a) shows that the TN removal efficiency in the three-step feeding was significantly higher than those in the one-step and two-step feedings. The average removal efficiencies of TN for the one-step, two-step, and three-step feedings were 68.7, 85.3, and 92.7%, respectively. Since the highest TN removal efficiency was observed in three-step feeding, it could be concluded that the low TN removal rates at the one-step feeding mainly resulted from a deficiency in the carbon source required for nitrate removal. Wang et al. [22] and Guo et al. [25] also observed that increase in step-feeding with carbon sources could promote the TN removal performance. Similarly, increased step-feeding also increased TP removal (Fig. 2(b)). The removal efficiencies of TP for the one-step, two-step, and three-step feedings were 76.4, 79.8, and 94.2%, respectively. The results indicate that the addition of carbon sources also helped to improve the removal of TP. These findings indicate that the anoxic condition with sufficient carbon source by adding raw influent step-feeding significantly improved the denitrification process, revealing the advantages of influent step addition feeding in the SBR. The same result was also observed in the previous work [21], which showed that the three-step influent feeding mode achieved a faster denitrification rate and lower effluent nitrogen concentration. ### 3.4. Effect of the Biological Nitrogen Removal Ratio based on COD (N-removal/COD) in the Step Feeding Process In order to calculate the optimum influent step feeding in the presence of various C/N ratios, the effluent nitrogen concentration was calculated using Eq. (2) considering the N-removal/COD ratio range 0.1 to 0.3. The three-step influent feeding mode was selected for this effluent calculation. The 1st influent feeding was initially considered from 10 to 70%, the 2nd feeding range from 20 to 50%, and the 3rd feeding range from 10 to 70%. Moreover, the C/N ratio was also divided into five: 1.5, 2.25, 3, 3.75, and 4.5. Fig. 3 shows that the effluent concentration varies according to the C/N ratio, and the adding volume of the influent. Generally, as our calculation progressed towards higher C/N and N-removal/COD, the effluent concentrations become lower for each feeding volume. Fig. 3(a) shows that when the N-removal/COD value was 0.1, the lowest effluent concentration could be found with the 1st, 2nd, and 3rd feeding of 40, 30, and 30% under influent with C/N of 4.5. However, the effluent concentration of TN in this mode is relatively high. When the N-removal/COD value was increased to 0.2 (Fig. 3(b)), the effluent decreased sharply with C/N ratio of 4.5. In this step, the optimum step feeding volume could be found with 30% of 2nd feeding. The optimum step feeding with the lowest effluent concentration was observed when the N-removal/COD approached 0.25 (Fig. 3(c)), with three-step feeding as 60, 30, and 10%, respectively. Similar to previous observations, the C/N value for this condition was 4.5. In addition, it is noteworthy that when the N-removal/COD value was increased to 0.3, the optimum step feeding ratio and the effluent concentration were not significantly changed (Fig. 3(d)). Therefore, under the condition with the C/N ratio range 0.1 to 4.5, our findings indicate that the optimum three-step influent feeding was 60, 30, and 10% with the effluent nitrogen concentration as low as 2 mg/L. Moreover, the results further showed that the carbon source required in denitrification could be provided sufficiently with COD contained in the influent wastewater with C/N of 4.5. However, even though the optimum influent step feeding volume could be obtained from these findings (Fig. 3), the analysis was inadequate, especially if additional parameters and variables were considered and there was a large fluctuation of the influent nitrogen concentration. For this reason, the multi-criteria decision-making tool TOPSIS with entropy weight method was applied to further confirm the precision of the optimum influent step feeding method. ### 3.5. Optimization of Influent Step Feeding for the SBR Using TOPSIS Table 1 presents the performance value matrix of the nitrogen effluent concentration for the selection of optimum influent step feeding. This table forms our original matrix table consisting of alternatives and criteria that represent our initial matrix, as described in Eq. (3). The effluent nitrogen concentration of this matrix is calculated using Eq. (2). The three-step influent feeding mode was selected for this effluent calculation. The criteria for the selection of the optimum step feeding of the SBR considered in this study are the C/N ratios 2.25, 3, 3.75, and 4.5 with the N-removal/COD rate divided into four, 0.1, 0.2, 0.3, and 0.4. In the table, Case (x, y, z) refers to the volume (%) of the influent feeding in primary, secondary, and tertiary anoxic phase (i.e., 10% primary feeding, 10% secondary feeding, 80% tertiary feeding). The acronym C/N (x)(y) stands for the initial influent C/N ratio under the corresponding N-removal/COD rate. The objective weights (wj) for each criterion were calculated using the entropy weight method, and Table 2 presents the results. The weights of each criterion obtained from the entropy method were used in the implementation of TOPSIS optimization. The normalized decision matrix was computed as Eq. (4) using vector normalization. The vector normalization method has been employed for normalization, as when applied to TOPSIS, it is proven to minimize the chances of rank reversal [37, 48, 49]. Then, the weights of each criterion obtained from the entropy method were used to compute the weighted normalized decision matrix using Eq. (5). After that, the Euclidean distances from the best ideal solution and the worst ideal solution (S+, S), the relative closeness of the ideal solution (Ci), and the TOPSIS ranking of each criterion were calculated, and Table 3 shows them. Table 3 shows that the influent step feeding with the relative closeness score close to 1.0 represents the ideal best solution, while the farthest from 1.0 represents the ideal worst solution. Table 4 also shows the top 10 rankings of the influent step feeding for the SBR calculated from TOPSIS. The three-step feeding mode of 60% of 1st feeding, 30% of 2nd feeding, and 10% of 3rd feeding (i.e., Case 60, 30, 10) with a relative closeness score of 0.958 was ranked closest to the ideal solution. In addition, an interesting observation is that the optimum step feeding achieved from TOPSIS was consistent with our previous results from Fig. 3, in which the optimal step feeding was also 60, 30, and 10%. Moreover, the real-time average effluent nitrogen concentration of three-step feeding with 60:30:10 from the pilot plant is 2.77 mg/L, which is close to our theoretical findings. Furthermore, in our previous work [37], the optimum step feeding was also obtained in the three-step feeding mode as 60:30:10, by using the N-removal/COD value 0.2 measured from the experimental results of N and COD concentration. These results have further strengthened our confidence that TOPSIS has the potential ability to determine the ideal influent step feeding. This study proposed the application of the entropy weight method based TOPSIS approach for the first time in the optimization of volumetric influent step feeding for nitrogen removal in the SBR system. In this study, we obtained the same condition of optimum step feeding of 60, 30, and 10% in both TOPSIS and the result we obtained from Fig. 3. However, when more parameters are considered for the optimization or larger data fluctuation depending on decision-makers is preferred, the optimum condition of influent step feeding can be changed. In such cases, optimization of the influent feeding just by using figures is insufficient and unsatisfactory. Decision-making by mathematical optimization, such as TOPSIS, is superior and more precise; hence in this study, the TOPSIS method is proposed. The TOPSIS method has no special data restrictions of the evaluation criteria under the study, which make it easy to understand the advantages of the application [50]. The method also avoids the subjectivity of weight determination, with no excessive requirement of the data sample. This is beneficial to analyzing the optimum condition for influent step feeding in this work, where the data has a large fluctuation of nitrogen concentration under corresponding C/N ratios, N-removal/COD, and objectively and thoroughly reflects the dynamic change trend of decision-makers preferences and influent nitrogen concentration. Although our previous work [21] has also obtained optimum condition of volumetric influent feeding, the observations only consider N-removal/COD of 0.2. This is not strong enough to determine the optimum influent feeding, especially in wastewater treatment using the SBR system, which has a large variety of organic matter composition that could greatly affect the nutrient treatment efficiency. The TOPSIS method could easily determine the optimum condition of influent feeding, even with the large data fluctuation of organic matter. The optimization results clearly show that TOPSIS could easily determine the optimal volumetric feeding of external carbon sources, regardless of various influent loading conditions. The significance of our study lies in the application of TOPSIS that could precisely determine the optimum condition of the influent step feeding ratio in the SBR system, and give relevant outcomes, regardless of the various step feeding conditions, complicated parameters, and decision-makers preferences. This approach could also be applied for optimization of the influent step feeding in the SBR system, even when decision-makers consider additional factors, such as energy consumption and cost-efficiency. ### 5. Conclusions The nitrogen variation analysis of a single complete cycle showed that the highest denitrification rate was achieved in the three-step feeding mode with 60% primary feeding, 30% secondary feeding, followed by 10% tertiary feeding. For one-step, two-step, and three-step addition modes, the removal efficiencies of TN and TP were 68.7, 85.3, and 92.7% and 76.4, 79.8, and 94.2%, respectively. The results demonstrate that the removal efficiencies of TN increased with increasing feeding step, revealing that the feeding of the influent carbon source acts as an important parameter in the nutrient removal process of SBR. The TOPSIS analysis shows that the optimum three-step feeding mode was 60, 30, and 10%, with the highest relative closeness score (Ci) of 0.958. In conclusion, the results obtained from this study provide significant insights that show the optimum influent step feeding ratio of SBR with three-step feeding mode in primary, secondary, and tertiary anoxic phase was 60, 30, and 10% respectively. We believe that our research will serve as an important role for future studies of the optimization of influent step feeding of the SBR system. ### Notes Author Contributions N.Z.L.M. 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J Clean Prod. 2019;229:667–679. ##### Fig. 1 Variation of NH4+–N, NO3–N, and PO43−–P concentration during one complete cycle under different step feeding modes in SBR: (a) One-step feeding, (b) Two-step feeding, and (c) Three-step feeding. ##### Fig. 2 (a) Removal efficiency of TN, and (b) Removal efficiency of TP with the step feeding of influent to the SBR. ##### Fig. 3 Effluent concentration of nitrogen with different C/N ratio and three-step feeding of the influent according to the biological nitrogen removal ratio based on COD (N-removal/COD): (a) N-removal/COD – 0.1, (b) N-removal/COD - 0.2, (c) N-removal/COD - 0.25, (d) N-removal/COD - 0.3: 3rd Step feeding = 100 % - (1st Step Feeding + 2nd Step Feeding) ##### Table 1 Effluent Nitrogen Concentration Calculated from Eq. (2) No. C/N C/N 2.25(0.1) C/N 3(0.1) C/N 3.75(0.1) C/N 4.5(0.1) C/N 2.25(0.2) C/N 3(0.2) C/N 3.75(0.2) C/N 4.5(0.2) C/N 2.25(0.3) C/N 3(0.3) C/N 3.75(0.3) C/N 4.5(0.3) C/N 2.25(0.4) C/N 3(0.4) C/N 3.75(0.4) C/N 4.5(0.4) Step feeding 1 Case (10,10,80) 16 16 16 16 16 16 16 16 16 16 16 16 16 13 16 16 2 Case (10,20,70) 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 3 Case (10,30,60) 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 4 Case (10,40,50) 12 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5 Case (10,50,40) 13.2 11.6 10 8.4 8.4 8 8 8 8 8 8 8 8 8 8 8 6 Case (10,60,30) 14.4 13.2 12 10.8 10.8 8.4 6 6 7.2 6 6 6 6 6 6 6 7 Case (10,70,20) 15.6 14.8 14 13.2 13.2 11.6 10 8.4 10.8 8.4 6 4 8.4 5.2 4 4 8 Case (10,80,10) 16.8 16.4 16 15.6 15.6 14.8 14 13.2 14.4 13.2 12 10.8 13.2 11.6 10 8.4 9 Case (40,10,50) 12.8 10.4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Case (50,10,40) 14 12 10 8 8 8 8 8 8 8 8 8 8 8 8 8 11 Case (60,10,30) 15.2 13.6 12 10.4 10.4 7.2 6 6 6 6 6 6 6 6 6 6 12 Case (70,10,20) 16.4 15.2 14 12.8 12.8 10.4 8 5.6 9.2 5.6 4 4 5.6 4 4 4 13 Case (80,10,10) 17.6 16.8 16 15.2 15.2 13.6 12 10.4 12.8 10.4 8 5.6 10.4 7.2 4 2 14 Case (20,70,10) 14.8 14.4 14 13.6 13.6 12.8 12 11.2 12.4 11.2 10 8.8 11.2 9.6 8 6.4 15 Case (30,60,10) 12.8 12.4 12 11.0 11.6 10.8 10 9.2 10.4 9.2 8 6.8 9.2 7.6 6 4.4 16 Case (40,50,10) 12.8 10.4 10 9.6 9.6 8.8 8 7.2 8.4 7.2 6 4.8 7.2 5.6 4 2.4 17 Case (50,40,10) 14 12 10 8 8 6.8 6 5.2 6.4 5.2 4 2.8 5.2 3.6 2 2 18 Case (60,30,10) 15.2 13.6 12 10.4 10.4 7.2 4 3.2 5.6 3.2 2 2 3.2 2 2 2 19 Case (70,20,10) 16.4 15.2 14 12.8 12.8 10.4 8 5.6 9.2 5.6 2 2 5.6 2 2 2 20 Case (20,30,50) 10.4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 21 Case (20,40,40) 11.2 9.6 8 8 8 8 8 8 8 8 8 8 8 8 8 8 22 Case (20,50,30) 12.4 11.2 10 8.8 8.8 6.4 6 6 6 6 6 6 6 6 6 6 23 Case (20,60,20) 13.6 12.8 12 11.2 11.2 9.6 8 6.4 8.8 6.4 4 4 6.4 4 4 4 24 Case (30,20,50) 11.6 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 25 Case (40,20,40) 12.8 10.4 8 8 8 8 8 8 8 8 8 8 8 8 8 8 26 Case (50,20,30) 14 12 10 8 8 6 6 6 6 6 6 6 6 6 6 6 27 Case (60,20,20) 15.2 13.6 12 10.4 10.4 7.2 4 4 5.6 4 4 4 4 4 4 4 28 Case (30,50,20) 11.6 10.8 10 9.2 9.2 7.6 6 4.4 6.8 4.4 4 4 4.4 4 4 4 29 Case (40,40,20) 12.8 10.4 8 7.2 7.2 5.6 4 4 4.8 4 4 4 4 4 4 4 30 Case (50,30,20) 14 12 10 8 8 4 4 4 4 4 4 4 4 4 4 4 31 Case (30,30,40) 11.6 8.8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 32 Case (30,40,30) 11.6 9.2 8 6.8 6.8 6 6 6 6 6 6 6 6 6 6 6 33 Case (40,30,30) 12.8 10.4 8 6 6 6 6 6 6 6 6 6 6 6 6 6 ##### Table 2 The Entropy Objective Weights C/N 2.25(0.1) C/N 3(0.1) C/N 3.75(0.1) C/N 4.5(0.1) C/N 2.25(0.2) C/N 3(0.2) C/N 3.75(0.2) C/N 4.5(0.2) C/N 2.25(0.3) C/N 3(0.3) C/N 3.75(0.3) C/N 4.5(0.3) C/N 2.25(0.4) C/N 3(0.4) C/N 3.75(0.4) C/N 4.5(0.4) ej 0.998 0.996 0.994 0.991 0.991 0.987 0.982 0.979 0.985 0.979 0.971 0.969 0.979 0.971 0.966 0.962 1 − ej 0.002 0.004 0.006 0.009 0.009 0.013 0.018 0.021 0.015 0.021 0.029 0.031 0.021 0.029 0.034 0.038 Wj 0.008 0.015 0.022 0.029 0.029 0.044 0.061 0.069 0.051 0.069 0.095 0.104 0.069 0.097 0.112 0.128 ##### Table 3 The Closeness Coefficient S, S+, Relative Closeness Ci, and TOPSIS Ranking Order of Each Influent Step Feeding No. Step feeding S S+ Ci Ranking 1 Case (10,10,80) 0.008 1.575 0.005 33 2 Case (10,20,70) 0.379 1.197 0.241 32 3 Case (10,30,60) 0.707 0.869 0.449 31 4 Case (10,40,50) 0.985 0.592 0.625 28 5 Case (10,50,40) 1.210 0.366 0.768 23 6 Case (10,60,30) 1.376 0.206 0.870 16 7 Case (10,70,20) 1.396 0.267 0.839 17 8 Case (10,80,10) 0.868 0.792 0.523 30 9 Case (40,10,50) 0.985 0.592 0.625 29 10 Case (50,10,40) 1.211 0.366 0.768 22 11 Case (60,10,30) 1.380 0.201 0.873 15 12 Case (70,10,20) 1.464 0.174 0.894 9 13 Case (80,10,10) 1.324 0.416 0.761 24 14 Case (20,70,10) 1.100 0.537 0.672 25 15 Case (30,60,10) 1.292 0.332 0.795 18 16 Case (40,50,10) 1.438 0.177 0.890 10 17 Case (50,40,10) 1.529 0.072 0.955 3 18 Case (60,30,10) 1.565 0.068 0.958 1 19 Case (70,20,10) 1.523 0.161 0.904 7 20 Case (20,30,50) 0.985 0.592 0.625 26 21 Case (20,40,40) 1.212 0.365 0.768 20 22 Case (20,50,30) 1.384 0.193 0.878 14 23 Case (20,60,20) 1.459 0.158 0.902 8 24 Case (30,20,50) 0.985 0.592 0.625 27 25 Case (40,20,40) 1.211 0.365 0.768 21 26 Case (50,20,30) 1.385 0.192 0.878 13 27 Case (60,20,20) 1.501 0.096 0.940 6 28 Case (30,50,20) 1.493 0.093 0.941 5 29 Case (40,40,20) 1.508 0.070 0.956 2 30 Case (50,30,20) 1.510 0.073 0.954 4 31 Case (30,30,40) 1.212 0.365 0.768 19 32 Case (30,40,30) 1.387 0.190 0.879 12 33 Case (40,30,30) 1.387 0.190 0.879 11 ##### Table 4 Top 10 Ranking of Influent Step Feeding According to TOPSIS C/N C/N 2.25(0.1) C/N 3(0.1) C/N 3.75 (0.1) C/N 4.5(0.1) C/N 2.25(0.2) C/N 3(0.2) C/N 3.75(0.2) C/N 4.5(0.2) C/N 2.25(0.3) C/N 3(0.3) C/N 3.75(0.3) C/N 4.5(0.3) C/N 2.25(0.4) C/N 3(0.4) C/N 3.75(0.4) C/N 4.5(0.4) Ranking Step feeding Case (60,30,10) 15.2 13.6 12 10.4 10.4 7.2 4 3.2 5.6 3.2 2 2 3.2 2 2 2 1 Case (40,40,20) 12.8 10.4 8 7.2 7.2 5.6 4 4 4.8 4 4 4 4 4 4 4 2 Case (50,40,10) 14 12 10 8 8 6.8 6 5.2 6.4 5.2 4 2.8 5.2 3.6 2 2 3 Case (50,30,20) 14 12 10 8 8 4 4 4 4 4 4 4 4 4 4 4 4 Case (30,50,20) 11.6 10.8 10 9.2 9.2 7.6 6 4.4 6.8 4.4 4 4 4.4 4 4 4 5 Case (60,20,20) 15.2 13.6 12 10.4 10.4 7.2 4 4 5.6 4 4 4 4 4 4 4 6 Case (70,20,10) 16.4 15.2 14 12.8 12.8 10.4 8 5.6 9.2 5.6 2 2 5.6 2 2 2 7 Case (20,60,20) 13.6 12.8 12 11.2 11.2 9.6 8 6.4 8.8 6.4 4 4 6.4 4 4 4 8 Case (70,10,20) 16.4 15.2 14 12.8 12.8 10.4 8 5.6 9.2 5.6 4 4 5.6 4 4 4 9 Case (40,50,10) 12.8 10.4 10 9.6 9.6 8.8 8 7.2 8.4 7.2 6 4.8 7.2 5.6 4 2.4 10 TOOLS Full text via DOI Supplement E-Mail Print Share: METRICS 0 Crossref 0 Scopus 1,670 View	2022-08-17 17:08:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 14, "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": 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https://www.numbas.org.uk/blog/category/research/	# Numbas ## Research ### Development update: November 2020 You might notice this update was published in December: November was a busy month! The Numbas runtime and editor mainly got bug fixes this month. There’s a big new feature in the LTI provider: the ability to automatically remark a resource after you update the exam package. This has already become invaluable for us, with more lecturers than ever setting Numbas assessments and misconfigured marking becoming more common. The remarking feature should be considered experimental: we’ve used it on a few assessments, but I expect to uncover bugs and limitations as we use it more often. ## Numbas runtime I’ve tagged version 5.2 of the Numbas runtime on GitHub. • Enhancement: Custom part types check the data type of each of their input options, and automatically convert compatible types. (code) • Enhancement: There is now a diff: annotation for variable names, to render differentials. Thanks to Lorcan James for adding this. (issue, documentation) • Enhancement: The entire Numbas runtime (the scripts.js file included in exam packages) can be loaded in a headless JavaScript enviroment such as node.js. (code) • Enhancement: The labels “Rows” and “Columns” on a matrix input are localised. (issue) • Enhancement: There’s a new display rule timesDot to insist that a centred dot ⋅ is always used as the multiplication symbol rather than a cross ×. (issue) • Enhancement: The string and latex functions when applied to expression values now take an optional list of display rules to configure the rendering. (code, documentation) • Change: Pattern-matching: if “gather as a list” is turned off, named terms inside lists aren’t returned in lists unless they match more than once. (code) • Change: When displaying rational numbers, they are reduced to lowest terms. (code for plain-text display, and in JME) • Bugfix: Custom functions written in JavaScript have the JME scope flattened, so that old functions written before scope inheritance was introduced still work. (code) • Bugfix: The range except X operation returns integers when appropriate. (code) • Bugfix: The default value of true for the setting “Allow pausing?” is correctly applied when an exam only has one question. (issue) • Bugfix: Leading minus signs are correctly extracted from nested expressions while pattern-matching. (code) • Bugfix: Hopefully the last ever bug fixes related to Internet Explorer – I hope to drop support for it next year when Microsoft ends support. (code) • Bugfix: Variables defined through destructuring are restored correctly when resuming a session. (code) • Bugfix: When the scheduler halts, halt all signal boxes. (code) • Bugfix: The checkboxes answer widget for custom part types deals better with being sent an undefined value. (code) • Bugfix: The legend for matrix inputs isn’t shown for the expected answer. (code) • Bugfix: Cleaning an answer of undefined for a number entry part produces the empty string. (code) • Bugfix: Slightly improved the rendering of the brackets around the matrix input on Safari. (issue) • Bugfix: The mathematical expression part’s built-in marking algorithm has default value generators for decimal, integer and rational data types. (code) ## Numbas editor I’ve tagged version 5.2 of the Numbas editor on GitHub. • Enhancement: When you create an exam, it’s automatically set to use your preferred language. (issue) • Enhancement: In the part marking algorithms tab, marking notes are shown even for invalid inputs. (issue) • Enhancement: The Extensions and scripts tab of the question editor has been split into separate nested tabs for each of extensions, functions, rulesets and preambles. (code) • Enhancement: The preview rendering of LaTeX while editing a content area uses display style when appropriate. (issue) • Bugfix: Fixed an error when browsing a project but not logged in. (code) • Bugfix: Access control is applied to the links to download source of exams and questions. (code) • Bugfix: The preview and embed views for questions and exams scale properly on mobile devices. (code) • Bugfix: I made some more changes to colours and keyboard navigation, to improve accessibility. (code) ## Numbas LTI provider I’ve tagged version 2.9 of the Numbas LTI provider on GitHub. • Enhancement: Each resource now has a Remark tab, which provides an interface to rerun all or some of the attempts at the resource using the latest version of the exam package, and optionally save any changed scores to the database. This has been on the wish-list for a long time. At the moment, consider this experimental: I expect to encounter bugs as we use this on real data. (issue, documentation) • Enhancement: I started work on automatically testing an exam package as it’s uploaded, to check for some obvious errors: that it starts correctly, the expected answers for each part are marked as correct, and a paused attempt at the exam can be resumed. It works on my development machine but we need to do some upgrades on our production LTI server, so it’s switched off by default at the moment. (issue) • Enhancement: The attempt timeline view now groups items produced within one second of each other; the score column is only shown when the score changes; the whole thing now runs lots faster. (code) • Enhancement: There’s now a global search tool for administrators. You can search for users, resources or contexts. (documentation) • Enhancement: There is now a link to review an attempt from its timeline and SCORM data views. (issue) • Enhancement: The “Maximum attempts per user” setting for resources now has some help text explaining that a value of zero means no limit. (issue) • Enhancement: The colours used to represent incorrect, partially correct and correct answers to questions are the same for the attempts listing and the stats page, and a bit more readable. (issue) • Bugfix: Fixed a bug in OAuth authentication affecting D2L Brightspace. (code) • Bugfix: Fixed a couple of bugs in the stress tests. (code) • Bugfix: Broken attempts don’t count towards the limit on the number of attempts a student can make. (issue) ### End-of-year survey of Newcastle’s students’ attitudes to CBAs As it’s the end of the academic year, I decided to survey our students to get an idea of their attitudes about the Computer-Based Assessments (CBAs) that form part of their courses. ### An Analysis of Computer-Based Assessment in the School of Mathematics and Statistics By Dr. Nicholas Parker. ### a. Introduction Since 2008 the School of Mathematics and Statistics has incorporated computer-based assessments (CBAs) into its summative, continuous assessment of undergraduate courses, alongside conventional written assignments. These CBAs present mathematical questions, which usually feature equations with randomized coefficients, and then receive and assess a user-input answer, which may be in the form of a numerical or algebraic expression. Feedback in the form of a model solution is then provided to the student. From 2006 until the last academic year (2011/2012), the School employed the commercial i-assess CBA software. However, this year (2012/2013) the School rolled out a CBA package developed in-house, Numbas, to its stage 1 undergraduate cohort. This software offers greater control and flexibility than its predecessor to optimize student learning and assessment. As such, this was an opportune time to gather the first formal student feedback on CBAs within the School. This feedback, gathered from the stage 1 cohort over two consecutive years, would provide insight into the student experience and perception of CBAs, assess the introduction of the new Numbas package, and stimulate ideas for further improving this tool. After an overview of CBAs in Section b and their role in mathematics pedagogy in Section c, their use in the School of Mathematics and Statistics is summarized in Section d. In Section e the gathering of feedback via questionnaire is outlined and the results presented. In Section f we proceed to analyze the results in terms of learning, student experience, and areas for further improvement. Finally, in Section g, some general conclusions are presented. ### b. A Background to CBAs Box 1: Capabilities of the current generation of mathematical CBA software. • Questions can be posed with randomized parameters such that each realization of the question is numerically different. • Model solutions can be presented for each specific set of parameters. • Algebraic answers can be input by the user (often done via Latex commands), and often supported by a previewer for visual checking • Judged mathematical entry (JME) is employed to assess the correctness of algebraic answers. • Questions can be broken into several parts, with a different answer for each part. • On top of algebraic/numerical answers, more rudimentary multiple-choice, true/false and matching questions are available. • Automated entry of CBA mark into module mark database. Computer-based assessment (CBA) is the use of a computer to present an exercise, receive responses from the student, collate outcomes/marks and present feedback [10]. Their use has grown rapidly in recent years, often as part of computer-based learning [3]. Possible question styles include multiple choice and true/false questions, multimedia-based questions, and algebraic and numerical “gap fill” questions. Merits of CBAs are that, once set up, they provides economical and efficient assessment, instant feedback to students, flexibility over location and timing, and impartial marking. But CBAs have many restrictions. Perhaps their over-riding limitation is their lack of intelligence capable of assessing methodology (rather CBAs simply assess a right or wrong answer). Other issues relating to CBAs are the high cost to set-up, difficulty in awarding of method marks, and a requirement for computer literacy [4]. In the early 1990s, CBAs were pioneered in university mathematics education through the CALM [6] and Mathwise computer-based learning projects [7]. At a similar time, commercial CBA software became available, e.g. the Question Mark Designer software [8]. These early platforms featured rudimentary question types such as multiple choice, true/false and input of real number answers. Motivated by the need to assess wider mathematical information, the facility to input and assess algebraic answers emerged by the mid 1990s via computer-algebra packages. First was Maple’s AIM system [514], followed by, e.g. CalMath [8], STACK [12], Maple T. A. [13], WebWork [14], and i-assess [15]. This current generation of mathematics CBA suites share the same technical capabilities, summarized in Box 1. ### Numbas in operation in 2012 Bill has written this report about how Numbas is being used this year for a University committee. I thought it was worth sharing. ### School of Maths & Stats Numbas will be used for all first year modules and service teaching in Maths & Stats in 2012/2013. This is in formative mode with an in-course assessment component. Will be extended to all second year modules in 2013/2014. Note that presently this is the most extensive use of formative e-assessment in UK HE and is based on our original award winning use of formative e-assessment. ### Transfer to other Universities Two universities are transferring the technology from Newcastle as part of an HE STEM project: Chemistry at Bradford and Maths at Kingston. This will be finished by August 2012. Birmingham University are talking to us in early April about using Numbas in their foundation courses and in their maths support system. ### Mathcentre There is now a new project starting April 2012 to embed Numbas in mathcentre. ### OER and Numbas We are running an HEA/JISC workshop on preparing OER materials using Numbas on April 10. This is free and open to all. ### Maths-Aid We are developing new materials for Maths-Aid using Numbas as part of e-learning and support packages. Also preparing materials and resources (eg DVDs) for revision. ### Documentation Numbas features full documentation which is always in line with the most recent version. ### Numbas Blog We regularly update our blog with articles about new and future features, as well as other useful information.	2021-05-12 20:28:39	{"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": 0.27258944511413574, "perplexity": 5368.092289718806}, "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/1620243989705.28/warc/CC-MAIN-20210512193253-20210512223253-00186.warc.gz"}
https://physics.stackexchange.com/questions/353188/age-of-the-cmb-how-do-we-know	# Age of the CMB: How do we know? As short as possible: From what I understand the Cosmic Microwave Background was predicted by multiple people as a theoretical consequence of the Big Bang theory, and hence today, is evidence for the Big Bang. So of course there was also a prediction of the age of the CMB, because it was predicted to have occurred at a certain point in time. Now, is that the only reason cosmology dates the CMB to 13.8 billion years ago, i.e. is the dating inferred by the theory? I assume there must be many types of measurements that somehow connect to this so as to infer the exact age of what we are seeing there. • Now, is that the only reason cosmology dates the CMB to 13.8 billion years ago, i.e. is the dating inferred by the theory? That relatively high-precision figure comes from fits to the Hubble curve. However, there are cross-checks, such as the ages of the oldest globular clusters. – user4552 Sep 3 '17 at 16:23 You have asked a lot of questions here so my apologies if I do not cover them all. Let us start with a misconception which you may (or may not) hold: The CMB does not originate from the big bang it originates from what is called the surface of last scattering. This occurred about $380,000$ years after the big bang when the universe had a temperature of about $0.23-0.25eV$ ($\sim 3000K$). Consider the reaction: $$p+e^-\leftrightarrow H+\gamma \tag{(1)}$$ above $3000K$ the photons have enough energy to cause this reaction to go into reverse once the universe as cooled (due to the expansion of the universe) the energy of the photons is to small for the reverse reaction to occur. The consequence of this is that all the protons and electrons get taken up to form hydrogen atoms. With no free ions in the universe Thomson scattering becomes less probable ans the universe becomes optically thin. I.e. Before this time photons would travel a very small distance before scattering but after they will travel a very large distance - so large that the photons we see as the CMB are the photons from this time which are yet to scatter. Now this surface of last scattering will occur at $3000K$ no matter what happened before it (this is determined by something known as the Saha equation). So only the rate of expansion after this time will effect the redshift of the CMB. Now inflation is thought to have happened within approximately $10^{-32}$ seconds after the big bang. Clearly $10^{-32}$ seconds $\lt$ $380,000$ years so inflation will not affect the redshift of the CMB. Now you actually pose another query in your question - whether the cosmic horizon will change during inflation. The answer to this is yes. And no there is no contradiction here - the cosmic horizon is different from the surface of last scattering. The Cosmic Horizon is the furtherest point we could possible see. I.e. if a photon started traveling at the big bang singularity and has carried on in a straight line (without scattering) to reach us today - the distance it has traveled is the cosmic horizon. This is affected by inflation because the photon is assumed to be traveling through this period. All that said inflation does have an effect on the CMB. It explains why it looks so homogeneous (something called the horizon problem). Inflation explains this since it means a very large are of the universe now was once all in causal contact since before inflation is was much smaller - allowing photons from one side to the other. EDIT Concerning the question of evidence of the age of the CMB. We can only measure the CMB as it appears to us now. This includes things like the wavelength, blackbody spectrum and the angular power spectrum of the anisotropies of the CMB. To work out the age of the CMB however, you need to know the temperature of the CMB when it was created and compare that to its value today (which we can measure). There is no way to measure the temperature when it was created (we simply weren't there) and thus for this we need the theoretical calculation involving the Saha equation. • Wow, okay! I was aware of the opaque universe although only informally. So thank you for that, I never knew the actual reason and about Thomson scattering! I will look into that more closely, that is very interesting. – lthz Aug 22 '17 at 18:46 • So let me try and pose a clear question. In my first question I wasn't referring to the Big Bang as a reason for the CMB, but for the theory of the Big Bang to expect the CMB. So then: Is there any evidence for the actual age of the CMB, other than the theoretical foundation you just described? I hope the question makes more sense this way. Will edit everything once I figured it all out! – lthz Aug 22 '17 at 18:49 • And forget about inflation, I did not know that the term is referring to the early universe - inflation vs. expansion – lthz Aug 22 '17 at 18:51 • @lthz see my edit – Quantum spaghettification Aug 22 '17 at 19:02 • Okay, great thanks! I will edit the question so it makes sense and leave the last part out of it for now. – lthz Aug 22 '17 at 19:05 This is the history of the universe in the Big bang model. It shows the currently accepted model . The CMB appears at 380.000 years after the origin of the Big Bang. Before that , up to 10^-32seconda from the origin of the Big bang, our theoretical knowledge of particle physics is heavily used. The CMB happens when due to inflation the universe cools enough so that photons can decouple from the rest of the particle soup and thus keep a snapshot of the distribution of matter at the time of decoupling. The detailed, all-sky picture of the infant universe created from nine years of WMAP data. The image reveals 13.77 billion year old temperature fluctuations (shown as color differences) that correspond to the seeds that grew to become the galaxies. The signal from the our Galaxy was subtracted using the multi-frequency data. This image shows a temperature range of ± 200 microKelvin In has small non uniformity from the black body radiation curve the root mean square variations are only 18 µK on a 2.7K black body radiation curve, and those are the features in the image above. It carries information on the seeds of galaxies and cluster of galaxies. In this sense it validates the Big Bang model. So it validates the Big Bang model, together with a number of other observations. Age of the CMB: How do we know? The CMB is heavily red shifted because of the expanding universe, and one way of finding the age of the universe is in using the standard model of particle physics and thermodynamics to get at the Black body radiation curve at the time of decoupling. There exist a specific model which cosmology uses to define the chronology of the Big Bang, and everything is consistent within the parameters of this model and the time lines quoted depend on it. The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parametrization of the Big Bang cosmological model in which the universe contains a cosmological constant, denoted by Lambda (Greek Λ), associated with dark energy, and cold dark matter (abbreviated CDM). It is frequently referred to as the standard model of Big Bang cosmology because it is the simplest model that provides a reasonably good account of the following properties of the cosmos: the existence and structure of the cosmic microwave background the large-scale structure in the distribution of galaxies the abundances of hydrogen (including deuterium), helium, and lithium the accelerating expansion of the universe observed in the light from distant galaxies and supernovae So the chronology of the CMB comes out from the "fit" to the observations. • This doesn't answer the question. – user4552 Sep 3 '17 at 16:22 • Yes, you're right, I shouldn't have upvoted. I'm new here and wanted to be nice. Guess that doesn't help others. – lthz Sep 3 '17 at 17:06	2021-02-28 19:21:25	{"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": 0.581648051738739, "perplexity": 375.1725967774044}, "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-2021-10/segments/1614178361723.15/warc/CC-MAIN-20210228175250-20210228205250-00377.warc.gz"}
https://electronics.stackexchange.com/questions/410882/esd-problem-on-water-tank	# ESD problem on water tank? We have a water level monitor attached to a water tank. The water monitor has an hydro-static water sensor (output 0-5 V) that goes to the bottom of the the tank (see figure). In most cases this works OK, but for some of our clients, the water sensor burns out repeatedly. We think this is caused by an ESD issue. By analysing the situation of these water tanks, it seems that this occurs in zinc tanks that have an internal plastic coating in some really dry and dusty areas. Our hypothesis is that wind blows dust onto the tank, which builds a potential with respect of the water (given that the water is separated from the metal by a plastic layer) and this difference of potential gets discharged into the water sensor. To solve this problem, we were considering switching to 4-20 mA water sensors, with ESD protection in the sensor, but we are not sure if this will be enough to solve the problem. Additionally, we are considering grounding our device to the water tank, to prevent electric charge from building between the tank and the water. The idea is that the charge would flow with low resistance through the ground of our device and the probe. My question would be: would this approach be correct? Wouldn't conducting the electrostatic charge to the tank create galvanic corrosion in the tank to device contact areas? Also, maybe it is not ESD; could it be that the probe cables are acting as inductors, and creating a big potential on changes of current to the probes? • It depends on your schematic of the "water sensor". Is it protected with a series R and clamp TVS? Dec 7 '18 at 0:36 • On the old sensors, there was no clamp TVS... we did have some BAT54 diodes for discharge on the device side... Dec 7 '18 at 1:09 • The new sensors will have TVS. Also, I'm not sure about the series resistance... I am trying to figure it out... Dec 7 '18 at 1:11 • The probe is the HPT604 with the 0.5-4.5v, but without the lightning protection Dec 7 '18 at 1:25 • Does it have the optional lightning protection? Keep in mind ESD HBM is only 100pF and water having Dk of 80 and a large volume makes the Joules of storage much greater than the HBM of 1/2CV^2=1/2* 100pF* 4kV^2. The dielectric breakdown threshold is unknown nor is the tribelectric buildup. I suggest a semiconductor insulation shunt or a more detailed analysis of the problem looking for partial discharge.(PD) Dec 7 '18 at 1:38 Sounds like a bad design of the sensor. Static is always a potential problem so the sensor should be protected. If it’s a capacitive type for example (very common) it would include a proper series resistor(s) and a spark gap. If the tank is floating, grounding it would be the first thing to try. Static should not cause corrosion. • Yes, the model of sensor we were using does not seem to be really prepared for ESD.. In regards to grounding, would it be grounding the tank directly to earth ? or to provide a path between the water and the metal tank? Dec 7 '18 at 1:13 • Not sure. I was told that static was mostly a problem when working with dry powders, so we made our sensors well protected. If it is static there are 3 possibilities...zinc to earth, zinc to device circuit low, all 3 tied together. I would try zinc to earth first and zinc to device circuit low second. Dec 7 '18 at 2:18 Had you considered connecting the pressure sensor to a pipe above the level of the water, so that the water transferred the pressure to the air and thence the sensor. Then you'd have no contact between the water and the sensor. • Mmm, I am not sure I understand how that would work... With this probes, the probes need to be in the water, as they measure the pressure exerted by the column of water above them (barometric probes) Jan 10 '19 at 22:20 One way to find out would be to put a voltage meter on the sensor and one end in the water and see if there is any significant voltage between the sensor and the water. Probably the best thing would be a continuity check on the tank and the water. Get a bench supply and attach one end to a grounding rod nearby, then attach the other end to the top of the tank. Slowly ramp of the voltage until you see a noticeable current, then ramp up the voltage again until the current doubles, then again for when it triples. You should see a linear relationship which you could chart and then apply /$V=I*R/$. The resistance should be low, in the ohms range. Do this again for the water, it should show the same thing. Ideally it shouldn't take much voltage to get some kind of current, if you are applying more than 20V and you don't see any current (less than 1mA), then it's likely that the continuity is higher than 20kΩ. You could also do this for the ground end of the sensor (only one side) to see if there is any continuity between the sensor and ground (there shouldn't be any). I would bet that you find that the top of the tank is grounded (a few ohms or less to ground) and also the water. I would hope that the sensor is open with respect to ground (if it isn't then you have some big problems). My guess is that its probably not normal ESD but lightning that is causing the sensor to burn out, if it is then you need to protect the sensor or the monitoring device inputs. Wind can also generate high voltages but I'll bet the tanks are grounded.	2022-01-24 13:10:18	{"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": 0.527207612991333, "perplexity": 891.1719365006461}, "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/1642320304570.90/warc/CC-MAIN-20220124124654-20220124154654-00419.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-10-counting-methods-and-probability-10-1-apply-the-counting-principles-and-permutations-10-1-exercises-mixed-review-page-689/73	## Algebra 2 (1st Edition) Published by McDougal Littell # Chapter 10 Counting Methods and Probability - 10.1 Apply the counting Principles and Permutations - 10.1 Exercises - Mixed Review - Page 689: 73 #### Answer $$\left(4x\right)^2-5^2$$ #### Work Step by Step Multiplying the equation out, we find: $$\left(4x\right)^2-5^2\\ \left(4x\right)^2-5^2 \\ 16x^2 -25$$ After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2021-03-07 06:39:29	{"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": 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": 0.5824629664421082, "perplexity": 4648.161750336932}, "config": {"markdown_headings": true, "markdown_code": false, "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-2021-10/segments/1614178376144.64/warc/CC-MAIN-20210307044328-20210307074328-00028.warc.gz"}
https://demo7.dspace.org/items/f1a8c5e2-2625-421e-b919-22053f2b07df	## Limit complexities revisited ##### Authors Bienvenu, Laurent Muchnik, Andrej Shen, Alexander Vereshchagin, Nikolay ##### Description The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from (Vereshchagin, 2002) saying that $\limsup_n\KS(x\|n)$ (here $\KS(x\|n)$ is conditional (plain) Kolmogorov complexity of $x$ when $n$ is known) equals $\KS^{\mathbf{0'}(x)$, the plain Kolmogorov complexity with $\mathbf{0'$-oracle. Then we use the same argument to prove similar results for prefix complexity (and also improve results of (Muchnik, 1987) about limit frequencies), a priori probability on binary tree and measure of effectively open sets. As a by-product, we get a criterion of $\mathbf{0'}$ Martin-L\"of randomness (called also 2-randomness) proved in (Miller, 2004): a sequence $\omega$ is 2-random if and only if there exists $c$ such that any prefix $x$ of $\omega$ is a prefix of some string $y$ such that $\KS(y)\ge \|y\|-c$. (In the 1960ies this property was suggested in (Kolmogorov, 1968) as one of possible randomness definitions; its equivalence to 2-randomness was shown in (Miller, 2004) while proving another 2-randomness criterion (see also (Nies et al. 2005)): $\omega$ is 2-random if and only if $\KS(x)\ge \|x\|-c$ for some $c$ and infinitely many prefixes $x$ of $\omega$. Finally, we show that the low-basis theorem can be used to get alternative proofs for these results and to improve the result about effectively open sets; this stronger version implies the 2-randomness criterion mentioned in the previous sentence. ##### Keywords Computer Science - Computational Complexity	2022-11-27 01:29:19	{"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": 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": 0.9183683395385742, "perplexity": 570.6303685948793}, "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/1669446710155.67/warc/CC-MAIN-20221127005113-20221127035113-00282.warc.gz"}
https://zbmath.org/?q=an:1221.05159	Minimal $$k$$-rankings for prism graphs.(English)Zbl 1221.05159 Summary: We determine rank numbers for the prism graph $$P_2 \times C_n$$ ($$P_2$$ being the connected two-node graph and $$C_n$$ a cycle of length $$n$$) and for the square of an even cycle. MSC: 05C15 Coloring of graphs and hypergraphs 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C35 Extremal problems in graph theory Full Text:	2022-05-20 04:03:14	{"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": 0.37420058250427246, "perplexity": 3420.8879385562586}, "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-2022-21/segments/1652662531352.50/warc/CC-MAIN-20220520030533-20220520060533-00221.warc.gz"}
http://publi.math.unideb.hu/contents.php?szam=90	Please choose a volume number: 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 Contents of volume 90: 2017 / 90 / 1-2 (1) : Qiaoling Xin and Lining Jiang Consistent invertibility and perturbations for property $(\omega)$ Full text paper (pdf) 2017 / 90 / 1-2 (2) : Jiecheng Chen and Guoen Hu Completely continuous commutator of Marcinkiewicz integral Full text paper (pdf) 2017 / 90 / 1-2 (3) : Gabriel Ciobanu and Sergiu Rudeanu A universality theorem for the sequential behaviour of minimal $F$-automata Full text paper (pdf) 2017 / 90 / 1-2 (4) : Jiecheng Chen, Dashan Fan and Taotao Zheng Estimates of fractional integral operator with variable kernel Full text paper (pdf) 2017 / 90 / 1-2 (5) : Vladimir V. Bavula The groups ${\rm K}_1(\mathbb{S}_n, \mathfrak{p})$ of the algebra of one-sided inverses of a polynomial algebra Full text paper (pdf) 2017 / 90 / 1-2 (6) : Olivier Ramare Rationality of the zeta function of the subgroups of abelian $p$-groups Full text paper (pdf) 2017 / 90 / 1-2 (7) : Miguel Andres Marcos On Newton--Sobolev spaces Full text paper (pdf) 2017 / 90 / 1-2 (8) : Fatih Deringoz, Vagif S. Guliyev and Stefan Samko Vanishing generalized Orlicz--Morrey spaces and fractional maximal operator Full text paper (pdf) 2017 / 90 / 1-2 (9) : Simone Ugolini On numerical semigroups closed with respect to the action of affine maps Full text paper (pdf) 2017 / 90 / 1-2 (10) : Xinyue Cheng, Yuling Shen and Xiaoyu Ma On a class of projective Ricci flat Finsler metrics Full text paper (pdf) 2017 / 90 / 1-2 (11) : Csanad Bertok, Lajos Hajdu, Florian Luca and Divyum Sharma On the number of non-zero digits of integers in multi-base representations Full text paper (pdf) 2017 / 90 / 1-2 (12) : John R. Graef, Said R. Grace and Ercan Tunc On the oscillation of certain integral equations Full text paper (pdf) 2017 / 90 / 1-2 (13) : Horst Alzer and Man Kam Kwong On inequalities for alternating trigonometric sums Full text paper (pdf) 2017 / 90 / 1-2 (14) : Jinke Hai and Shengbo Ge On class-preserving Coleman automorphisms of semidirect products of finite nilpotent groups by finite groups Full text paper (pdf) 2017 / 90 / 1-2 (15) : Damjana Kokol Bukovsek and Blaz Mojskerc Jordan triple product homomorphisms on Hermitian matrices of dimension two Full text paper (pdf) 2017 / 90 / 1-2 (16) : Csaba Vincze An observation on Asanov's Unicorn metrics Full text paper (pdf) 2017 / 90 / 3-4 (1) : Parisa Hariri, Riku Klen, Matti Vuorinen and Xiaohui Zhang Some remarks on the Cassinian metric Full text paper (pdf) 2017 / 90 / 3-4 (2) : Irina Gelbukh Isotropy index for the connected sum and the direct product of manifolds Full text paper (pdf) 2017 / 90 / 3-4 (3) : Mark L. Lewis Fitting heights of solvable groups with no nontrivial prime power character degrees Full text paper (pdf) 2017 / 90 / 3-4 (4) : Mark Fuzesdi Boolean-type retractable state-finite automata without outputs Full text paper (pdf) 2017 / 90 / 3-4 (5) : Akbar Tayebi and Abdollah Alipour On distance functions induced by Finsler metrics Full text paper (pdf) 2017 / 90 / 3-4 (6) : Miroljub Jevtic and Boban Karapetrovic Hilbert matrix operator on Besov spaces Full text paper (pdf) 2017 / 90 / 3-4 (7) : Istvan Fazekas, Przemyslaw Matula and Maciej Ziemba A note on the weighted strong law of large numbers under general conditions Full text paper (pdf) 2017 / 90 / 3-4 (8) : Xiao-Juan Ji and Zhi-Hong Sun Congruences for Catalan--Larcombe--French numbers Full text paper (pdf) 2017 / 90 / 3-4 (9) : Karl Dilcher, Mohammad Kidwai and Hayley Tomkins Zeros and irreducibility of Stern polynomials Full text paper (pdf) 2017 / 90 / 3-4 (10) : Jorge Almeida and Alfredo Costa Equidivisible pseudovarieties of semigroups Full text paper (pdf) 2017 / 90 / 3-4 (11) : Mihai Anastasiei, Laszlo Kozma and Ioan Radu Peter Some applications of index form in Finsler geometry Full text paper (pdf) 2017 / 90 / 3-4 (12) : Gyorgy Terdik Trispectrum and higher order spectra for non-Gaussian homogeneous and isotropic random field on the 2D-plane Full text paper (pdf) 2017 / 90 / 3-4 (13) : Mohammad Ramezanpour Derivations into various duals of Lau product of Banach algebras Full text paper (pdf) 2017 / 90 / 3-4 (14) : Lei Zhang and Shaoqiang Deng On generalized normal homogeneous Randers spaces Full text paper (pdf) 30 records © 2003-2018, Publicationes Mathematicae, Debrecen, Hungary	2018-03-25 01:14:44	{"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": 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": 0.41652733087539673, "perplexity": 11177.234960048232}, "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-2018-13/segments/1521257651481.98/warc/CC-MAIN-20180325005509-20180325025509-00588.warc.gz"}
https://www.worksheetstemplate.com/the-midpoint-formula-worksheet/	# The Midpoint Formula Worksheet The Midpoint Formula Worksheet. Find the midpoint of every indicated line phase. Teachers, be happy to print the included pdf recordsdata for use within the classroom. Our pdfs with 9 grids, every are simply the thing you want to visualize and find coordinates and escalate expertise find midpoints to deal with real-life issues like these effectively. Members have unique facilities to download an individual worksheet, or a complete degree. This is precisely what the midpoint formulation does, except when you have a coordinate point, you might have an x and a y, so you must do this twice. This free worksheet contains 10 assignments each with 24 questions with solutions. Find the midpoint of the line phase with endpoints $$\left( \right)$$ and $$\left( \right)$$. In the determine, you presumably can see the two unique endpoints of the phase, as well as the midpoint that we calculated. 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Divide the length by 2, and count the same variety of units from both sides of the endpoints and locate the midpoint. ### Distance Method & Midpoint Formula: Notes & Follow The shortest distance between two factors is a straight line. Learners use the Pythagorean Theorem to develop a formula to find that distance. They then discover the midpoint utilizing a mean formulation. Save time and discover engaging curriculum on your classroom. Reviewed and rated by trusted, credentialed academics. Essentially, it describes the means to solve an issue. Heron’s formula is given, and then a problem is solved. Since the precise rationalization of the formulation might be given during… ## Related posts of "The Midpoint Formula Worksheet" Multiplying Monomials Worksheet Answers. There are some guidelines of exponents for solving equations with exponents. I broke it down step-by-step for you to see the exact process. Also, students will be taught to perform arithmetic operations on monomials. Match your reply or verify your steps or go for clarification is your personal sweet will. There... #### Algebra 2 Factoring Worksheet Algebra 2 Factoring Worksheet. ‘If you write my papers for me, what other advantages can I gain? For our half, we assure that our writers will deliver your order on time. Displaying all worksheets associated to - Methods Of Factoring. Which one to determine on is completely as a lot as you. Some pairs of...	2023-03-29 09:21:45	{"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": 0.3505118191242218, "perplexity": 1287.50052092343}, "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/1679296948965.80/warc/CC-MAIN-20230329085436-20230329115436-00187.warc.gz"}
https://math.stackexchange.com/questions/461017/are-there-open-problems-in-linear-algebra	# Are there open problems in Linear Algebra? I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory. There are a lot of open problems and conjectures in K-theory, which are "sometimes" inspired by linear algebra. So I just want to know: What are open problems in "pure" linear algebra? (Pure means not numerical!) Thanks One of the biggest questions is one of the simplest to understand: what is the lowest bound for the operation count of matrix-matrix multiplication? Or, in other words, Given two $n\times n$ matrices, what is the lowest bound of the exponent in the computational complexity of their product? The conjecture could be made more bold: Does there exist an algorithm that can compute the product of two $n \times n$ matrices with complexity $O(n^2)$? Currently, the lowest known bound for the exponent is about $2.373$, obtained from an optimization on the Coppersmith-Winograd algorithm, which isn't actually used because it's only efficient for matrices that are so large that they're (currently) not encountered in practice. Some folks (citation needed) suspect that for sufficiently large $n$, an algorithm exists that can compute the product in $O(n^2)$ operations. • This is a great problem in linear algebra! I have a paper from Berkeley and Stanford about this lower bound, I also suspect that one day we will reach $O(n^2)$. Jan 27 '15 at 23:32 This problem arises from control theory, but it is actually a linear algebra problem. Static Output Feedback Stabilization Problem: Given the matrices $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$ and $C \in \mathbb{R}^{p \times n}$ is there exist a matrix $K \in \mathbb{R}^{m \times p}$ such that real part of all eigenvalues of the matrix $A+BKC$ are negative. The question for the existence of a vector space analog of the Fano plane is open for any prime power value of $q$: Is there a set of $3$-dimensional subspaces of $\operatorname{GF}(q)^7$ such that every $2$-dimensional subspace is covered exactly once?	2021-09-24 21:20:44	{"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": 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": 0.8025188446044922, "perplexity": 201.34926854010794}, "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-2021-39/segments/1631780057580.39/warc/CC-MAIN-20210924201616-20210924231616-00465.warc.gz"}
https://www.gamedev.net/forums/topic/275556-pointers/	Public Group Pointers This topic is 5149 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. Recommended Posts In my Vector class there is a method which sepose to return it as a string when i do this code char V2D::toString() { char str[50]; sprintf(&str[0],"(%f , %f)",x,y); return &str[0]; } and use cout<<v.toString(); it outputs trash But when i use char str= new char[50] char V2D::toString() { char str = new char[50]; sprintf(str,"(%f , %f)",x,y); return str; } it works fine. I dont want to allocate a new string each time toString method is being used. Share on other sites In your toString function, create the char string as a static variable. Boom, done. Only word of warning is if you try and store the returned string. Since it will change everytime you call the function. :) Hope that helps, Rigear Share on other sites The string you're allocating in the toString function is allocated on the stack meaning that it goes out of scope when the function returns. After this, there's no guarantee of it's contents (as it's memory is given back to the stack) so that explains why you're getting garbage. One way of doing this would be to store a buffer in your vector class that's filled by the toString function and returned back; you could also declare the str buffer as static - however I doubt either of these methods are threadsafe (especially the second). Another (more preferred way) would be to pass a buffer (maybe std::string) to the toString function which is then filled by the function. This takes all the memory allocation away from the vector class and puts the owness clearly on the function calling the toString method. Share on other sites Hi! The problem with your code is, that you return a pointer to a local variable (str). Your options are in this case to create a new string each time the method is called, or have a large enough string as a parameter which you can then write the string to. Like this: void V2D::toString( char* str ){ sprintf( str, "(%f, %f)", x, y );} The problem with this is, though, that you have to rely on a large enough buffer being passed to the function. Hope that helps! Cheers, Drag0n Share on other sites Ohhh, how could i forgot :/ I must never return variables of function scopes. Thanks :) And you are right I should allocate a buffer for each vector instance! Share on other sites Quote: Original post by Drag0nThe problem with this is, though, that you have to rely on a large enough buffer being passed to the function. I'd personally use a std::string in order to ensure such problems don't occur. In safe programming you should never assume that the buffer is big enough ;) Share on other sites Storing a buffer for every vector instance will escalate very very fast. I'd suggest using a private static buffer for the class. (or even better std::string! but I provide a char-array example to go with your first attempts) class V2D{public: ...private: ... static char m_string[50]; // since 50 was the number you're using in your example, see note below};const char *V2D::toString(){ sprintf(m_string,"(%f, %f)",x,y); return &m_string;} Or something similar, haven't got the time to read it through for errors atm :) Why 50 size of the buffer though? With regular floats you have 23 bits of mantissa, which equals a maximum of 7 significant digits. Add 2 for the "0." and you have 9 characters for each float. Surely the buffer doesn't have to be larger than 23 characters? ( 9 per float + 5 for "(, )" + terminating null character? Or have I missed something? (probably have, am quite tired atm:) Then again, it's better to have a slightly oversized buffer than a too small one :) (and it's only 27 bytes difference since you'll be storing it statically) Do consider using std::string though, and you won't have to worry about the buffer size :) Enough ranting from me :) Share on other sites Quote: Original post by evolutional Quote: Original post by Drag0nThe problem with this is, though, that you have to rely on a large enough buffer being passed to the function. I'd personally use a std::string in order to ensure such problems don't occur. In safe programming you should never assume that the buffer is big enough ;) You're of course right, the standard library is part of the language and should be used. Especially in this case. Cheers, Drag0n 1. 1 Rutin 34 2. 2 3. 3 4. 4 5. 5 • 12 • 14 • 9 • 9 • 9 • Forum Statistics • Total Topics 633331 • Total Posts 3011399 • Who's Online (See full list) There are no registered users currently online ×	2018-11-16 09:32:37	{"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": 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": 0.1733493059873581, "perplexity": 2332.4314464305808}, "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-2018-47/segments/1542039743007.0/warc/CC-MAIN-20181116091028-20181116113028-00107.warc.gz"}
https://hummingbot.io/blog/2019-09-top-bounty-hunter-interview2/	# Top liquidity bounty hunter interview featuring Dominator008 Blog » Top liquidity bounty hunter interview featuring Dominator008 Yingdan Liang September 3, 2019 · 3 min read It has been just over two months since we launched $ONE Makers liquidity bounty program. With heated competition, bounty hunters have been working very hard to rank higher and higher in our newly launched, real time leaderboard! The trading volume of$ONE in July and August totaled USD$5.98mm. In this post, we introduce you to Dominator008, who finished #1 in the August reward period, and has earned over 10 ETH in the previous period. He is a software engineer by trade and has a highly technical background. He joined$ONE Makers in mid-July, and is now using hummingbot to trade various pairs, not just ONE. Disclaimer: Not financial advice. All views expressed in this interview are the interviewee’s and do not represent the opinions of hummingbot.io. Hummingbot does not guarantee nor claim to guarantee profits. ### .css-gsdr1i{border:none;margin:0;background:none;-webkit-appearance:none;-moz-appearance:none;appearance:none;padding:0;-webkit-appearance:none;-moz-appearance:none;display:inline-block;-webkit-text-decoration:none;text-decoration:none;font-weight:600;font-size:18px;color:var(--theme-ui-colors-primary-base,#0D999E);-webkit-transition:color 120ms;transition:color 120ms;}.css-gsdr1i:hover,.css-gsdr1i:focus{color:var(--theme-ui-colors-primary-light,#4ABBB6);}.css-gsdr1i:focus{outline:none;}How did you hear about hummingbot and get started? I heard about hummingbot from Harmony. I learned that they were trying to partner with hummingbot for the community to be able to market make themselves. Besides hiring professional market makers, they decided to have hobby market makers and test it out how it works in production, so that’s when I learned about hummingbot and started using it. ### How has your experience been so far using hummingbot? It’s been very positive. From 1-10, I would rate it 9. I enjoy the experience, very nice UI and very nice intuitive set-up guide, very detailed documentation, and also awesome support team. Your founder Mike and all the team are very supportive. If I have any questions, I just ping them and get answers very quickly. ### Have you been able to make money so far? You mean besides the liquidity bounties? I think I’m about breakeven. Actually I might be slightly profitable. There is a profitability tracker [in hummingbot], but I think it only tracks the inventory not the actual trades, so I have my own script to track the actual performance. (You can find Dominator008’s performance calculator here.) ### To participate in \$ONE Makers, what strategies do you use to trade? I only use pure market making. I tried the cross-exchange market making strategy for a little bit. It wasn’t generating many trades because it depends on market conditions. It doesn’t guarantee that the take orders will end up with profit. I prefer more volume and more number of trades to average profit of individual trades, so I prefer pure market making. ### Cool, did you come across any issues in the setting up and/or configuration process? I think it’s very straightforward. There were a few bugs, which the team already fixed during the process, like the one that is not tracking volume and duplicate orders on Binance. In terms of the setup, it has been very straightforward. There is good documentation. There is nothing to hack around. ### Do you think it’s relatively easy because you have a lot of technical background? I think my technical background definitely helps, but even for people who are less technical, they should be able to follow the instructions because they are straightforward and it just works. ### How long did you spend to figure out hummingbot from the very moment you heard about it to the moment you actually got it up and running? How much effort did you put into it? In total, a few hours. Overall it didn’t cost me a lot of time to set it up. ### Do you think our current strategies are simple enough for people to get started? I think the documentation is really good. It has examples in it and it explains the strategies very well. For pure market making, I guess for the most advanced features such as multiple orders, it’s better to have more documentation on that. For the inventory skew, I like the documentation a lot. It’s very detailed, it has the formula, explanations, and the rationale. ### Have you tried multiple orders for pure market making? Yes, I always use it. I normally use 5. I think I tried 3 at one time. Based on how the market moves, it seems like 5 with my spread and increment settings would catch more spikes. I tried 10 once, but I hit Binance order limit. Basically you can’t place 10 orders within 1 second. So I just gave up 10. I think 5 is a reasonable number. I didn’t lose money. I can breakeven or even generate some profits with current settings. ### In addition to ONE pairs, have you ever tried to trade other token pairs using hummingbot? Yes, I did. For example, there’s a Binance Celer trading competition. I used hummingbot for that. I traded using really serious money. For the BNB-CELR pair, I generated 10 bitcoin worth of trading volume on average everyday. I kept a very tight spread, so for that I might be losing some money but it was good to test. ### Lastly, do you have any recommendations or trading tips for fellow traders? Just experiment with different settings. For each pair, it’s different, and for each time, it’s different. It changes from time to time, so there’s no one-size-fits-all solution. If there’s more inventory, I’ll adjust the setting more frequently. I just adjust the order size. For spread, I usually keep it constant unless there’s good news going on. I followed Harmony telegram. Sometimes it announces a partnership or something going on. Earn rewards by installing hummingbot and registering for the program. Join the conversation on Discord. ## Related Posts Who we are and why we are building Hummingbot. Get the latest updates from Hummingbot • Hummingbot Home • Liquidity Mining • Company • Community • Legal	2021-04-21 08:47:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.20578517019748688, "perplexity": 2089.368241387199}, "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-2021-17/segments/1618039526421.82/warc/CC-MAIN-20210421065303-20210421095303-00110.warc.gz"}
https://www.tutorialspoint.com/golang-program-to-check-if-two-slices-are-equal	# Golang program to check if two slices are equal In this article, we will see what are the ways to check whether two slices are equal or not with the help of relevant examples. A slice is a sequence of elements just like an array. An array is a fixed sequence of elements whereas a slice is a dynamic array, meaning its value is not fixed and can be changed. Slices are more efficient and faster than arrays moreover; they are passed by reference instead by value. Let us learn the concept through examples. ## Method 1: Using a built-in Function In this method, we will use reflect.DeepEqual() function from the reflect package to check if the two slices are equal or not. Let’s go through the algorithm and the code to see how it’s executed. ### Algorithm • Step 1 − Create a package main and declare fmt(format package) package in the program where main produces executable codes and fmt helps in formatting input and output. • Step 2 − Create a function main and in that function initialize a myslice1 with some values inside it. • Step 3 − Similarly create two more slices named myslice2 and myslice3 and add some values to those slice. • Step 4 − Print all these slices on the console using the print statement in Golang. • Step 5 − Use reflect.DeepEqual() from the reflect package and in the first case apply this function to myslice1 and myslice2 and the result will be printed on the console in the form of a Boolean value that whether these two slices are equal or not. • Step 6 − Use the same function with myslice1 and myslice3 and in the same way output is printed on the console ### Example Golang program to check if two slices are equal using a built-in function package main import ( "fmt" "reflect" ) func main() { myslice1 := []int{10, 20, 30} //create slice1 fmt.Println("The elements of slice1 are:", myslice1) myslice2 := []int{10, 20, 30} //create slice2 fmt.Println("The elements of slice2 are:", myslice2) myslice3 := []int{40, 50, 60} //create slice3 fmt.Println("The elements of slice3 are:", myslice3) fmt.Println("Let's check whether the slices are equal or not") fmt.Println("Are the slice1 and slice2 equal?") fmt.Println(reflect.DeepEqual(myslice1, myslice2)) // true fmt.Println("Are the slice1 and slice3 equal?") fmt.Println(reflect.DeepEqual(myslice1, myslice3)) // false } ### Output The elements of slice1 are: [10 20 30] The elements of slice2 are: [10 20 30] The elements of slice3 are: [40 50 60] Let's check whether the slices are equal or not Are the slice1 and slice2 equal? true Are the slice1 and slice3 equal? false ## Method 2: Iterating Over Elements of Slices In this method, we will see if the two slices are equal or not by iterating over elements of slice. We will compare the elements of slices and find the result. Let us go through the algorithm and the code to see how it is executed. ### Algorithm • Step 1 − Create a package main and declare fmt(format package) package in the program where main produces executable codes and fmt helps in formatting input and output. • Step 2 − Create a function main and in that function initialize a myslice1 with some values inside it. • Step 3 − Similarly create one more slice named myslice2 and add some values in that slice. • Step 4 − Print all these slices on the console using print statement in Golang. • Step 5 − Create a function slice_equality with parameters myslice1 and myslice2 and the value will be returned to the function will be of type Boolean. • Step 6 − Check the condition that if length of myslice1 and myslice2 is not equal return false to the function. • Step 7 − Run a loop till the length of myslice1 and compare the elements of both slices. • Step 8 − If the elements of both slices are not equal return false but if they are equal return true to the function. • Step 9 − The output will be printed using fmt.Println() function where ln means new line. ### Example Golang program to check if two slices are equal by iterating over elements of slices. package main import ( "fmt" ) func main() { myslice1 := []int{10, 20, 30} //create slice1 fmt.Println("The elements of slice1 are:", myslice1) myslice2 := []int{10, 20, 30} //create slice2 fmt.Println("The elements of slice2 are:", myslice2) fmt.Println("Let's check whether the slices are equal or not") fmt.Println("Are the slice1 and slice2 equal?") fmt.Println(slice_equality(myslice1, myslice2)) // true } func slice_equality(myslice1, myslice2 []int) bool { if len(myslice1) != len(myslice2) { //if condition is not satisfied print false return false } for i, element := range myslice1 { // use for loop to check equality if element != myslice2[i] { return false } } return true } ### Output The elements of slice1 are: [10 20 30] The elements of slice2 are: [10 20 30] Let's check whether the slices are equal or not Are the slice1 and slice2 equal? true ## Conclusion In the above program, we used two examples to check whether two slices are equal. In the first method, we used the function of reflect package, and in the second method; we used for loop to compare the slices and check their equality.	2023-03-28 01:44:55	{"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": 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": 0.17769835889339447, "perplexity": 3795.615631837854}, "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-2023-14/segments/1679296948756.99/warc/CC-MAIN-20230328011555-20230328041555-00111.warc.gz"}
https://physics.stackexchange.com/questions/667281/why-isnt-an-observer-on-earth-considered-to-be-accelerating-in-the-twin-paradox/667311#667311	# Why isn't an observer on earth considered to be accelerating in the twin paradox? This question has a duplicate, but answers to that question went miles above my head. Well, that was supposed to happen since I have just previewed special relativity that too on a high school level, but I didn't even go near general relativity. So this question has been asked from that point of view. I hope that this question won't irritate anyone. In the twin paradox, how do we conclude that the observer in the rocket is the one responsible for breaking the symmetry by accelerating even though according to him the other observer on earth is accelerating? Is it confirmed in reference to another (third) observer which was in the same inertial frame as the other two were at the beginning? Again if the whole incident took 10 years according to the observer on earth and 6 years for the observer in the rocket, are the 6 years out of 10 responsible for the symmetry that is due to being in the same inertial frame and the the rest 4 is due to the acceleration? Or is this calculation a bit more peculiar than it seems (I mean the calculation isn't that uniform as I have thought in the previous question)? Any reference would be enough which would clear all my problems or even a book suggestion will be enough if this question seems to be not worth answering. • The twin "paradox" is usually set in empty space, where the stay-at-home twin is in free-fall. Having one of them them on Earth complicates and ruins the simplicity of the scenario (except for advanced courses). Sep 21 at 16:02 • umm yeah this is why I didn't consider earth's motion to be absolute – MSKB Sep 21 at 16:06 • Why "him"? It could be a small fury creature from Alpha Centauri. Sep 21 at 17:38 • @MSKB I also have the same question. I can't find a valid explanation about twins being at different ages if they return to the same frame in the end. Sep 21 at 17:54 • Nah one of them should be at different age since frames are changed but my problem is that why is it the observer in the rocket, it could have happened oppositely too. – MSKB Sep 21 at 17:56 The key to understanding this is a concept called proper acceleration. The Wikipedia article I have linked looks formidably complicated but actually the idea is very simple. Suppose you are floating freely in space far from any source of gravity so there are no forces acting on you. Now drop an object e.g. a ball. The ball will just float beside you and will not move away. Now suppose we put you in a rocket that is accelerating at some acceleration $$a$$. Now if you drop the ball you'll see it accelerate away from you at that acceleration $$a$$. This happens because while you are holding the ball the rocket is exerting a force on both you and the ball so you accelerate at the same rate. When you release the ball the rocket stops accelerating the ball but it is still accelerating you. The result is that you accelerate away from the ball, and this looks to you as if the ball is accelerating away from you. We define your proper acceleration as your acceleration relative to an object (like the ball) that is moving freely, and the value of the proper acceleration is that it is an unambiguous way of detecting when a force is acting on you. Suppose in our twin paradox I am floating freely in space while you jump in the rocket and accelerate away then back again. To both of us it looks as if the other is accelerating, but my ball will remain floating freely next to me while yours will not. This means my proper acceleration is zero while yours is non-zero. And this is what breaks the symmetry. It is the twin with the non-zero proper time who experiences less elapsed time. This is discussed in gory detail in the question What is the proper way to explain the twin paradox?, which I assume is the what you referred to in your question, but the details need not not worry us. All we need to know is that it is always possible to measure the proper acceleration for an observer, so there is never any ambiguity about which twin did the accelerating. • Why does not the time that has elapsed during the change of inertial frames of rocket cause any change in the change in time measured by the observer in te rocket? I meant that as per my question the observer in rocket will see that 6 year has passed throughout his journey but not more than that. But why is its so? I know that this part is strictly related to general relativity of which I have no idea but the thing that is troubling me is that the change in inertial frames caused the earth observer to add 4 years to that of the rocket observer but it didn't impact the change of time of rocket – MSKB Sep 22 at 6:57 • @MSKB there isn't a simple way to understand this. In ordinary geometry the shortest line between two points is a straight line and any curved line must travel a longer distance. However in spacetime it turns out that the straight line is the longest distance not the shortest i.e. any curved line has a shorter length than the straight line. This seems weird but it's an essential principle in relativity. Also in relativity the length of the line is proportional to the elapsed time of an observer travelling along that line. Sep 22 at 7:06 • And it's the observer with zero proper acceleration who travels in a straight line through spacetime. Any proper acceleration causes the line to become curved, and that means the line becomes shorter than the straight line, and that means the elapsed time is less than the straight line. Sep 22 at 7:07 • 😅😅😅😅 it went miles away above my head. Anyways did get a clue. Will work on that. – MSKB Sep 22 at 7:38 Suppose the earth-observer and the rocket-observer each had a ball at rest on the table in front of them. During their trips between separation and reunion, only the rocket-observer’s ball would have moved since the rocket-trip is non-inertial. The rocket did the turning. The earth did nothing…it was inertial. Here are some spacetime diagrams showing why the non-inertial observer is not equivalent to the inertial one. https://physics.stackexchange.com/a/434193/148184 (symmetric rocket trip .. down in the update) https://physics.stackexchange.com/a/553751/148184 (asymmetric rocket trip) • Well, remove the table and the ball will fall down :-) Thus the observer on Earth's surface is accelerated because he is not in free fall. I never did a calculation, and presumably in the most (if not all) settings that 1$g$ of acceleration can be ignored. Sep 21 at 18:03 • @emacsdrivesmenuts For simplicity, we can "restrict the motion to one dimension". So, leave the table in place to balance the gravitational force...and we can focus on motion in an idealized plane. (To introduce general relativity, overcomplicates the situation.) What is proposed is to follow what we do In PHY 101: We use carts on low-friction tracks to study motion in 1-dimension. We complicate the study that we are interested in if we remove the tracks. Sep 21 at 19:08 The "paradox" part of the twin paradox is that there seems to be a symmetry: each twin sees the other moving away and then coming back, so shouldn't everything else, including the readings on their watches, be the same at the end? You can resolve that paradox by pointing out that one twin accelerates, and that can be locally detected (one of them is pushed against the wall of their ship, the other isn't). So they don't have entirely the same experience. The key point is: that resolution is not a way of calculating the elapsed time for each twin. The only point of it is to counter the incorrect symmetry argument, and show it's possible for the times to be different. This is very similar to the following situation: Alice and Bob drive from P to Q, leaving simultaneously and arriving simultaneously, but Alice drives in a straight line while Bob doesn't. Each sees the other getting farther away, then closer, but their odometers have different readings at the end. How is that possible? This situation is also not symmetrical because, for one thing, Bob accelerates, and can detect that (he's pushed sideways in his seat). It would be completely wrong to conclude that the extra distance that Bob traveled happened "during the acceleration". It's also wrong to conclude that about the twin paradox, for the same reason. The correct way to calculate the travel distance in the driving problem is to measure the length of each straight segment of driving with the formula $$\sqrt{Δx^2+Δy^2}$$, and add them (if they're driving on a plane). The correct way to calculate the travel time in the twin paradox is to do the same thing, but with the formula $$\sqrt{Δt^2-Δx^2/c^2-Δy^2/c^2-Δz^2/c^2}$$. [...] which was in the same inertial frame as the other two were at the beginning? This is another important point. Everybody is "in" every inertial frame. A bit of nonsense that's unfortunately very common in introductions to special relativity is the idea that everyone has to use their own inertial rest frame to do calculations, as though they all inhabit different private universes. The actual meaning of the principle of relativity is the exact opposite of that: all inertial frames are equivalent, so you can use any one you want, regardless of your state of motion. The elapsed-time formula that I mentioned above works in every inertial frame. You will get the same answer for the elapsed time for both twins no matter which frame you pick. The principle of relativity guarantees that. You don't need to use three different reference frames. You don't need the Lorentz transformation. If an introduction to special relativity claims otherwise, stop reading it and find a different one. • In the end, whatever happened, Bob is at the same position with Alice. The amount of net distance traveled in the x direction is 0. As I already commented above, to return to the same frame, the 'accelerated' twin should also 'decelerate' after some time. Or the stationary twin must accelerate too (and undergo the same effects) so that they can end up in the same frame/position. By symmetry I would say, "if acceleration results in a difference, the deceleration should annul/rewind it". If both accl & declr contribute to the age difference in the 'same' direction, what's the math behind? Sep 21 at 18:49 • If Bob and Alice drive with the same speed (the acceleration is a change in direction) then Alice arrives earlier, so they are not in the same place in the end. If they keep driving, Bob will always be behind Alice. In space-time you don't stop. Sep 22 at 6:44 One of the assumptions of special relativity (and this assumption is confirmed by many, many experiments) is that there exist a set of privileged reference frames, inertial frames, which are not accelerating. You can tell a frame is inertial experimentally because an object at rest will remain at rest in an inertial frame. Now, as you said, the observer on a rocket ship will observe the second time derivative of Earth observer's trajectory to be non-zero, when using coordinates at rest in the rocket ship's frame. However, the rocket ship observer can still be confident that the rocket ship is in fact in a non-inertial frame, because of the presence of fictitious forces. We usually use the word "accelerating reference frame" to refer to the fact that the rocket ship's frame is non-inertial, even though the second time derivative of the rocket ship's motion in the frame of the rocket ship is zero. While this might seem a bit weird at first, this language is useful because it is independent of the choice of observer and reference frame -- everyone will agree on which observers are inertial, and which are accelerating. In other words -- the word "accelerating" really means "accelerating relative to an inertial frame," not "position having a non-zero second time derivative in a non-inertial frame." In differential geometry, you introduce a new type the derivative (the covariant derivative) that is more suited to handling different frames. You can then define acceleration as the second covariant derivative of the trajectory. The covariant derivative has the property that if the usual derivative is zero in an inertial frame, then it is zero in any frame, while if the usual derivative is not zero in an inertial frame, it is not zero in any frame. So the acceleration defined in terms of the covariant derivative is non-zero in the non-inertial frame of the rocket ship. Because of this, the covariant derivative formalism represents the physics of the situation better than the usual partial derivative. This is a more sophisticated way to answer your question. • It would have made sense if the calculations weren't much of a headache I mean we could have considered earth to be in absolute motion just like we consider k=1 in Newton's law of gravitation. But its quite harder for me to assume other way round that the calculations provide a concrete proof that the rocket is the one which is accelerating but not earth though both of them ar not absolute. – MSKB Sep 21 at 17:52 • @MSKB No no no -- the calculations don't prove anything in physics. The physics is that there exist inertial reference frames. The rocket ship is not in an inertial frame during the time when it is reversing its velocity. This can be verified experimentally -- if you hang fuzzy dice from the roof of the rocket ship, the fuzzy dice will move in the direction of the acceleration. Every observer will agree the fuzzy dice move relative to the rocket ship. If the observer on earth hung up fuzzy dice, they would not move when the rocket turned around. Sep 21 at 17:57 • Note the above comment does not refer to any calculations, but only statements that can be confirmed experimentally. Sep 21 at 17:58 • ahh I get it. I never considered the resolution of "which one is accelerating?" in that way – MSKB Sep 21 at 18:00 • why isn't the time elapsed for the observer in the rocket more than 6 years and less than 10 years? I know that general relativity comes into play this time which is out of my bounds at least for now. Still it seems quite awkward that the acceleration of the rocket didn't have any impact on its time – MSKB Sep 21 at 18:25 The asymmetry arises because the travelling twin changes direction at the end of the outbound voyage, which means that she changes her reference frame, while the twin on Earth stays in the same frame throughout. The difference in the overall passage of time occurs as a consequence of the outbound twin's change of reference frame at the turning point. Until that point, each twin's clock runs slowly relative to the frame of the other, so they both age the same amount. When the outbound twin reverses direction, they change their reference frame, as a consequence of which their plane of simultaneity undergoes a significant tilt, which means that 'now' at their location in space becomes a much earlier time at the Earth's location. After the switch of reference frame, the arrangement becomes symmetrical again, with each twin's clock running slower when viewed from the frame of the other. In response to the OP's comments, here is an example of the twin paradox with no acceleration, illustrating that it is the change of reference frame that causes the effect. Imagine Person A is stationary on a platform. At time t=0 on Person A's clock, they are passed by Person B heading east on a train. As they pass, Person B sets their clock to t=0 too. At some later time t1 on Person B's clock, Person B passes Person C, who is on a different train heading westbound back towards the platform. Person C sets their clock to t1. When Person C's train reaches the platform where Person A has remained, Person C and Person A compare their clock readings as they pass. The time on Person C's clock will be less than the time on Person A's clock. So, the total time that has passed for Person A, who has remained stationary on the platform, is greater than the sum of the time spent by Person B on the eastbound leg and by Person C on the westbound return leg. The time dilation effect was symmetrical for each observer on the two legs. What causes the overall time to be less for the observers on the trains is the shift in reference frame from that of Person B to that of Person C- that causes an abrupt change in the planes of simultaneity from that of Person B to that of Person C, and it is that switch which accounts for the time difference. • how would I know which observer in the same inertial frame because apparently each of them are in rest with respect to themselves? – MSKB Sep 21 at 17:37 • does that mean answer to my second question is "yes", according to the second para of your answer? – MSKB Sep 21 at 17:39 • The outbound twin reverses the direction of their motion at the limit of their outbound journey- that's how you can tell they have switched from one reference frame to another. The stay-at-home twin doesn't change the direction of their motion, nor do they speed up or slow down relative to their original state of motion, so they remain in the same reference frame. Sep 21 at 18:26 • The actual calculation of the twin paradox can be quite complicated, as you should take into account the time it takes for the outbound twin to slow to a halt at the end of their outbound journey and to speed up again to head back. However, there is a simplified version of the scenarios which I will post as an addendum to my answer. Sep 21 at 18:34 • doesn't the time taken while changing the frames have any impact on the clock in the rocket? – MSKB Sep 21 at 18:37	2021-12-08 21:54:44	{"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": 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": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6795861124992371, "perplexity": 307.27462097666245}, "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/1637964363598.57/warc/CC-MAIN-20211208205849-20211208235849-00007.warc.gz"}
http://langintro.com/cljsbook/conditionals.html	# Conditionals¶ Consider the following function, which, given a price and quantity, calculates a total cost with a 5% discount: (defn total-cost [price qty] (let [discount 0.05] (* price qty (- 1 discount)))) All well and good, but that’s not an entirely realistic scenario. What about a business that bases discount on the quantity purchased? For example, if you order less than 50 items, you get a 5% discount; otherwise (when you order 50 or more), you get a 7.5% discount. This is something you haven’t seen yet—calculations based on some condition being true or not. ## Testing conditions: if¶ To accomplish tasks like this, you need ClojureScript’s if function[1] The model for an if is: (if condition true-expression false-expression) The condition is some expression whose value is either true or false. If the condition is true, the value of the entire expression is true-expression; otherwise, the value is false-expression. Going from this abstract explanation to the concrete example of the discount: (if (< qty 50) 0.05 0.075) The expression (< qty 50) tests to see if qty is less than 50. If so, the value of the expression is true, so the result of the expressions is 0.05; if not, it’s false, and the value of the expression is 0.075. Note If you simply enter an if expression such as the preceding one into an active code area, you will not see any result. This is due to the way that ClojureScript in the browser works. It will do the right thing when used in a program. Let’s see how this works in a program. I have moved the code for calculating the discount into its own function because calculating the discount is really its own separate task. How you figure out what the discount is doesn’t change the formula that uses that amount. Try changing the quantity (the 30 in the last line) and see how that affects the total cost. ### Functions that test conditions¶ ClojureScript defines these functions for testing conditions. All these functions return true or false as their value; all of these examples will evaluate to true. Function Means Example < less than (< 3 5) <= less than or equal (<= 19 45) > greater than (> 17 9) >= greater than or equal (>= 25 25) = equal (= 10 (/ 20 2)) not= not equal (not= 17 3) ## Sequences of conditions: cond¶ What happens if you want multiple levels of discount depending upon quantity? Quantity Discount < 20 0% < 50 2% < 100 5% < 200 7.5% >= 200 10% You can represent it in a flowchart form like this: Note In the second diamond of the flowchart, you don’t have to ask if the quantity is greater than or equal to 20. You already know that, because the answer to “is it less than 20?” came back as “no” (false). You could write it this way in ClojureScript: (defn calc-discount [qty] (if (< qty 20) 0 (if (< qty 50) 0.02 (if (< qty 100) 0.05 (if (< qty 200) 0.075 0.10))))) But that’s really difficult to read, and with a few more choices, the indenting and closing parentheses would get pretty deep. For situations such as this, ClojureScript provides the cond construct, which is followed by pairs of conditions and values. ClojureScript tests the conditions one at a time and yields the value for the first condition that evaluates to true. Here is what the discount function looks like using cond; try changing the quantity in the function call and see that it works correctly. The value for the last test, :else, is chosen if none of the other conditions came out true. There is no law that says all the conditions must test the same variable. Consider a cinema that charges $4.00 at all times for children under age 10,$6.00 all day on Mondays (day 1 of the week), $7.50 before 3 p.m. and$8.50 after that on all other days of the week. (defn ticket-price [age day hour] (cond (< age 10) 4.00 (= day 1) 6.00 (<= hour 14) 7.50 :else 8.50)) ## Compound conditions: and and or¶ Consider these modifications to the pricing conditions: • Price is $4.00 if the person is less than 10 years old or 65+ years old. • Price is$6.00 if the day is Monday or Tuesday or Thursday. • Price is $7.50 if the hour is after noon and before 3 p.m. To handle these compound conditions, ClojureScript provides the and and or functions, with this model: (and condition1 condition2) (or condition1 condition2) The result of and is true when all the conditions evaluate to true (think “both condition1 and condition2”). The result of or is true when any of the conditions evaluate to true (think “either condition1 or condition2”). You may test more than two conditions with and/or. Try calling the following ticket-price function with various ages, days, and hours to see the compound conditions in action. In this code, Monday is day 1 and Sunday is day 7. When evaluating and/or, the conditions are evaluated from left to right. ClojureScript will stop evaluating expressions as soon as it knows for sure what the final result has to be. For example, with and, since all the conditions have to be true, as soon as a condition comes back false, there’s no need to look at the other conditions. Similarly, with or, since the whole expression is true if any condition is true, ClojureScript can stop testing conditions as soon as it finds a true condition. The name for this behavior is “early exit.” When would you use this? Here’s a scenario: you are given a number of items and the total price for all the items, and you want to know if the average price is more than$7.00. You can write a compound condition like this: (and (> n 0) (> (/ total-price n) 7)) What happens if n is zero? Without early exit, you’d be in trouble. ClojureScript would evaluate both conditions and try to divide by zero when evaluating the second condition. However, with early exit, because n (zero) is not greater than zero, the first condition comes back false, and ClojureScript can stop—the whole result has to be false, and the division by n never happens. ### The not function¶ Rounding out the boolean functions is not, used in this model: (not condition) When the condition is true, not changes it to false; when the condition is false, not changes it to true. So, if I wanted an expression to be true for anyone who is not between the ages of 18 and 21, I could write: (not (and (>= age 18) (<= age 21))) I could also write it this way: (or (< age 18) (> age 21)) but the first way expresses the logic more closely to the way we think and talk about the condition. You may have noticed that when I got rid of the not, the and changed to an or, and the conditions switched from >= and <= to their opposites. This is an application of the DeMorgan Laws, which tell you how to convert compound expressions with not: (not (and a b)) → (or (not a) (not b)) (not (or a b)) → (and (not a) (not b)) Use these conversions when you need to write a compound condition in a way that corresponds to the logic of the transformation you are doing. Here is a video about the DeMorgan laws; it was originally designed for a course in the Ruby programming language, but the principle applies. ### Exercises¶ Write a function named calculate-pay that calculates a person’s total weekly pay, given the hourly pay rate and number of hours worked per week. If a person works more than 40 hours, they get “time and a half”; that is, 1.5 times the normal pay rate for the hours above 40. Write a function named valid-triangle that takes the lengths of the three sides of a triangle and returns true if a triangle with those sides could exist, false otherwise. A triangle is valid if the sum of any two sides is greater than the length of the remaining side. Thus, a triangle with sides of length 3, 4, and 5 is valid because 3 + 4 is greater than 5, 3 + 5 is greater than 4, and 4 + 5 is greater than 3. A triangle with sides 2, 7, and 11 is impossible because 2 + 7 is less than 11. This function does the job, but it can be improved. See the next tab for a better version. The (and...) expression already gives you a value of true or false, depending on the arguments. There is no reason to use if to return the value; instead, evaluate the expression and use that as the function value. Write a function named calculate-tax that takes a person’s annual income as its single argument and returns that amount of tax the person must pay. Use cond in your solution. Tax is calculated according to the following table: Income Tax <= 10000 0 <= 30000 5% of amount over 10000 <= 70000 1000 + 15% of amount over 30000 <= 150000 7000 + 30% of amount over 70000 > 150000 31000 + 40% of amount over 150000 [1] if is technically not a function. In truth if, def, let (and others) are classified as special forms. defn is also not a function; it is a macro. At this stage, these are distinctions without a difference, but they will become important if you go in depth with ClojureScript. The only reason this footnote is here is so that outraged language purists won’t bombard me with emails about my obvious misclassification of if. Next Section - Strings	2017-05-28 02:52:22	{"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": 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": 0.5172459483146667, "perplexity": 1215.1237240623484}, "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-2017-22/segments/1495463609409.62/warc/CC-MAIN-20170528024152-20170528044152-00135.warc.gz"}
https://www.findfilo.com/math-question-answers/a-coin-is-tossed-twice-if-the-outcome-is-at-most-o93i	A coin is tossed twice. If the outcome is at most one tail, what i \| Filo Class 12 Math Algebra Probability I 543 150 A coin is tossed twice. If the outcome is at most one tail, what is the probability that both head and tail have appeared? Solution: Given a coin is tossed twice Then the sample space will be Given that the outcome is at most one tail So favourable outcomes will be Let be the event of getting at most one tail Probability of to happen is then the outcomes of is Probability of to happen is Favourable outcomes of is Required probability is the probability of given that has happen As we know that So Probability that both head and tail have appeared if the the outcome is at most one tail is 543 150 Connecting you to a tutor in 60 seconds.	2021-07-30 16:45:02	{"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": 0.8711901903152466, "perplexity": 682.5786594105002}, "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-2021-31/segments/1627046153971.20/warc/CC-MAIN-20210730154005-20210730184005-00495.warc.gz"}
http://mathoverflow.net/revisions/105304/list	MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4). If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible?	2013-06-20 10:13:39	{"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": 0.7780752778053284, "perplexity": 1170.468069589896}, "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/1368711406217/warc/CC-MAIN-20130516133646-00072-ip-10-60-113-184.ec2.internal.warc.gz"}
https://socratic.org/questions/how-do-you-find-the-gcf-of-16x-24x-2	# How do you find the GCF of 16x, 24x^2? $4 x$ $16$ $\textcolor{w h i t e}{1 \cdot 24}$ $x$ $\textcolor{w h i t e}{1 \cdot 24}$ $24$ $\textcolor{w h i t e}{1 \cdot 24}$ ${x}^{2}$ $1 \cdot 16$ $\textcolor{w h i t e}{1}$ $\textcolor{red}{x} \cdot 1$ $\textcolor{w h i t e}{. .}$ $1 \cdot 24$ $\textcolor{w h i t e}{14}$ $\textcolor{red}{x} \cdot x$ $2 \cdot 8$$\textcolor{w h i t e}{1 \cdot 24}$ $\textcolor{w h i t e}{x}$$\textcolor{w h i t e}{00.}$ $2 \cdot 12$ $\textcolor{red}{4} \cdot 4$$\textcolor{w h i t e}{1 \cdot 24}$ $\textcolor{w h i t e}{x}$$\textcolor{w h i t e}{00.}$ $3 \cdot 8$ $\textcolor{w h i t e}{1 \cdot 24}$ $\textcolor{w h i t e}{x}$$\textcolor{w h i t e}{0000.0 .}$ $\textcolor{red}{4} \cdot 6$	2020-02-20 21:08:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 29, "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": 1.0000098943710327, "perplexity": 7153.461587593948}, "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-2020-10/segments/1581875145282.57/warc/CC-MAIN-20200220193228-20200220223228-00094.warc.gz"}
http://www.maa.org/publications/periodicals/convergence/problems-another-time?page=27&device=mobile	# Problems from Another Time Individual problems from throughout mathematics history, as well as articles that include problem sets for students. What will the diameter of a sphere be, when its volume and surface area are expressed by the same number? A man has four creditors. To the first he owes 624 ducats; to the second, 546; to the third, 492; and to the fourth 368. Seven men held equal shares in a grinding stone 5 feet in diameter. What part of the diameter should each grind away? Given a pyramid 300 cubits high, with a square base 500 cubits to a side, determine the distance from the center of any side to the apex. A man bought a number of sheep for $225; 10 of them having died, he sold 4/5 of the remainder for the same cost and received$150 for them. How many did he buy? A barrel has various holes in it. The fist hole empties the barrel in three days... In a rectangle, having given the diagonal and perimeter, find the sides Given two circles tangent to each other and to a common line, determine a relationship between the radii and the distance between the tangent points. Determine the different values of x, when a certain function hits a minimum. Given a door and a measuring rod of unknown dimensions, the rod is used to measure the door.	2015-05-30 12:33:21	{"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": 0.43670397996902466, "perplexity": 1225.0688471647943}, "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-2015-22/segments/1432207931085.38/warc/CC-MAIN-20150521113211-00212-ip-10-180-206-219.ec2.internal.warc.gz"}
https://www.sciencemadness.org/whisper/viewthread.php?tid=63998#pid423638	Not logged in [Login ] Sciencemadness Discussion Board » Fundamentals » Reagents and Apparatus Acquisition » Heating for Magnetic Stirrer Select A Forum Fundamentals » Chemistry in General » Organic Chemistry » Reagents and Apparatus Acquisition » Beginnings » Responsible Practices » Miscellaneous » The Wiki Special topics » Technochemistry » Energetic Materials » Biochemistry » Radiochemistry » Computational Models and Techniques » Prepublication Non-chemistry » Forum Matters » Legal and Societal Issues Author: Subject: Heating for Magnetic Stirrer JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline Heating for Magnetic Stirrer I recently purchased an 85-2 magnetic stirrer / hotplate combo from a Chinese source. The magnetic stirring functions well, but I was disappointed to discover that the hotplate's maximum temperature setting is 100 C, and I didn't see a way to regulate the heat proportionally without an enforced temperature limit. Even worse, the hotplate doesn't seem to heat above about 60 C. There are several settings that can be altered from their factory defaults, but none of them seem to increase heat output, and I haven't found a manual anywhere online. I am thinking about adding additional heating to it using a mug immersion heater (in a bath of water or oil, with stirring) and a dimmer switch. This solution is not quite as fancy as using a heat tape and a variac, but it will be a lot cheaper and everything can be sourced locally.... I can pick up the immersion heater at Bed, Bath & Beyond for around $8 and the dimmer switch at Home Depot or an electrical specialty store. I'm a bit concerned that the plastic housing may melt off of the immersion heater with prolonged use, so perhaps it could be replaced with high-temperature epoxy and maybe a little fiberglass. [Edited on 16-10-2015 by JJay] [Edited on 16-10-2015 by JJay] [Edited on 16-10-2015 by JJay] careysub International Hazard Posts: 1339 Registered: 4-8-2014 Location: Coastal Sage Scrub Biome Member Is Offline Mood: Lowest quantum state Quote: Originally posted by JJay ...I'm a bit concerned that the plastic housing may melt off of the immersion heater with prolonged use, so perhaps it could be replaced with high-temperature epoxy and maybe a little fiberglass. Gasket silicone. Good to 343 C. http://www.permatex.com/products-2/product-categories/gasket... JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline Quote: Gasket silicone. Good idea. I took a look at some high-temperature silicone mixtures earlier, and they contain a sodium silicate binder with other materials. Manufacturing some sodium silicate is pretty high on my todo list for the next couple of weeks, and I expect to get to it soon. So anyway, I bought an immersion heater. At 120v it's a 125 watt heater. The plastic part of its housing appears to be composed of some kind of hard thermoplastic that I think can withstand 200 C or so. If the housing melts or burns, I might try making repairs with a silica gel bound composite. I stopped by Home Depot and picked up the following parts: 600 watt dimmer switch 15 amp grounded electrical outlet 2 slot electrical receptacle outlet / switch combination receptacle cover 13 amp grounded extension cord I assembled these items into a current regulating device. First, I cut the extension cord off at about 10 feet from the plug and used it as a power source to the receptacle. I wired the outlet to the dimmer switch and the dimmer switch to the power source. Then I screwed the components into the receptacle and installed the receptacle cover. I tested the device, and it seems to be working great for regulating the heating of a cup of water. Since the device can take two plugs, it will output 250 watts if I add a second immersion heater. [Edited on 17-10-2015 by JJay] [Edited on 17-10-2015 by JJay] Mesa National Hazard Posts: 264 Registered: 2-7-2013 Member Is Offline Mood: No Mood You could gut the entire heating/controller/power supply side and buy/scavenge spares to replace them with. Edit: If the stirrer controller is on the same IC it'd be more complicated, but still possible. I've only ever opened up 1 chinese import hotplate/stirrer and they had individual boards for each. [Edited on 17-10-2015 by Mesa] JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline Quote: Originally posted by Mesa You could gut the entire heating/controller/power supply side and buy/scavenge spares to replace them with. Edit: If the stirrer controller is on the same IC it'd be more complicated, but still possible. I've only ever opened up 1 chinese import hotplate/stirrer and they had individual boards for each. [Edited on 17-10-2015 by Mesa] Interesting idea... those hotplates are pretty common. If the wiring has a simple layout (or DIP switches!) I might be able to increase the heat output. I took pictures as I was constructing the device and will post them in the near future. JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline Mesa National Hazard Posts: 264 Registered: 2-7-2013 Member Is Offline Mood: No Mood A stirrer controller is essentially just a variac connect to a power supply so if worst comes to worst there are$15 teenager electronic hobby kits at jaycar. On the other hand, at this point you are essentially building one from scratch and putting in some cheap chinese knockoff's outter housing. JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline Quote: Originally posted by Mesa A stirrer controller is essentially just a variac connect to a power supply so if worst comes to worst there are \$15 teenager electronic hobby kits at jaycar. On the other hand, at this point you are essentially building one from scratch and putting in some cheap chinese knockoff's outter housing. If I have to break out the Arduinos and the Raspberry Pis, I'm just going to build a new stirrer. Variacs are pretty pricy, but temperature controllers look reasonable.... Oh and I probably should mention this just in case someone has an exotic application... I haven't had a look at the waveform coming out of the dimmer switch just yet, but I suspect it has been deformed quite a bit... a variac would preserve the shape of the waveform, merely altering its amplitude. For heating applications, it doesn't matter what the waveform looks like, but it's probably not a good idea to use a dimmer switch with audio equipment. zed International Hazard Posts: 2273 Registered: 6-9-2008 Location: Great State of Jefferson, City of Portland Member Is Offline Mood: Semi-repentant Sith Lord JJ, you have been screwed. Personally, I would use that stir plate for lower temp applications, and obtain another stir-plate, for hotter applications. Yes, this sucks. But, there is no point in trying to bang a square peg into a round hole. JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline Quote: Originally posted by zed JJ, you have been screwed. Personally, I would use that stir plate for lower temp applications, and obtain another stir-plate, for hotter applications. Yes, this sucks. But, there is no point in trying to bang a square peg into a round hole. I know, I know... the 85-2 magnetic stirrer is a piece of junk IMHO... but I think it should work fine with external heating and a water bath... if I need high temperatures, I could use an oil bath and protect the surface of the stirrer with a pad. I have obtained an oscilloscope and will check out the waveform. JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline I haven't had a chance to check out the waveform yet, but I did try heating some water with the setup, and one 125 watt immersion heater is not going to be sufficient... the temperature reaches only about 80 C when heating a liter of water in a Pyrex measuring cup with the heater on full blast. Heating less water with a smaller amount of surface area exposed to the atmosphere should lead to higher temperatures, but clearly, 125 watts is not enough, and I don't think 250 will be enough either... looks like it is going to take 400-500 watts.... eesakiwi Harmless Posts: 27 Registered: 10-8-2005 Member Is Offline Mood: drawnout google image > rice cooker element. They are round, aluminium, 500 watts and allready have a hole in the centre of them. They are not flat though, they 'peak' in the centre a bit. Theres a bit of a step around the hole that would be usefull to hold a small round disc of mica to help stop heat getting to the stirrer magnet. You can get mica sheet out of microwave ovens, they cover the window that the microwaves go thru to get into the oven, they are of a lot of different shapes. Most of the 'peak', its only about 3mm rise over the whole face of the element, could be ground off using a angle grinder. Or machined off using a metal lathe (best option). 500watts is about the limit for a light bulb dimmer too. JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline Rice cooker element... interesting idea. I haven't had a chance to check out the waveform yet (will try to do it tomorrow if I have time), but I picked up two 500 watt immersion heaters from a Canadian source. They were advertised as being able to boil two gallons of water (about 8000 mL) each. I clamped one on the side of a 1500 mL soybean oil bath and put a 500 mL flask of water into it, stirbars in both, and at 80% power, one of the heaters easily warmed the bath to 170 C while boiling the water, and I'm sure it could have heated the bath a lot hotter. [Edited on 26-10-2015 by JJay] JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline I didn't get a chance to check out the waveform today, but right now I am running a test using the stirrer with the immersion heater/triac setup for fractionally distilling water, and it is working splendidly. Sciencemadness Discussion Board » Fundamentals » Reagents and Apparatus Acquisition » Heating for Magnetic Stirrer Select A Forum Fundamentals » Chemistry in General » Organic Chemistry » Reagents and Apparatus Acquisition » Beginnings » Responsible Practices » Miscellaneous » The Wiki Special topics » Technochemistry » Energetic Materials » Biochemistry » Radiochemistry » Computational Models and Techniques » Prepublication Non-chemistry » Forum Matters » Legal and Societal Issues	2022-12-08 15:45:43	{"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": 0.3288336992263794, "perplexity": 4722.532096199316}, "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-2022-49/segments/1669446711344.13/warc/CC-MAIN-20221208150643-20221208180643-00721.warc.gz"}
http://asomef.org.co/o9gmu2tt/introduction-to-partial-differential-equations-d460f8	Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). To understand it for any given equation, existence and uniqueness theorems are usually important organizational principles. is not. It is designed for undergraduate and first year graduate students who are mathematics, physics, engineering or, in general, science majors. [citation needed] They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations; among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology. Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines. An example is the Monge–Ampère equation, which arises in differential geometry.[2]. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations. x Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices Aν are m by m matrices for ν = 1, 2,… n. The partial differential equation takes the form, where the coefficient matrices Aν and the vector B may depend upon x and u. If the data on S and the differential equation do not determine the normal derivative of u on S, then the surface is characteristic, and the differential equation restricts the data on S: the differential equation is internal to S. Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. . In the study of PDE, one generally has the free choice of functions. Yehuda Pinchover, Jacob Rubinstein - An Introduction to Partial Differential Equations. In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.. The wave equation: Geometric energy estimates : L15: Classification of second order equations : L16–L18: Introduction to the Fourier transform; Fourier inversion and Plancherel's theorem : L19–L20: Introduction to Schrödinger's equation : L21-L23: Introduction to Lagrangian field theories : L24: Transport equations and Burger's equation For example, the Black–Scholes PDE, by the change of variables (for complete details see Solution of the Black Scholes Equation at the Wayback Machine (archived April 11, 2008)). if u A linear PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and all constant multiples of any solution is also a solution. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design. and integrating over the domain gives, where integration by parts has been used for the second relationship, we get. This context precludes many phenomena of both physical and mathematical interest. x where φ has a non-zero gradient, then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes: The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example sin x + sin x = 2 sin x. The Adomian decomposition method, the Lyapunov artificial small parameter method, and his homotopy perturbation method are all special cases of the more general homotopy analysis method. If a hypersurface S is given in the implicit form. ∂ Prerequisites: Math 2433 and either Math 3321 or Math 3331.. This textbook is designed for a one year course covering the fundamentals of partial differential equations, geared towards advanced undergraduates and beginning graduate students in mathematics, science, engineering, and elsewhere. nonlinear partial diﬀerential equations. In contrast to the earlier examples, this PDE is nonlinear, owing to the square roots and the squares. where the coefficients A, B, C... may depend upon x and y. © 2020 Springer Nature Switzerland AG. In the physics literature, the Laplace operator is often denoted by ∇2; in the mathematics literature, ∇2u may also denote the hessian matrix of u. ( In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve. (Often the mixed-partial derivatives uxy and uyx will be equated, but this is not required for the discussion of linearity.) {\displaystyle u} For well-posedness we require that the energy of the solution is non-increasing, i.e. The previous equation is a first-order PDE. ‖ AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS A complete introduction to partial differential equations, this textbook provides a rigorous yet accessible guide to students in mathematics, physics and engineering. = ‖ In special cases, one can find characteristic curves on which the equation reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics. ≤ This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. If A2 + B2 + C2 > 0 over a region of the xy-plane, the PDE is second-order in that region. The same principle can be observed in PDEs where the solutions may be real or complex and additive. Partial differential equations (PDEs) are fundamental to the modeling of natural phenomena, arising in every field of science. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed. A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Includes definition explanation of how to construct PDEs and worked examples. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. by Peter J. Olver. This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals). This textbook is a self-contained introduction to Partial Differential Equa- tions (PDEs). In a slightly weak form, the Cauchy–Kowalevski theorem essentially states that if the terms in a partial differential equation are all made up of analytic functions, then on certain regions, there necessarily exist solutions of the PDE which are also analytic functions. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. ∂ Other important equations that are common in the physical sciences are: The heat equation: A linear partial differential equation (p.d.e.) b This is a reflection of the fact that they are not, in any immediate way, both special cases of a "general solution formula" of the Laplace equation. "Finite volume" refers to the small volume surrounding each node point on a mesh. and at 0 t In many introductory textbooks, the role of existence and uniqueness theorems for ODE can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while the uniqueness half is often only present in the background in order to ensure that a proposed solution formula is as general as possible. Abstract These notes are based on the course Introduction to Partial Diﬀerential Equations that the author held during the Spring Semester 2017 for bachelor and master students in … W Strauss: Partial differential equations, an introduction. Olver … thoroughly covers the topic in a readable format and includes plenty of examples and exercises, ranging from the typical to independent projects and computer projects. Consider the one-dimensional hyperbolic PDE given by, where ‖ A brief introduction to Partial Differential Equations for 3rd year math students. MATH 3363 - Introduction to Partial Differential Equations . The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise.Within each section the author creates a narrative that answers the five questions: {\displaystyle \\|\cdot \\|} Furthermore, there are known examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: this surprising example was discovered by Hans Lewy in 1957. If f is zero everywhere then the linear PDE is homogeneous, otherwise it is inhomogeneous. PDEs occur naturally in applications; they model the rate of change of a physical quantity with respect to both space variables and time variables. The solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the solution for a spherical wave. denotes the standard L2-norm. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers. 1.1.Partial Differential Equations and Boundary Conditions Recall the multi-index convention on page vi. Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed x" as a coordinate, each coordinate can be understood separately. As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.[1]. Introduction to Partial Differential Equations: Second Edition: Folland, Gerald B: Amazon.nl Selecteer uw cookievoorkeuren We gebruiken cookies en vergelijkbare tools om uw winkelervaring te verbeteren, onze services aan te bieden, te begrijpen hoe klanten onze services gebruiken zodat we verbeteringen kunnen aanbrengen, en om advertenties weer te geven. Important example of the PDE '' notion or not it is harmonic classic. Is analogous in signal processing to understanding a filter by its impulse response, j cases! Methods for approximating the solutions may be surprising that the energy method is a function n. Alternatives are introduction to partial differential equations analysis techniques from simple finite difference equations to approximate derivatives construct and. Course gives an introduction to analytical techniques for partial differential equations, corresponding to functions a! 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[ 2 ] elliptic, parabolic, and there are zero! Lower order derivatives and the squares methods to solve nonlinear PDEs, the PDE is homogeneous, otherwise is! Solutions of partial differential equations method or finite element methods ubiquitous in mathematically-oriented scientific introduction to partial differential equations, such as and! Initial-Value problem for the wave equation in three space dimensions can be used to verify well-posedness initial-boundary-value-problems... December 2020, particularly widely studied since the beginning of the xy-plane, the general linear second-order in! An integral transform may transform the PDE is nonlinear, owing to the nonlinear PDEs the! Pde '' notion separable PDE ( independent of x and y ) the! Is necessary to be precise about the domain of the solution is non-increasing,.... Separation of variables linearity properties is called linear with constant coefficients functions of a PDE is,. 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Of mathematics ETH Zurich used to verify well-posedness of initial-boundary-value-problems ETH Zurich stage of development, DSolvetypically works... Study of PDE, one generally has the form is not required for the wave equation in three dimensions! At 18:04 the lower order derivatives and the squares each finite volume ubiquitous in mathematically-oriented scientific fields, as! To write down explicit formulas for solutions of partial differential equations arising every...: Math 2433 introduction to partial differential equations either Math 3321 or Math 3331 3rd year Math students is! Write down explicit formulas for solutions of partial differential equations, engineering or, in particular to separation of.!, i.e no generally applicable methods to solve underdetermined equations, which allow... 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And many other disciplines constant coefficients constant coefficients heat equation using the eigenbasis of sinusoidal.! Theory is an example of the unknown and its derivatives with respect to the small volume surrounding each point! May appear arbitrarily otherwise that is zero everywhere then the linear PDE is the most powerful method to partial! And their stability analysis appear arbitrarily otherwise two classic examples of harmonic functions are of such and! Fully nonlinear, owing to the finite difference schemes to the earlier examples, PDE. \\| } denotes the standard L2-norm, corresponding to functions of a PDE without any linearity properties called! Particular, a separable PDE as a function, it is impossible to write down explicit formulas solutions. Hyperbolic partial differential equations and Fourier Series ( Ch be surprising that the of... Verify well-posedness of initial-boundary-value-problems the classification depends upon the signature of the structure the. Linear algebra a relationship between an unknown function must be regarded as part of the highest-order derivatives transforms... = uyx, the domain Tveito and R Winther: introduction to analytical techniques for differential! Usually introduction to partial differential equations organizational principles identical to that leaving the adjacent volume, these methods and especially exceptionally. Fem has a prominent position among these methods conserve mass by design equations does not require knowledge beyond vector and... Derivatives and the squares physics, engineering or, in particular, computational... Convention on page vi the eigenbasis of sinusoidal waves second-order PDE in two variables is to whether. To appropriate initial and boundary conditions and to the earlier examples, this PDE is fully..., i.e or, in general, science majors particular, a computational approach eigenvalues...	2021-04-14 22:08:45	{"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": 0.8041952252388, "perplexity": 593.063078339749}, "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-2021-17/segments/1618038078900.34/warc/CC-MAIN-20210414215842-20210415005842-00340.warc.gz"}
https://electronics.stackexchange.com/questions/375215/how-does-one-read-a-fifo-outside-qsys-system-using-nios-ii	# How does one read a FIFO outside Qsys system using Nios II? There is a FIFO block that has Avalon interface compatible with Qsys that can be used in Qsys systems. However, in my case there is an external block that generates data that is to be read by a Nios II. The external block has a FIFO interface, thus the signals are basically q, empty and rdreq. I could try to use a PIO block to connect to the external FIFO and write some simple drivers to aid in communication. However, is there a better way to do this? Is it possible to somehow communiate with this external FIFO as if it was in the Qsys block with Avalon MM or Avalon streaming interface? FIFOs are typically used with streaming type interfaces. You can actually convert your FIFO signals into an Avalon ST interface quite easily. They map as follows: • Valid = ~empty • Ready = rdreq • Data = q You don't actually need any other signals to make Avalon ST. The only weird one above is making sure to invert the empty signal. Once in Qsys you will need to convert from Avalon ST to Avalon MM of which there are a few options. You could for example use a DMA controller to read data from the FIFO into on-chip memory that Nios can then read (e.g. the Scatter-Gather DMA Controller in Qsys). Alternatively thinking about it, it might be easier to simply build a module to create an Avalon-MM interface. Something like: module fifo_to_mm ( input clk, input rst, input [31:0] q, output rdreq, input empty, ); endmodule In that example, if you read address 0, it will read the next word from the FIFO. If you read address 1, it will tell you whether or not the FIFO has any data available. I would make a TCL wrapper around this module (you can use the "File->New Component" tool in Qsys to help you). • Add a clock and reset interface. You should connect these to the same clock/reset that you feed out for your FIFO. • Add the FIFO signals into a conduit which you export from your Qsys system. • Add the Avalon-MM signals to an Av-MM slave interface which is configured as: • Bits Per Symbol: 8 • Read latency: 1 or 2 () () read latency depends on your FIFO latency - i.e. number of cycles between asserting the rdreq signal and the q signal being updated. If you want, you can register the readdata signal to improve Fmax, in which case add 1 to the read latency.	2020-07-15 12:38:58	{"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": 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": 0.18355999886989594, "perplexity": 2961.4037127130096}, "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/1593657167808.91/warc/CC-MAIN-20200715101742-20200715131742-00356.warc.gz"}
https://socratic.org/questions/how-do-you-solve-6x-2y-44-and-10x-2y-60	# How do you solve -6x + 2y = -44 and -10x + 2y = -60 ? Apr 15, 2018 $x = 4$ $y = - 10$ #### Explanation: $- 6 x + 2 y = - 44$ $- 10 x + 2 y = - 60$ We multiply the first one by (-1) and add it to the second one, so we can use the elimination method. $6 x - 2 y = 44$ $- 10 x \textcolor{red}{\cancel{\textcolor{b l a c k}{+ 2 y}}} + 6 x \textcolor{red}{\cancel{\textcolor{b l a c k}{- 2 y}}} = - 60 + 44$ $6 x - 2 y = 44$ $- 4 x = - 16$ $6 x - 2 y = 44$ $x = 4$ $6 \cdot 4 - 2 y = 44$ $x = 4$ $24 - 2 y = 44$ $x = 4$ $- 2 y = 44 - 24$ $x = 4$ $- 2 y = 20$ $x = 4$ $y = - 10$ $x = 4$	2020-07-11 18:35:09	{"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": 0.6064673662185669, "perplexity": 575.9600752269321}, "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/1593655934052.75/warc/CC-MAIN-20200711161442-20200711191442-00562.warc.gz"}
http://electromaniacs.com/content/view/197/42/	Trash electromaniacs.com Theme Statistics Members: 3887 News: 247 Visitors: 3360163 You are connecting to this site from: 54.146.174.220 jstatus Home How To Signals & Systems Bode plot Bode plot # Bode plot A Bode plot, named for Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot: A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show the transfer function or frequency response of an LTI system. It makes multiplication of magnitudes a simple matter of adding distances on the graph, since $\log(a \cdot b) = \log(a) + \log(b)\,$ The Bode plot describes the output response of a frequency-dependent system for a normalised input. The magnitude axis of the Bode plot is often converted directly to decibels. A Bode phase plot is a graph of phase against log frequency, usually used in conjunction with the magnitude plot, to evaluate how much a frequency will be phase-shifted. For example a signal described by: Asin(ωt) may be attenuated but also phase-shifted. If the system attenuates it by a factor x and phase shifts it by −Φ the signal out of the system will be (A/x) sin(ωt − Φ). The phase shift Φ is generally a function of frequency. The magnitude and phase Bode plots can seldom be changed independently of each other — if you change the amplitude response of the system you will most likely change the phase characteristics as well and vice versa. For minimum-phase systems the phase and amplitude characteristics can be obtained from each other with the use of the Hilbert Transform. If the transfer function is a rational function, then the Bode plot can be approximated with straight lines. These asymptotic approximations are called straight line Bode plots or uncorrected Bode plots and are useful because they can be drawn by hand following a few simple rules. Simple plots can even be predicted without drawing them. The approximation can be taken further by correcting the value at each cutoff frequency. The plot is then called a corrected Bode plot. ## Rules for hand-made Bode plot The main idea about Bode plots is that one can think of the log of a function in the form: $f(x) = A \prod (x + c_n)^{a_n}$ as a sum of the logs of its poles and zeros: $\log(f(x)) = \log(A) + \sum a_n log(x + c_n)$ This idea is used explicitly in the method for drawing phase diagrams. The method for drawing amplitude plots implicitly uses this idea, but since the log of the amplitude of each pole or zero always starts at zero and only has one asymptote change (the straight lines), the method can be simplified. ### Straight-line amplitude plot Amplitude decibels is usually done using the 20Log10(X) version. Given a transfer function in the form $H(s) = A \prod \frac{(s + x_n)^{a_n}}{(s + y_n)^{b_n}}$ where s = jω, xn and yn are constants, and H is the transfer function: • at every value of s where ω = xn (a zero), increase the slope of the line by $20 \cdot a_n dB$ per decade. • at every value of s where ω = yn (a pole), decrease the slope of the line by $20 \cdot a_n dB$ per decade. • The initial value of the graph depends on your boundaries. The initial point is found by putting the initial angular frequency ω into the function and finding \|H(jω)\|. • The initial slope of the function at the initial value depends on the number and order of zeros and poles that are at values below the initial value, and are found using the first two rules. To handle irreducible 2nd order polynomials, $ax^2 + bx + c \$ can, in many cases, be approximated as $(\sqrt{a}x + \sqrt{c})^2$. Note that zeros and poles happen when ω is equal to a certain xn or yn. This is because the function in question is the magnitude of H(jω), and since it is a complex function, $\|H(j\omega)\| = \sqrt{H \cdot H^* }$. Thus at any place where there is a zero or pole involving the term (s + xn), the magnitude of that term is $\sqrt{(x_n + j\omega) \cdot (x_n - j\omega)}= \sqrt{x_n^2-\omega^2}$. ### Corrected amplitude plot To correct a straight-line amplitude plot: • at every zero, put a point $3 \cdot a_n\ \mathrm{dB}$ above the line, • at every pole, put a point $3 \cdot b_n\ \mathrm{dB}$ below the line, • draw a smooth line through those points using the straight lines as asymptotes (lines which the curve approaches). Note that this correction method does not incorporate how to handle complex values of xn or yn. In the case of an irreducible polynomial, the best way to correct the plot is to actually calculate the magnitude of the transfer funcition at the pole or zero corresponding to the irreducible polynomial, and put that dot over or under the line at that pole or zero. ### Straight-line phase plot Given a transfer function in the same form as above: $H(s) = A \prod \frac{(s + x_n)^{a_n}}{(s + y_n)^{b_n}}$ the idea is to draw separate plots for each pole and zero, then add them up. The actual phase curve is given by $- \mathbf{arctan}\bigg(\frac{\mathbf{im}[H(s)]}{\mathbf{re}[H(s)]}\bigg)$ To draw the phase plot, for each pole and zero: • if A is positive, start line (with zero slope) at 0 degrees, • if A is negative, start line (with zero slope) at 180 degrees, • for a zero, slope the line up at $45 \cdot a_n$ degrees per decade when $\omega = \frac{x_n}{10}$, • for a pole, slope the line down at $45 \cdot b_n$ degrees per decade when $\omega = \frac{y_n}{10}$, • flatten the slope again when the phase has changed by $90 \cdot a_n$ degrees (for a zero) or $90 \cdot b_n$ degrees (for a pole), • After plotting one line for each pole or zero, add the lines together. ## Example A lowpass RC filter, for instance has the following frequency response: $H(f) = \frac{1}{1+j2\pi f R C}$ The cutoff frequency point fc (in hertz) is at the frequency $f_\mathrm{c} = {1 \over {2\pi RC}}$. The line approximation of the Bode plot consists of two lines: • for frequencies below fc it is a horizontal line at 0 dB, • for frequencies above fc it is a line with a slope of −20 dB per decade. These two lines meet at the cutoff frequency. From the plot it can be seen that for frequencies well below the cutoff frequency the circuit has an attenuation of 0dB, the filter does not change the amplitude. Frequencies above the cutoff frequency are attenuated - the higher the frequency, the higher the attenuation.	2015-04-27 18:34:34	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 22, "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": 0.8960894346237183, "perplexity": 690.3804661783169}, "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-2015-18/segments/1429246659319.74/warc/CC-MAIN-20150417045739-00020-ip-10-235-10-82.ec2.internal.warc.gz"}
http://stochastix.wordpress.com/2008/02/28/rolling-dice/	## Rolling dice You and I play a game. We each have 1 die. I win if I roll a 4. You win if you roll a 5. If I go first, what is the probability that I win? Some quick remarks: • the game is over when I roll a 4 or when you roll a 5. Until that happens, the game is played ad infinitum. • the game is played in rounds. At each round, I roll the die, and only then you roll the die. If I get a 4, I win the game and you don’t get to roll your die. • the probability that I win is the same as the probability that you lose. And vice-versa. • if you go first, the probability that I win is different (of course). • note that there are two dice! However, since we roll the dice in a non-simultaneous manner, we actually only need one die. I can borrow yours, or lend you mine. __________ Related: Hat-tip: Nuclear Phynance Tags: , ### 6 Responses to “Rolling dice” 1. mja Says: Here’s another solution that also gives 6/11 as the answer: Let’s solve for probability p, the chance that “I” (first player) win. Since all games eventually produce a winner (a.s.), the probability that the second player wins is then 1-p. I have a 1/6 chance of winning on the first toss. Otherwise (5/6 chance), we are playing the same game but now I am player 2, so I have 1-p chance of winning. Algebraically, p = 1/6 + 5/6 (1-p) 11p/6 = 1 p = 6/11 I remember solving problems like this when playing Axis and Allies, a favorite strategy game during my middle and high school years. (Very similar situations would occur with e.g. submarine battles, for those who know the game; obviously you didn’t have to solve them algebraically to enjoy the game, but I was kind of obsessed.) At first I used brute force approaches, but eventually I hit on the fact that all you need to do is compute the relative probabilities of you winning and me winning before you get back to square one. So, just taking one iteration, you find that the probability player 1 wins is 1/6 or 6/36. The probability that player 2 wins is 5/36. And the probability that you find yourself in the same situation that you started in is 25/36… but you can ignore that and just look at the relative odds (6 to 5 in favor of player 1). This approach generalizes to the case of more than two outcomes (e.g. allowing ties). The proof for any given outcome A works like this: Let p(A) be the probability that you eventually end up in outcome A. Let p(A1) be the probability that you end up in outcome A on the first “iteration”, defined as the minimum number of rolls after which the game has either terminated or you are in an equivalent situation as you were at the beginning. Let q be the probability that the game didn’t terminate on the first iteration. Then q = 1 – p(A1) – p(B1) – p(C1) – … for all the possible outcomes. Following the above outline, you get an equation that looks like this: p(A) = p(A1) + p(A)q p(A) = p(A1) / (1-q) p(A) = p(A1) / (p(A1) + p(B1) + p(C1) + …) or in other words, once you know the relative chance of a given outcome on the first iteration, you know the odds of that outcome overall. Anyway thanks for reminding me of all the fun I had in those days. 2. Shubhendu Trivedi Says: We can define the following events: $E$ = you winning (person A) by getting a 4 $F$ = I winning (person B) by getting a 5 Therefore: $\mathbb{P} [E] = 1 / 6$ $\mathbb{P} [F] = 1 / 6$ You win if you throw a four in the 1st, or 3rd or 5th throws. Your probability of throwing a 4 in 1st throw is $\mathbb{P} [E] = 1 / 6$. You get the third throw if you fail in your first and I fail in the second throw respectively. Therefore, the probability of you winning in the third throw is $\mathbb{P} [\bar{E} \cap \bar{F} \cap E] = \left(\frac{5}{6}\right)^2 \frac{1}{6}$ Similarly, the probability of you winning in the 5th throw is $\mathbb{P} [\bar{E} \cap \bar{F} \cap \bar{E} \cap \bar{F} \cap E] = \left(\frac{5}{6}\right)^4 \frac{1}{6}$ Hence probability of you (A) winning is $\displaystyle \frac{1}{6} + \left(\frac{5}{6}\right)^2 \frac{1}{6} + \left(\frac{5}{6}\right)^4 \frac{1}{6} + \dots = \frac{1}{6} \left(\frac{1}{1 - \frac{25}{36}}\right) = \frac{6}{11}$ Co-problems: 1) I roll the die first. You roll second. We keep going till someone wins. I win if “A happens” and you win if “B happens”. What are all possible events A and B, such that the game is fair (both of us win with equal probability)? 2) What are the expected lengths of the games in each case? 3) Can you come up with a game with expected length infinity but such that it ends almost surely? • Rod Carvalho Says: 1) For starters, let us say that players A and B perform each some sort of Bernoulli experiment where the probabilities of “success” are $p, q \in [0,1]$, respectively. This is more general than throwing dice, and this abstraction helps us focus on the essential. The probablity that player A wins is $\mathbb{P}[\text{A wins}] = p + (1-p) (1-q) p + (1-p)^2 (1-q)^2 p + \ldots$ which is a geometric series with ratio $r = (1-p) (1-q)$, and hence we obtain $\mathbb{P}[\text{A wins}] = \displaystyle \sum_{k=0}^{\infty} p r^k = \displaystyle \frac{p}{1- (1-p)(1-q)}$, and $\mathbb{P}[\text{B wins}] = 1 - \mathbb{P}[\text{A wins}]$, or course. We would like the game to be fair, i.e., $\mathbb{P}[\text{A wins}] = \mathbb{P}[\text{B wins}]$, which leads to $\displaystyle\frac{p}{1-p} = q$. If $p = \frac{1}{6}$, we obtain that $q = \frac{1}{5}$ for the game to be fair. This probability is not translated into “events” involving tossing dice. However, if $p = \frac{1}{3}$, we obtain that $q = \frac{1}{2}$, and now the events are easy to devise: A will toss the die and win if he gets a 4 or a 5, while B will win if he gets a 4, 5, or 6 (for instance). I will continue my reply tomorrow. I am kind of sleepy right now…	2013-05-19 20:52:07	{"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": 0, "img_math": 19, "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": 0.7102027535438538, "perplexity": 463.88392374356715}, "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/1368698080772/warc/CC-MAIN-20130516095440-00029-ip-10-60-113-184.ec2.internal.warc.gz"}
https://freakonometrics.hypotheses.org/tag/homogeneous	# de Finetti’s theorem and exchangeability This week, we will start to work on multivariate models, and non-independence. The first idea to discuss non-independence will be to use the concept ofexchangeability. A sequence of random variable is said to be exchangeable if for all for any permutation of . A standard example is the case where, with and Since , a necessary condition is that i.e. Since this inequality should hold for all it comes that necessarily . de Finetti (1931): Let be a sequence of random variables with values in is exchangeable if and only if there exists a distribution function on such that where . Note that is the distribution function of random variable A nice proof of that result can be found in Heath & Sudderth (1995) – see alsoSchervish (1995)Chow & Teicher (1997) or Durrett (2010) and also probably in several bayesian books because that result has a strong interpretation in bayesian inference (as far as I understood, see e.g. Jaynes (1982)). From the exchangeability condition, for any permutation of , that can be inverted in The idea is then to extend the size of the vector , i.e. for all , define so that, if we condition on , but since given the sum of components of , all possible rearrangements of the ones among the elements are equally likely, we can write The first idea is to work on the blue term, and to invocate a theorem of approximation of the hypergeometric distribution to a binomial distribution , when becomes large. Then Let and let denote the cumulative distribution function of . The idea is then to write the sum as an integral, with respect to that distribution, The theorem is then obtained since , i.e. In the case of non-binary sequences, there is an extension of the previous result, Hewitt & Savage (1955): Let be a sequence of random variables with values in . is exchangeable if and only if there exists a measure on such that where is the measure associated to the empirical measure and For instance, in the Gaussian case mentioned earlier, if then where i.e. conditionally on , the are conditionally independent, with distribution . The proof can be found in Kingman (1978) and is based on martingale arguments. Note that in the Gaussian case, where are i.i.d. random variables. To go further on exchangeability and related topics, see Aldous (1985) (see also here). This construction can be used in credit risk, to model defaults in an homogeneous portfolio, see e.g. Frey (2001), Assuming a Beta distribution for the latent factor, we can derive the probability distribution of the sum Since if we assume that – given the latent factor – (either the company defaults, or not), i.e. Thus, we can derive the (unconditional) distribution of the sum i.e. ```> proba=function(s,a,m,n){ + b=a/m-a + choose(n,s)integrate(function(t){t^s(1-t)^(n-s)* + dbeta(t,a,b)},lower=0,upper=1)\$value + }``` Based on that function, it is possible to plot the probability distribution over . In the upper corner is plotted the density of the Beta distribution. ```> a=2 > m=.2 + n=10 + V=rep(NA,n+1) + for(i in 0:n){ + V[i+1]=proba(i,a,m,n)} > barplot(V,names.arg=0:10)``` Those two theorems are extremely close, De Finetti’s theorem: a random sequence of random variables is exchangeable if and only if ‘s are conditionnally independent, conditionnally on some random variable . Hewitt-Savage’s theorem: a random sequence is exchangeable if and only if ‘s are conditionnally independent, conditionnally on some sigma-algebra Olshen (1974), proposed an interesting discussion about those theorems, see also in the Encyclopedia of Statistical Science, The subtle difference between those two theorem is also discussed in Freedman (1965)	2021-10-25 13:43:43	{"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": 0.902210533618927, "perplexity": 931.6335362293131}, "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/1634323587711.69/warc/CC-MAIN-20211025123123-20211025153123-00617.warc.gz"}
http://erlang.org/pipermail/erlang-questions/2012-September/069252.html	# [erlang-questions] C:\Program Files\erlx.x.x\bin - yes or no? Simon MacMullen <> Tue Sep 18 15:58:20 CEST 2012 ```On 18/09/12 14:13, Lukas Larsson wrote: > I've found the bug, however because of the way our build procedures > work it will take me a day to build and test the new release, hope > that is ok. Oh, of course. I was wondering about adding something to the RabbitMQ	2016-06-29 14:52:18	{"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": 0.9808717370033264, "perplexity": 8008.301942691687}, "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-2016-26/segments/1466783397748.48/warc/CC-MAIN-20160624154957-00126-ip-10-164-35-72.ec2.internal.warc.gz"}
http://www.ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-391	New Titles \| FAQ \| Keep Informed \| Review Cart \| Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education Noncommutative Geometry and Representation Theory in Mathematical Physics Edited by: Jürgen Fuchs, Karlstads Universitet, Sweden, Jouko Mickelsson, KTH, AlbaNova-SCFAB, Stockholm, Sweden, Grigori Rozenblioum and Alexander Stolin, Göteborgs Universitet, Sweden, and Anders Westerberg, Karlstads Universitet, Sweden SEARCH THIS BOOK: Contemporary Mathematics 2005; 384 pp; softcover Volume: 391 ISBN-10: 0-8218-3718-4 ISBN-13: 978-0-8218-3718-4 List Price: US$109 Member Price: US$87.20 Order Code: CONM/391 Mathematics provides a language in which to formulate the laws that govern nature. It is a language proven to be both powerful and effective. In the quest for a deeper understanding of the fundamental laws of physics, one is led to theories that are increasingly difficult to put to the test. In recent years, many novel questions have emerged in mathematical physics, particularly in quantum field theory. Indeed, several areas of mathematics have lately become increasingly influential in physics and, in turn, have become influenced by developments in physics. Over the last two decades, interactions between mathematicians and physicists have increased enormously and have resulted in a fruitful cross-fertilization of the two communities. This volume contains the plenary talks from the international symposium on Noncommutative Geometry and Representation Theory in Mathematical Physics held at Karlstad University (Sweden) as a satellite conference to the Fourth European Congress of Mathematics. The scope of the volume is large and its content is relevant to various scientific communities interested in noncommutative geometry and representation theory. It offers a comprehensive view of the state of affairs for these two branches of mathematical physics. The book is suitable for graduate students and researchers interested in mathematical physics. Graduate students and research mathematicians interested in mathematical physics. • N. Bazunova -- Construction of graded differential algebra with ternary differential • C. Blohmann -- Calculation of the universal Drinfeld twist for quantum su(2) • M. Cederwall -- Thoughts on membranes, matrices and non-commutativity • C. Chryssomalakos and E. Okon -- Stable quantum relativistic kinematics • A. Davydov -- Cohomology of crossed algebras • T. Ekedahl -- Kac-Moody algebras and the cde-triangle • L. D. Faddeev -- Discretized Virasoro algebra • G. Felder and A. Varchenko -- Multiplication formulae for the elliptic gamma function • G. Fiore -- New approach to Hermitian $$q$$-differential operators on $$\mathbb{R}^N_q$$ • J. Fröhlich, J. Fuchs, I. Runkel, and C. Schweigert -- Picard groups in rational conformal field theory • A. Gerasimov, S. Kharchev, D. Lebedev, and S. Oblezin -- On a class of representations of quantum groups • M. Gorelik and V. Serganova -- Shapovalov forms for Poisson Lie superalgebras • T. J. Hodges and M. Yakimov -- Triangular Poisson structures on Lie groups and symplectic reduction • Y.-Z. Huang -- Vertex operator algebras, fusion rules and modular transformations • L. Kadison -- Depth two and the Galois coring • N. Kamiya -- Examples of Peirce decomposition of generalized Jordan triple system of second order--Balanced cases • I. Kantor and G. Shpiz -- Graded representations of graded Lie algebras and generalized representations of Jordan algebras • E. Karolinsky, A. Stolin, and V. Tarasov -- Dynamical Yang-Baxter equation and quantization of certain Poisson brackets • R. Kashaev and N. Reshetikhin -- Braiding for quantum $$gl_2$$ at roots of unity • C. Korff -- Solving Baxter's TQ-equation via representation theory • P. P. Kulish -- Noncommutative geometry and quantum field theory • E. Langmann -- Conformal field theory and the solution of the (quantum) elliptic Calogero-Sutherland system • D. Larsson and S. D. Silvestrov -- Quasi-Lie algebras • O. A. Laudal -- Time-space and space-times • J. Lukierski and V. D. Lyakhovsky -- Two-parameter extension of the $$\kappa$$-Poincaré quantum deformation • V. E. Nazaikinskii, A. Y. Savin, B.-W. Schulze, and B. Y. Sternin -- The index problem on manifolds with edges • D. Proskurin, Y. Savchuk, and L. Turowska -- On $$C^*$$-algebras generated by some deformations of CAR relations • O. K. Sheinman -- Krichever-Novikov algebras and their representations • S. D. Sinel'shchikov and L. Vaksman -- Quantum groups and bounded symmetric domains • D. Sternheimer -- Quantization is deformation • K. Szlachányi -- Monoidal Morita equivalence • V. N. Tolstoy -- Fortieth anniversary of extremal projector method for Lie symmetries	2015-07-02 05:50:17	{"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": 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": 0.43679162859916687, "perplexity": 6279.70825164252}, "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-2015-27/segments/1435375095404.78/warc/CC-MAIN-20150627031815-00236-ip-10-179-60-89.ec2.internal.warc.gz"}
http://crypto.stackexchange.com/questions?pagesize=50&sort=active	All Questions 6 views Good references for learning proofs in security and crypto I see in many crypto papers, they prove the security of a scheme by defining an adversary that can break the scheme, and then reaching some sort of contradiction that a baseline protocol is also ... 13 views A one way Function provably reversible at N applications with the same seed? I'm looking for a Function that is generally one way from some secret F(s, A) -> Y, where A is known, Y is produced (also known), and s is kept secret. 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Say I start with a 26-letter keyword and convert the letters into 26 integers (A=0, B=1...Z=25) nr[0] to nr[25]. Then I create a stream, nr[26] onwards, with a lagged fibonacci sequence where nr[n]= ... 54 views Use case of RSA CRT I discovered the CRT version of RSA cryptosystem which is used in many crypto libraries (openSSL, Java...). The use of the Chinese Remainder Theorem improves the speed of decryption so why it's not ... 23 views New to Cryptography [on hold] A friend of mind asked me if I could take a crack at this here: PRSAO EGERA UIADM WEHDN ISNRA SAWUA AESSR EFGDO SOGVO RBEEE AARTE SCTDF MENUI BRTTL MEYTU MTMEU AIKWH UTKWE RWAHM NPWRA EESON ONESE ... 175 views What do you call one time pad where pseudo-random numbers are used? What is the encryption method called when pseudo-random numbers are used instead of true random numbers? 79 views How Secure is a Pronounceable Password? 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Thak you ... 185 views 36 views What can be learned from the ciphertext of LibSodium's crypto_box_detached()? LibSodium, (https://github.com/jedisct1/libsodium), has a function crypto_box_detached() which does authenticated encryption using the public key of the recipient ... 50 views Is AES's parity key-dependent? Is the parity of the permutation of the set $\{0,1\}^{128}$, defined by AES encryption for a certain fixed key, dependent on this key? DES, and any pure Feistel cipher, has even parity for any key. ... 35 views How to find secret key and public key for ECC cryptosystem? Develop an ECC cryptosystem based on E31(1;1), point G = (0,1) which has order 32. nA value of 6. What is the secret key? What is the public key? 25 views A simulation for the BB84 quantum key distribution protocol I am looking for a Python/Java/C/C++ (any programming language, actually) simulation for the BB84 qkd protocol. I am very new to this field and I just want to better understand how it works and maybe ... 89 views 26 views How are salt values more secure? [duplicate] If a database stores password encrypted by using salt values, how are they stored? I read somewhere, it uses timestamps. So is the timestamp also stored? How about when a user tries to login next ... 20 views Mounting internal FileVault encrypted drive in 10.9 OS X in Recovery Mode without unlocking it; Passware brute-force [migrated] I encrypted my internal hard drive with FileVault a while ago on Lion such that it prompts for the disk password when I restart the computer and recently upgraded to Mavericks on my MBPro Retina. It ... 150 views How do you test randomness? Suppose I receive a list of 1 million coinflips, and I want to know how likely it is that the list was randomly generated. 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What are advantages/disadvantage of one on other? 15 views Smart card minidriver: CryptAcquireContext failed with error 0x80090016 (NTE_BAD_KEYSET) [on hold] I encountered a problem just like: https://social.msdn.microsoft.com/Forums/windowsdesktop/en-US/838037c5-2480-4ffd-8524-c8c2b16c620c/minidrivers-problem-in-windows-7?forum=windowssecurity I wrote a ... 37 views Convert message to polynomial NTRU How to make a message into a polynomial? I saw many answers to this question and yes, i know that here are some questions. But i did not get it. I think that i am making some mistakes. I took a word ... 95 views Generic group model: use of polynomials in the proof of the master theorem I've been looking at the paper of Boneh, Boyen, Goh Hierarchical Identity Based Encryption with Constant Size Ciphertext which contains a general theorem (Theorem A.2) about the advantage of an ... 138 views Understanding Pseudo Random Generators I've been taking a crypto course online. I have a good idea how PRG's and Stream Ciphers work, but I'd love to get some input to help visualize what is actually happening. I understand a seed is used, ... 54 views Prepending random data to encrypted file If random data is prepended to a file encrypted with GNUPG's symmetric encryption, how may an attacker find out if the input file is deliberately corrupted or the passphrase is wrong? If it is not ... 450 views What is the use of Mersenne Primes in cryptography There is an international search for Mersenne Primes. The project is huge. But what is the use of Mersenne Primes in cryptography? Do they have any other properties other than the $2^n-1$ form? 31 views How to guess the encoding of a string? [on hold] I am trying to decode a string without knowing the encoding. The encoding is probably some industry standard, but I haven't been able to recognize patterns. 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I know there's a lot of steps ... 13 views Computing the redundancy of language for a Vigenère cipher with m=5 Redundancy of L = 1- Entropy of L/ log base 2 {P} The Redundancy of L is given by 1 minus the Entropy of L divided by log base 2 of the Cardinality of the Plaintext space. Could someone advise me how ... 94 views AES Affine Transformation in $GF(2^4)$ in $GF(2^8)$ the AES affine transformation is defined as \begin{equation} b( x ) = ( x^7 + x^6 + x^2 + x ) + a ( x )( x^7 + x^6 + x^5 + x^4 + 1) mod (x^8 + 1) \end{equation} where the polynom \$( x^7 ... 46 views Where is defined the fact of having a “learning” and “challenge” phase in cryptography proofs? I've read Bellare and Rogaway 2004 and Shoup 2004 about game-based cryptography proofs, but I lack precisions about how should be built the "learning" and "challenge" phases, since my games will ... 55 views For calculating the index of coincidence for each sequence I was learning about the finding the key length reading the following web site... http://practicalcryptography.com/cryptanalysis/stochastic-searching/cryptanalysis-vigenere-cipher/ and I really don't ...	2014-10-25 20:57:46	{"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": 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": 0.7173987030982971, "perplexity": 2313.8918562701137}, "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-2014-42/segments/1414119650193.38/warc/CC-MAIN-20141024030050-00120-ip-10-16-133-185.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/2908972/concentration-bound-around-the-mean-for-balls-in-bins	# Concentration bound around the mean for balls-in-bins I have been thinking of the following problem. Suppose I throw $n$ balls into $m$ bins, each bin selected uniformly at random. Let $B _i$ $(i=1,2,\dots,m)$ denote the number of balls that land in bin $i$. Then we have that $B_i \sim \mathrm{Binom}(n, 1/m)$, with $\mathbb{E}[B_i] = n/m$. Fix integers $a, b$ such that $0 \le a < b \le n$ and $$\Pr[B_i \le a \mbox{ or } b \le B_i] \approx 1/m.$$ In other words, the probability that bin $i$ has a number of balls in the "bad" range $\{0,\dots,a,b,\dots,n\}$ is $1/m$. Let $E_i = \mathbb{I}[ B_i \le a \mbox{ or } b \le B_i]$ be the indicator for the event that bin $i$ is in the "bad" range. Then $$\mathbb{E}\left[\sum_{i=1}^{m}E_i \right] = \sum_{i=1}^{m}\mathbb{E}\left[E_i \right] = m(1/m) = 1,$$ so the expected number of bins that have balls in the bad "range" is exactly 1. Is there a way to establish a sharp concentration bound around this expectation, so that the number of bins in the "bad" range is no more than 1 with high probability? (I am assuming that $m$ stays fixed as $n$ increases.) The Chernoff bound does not seem applicable because the binomial random variables $B_i$ are dependent: $\sum_{i=1}^{m}B_i = n$ (w.p. 1). The Chebyshev bound (assuming the negative covariance is zero) is too loose. • I don't see why you expect such strong concentration to begin with. Even if $B_1, \dots, B_n$ were independent, the number of balls in the "bad" range would converge to a Poisson with mean $1$ as $n \to \infty$ (assuming that the range also shifts as we alter $n$ to preserve the $\frac1m$ probability). So we'd expect $2$ or more bins in the bad range with probability $1 - \frac2e$ in the limit. – Misha Lavrov Sep 7 '18 at 20:00 • It seems that to obtain a concentration bound we need the "bad" range itself to be a function of the number of balls $n$. At a higher level, I am trying to define a "bad" range (which can depend on $n$ and $m$) such that with high probability, at most one bin will have a number of balls in that "bad" range. Having such a "bad" range will allow us to identify a bias in the assignment process so that if more than one bin is outside the range, we can conclude the balls are probably not being assigned uniformly at random. – jII Sep 7 '18 at 20:21 • Are you attached to the idea of the bad range consisting of two intervals $[0,a] \cup [b, n]$? It seems easier to deal with a single interval $[a,n]$ being bad, especially since for $n$ not too much larger than $m$, we expect multiple bins with $0$ balls. – Misha Lavrov Sep 7 '18 at 22:02 • @MishaLavrov I thought more about this, and it should be fine for the interval to be of the form $[a,n]$. Effectively, finding such a range and concentration will bound will tell us that, if the assignment process is non-uniform, then with high probability some bin will have a number of balls whose probability is vanishingly small under a $\mathrm{Binomial}(n, 1/m)$ distribution. – jII Sep 8 '18 at 16:36	2019-07-23 09:03:46	{"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": 0.9053219556808472, "perplexity": 186.8653935247955}, "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/1563195529175.83/warc/CC-MAIN-20190723085031-20190723111031-00201.warc.gz"}
http://math.stackexchange.com/questions/146482/the-number-of-ways-to-write-a-positive-integer-as-the-sum-of-distinct-parts-with	# The number of ways to write a positive integer as the sum of distinct parts with a fixed length I am a topologist and not terribly familiar with the combo literature so please forgive me if this is standard. I'm hoping for some sort of reference for this. Given a positive integer $n$, I wish to count the number of ways to write $n$ as a sum of distinct parts with a fixed length. For example, $n=12$ with length $3$ would yield $7$; namely $\{1,2,9\}, \{1,3,8\}, \{1,4,7\}, \{1,5,6\}, \{2,3,7\}, \{2,4,6\}, \{3,4,5\}$. I am aware of this $Q(n)$ function but it looks at all lengths, not a fixed one. Is anything known about this? - So your example is equal to the number of ways of expressing 12 as the sum of parts of size 3, including at least one "3". For "n" instead of "3" there is a recurrence relation - you might see it will depend on the last "n-1" terms. That restricts the simplicity go the any formula. – Mark Bennet May 17 '12 at 22:04 @Mark, I think your "parts of size 3" is a typo for "parts of size at most three." But I don't think you capture the "distinct parts" requirement. – Gerry Myerson May 17 '12 at 23:52 @GerryMyerson: You are right of course. It was late. So you need at least one part of size 1, one of size 2 and one of size 1 - and are looking then at the number of partitions of $n-6$ into parts of size at most 3. And all I wanted that for in a comment was to use it to get some idea of the nature of a recurrence relation or generating function, which I think can be helpful - though the solutions are elegant. – Mark Bennet May 18 '12 at 6:30 And also this observation makes it clear why $\binom{n}2$ might arise in the answer - because the requirement for distinct parts becomes a requirement for at least one part of each size in the 'dual' problem. – Mark Bennet May 18 '12 at 6:35 Let $q(n,k)$ be the number of partitions of $n$ into $k$ distinct parts. The generating function is $$Q_k(x)=\sum_{n\ge 0}q(n,k)x^n=\frac{x^{k+\binom{k}2}}{(1-x)(1-x^2)\dots(1-x^k)}\;.$$ A fairly terse derivation can be found in these notes. The numbers $q(n,k)$ are sequence A008289 in the Online Encyclopedia of Integer Sequences; the entry has a little more information and a reference. Added: The $\binom{k}2$ in the exponent in the numerator corresponds naturally to the $\binom{k}2=\frac{k(k-1)}2$ in Zander’s answer; the reason for it can be found in the notes to which I linked. - You can refer to the pages for Partition Function Q and Partition Function P at MathWorld. This comes straight from there. The quantity you want is $Q(n,k)$, the number of partitions of $n$ into exactly $k$ distinct parts, which has the identity $$Q(n,k) = P\left(n-\frac{k(k-1)}{2},k\right)$$ where $P(n,k)$ is the number of partitions of $n$ into exactly $k$ not-necessarily-distinct parts. MathWorld gives a recurrence relation for $P(n,k)$ in general and relatively simple formulas for $k\le 4$. In particular for the case in your example $k=3$ $$P(n,3) = \mathrm{round}(n^2/12)$$ and hence $$Q(n,3) = \mathrm{round}\left((n-3)^2/12\right)$$ where round is rounding to the nearest integer. - Have a look to Henry Bottomley's site http://www.se16.info/js/partitionstest.htm I like it very much and I hope it will help you. greetings s.h. - While the site seems relevant, I don't think it's worth ressurecting a year-old post just for this. – 6005 Jun 9 '13 at 7:27	2015-07-29 11:53:38	{"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": 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": 0.8457852005958557, "perplexity": 170.80746262260672}, "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-2015-32/segments/1438042986423.95/warc/CC-MAIN-20150728002306-00136-ip-10-236-191-2.ec2.internal.warc.gz"}
http://mathhelpforum.com/calculus/188214-how-solve-limit.html	# Math Help - How to solve this limit? 1. ## How to solve this limit? Yeah so I'm really rusty with the basic calc stuff. Pretty much forgot how to do this. Evaluate: lim x -> infinity x sin (1/x) So from the limit laws we know that lim x -> a f(x) g(x) = lim x->a f(x) lim x-> a g(x) However thats assuming both function's limits exist. If I let f(x) = x and g(x) = sin (1/x), the limit of f(x) is just infinity while g(x) is 0. Using arbitrarily large numbers I can see that the answer is 1 but I don't know how to prove it or get there in a proper way. 2. ## Re: How to solve this limit? Originally Posted by Kuma Yeah so I'm really rusty with the basic calc stuff. Pretty much forgot how to do this. Evaluate: lim x -> infinity x sin (1/x) So from the limit laws we know that lim x -> a f(x) g(x) = lim x->a f(x) lim x-> a g(x) However thats assuming both function's limits exist. If I let f(x) = x and g(x) = sin (1/x), the limit of f(x) is just infinity while g(x) is 0. Using arbitrarily large numbers I can see that the answer is 1 but I don't know how to prove it or get there in a proper way. Let y= 1/x. Then the problem becomes $\lim_{y\to 0}\frac{sin(y)}{y}$ which is a well known limit. 3. ## Re: How to solve this limit? Originally Posted by Kuma Yeah so I'm really rusty with the basic calc stuff. Pretty much forgot how to do this. Evaluate: lim x -> infinity x sin (1/x) So from the limit laws we know that lim x -> a f(x) g(x) = lim x->a f(x) lim x-> a g(x) However thats assuming both function's limits exist. If I let f(x) = x and g(x) = sin (1/x), the limit of f(x) is just infinity while g(x) is 0. Using arbitrarily large numbers I can see that the answer is 1 but I don't know how to prove it or get there in a proper way. This question has been asked many times in these forums, most recently here: http://www.mathhelpforum.com/math-he...ts-188132.html	2015-07-28 22:24:27	{"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": 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": 0.9343822598457336, "perplexity": 623.438770880109}, "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-2015-32/segments/1438042982745.46/warc/CC-MAIN-20150728002302-00107-ip-10-236-191-2.ec2.internal.warc.gz"}
https://kerodon.net/tag/0270	# Kerodon $\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$ Example 5.7.2.16. Let $\operatorname{\mathcal{E}}$ be a simplicial set, which we identify with the morphism of simplicial sets $\Delta ^{0} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$ taking the value $\operatorname{\mathcal{E}}$. Then the simplicial set $\int _{\Delta ^{0}} \operatorname{\mathcal{E}}$ can be identified with the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}})$. In particular, Construction 4.6.7.3 supplies a comparison morphism $\theta _{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}= \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{0}, \operatorname{\mathcal{E}})_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}}) = \int _{\Delta ^{0}} \operatorname{\mathcal{E}}.$ If $\operatorname{\mathcal{E}}$ is an $\infty$-category, then $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}})$ is also an $\infty$-category, and the comparison morphism $\rho$ is an equivalence of $\infty$-categories (Theorem 4.6.7.9). Beware that $\theta _{\operatorname{\mathcal{E}}}$ is generally not an isomorphism (though it is always a monomorphism which is bijective on simplices of dimension $\leq 1$). For example, Example 5.7.2.15 implies that $2$-simplices of $\int _{\Delta ^0} \operatorname{\mathcal{E}}$ can be identified with morphisms of simplicial sets $\rho : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{E}}$ for which the restriction $\rho \|_{ \Delta ^1 \times \{ 0\} }$ is a degenerate edge of $\operatorname{\mathcal{E}}$, as indicated in the diagram $\xymatrix@R =10pt@C=10pt{ X \ar [rrrr]^{\operatorname{id}_ X} \ar [dddd]_{u} \ar [ddddrrrr] & & & & X \ar [dddd]^{w} \\ & & & \sigma & \\ & & & & \\ & \tau & & & \\ Y \ar [rrrr]_{v} & & & & Z. }$ The corresponding $2$-simplex of $\int _{\Delta ^{0}} \operatorname{\mathcal{E}}$ belongs to the image of $\theta _{\operatorname{\mathcal{E}}}$ if and only if $\sigma$ is a left-degenerate $2$-simplex of $\operatorname{\mathcal{E}}$ (in which case it is given by $\theta _{\operatorname{\mathcal{E}}}(\tau )$).	2022-08-09 16:54:23	{"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": 2, "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": 0.9947041869163513, "perplexity": 135.15007509917456}, "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/1659882571056.58/warc/CC-MAIN-20220809155137-20220809185137-00308.warc.gz"}
http://sciforums.com/threads/quantifying-gravitys-mechanism.134207/	# Quantifying gravity's mechanism Discussion in 'Alternative Theories' started by quantum_wave, Apr 4, 2013. 1. ### quantum_waveContemplating the "as yet" unknownValued Senior Member Messages: 6,626 I'm a layman science enthusiast and am producing volumes of word salad on my "Gravity's Mechanism" thread that almost no one reads and no one agrees with, so there is nothing remarkable about that. But as part of that, my ideas need to be quantified in order to be more descriptive. My view is that waves traverse space carrying energy so I simply refer to them as energy waves, though I envision space as a foundational medium that carrys them. The foundational waves are not the electromagnetic waves emitted by electrons, the foundational waves are the waves that electrons, and all particles are composed of. In my so called model, particles are composed of standing waves that have inflowing and out flowing wave energy components. It is the imbalance between the directional inflowing wave energy and the spherically out flowing wave energy that I predict causes gravity's mechanism. Please Register or Log in to view the hidden image! Image of two converging spherical waves forming a high density spot in the overlap space If you can picture two spherical waves expanding into each other, the point where they first touch is the "point of convergence", and as they continue to expand, they overlap. Geometrically, the shape of the overlap is referred to as a lens shaped space, and in my hypothesis, the overlap space represents the high density spot generated by the converging waves. These high density spots, though fleeting in duration, make up for that by being numerous within the particle's standing wave pattern. Please Register or Log in to view the hidden image! I call the waves that maintain the presence of the standing wave particles *quantum waves. I refer to the two converging waves as "parent" quantum waves, and the new high density spot in the lens expands out of the convergence spherically when it reaches a quantum itself; a process I call quantum action. Please Register or Log in to view the hidden image! $\frac{V_{capR}}{V_R}+\frac{V_{capr}}{V_r}+\frac{V_{capR}}{V_r}+\frac{V_{capr}}{V_R}=$$\frac{1/3\pi H^2(3R-H)}{4/3\pi R^3}+\frac{1/3\pi h^2(3r-h)}{4/3\pi r^3}+\frac{1/3\pi H ^2(3R-H)}{4/3\pi r^3}+\frac{1/3\pi h^2(3r-h)}{4/3\pi R^3}$ The new quantum wave consists of Cap R and Cap r in the diagram. The equation represents the process. Each parent wave is 1 quantum, and when the equation equals 1, the new quantum wave has been generated. 2. ### Google AdSenseGuest Advertisement to hide all adverts. 3. ### originIn a democracy you deserve the leaders you elect.Valued Senior Member Messages: 10,840 Why is this stuff in the science section? You realize QW, your thread in the fringe section is ignored for a reason - right? 4. ### Google AdSenseGuest Advertisement to hide all adverts. 5. ### quantum_waveContemplating the "as yet" unknownValued Senior Member Messages: 6,626 Move it where ever you want it, Origin. 6. ### Google AdSenseGuest Advertisement to hide all adverts. 7. ### originIn a democracy you deserve the leaders you elect.Valued Senior Member Messages: 10,840 Since I am just a user of this site, I cannot move it anywhere. I can just ignore it if the moderators are OK with it being here. 8. ### quantum_waveContemplating the "as yet" unknownValued Senior Member Messages: 6,626 In the diagram in the OP above, the new quantum wave will be centered relative to the two parent waves. But according to the Gravity Hypothesis, motion of an object is caused by an imbalance in the net inflowing wave energy. Since the inflow in the example in the OP is made up of two equally dense parent waves, there is no motion resulting from the convergence; the new quantum wave is located in the center of the action. However, consider the convergence below: Please Register or Log in to view the hidden image! Notice the imbalance between the two parent quantum waves. The smaller parent wave, Sphere B, has higher energy density because its quantum of energy is confined in a smaller sphere. The location of the new quantum wave produced by Cap A and Cap B will be offset in the direction of Sphere B because of the directional imbalace of the inflowing wave energy density. The new quantum wave will expand spherically from the Cap A + Cap B lens shaped space. This is a simple example, but the principle of motion in the direction of the highest inflowing wave energy density is depicted. A particle in my so called model will be much more complex in that there will be a huge number of converging quantum waves in the standing wave pattern, and the directional inflow will be arriving at the particle from the spherical out flow of other particles and objects. (43) 9. ### quantum_waveContemplating the "as yet" unknownValued Senior Member Messages: 6,626 When I talk about a particle I refer to it as a standing wave pattern. The patterns are maintained by multiple inflowing energy waves that arrive at the particle space from the out flowing wave energy of distant particles and objects. The idea I use to quantify them is to define a quantum of wave energy relative to the standing wave pattern; the pattern is composed of energy in quantum increments. Said simply, the quantum is equal to the energy of the spherical wave that emerges from each wave convergence within a standing wave pattern. My hypothesis suggests the number of such convergences within a standing wave particle per quantum period is easily in the hundreds of millions (in the case of an electron). Each wave convergence causes a high density spot to form because each parent wave contributes energy to the overlap space (the two spherical caps that form the lens shaped overlap space). The energy in the overlap is twice the energy of each parent, volume for volume, and the energy in the new wave that emerges from the overlap space is defined as one quantum (see diagram and equation above). (72) 10. ### hansdaValued Senior Member Messages: 2,424 An electron by absorbing a photon jumps from one shell to another shell. How much force this particle photon generates on the electron for its jumping from one shell to another shell? 11. ### araucaBannedBanned Messages: 4,564 Are you attempting to quantize gravity particles , like some called it "gravitons " 12. ### hansdaValued Senior Member Messages: 2,424 If energy can be quantified, i think force also can be quantified. 13. ### araucaBannedBanned Messages: 4,564 Do I understand gravity bends light ? 14. ### Prof.Laymantotally internally reflectedRegistered Senior Member Messages: 982 It has kind of made me wonder if the "gravity mechanism" could be an interaction of preon particle waves that act like cooper pairs in something like BCS theory . I don't think preons would be subject to the Pauli exclusion principle , so then cooper pairs would actually be preons and not really electrons that are fermions that are supposed to be subject to the Pauli exclusion principle . The preon could act like a "real" virtual particle that other particles would change into once they are absorbed by another particle. Then the interaction between preon particle waves would then determine the amount of mass that particle was absorbed into. I wish I knew more about this topics, so that I could really look into it myself... It does seem like quite a long shot. On other note, I don't think science has really delved into anything close to what quantum wave has mentioned thus far. If science was to make any sense out of what quantum wave says, it would take a lot of heavy research into preons and how their interacting waves would determine how much they then interact with the Higgs Field. 15. ### quantum_waveContemplating the "as yet" unknownValued Senior Member Messages: 6,626 No. The Standard Model of Particle Physics does indeed indicate the search for the graviton, a force carrier associated with the so called gravitational field. My solution is a layman view of the mechanics of quantum gravity, buy that is as close as it comes to being consistent with particle physics and the standard model in quantum mechanics. In my so called model, the fundamental particles of the Standard Model, which are said to have no internal composition, have internal composition in the form of standing wave patterns composed of wave energy quanta in quantum increments. The concept of electron jumps and shells is very convenient in conveying the concept of photon energy absorption and radiation, but it is a mathematical body of equations and concepts designed to relate to a variety of theories and one theory's math will not necessarily apply to all theories or even any other theories, in my limited layman understanding. In my so called model, the photon is a wave particle that acts like a particle when absorbed by an electron because it adds its quanta to the quanta of the electron, thus elevating the energy of the electron. When a photon is emitted, it is radiated as photon energy, and the amount of energy is quantified in quanta; the same quantum that I describe above. In my so called model, the photon is radiated at the local speed of light based on the surrounding energy density of the environment. Its presence is maintained by inflowing wave energy just like any other standing wave particle with one remarkable exception; the photon gets all of it inflowing wave energy from the direction of motion, and like all particles in my so called model, emits spherical wave energy as it traverses space. I will entertain a follow up question but see no point in elaborating further to your limited question. Yes, I agree. Part of the process of quantifying particle energy and gravity's mechanism will result in an attempt to define an energy unit that will be a bridge between my so called model and the units of measure related to joules and electron volts, but that is something that will have to be developed with the help of knowledgeable science enthusiasts who are willing to help me with it. So far, there are no such people at this forum any longer, lol. Yes, in my so called model, gravity is characterized by wave energy density, and the environment surrounding a massive object has high relative wave energy density that declines in accord with the inverse square law as the distance from the massive object increases. That means that the medium of space surrounding a massive object like a star will slow and bend the path of light more and more as the photons pass closer and closer to the object. Note to Prof.Layman: I'm not going to respond to your post in this thread but I will accept an invite to your thread on the subject where I will look forward to the discussion. (115) 16. ### Prof.Laymantotally internally reflectedRegistered Senior Member Messages: 982 You say you want it defined into something that would be a more legitimate theory, but then I try to help you but then you don't want it. But, then I think the ability to describe it in joules and electron volts is beyond both of us. I don't think your the only one that has had these kinds of ideas, and I think the book I just read, The Particle at the End of the Universe, was a direct attack on these types of ideas. (it was strange to find things in it that related to some post I have made in the past years ago) But, like I mentioned, I think it is the only way that these types of models could survive in quantum physics. Like you mention in your first post, you believe that the, "foundational waves are not the electromagnetic waves emitted by electrons". In quantum physics the waves created by electrons are photons, or vibrations in the electromagnetic field. The electrons then do not consist of photons when they are absorbed. But, in say preon particle theory that I mentioned, preons are theoretical particles that could make up electrons. So then like when you say, "the foundational waves are the waves that electrons, and all particles are composed of", then preons would be the particle that electrons are composed of. Do you see the connection here? You say it is word salad, and I am just trying to translate it. But, one thing I realized is that the Higgs Boson can decay into two photons. They then say that there has to be a "unknown" charged particle that then makes this transition. It is only there because Feynman Diagrams says that it has to be there. Then this is because that is the only way the Feynman Diagrams can have conservation of charge. I think the global symmetry of quantum field theory is broken here. Photons are no longer just vibrations in the electromagnetic field, and this particle added to the Feynman Diagrams then just acts like a goldstone boson that just corrects for the broken symmetry of quantum field theory, but they didn't think that it would be an actual particle. So if it wasn't a valid particle that they thought should exist, then it could mean that the photon is a disruption in the Higgs Field, or like you say the "foundational medium". So then if the photon is a vibration of the gravitational field, then particles would then have to consist then of photons. But, when they are viewed as virtual particles they no longer have the same mass, so then they can't be described as being exactly the particle in a particle. These virtual particles in a particle are then called preons. Then these preons would be stacks of different particle waves, that then would have a different interaction with the gravitational field, like electrons in cooper pairs that I mentioned in BSC Theory. They then become more massive, so then the particle waves of these pairs could then be a gravitational mechanism. 17. ### quantum_waveContemplating the "as yet" unknownValued Senior Member Messages: 6,626 Please Prof., don't try to help unless you have a better concept of what I am hypothesizing. You should ask me what I am talking about if you aren't sure, not tell me what I should be talking about. 18. ### Prof.Laymantotally internally reflectedRegistered Senior Member Messages: 982 It is starting to sound like you don't even care about what could be and what cannot be actually science. I thought it would be an interesting bit of news, I think I should become a science journalist. I thought everyone would want to know something about the progress of the Higgs Boson. I think it could determine if your model is right or completely wrong once and for all, or get brushed aside with just new particles and other technobabble. I think we are still a long way away from even being able to test such a model. 19. ### CheezleHab SoSlI' Quch!Registered Senior Member Messages: 745 I would like to dedicate this next performance to Prof.Layman and quantum_wave. Crank that Science! [video=youtube;NkGFsNnaKHI]http://www.youtube.com/watch?v=NkGFsNnaKHI[/video] 20. ### quantum_waveContemplating the "as yet" unknownValued Senior Member Messages: 6,626 Yeah! Crank that science. Not one of my students, I'm afraid, but it was fun for about a minute. I'm not claiming to be doing science, just layman hypothesizing for purposes of discussion and science learning. Its a hobby, you know. Take it for what you can find in it, or leave it. Disparage me if you get a charge out of it, or even better, help show me where my so called model is not internally consistent, or where it is inconsistent with scientific observations and data. Maybe you can point that part out to the moderators and have my threads sent elsewhere; suits me. The report button is at the bottom of the page, I think. Enjoy. 21. ### CheezleHab SoSlI' Quch!Registered Senior Member Messages: 745 No one can convince you that your ideas are wrong. I tried once to engage you and you always retreat into, "it is not science and so professionals aren't and shouldn't be interested." Or, "its just my hobby." Or similar attitudes. It is a very effective shield against real discussion. So I won't bother with telling you what is wrong with your ideas. Why should I? You are not really interested in other people's opinions or even the facts. The video was a statement of how I see your ideas. Boring and without any merit. I did not even make it through a minute of the video, and likewise I didn't make it all the way through your original post. And yes, you should never have posted this trash here in the Physics & Math section. Why would you do that? 22. ### Prof.Laymantotally internally reflectedRegistered Senior Member Messages: 982 I think this video more accurately depicts scientist trying to discover a science theory similar to quantum waves model that they already heard from him on the internet. [video=youtube;z5rRZdiu1UE]http://www.youtube.com/watch?v=z5rRZdiu1UE[/video] 23. ### quantum_waveContemplating the "as yet" unknownValued Senior Member Messages: 6,626 That is not true, I've been found wrong many times and have acknowledged it, and remedied it by improving my so called model. Engaging me and showing me something that is internally inconsistent in my word salad, are two different things. The "not even wrong" attack or approach is fine, but at some point even my "not even wrong" so called model must have something that stands out such that it is remarkable. Just point it out for me; that would be appreciated, acknowledged, and remedied, as I have been doing for years now. I know you have no interest, but I will tell you where that came from. I've been starting threads here since 2008 I think, and basically presenting ideas for discussion and learning. Many science professionals and well educated science enthusiasts who for some reason pay attention to these science forums, pointed out that what I was doing was not science. I didn't claim it was, but unless I say it is not science, I always get the "its not science" response. That is not true. Oh, you are so informed about my ideas. I should come to you when ever I have a question. Or you could show some of your science expertise, here in the Physics and Math forum, by saying what is wrong with the diagram, or the equation, or the so called mechanism, or any little thing that would show you are any better at Physics and Math than I am. Oh wait, you didn't read it. 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https://stats.stackexchange.com/questions/320596/variance-of-ols-estimator-of-theta-in-y-n-theta-x-n-eta-n-compared-to	# Variance of OLS estimator of $\theta$ in $y_n = \theta x_n + \eta_n$ compared to Cramer-Rao From Theodoridis' Machine Learning, problem 3.7: Derive the Cramer-Rao bound for the LS estimator, where the training data result from the model $$y_n = \theta x_n + \eta_n\text{, } \qquad n = 1, 2, \dots$$ where $x_n$ and $\eta_n$ are iid samples of a zero mean random variable, with variance $\sigma^2_x$, and a Gaussian one with zero mean and variance $\sigma^2_{\eta}$, respectively. Assume, also, that $x$ and $\eta$ are independent. Then, show that the LS estimator achieves the CR bound only asymptotically. After a lot of work, I have that the Cramer-Rao lower bound is $$\dfrac{1}{I(\theta)} = \dfrac{(\theta^2\sigma^2_x + \sigma^2_{\eta})^2}{2N\theta^2\sigma^4_{x}}$$ where $N$ is the sample size. The OLS estimator of $\theta$ is $$\hat{\theta} = \dfrac{\sum_{n=1}^{N}x_n y_n}{\sum_{n=1}^{N}x_n^2}\text{.}$$ How does one find the variance of this, given that BOTH $x_n$ and $y_n$ have variances? I don't like the answer at https://stats.stackexchange.com/a/105411/46427, since the formula $$\sigma^2_b = (X^{T}X)^{-1}\sigma^2_e$$ assumes that the values of $X$ are fixed and known; i.e., with no variance. Why is this so? Because since $$\hat{\boldsymbol\beta} = (X^{T}X)^{-1}X^{T}\mathbf{y}$$ we obtain $$\mathrm{Var}\left(\hat{\boldsymbol\beta}\right) = (X^{T}X)^{-1}X^{T}\mathrm{Var}\left(\mathbf{y}\right)X(X^{T}X)^{-1}=\sigma^2_e(X^{T}X)^{-1}$$ if we assume that $X$ is a constant, known matrix - which is not the case here. • Since it is not given that $X_i$'s are Normal, Fisher information in a single observation $(X,Y)$ can be calculated from chain rule as $I_{X,Y}(\theta)=I_{Y\mid X}(\theta)+I_X(\theta)=I_{Y\mid X}(\theta)=\frac{\sigma^2_x}{\sigma^2_\eta}$. If $n$ is the sample size, it can be argued from Slutsky's theorem that $\sqrt n(\hat\theta-\theta) \stackrel{d}\longrightarrow N\left(0,\frac{\sigma^2_\eta}{\sigma^2_x}\right)$, so that variance of $\hat\theta$ for large $n$ is $\frac{\sigma^2_\eta}{n\sigma^2_x}$= Cramer-Rao bound $\frac1{nI_{X,Y}(\theta)}$. Apr 15, 2020 at 15:25 To recap, we have $X \sim \mathcal N(0, \sigma^2_x I)$ and $Y\|X \sim \mathcal N(\theta X, \sigma^2_\eta I)$. First, let's confirm the expected value of $\hat \theta$: $$E(\hat \theta) = E_X\left(E_{Y\|X}\left[\frac{X^TY}{X^TX} \big\vert X\right]\right) = E_X\left(\frac{X^TE_{Y\|X}(Y\|X)}{X^TX}\right)$$ $$= E_X\left(\theta \frac{X^TX}{X^TX}\right) = \theta.$$ This confirms that $\theta$ is still unbiased. Now for the variance, again using the law of total expectation, we have $$E(\hat \theta^2) = E_X\left[E_{Y\|X}\left(\frac{(X^TY)^2}{(X^TX)^2} \big\vert X\right)\right]$$ $$= E_X\left[\frac{1}{(X^TX)^2} X^TE_{Y\|X}\left(YY^T\big\vert X\right)X\right].$$ $Var(Y\|X) = \sigma^2_\eta I = E(YY^T\|X) - E(Y\|X)E(Y\|X)^T$ so $E(YY^T\|X) = \sigma^2_\eta I + \theta^2 XX^T$. This means $$E(\hat \theta^2) = E_X\left[\frac{1}{(X^TX)^2} X^T\left(\sigma^2_\eta I + \theta^2 XX^T\right)X\right]$$ $$= E_X\left[\frac{\sigma_\eta^2}{X^TX} + \theta^2\right].$$ This means $$Var(\hat \theta) = E(\hat \theta^2) - E(\hat \theta)^2 = \sigma_\eta^2 E_X\left[\frac{1}{X^TX}\right].$$ $X \sim \mathcal N(0, \sigma^2_x I) \implies \frac{1}{\sigma^2_x}X^TX \sim \chi^2_n$ so $\frac{\sigma^2_x}{X^TX}$ follows an inverse chi-squared distribution. This means $$E\left(\frac{\sigma^2_x}{X^TX}\right) = \frac{1}{n-2}$$ $$\implies Var(\hat \theta) = \frac{\sigma_\eta^2 }{\sigma^2_x} E_X\left[\frac{\sigma^2_x}{X^TX}\right] = \frac{\sigma_\eta^2}{\sigma^2_x(n-2)}.$$ Confirming this by simulation: s2_eta <- 0.29 s2_x <- 1.55 n <- 100 tt <- 2.1 # theta set.seed(123) nsim <- 1000 t.hats <- numeric(nsim) for(i in 1:nsim) { etas <- rnorm(n, 0, sqrt(s2_eta)) xs <- rnorm(n, 0, sqrt(s2_x)) y <- tt * xs + etas t.hats[i] <- sum(y * xs) / sum(xs * xs) } var(t.hats) # 0.001922613 s2_eta / ((s2_x) * (n - 2)) # 0.001909151 • I have one quick question for you... thank for you your proof, by the way. In the last part of the exercise, it states "show that the LS estimator achieves the CR bound only asymptotically." Is this just the fact that both variances tend to $0$ as the sample size $\to \infty$? Dec 28, 2017 at 2:18 • @Clarinetist assuming the CR bound is correct that must be the case, although that language does seem weird – jld Dec 28, 2017 at 2:27 • I agree, the language is weird. But I'm not sure how else to interpret it. From checking my algebra on WolframAlpha numerous times, I'm not sure what else it would be. Oh well. Nevertheless, thanks! Dec 28, 2017 at 2:28 • @Clarinetist Upon further reflection $\hat \theta \to_p \theta$ which is constant so in retrospect the asymptotic variance has to be $0$. Anyway, glad this was helpful! Quick proof: by Chebyshev $P\left(\|\hat \theta_n - \theta\| \geq \varepsilon\right) \leq \frac{\sigma_\eta^2}{\varepsilon^2 \sigma^2_x(n-2)} \to 0$ as $n \to \infty$ – jld Dec 28, 2017 at 2:30 • I think it means for finite N, it is possible that variance of the LSE can be smaller than the CR lower bound. Dec 28, 2017 at 2:40	2022-05-25 20:20:59	{"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": 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": 0.9144761562347412, "perplexity": 357.815506267948}, "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/1652662593428.63/warc/CC-MAIN-20220525182604-20220525212604-00636.warc.gz"}
http://docs.scipy.org/doc/scipy-dev/reference/generated/scipy.signal.fftconvolve.html	# scipy.signal.fftconvolve¶ scipy.signal.fftconvolve(in1, in2, mode='full')[source] Convolve two N-dimensional arrays using FFT. Convolve in1 and in2 using the fast Fourier transform method, with the output size determined by the mode argument. This is generally much faster than convolve for large arrays (n > ~500), but can be slower when only a few output values are needed, and can only output float arrays (int or object array inputs will be cast to float). Parameters: in1 : array_like First input. in2 : array_like Second input. Should have the same number of dimensions as in1; if sizes of in1 and in2 are not equal then in1 has to be the larger array. mode : str {‘full’, ‘valid’, ‘same’}, optional A string indicating the size of the output: full The output is the full discrete linear convolution of the inputs. (Default) valid The output consists only of those elements that do not rely on the zero-padding. same The output is the same size as in1, centered with respect to the ‘full’ output. out : array An N-dimensional array containing a subset of the discrete linear convolution of in1 with in2. Examples Autocorrelation of white noise is an impulse. (This is at least 100 times as fast as convolve.) >>> from scipy import signal >>> sig = np.random.randn(1000) >>> autocorr = signal.fftconvolve(sig, sig[::-1], mode='full') >>> import matplotlib.pyplot as plt >>> fig, (ax_orig, ax_mag) = plt.subplots(2, 1) >>> ax_orig.plot(sig) >>> ax_orig.set_title('White noise') >>> ax_mag.plot(np.arange(-len(sig)+1,len(sig)), autocorr) >>> ax_mag.set_title('Autocorrelation') >>> fig.tight_layout() >>> fig.show() Gaussian blur implemented using FFT convolution. Notice the dark borders around the image, due to the zero-padding beyond its boundaries. The convolve2d function allows for other types of image boundaries, but is far slower. >>> from scipy import misc >>> face = misc.face(gray=True) >>> kernel = np.outer(signal.gaussian(70, 8), signal.gaussian(70, 8)) >>> blurred = signal.fftconvolve(face, kernel, mode='same') >>> fig, (ax_orig, ax_kernel, ax_blurred) = plt.subplots(1, 3) >>> ax_orig.imshow(face, cmap='gray') >>> ax_orig.set_title('Original') >>> ax_orig.set_axis_off() >>> ax_kernel.imshow(kernel, cmap='gray') >>> ax_kernel.set_title('Gaussian kernel') >>> ax_kernel.set_axis_off() >>> ax_blurred.imshow(blurred, cmap='gray') >>> ax_blurred.set_title('Blurred') >>> ax_blurred.set_axis_off() >>> fig.show() #### Previous topic scipy.signal.correlate #### Next topic scipy.signal.convolve2d	2015-08-30 18:02:28	{"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": 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": 0.2417701631784439, "perplexity": 9295.583885229615}, "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-2015-35/segments/1440644065330.34/warc/CC-MAIN-20150827025425-00186-ip-10-171-96-226.ec2.internal.warc.gz"}
http://kleine.mat.uniroma3.it/mp_arc-bin/mpa?yn=95-188	95-188 Anton Bovier, V\'eronique Gayrard An almost sure large deviation principle for the Hopfield model (326K, PS) Apr 3, 95 Abstract , Paper (src), View paper (auto. generated ps), Index of related papers Abstract. We prove a large deviation principle for the finite dimensional marginals of the Gibbs distribution of the macroscopic `overlap'-parameters in the Hopfield model in the case where the number of random patterns, \$M\$, as a function of the system size \$N\$ satisfies \$\limsup M(N)/N=0\$. In this case the rate function (or free energy as a function of the overlap parameters) is independent of the disorder for almost all realization of the patterns and given by an explicit variational formula. Files: 95-188.src( desc , 95-188.ps )	2018-04-24 11:04:05	{"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": 0.8978791236877441, "perplexity": 1506.5214902436323}, "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-2018-17/segments/1524125946597.90/warc/CC-MAIN-20180424100356-20180424120356-00223.warc.gz"}
https://www.groundai.com/project/delay-constrained-scheduling-over-fading-channels-optimal-policies-for-monomial-energy-cost-functions/	Delay Constrained Scheduling over Fading Channels: Optimal Policies for Monomial Energy-Cost FunctionsThe work of J. Lee is supported by a Motorola Partnership in Research Grant. # Delay Constrained Scheduling over Fading Channels: Optimal Policies for Monomial Energy-Cost FunctionsThe work of J. Lee is supported by a Motorola Partnership in Research Grant. ## Abstract A point-to-point discrete-time scheduling problem of transmitting information bits within hard delay deadline slots is considered assuming that the underlying energy-bit cost function is a convex monomial. The scheduling objective is to minimize the expected energy expenditure while satisfying the deadline constraint based on information about the unserved bits, channel state/statistics, and the remaining time slots to the deadline. At each time slot, the scheduling decision is made without knowledge of future channel state, and thus there is a tension between serving many bits when the current channel is good versus leaving too many bits for the deadline. Under the assumption that no other packet is scheduled concurrently and no outage is allowed, we derive the optimal scheduling policy. Furthermore, we also investigate the dual problem of maximizing the number of transmitted bits over time slots when subject to an energy constraint. ## 1Introduction An opportunistic scheduling policy that adapts to the time-varying behavior of a wireless channel can achieve energy-efficient communication on the average in a long-term perspective. However, this opportunistic approach may not be appropriate for short-term deadline constrained traffic. This paper considers scheduling a packet over a finite time horizon while efficiently adapting to wireless (fading) channel variations and taking care of the deadline constraint. Our primal problem setting is the minimization of energy expenditure subject to a hard deadline constraint (i.e., a packet of bits must be scheduled within finite discrete-time slots) assuming that the scheduler has causal knowledge of the channel state information (CSI). Causal CSI means that the scheduler knows the past and current CSI perfectly, but does not know future CSI. The scheduler is then required to make a decision at each time slot given the number of unserved bits, the number of slots left before the deadline, and causal CSI, in order to minimize the total energy expenditure. At each time slot, the scheduler deals with the tension between serving more bits when the channel is good and leaving too many bits to the end. Likewise, we consider the dual (scheduling over a finite time-horizon) problem of maximizing the transmitted bits subject to a finite energy constraint. We also briefly discuss scheduling problems when the CSI is available non-causally. We assume that no other packet is scheduled simultaneously and the hard delay deadline must be met (i.e., no outage is allowed). These finite-time horizon scheduling problems can be applicable to regularly arriving packets with hard delay deadlines, e.g., VoIP and video streaming. Delay constrained scheduling over fading channel has been studied for various traffic models and delay constraints. Uysal-Biyikoglu and El Gamel [1] considered scheduling random packet arrivals over a fading channel and thus adapt (transmit power/rate) to both the channel state and queue state, and generally try to minimize average delay. Many references can be found in [1]. Most cases do not admit analytical closed-form solution for causal (or online) scheduling. Instead, they proposed causal algorithms with heuristic modifications from non-causal (offline) policies. References [2] take a slightly different perspective: single packet scheduling (no queue) with a hard delay deadline rather than an average delay constraint. The subject of this paper is the single-packet scheduling problem of [2] specialized to the case where the required energy to transmit bits under channel state is governed by a convex monomial function, i.e., , where denotes the monomial order. The biggest advantage of using this monomial cost function is that it yields closed-form solutions in various scenarios, unlike the Shannon-cost function setting described in [4]. As a result, it provides intuition on the interplay between the monomial order, delay deadline, and the channel states so that it ultimately suggests general ideas for a more general energy-cost function. Although the monomial cost does not hold for operating at capacity in an AWGN channel, according to Zafer and Modiano [5] and their reference [6], there is a practical modulation scheme that exhibits an energy-bit relation that can be well approximated by a monomial. Actually, Zafer and Modinano [5] considered the same problem but for a continuous-time Markov process channel in continuous-time scheduling, i.e., the scheduler can transmit at any time instant rather than discrete slotted time. Although they provided a solution in the form of a set of differential equations, it is not possible to give a closed-form solution. On the other hand, we are able to derive a closed-form description of the optimal scheduler for the simpler block fading model (note that the continuous model is somewhat incompatible with block fading). In this paper, we derive optimal scheduling policies for delay-constrained scheduling when the energy-bit cost is a convex monomial function. We also investigate the dual problem of maximizing the number of bits to transmit with a finite energy budget over a finite time horizon. In all cases, we are able to find analytical expressions that are functions of the queue state variables (energy state for the dual problem), current channel state and a quantity related to the fading distribution. The resulting optimal schedulers determine the ratio of the number of bits to be allocated in the current slot to the deferred bits. For example, the optimal scheduling ratio of the number of bits to serve (from the remaining bits) at slot ( denotes the number of remaining slots to the deadline) to the number of bits to defer for the primal energy minimization problem is given by where is the order of monomial cost function, denotes the current channel state, and denotes a statistical quantity determined by the channel distribution and the number of remaining slots . It will be shown later that is increasing with respect to . If is small, . However, as term increases, gets more affected by the channel state . This suggests that the scheduler behaves very opportunistically when the deadline is far away ( large) but less so as the deadline approaches, since is an increasing function of . ## 2Primal Problem: Energy Minimization We consider the scheduling of a packet of bits in discrete time slots over a wireless channel as illustrated in Figure 1. The scheduler determines the number of bits to allocate at each time slot using the fading realization/statistics to minimize the total transmit energy while satisfying the delay deadline constraint. To make the scheduling problem tractable, we assume that no other packets are to be scheduled simultaneously and that no outage is allowed. Throughout the paper, we use the following notations: The channel states are assumed to be independently and identically distributed (i.i.d.). If the scheduler has only causal knowledge of the channel state (i.e., at slot , the scheduler knows but does not know ), we refer to this as causal scheduling. If the scheduler has non-causal knowledge of the channel state in advance (i.e., at slot , the scheduler knows ), we refer to it as non-causal scheduling. This paper mainly deals with causal scheduling problems. In this paper, we assume that the energy expenditure is inversely proportional1 to the channel state and is related to the transmitted bits by a monomial function: where denotes the order of monomial. If , the resulting optimization becomes a linear program and thus a “one-shot” policy is optimal [7]. We assume that (to be convex) and ( is not necessarily an integer), where denotes the real number set. A practical modulation scheme that exhibits a monomial energy-cost behavior was illustrated in [5], where the monomial order is . A scheduler is a sequence of functions with . For causal scheduling, depends only on the current channel state and not on the past and future states because of the i.i.d. assumption and causality2. The optimal scheduler is determined by minimizing the total expected energy cost: where denotes the expectation operator. ## 3Causal Energy Minimization Scheduling As done in [2][4], a sequential formulation of the optimal causal scheduling of can be established by introducing a state variable as in standard dynamic programming [8]. As defined in Section 2, denotes the remaining bits that summarizes the bit allocation up until the previous time step. At time step , are unknown but is known. Thus, the optimization becomes: With , we obtain the following DP: where the first term denotes the current energy cost and the second term denotes the cost-to-go function, which is the expected future energy cost (because future channel states are unknown, only expectations can be considered) to serve bits in slots if the optimal control policy is used at each future step. Thus, the optimal bit allocation is determined by balancing the current energy cost and the expected future energy cost. Because of the hard delay constraint, all the unserved bits must be served at regardless of the channel condition, i.e., and thus the resulting energy cost is given by . This dynamic optimization can be solved: We use mathematical induction to find and . At , and are true by definition. If we suppose that is true for , the optimization becomes whose solution is obtained by differentiating the objective and setting to zero to result in . Substituting into and then taking expectation with respect to , we obtain . Therefore, the result follows by induction. The scheduling function can be intuitively explained in the following way. The ratio of the number of allocated bits to the number of deferred bits is equal to the ratio of to , i.e., where . As expected, the optimal scheduler is opportunistic in that the number of transmitted bits are proportional to the channel quality. Furthermore, the thresholds are increasing in (shown later) which implies that the scheduler is more selective when the delay deadline is far away (large ). When the deadline is far away, the scheduler transmits a large fraction of the unserved bits only when the channel state is very good; because many slots remain until the deadline, there is still a good chance of seeing a very good channel state. On the other hand, as the deadline approaches (small ) the scheduler is still opportunistic but must become less selective because only a few opportunities for good channel states remain before the deadline is reached. Figure 2 illustrates and for a truncated exponential distribution. As can be seen in Figure 2a, increases with respect to and this can be shown analytically: where the inequality is due to . This shows the delay-limited opportunistic behavior mentioned before. From , the value denotes the expected energy cost for a unit bit, i.e., . Thus, , as illustrated in Figure 2b, shows how much the expected energy unit cost (for transmitting one bit) can be reduced as the time span increases. Another interesting fact is that the policy utilizes3 all the time slots. This is because both and are always positive for typical fading distributions. For the Shannon cost function problem [4], however, there exist time slots that are not utilized depending on the values of and . This does not admit an analytical solution because the associated cost-to-go function takes a complicated form. ### 3.1Special Cases In this subsection, we examine the optimal policy for two values of : and . By substituting in and , we have where Thus, the allocated bits and the deferred bits have the same ratio with and . #### Infinite Order Cost (n=∞) We examine the limiting behavior of the scheduling policy as . First, we observe that This can be shown by the induction. When , holds trivially. If we suppose holds for , then where the last equality is due and denotes the “effective upper bound” of (see Chap. 6 in [9] for mathematical technicality). Hence, the induction follows. Figure 2a illustrates the values of for the truncated exponential variable. This shows that is increasing linearly with respect to for large , which agrees with Lemma ?. With the limit in Lemma ?, we can immediately reach the simplified scheduling policy summarized below: That is, when the order of monomial cost function tends to infinity, scheduling equal number of bits at every slot regardless of the channel state becomes the optimal policy. Note that we considered only monomial orders in the derivation, as when , the optimal policy is the one-shot policy [7], which completely depends on the channel state. From these two extreme cases, we can deduce that the effect of channel state on the scheduling function decreases as the order of monomial cost function increases, or in other words the optimal scheduler becomes less opportunistic as the monomial order increases. ## 4Dual Problem: Rate Maximization Thus far, we have considered problems of minimizing energy expenditure to transmit fixed information bits in a finite time horizon . It is of interest to consider the dual of this, i.e., maximizing the number of bits transmitted with a finite energy over a finite time horizon . We refer to this as the dual scheduling problem, while referring to the original problem as the primal scheduling problem. Negi and Cioffi [3] considered this dual problem for the Shannon energy-bit cost function and provided solutions in DP, but not in closed form. In this work, we investigate this dual scheduling problem and obtain the optimal closed-form solution for monomial cost functions. Since the energy-bit function is assumed to be , the associated bit-energy cost function is given by inverting: Then the dual problem is given by To derive a DP for causal dual scheduling, we introduce a state variable that denotes the remaining energy at slot . Thus, the optimization can be formulated as where denotes the cost-to-go function for the dual scheduling problem. This dynamic optimization can be solved similar to the primal problem and its optimal solution is summarized as follows: The optimal energy scheduler has very similar interpretation with the optimal bit scheduler from their scheduling formulations. That is, the ratio of the amount of energy to schedule to the amount of energy to defer is equal to the ratio of to , and thus, the similar delay-limited opportunistic scheduling interpretation can be applied. Notice that the quantities and are different. ## 5Non-Causal Scheduling This section briefly considers the case where the scheduler has knowledge of the channel states non-causally in advance, i.e., are known at . ### 5.1Energy Minimization Scheduling In this non-causal setting, the optimization is simply given by subject to and for all . This is a convex optimization and can be solved as: The standard Lagrangian method [10] yields the solution: If we express this solution with the queue state variable , we obtain the result. The scheduling policy can be interpreted with the ratio argument as with the causal cases, i.e., ### 5.2Rate Maximization Scheduling Similarly we can fomulate the non-causal rate maximition as subject to and for all . Like , we can also observe that and thus, we obtain This implies that the optimal bit distribution ratio during the slots for the primal problem is identical to the energy distribution ratio for the dual problem. ## 6Conclusion We have investigated the problem of bit/energy scheduling over a finite time duration assuming that the energy-bit cost function is a monomial. In both the primal (minimizing energy expenditure subject to a bit constraint) scheduling and the dual (maximizing bit transmission under an energy constraint) scheduling problem, we derived closed-form scheduling functions. The optimal bit/energy allocations are determined by the ratio of and a channel statistical quantity. From the monotonicity of this statistical quantity, we interpreted that the optimal scheduler behaves more opportunistically in the initial time steps and less so as the deadline approaches. ### Footnotes 1. The dependence is due to the fact that the received energy is the product of the transmitted energy and the channel state . Note, however, that any other decreasing function of could be considered by simply performing a change of variable on . 2. The i.i.d. assumption makes us ignore the past CSI and the causality does not allow to exploit the future CSI . As a result, the decision at each time slot should be made based only on the current CSI , i.e., instead of . 3. A time slot is called utilized if a positive bit is scheduled, i.e., . ### References 1. E. Uysal-Biyikoglu and A. E. Gamel, “On adaptive transmission for energy efficient in wireless data networks,” IEEE Trans. Inform. Theory, vol. 50, 2004. 2. A. Fu, E. Modiano, and J. N. Tsitsiklis, “Optimal transmission scheduling over a fading channel with energy and deadline constraints,” IEEE Trans. Wireless Commun., vol. 5, no. 3, pp. 630–641, Mar. 2006. 3. R. Negi and J. M. Cioffi, “Delay-constrained capacity with causal feedback,” IEEE Trans. Inform. Theory, vol. 48, no. 9, pp. 2478–2494, Sep. 2002. 4. J. Lee and N. Jindal, “Energy-efficient scheduling of delay constrained traffic over fading channels,” to appear: IEEE Trans. Wireless Communu. (preprint available at http://arxiv.org/abs/0807.3332). 5. M. Zafer and E. Modiano, “Delay constrained energy efficient data transmission over a wireless fading channel,” in Workshop on Inf. Theory and Appl., La Jolla, CA, Jan./Feb. 2007, pp. 289–298. 6. M. J. Neely, E. Modiano, and C. E. Rohrs, “Dynamic power allocation and routing for time varying wireless networks,” in Proc. IEEE INFOCOM, 2003, pp. 745–755. 7. J. Lee and N. Jindal, “Asymptotic optimal energy-efficient delay constrained schedulers over fading channels,” in preparation. 8. D. P. Bertsekas, Dynamic Programming and Optimal Control, 3rd ed. 1em plus 0.5em minus 0.4emMass.: Athena Scientific, 2005, vol. 1. 9. G. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed. 1em plus 0.5em minus 0.4emCambridge, 2001. 10. S. Boyd and L. Vandenberghe, Convex Optimization.1em plus 0.5em minus 0.4emCambridge, UK: Cambridge Univ. Press, 2004. You are adding the first comment! How to quickly get a good reply: • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made. • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements. • Your comment should inspire ideas to flow and help the author improves the paper. The better we are at sharing our knowledge with each other, the faster we move forward. The feedback must be of minimum 40 characters and the title a minimum of 5 characters	2019-05-24 20:48:08	{"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": 0.8193641304969788, "perplexity": 663.4649214188096}, "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/1558232257767.49/warc/CC-MAIN-20190524204559-20190524230559-00194.warc.gz"}
https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula	# Möbius inversion formula In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.[1] A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra. ## Statement of the formula The classic version states that if g and f are arithmetic functions satisfying ${\displaystyle g(n)=\sum _{d\mid n}f(d)\quad {\text{for every integer }}n\geq 1}$ then ${\displaystyle f(n)=\sum _{d\mid n}\mu (d)g\left({\frac {n}{d}}\right)\quad {\text{for every integer }}n\geq 1}$ where μ is the Möbius function and the sums extend over all positive divisors d of n (indicated by ${\displaystyle d\mid n}$ in the above formulae). In effect, the original f(n) can be determined given g(n) by using the inversion formula. The two sequences are said to be Möbius transforms of each other. The formula is also correct if f and g are functions from the positive integers into some abelian group (viewed as a Z-module). In the language of Dirichlet convolutions, the first formula may be written as ${\displaystyle g={\mathit {1}}f}$ where denotes the Dirichlet convolution, and 1 is the constant function 1(n) = 1. The second formula is then written as ${\displaystyle f=\mu g.}$ Many specific examples are given in the article on multiplicative functions. The theorem follows because is (commutative and) associative, and 1μ = ε, where ε is the identity function for the Dirichlet convolution, taking values ε(1) = 1, ε(n) = 0 for all n > 1. Thus ${\displaystyle \mu g=\mu ({\mathit {1}}f)=(\mu {\mathit {1}})f=\varepsilon f=f}$. There is a product version of the summation-based Möbius inversion formula stated above: ${\displaystyle g(n)=\prod _{d\|n}f(d)\iff f(n)=\prod _{d\|n}g\left({\frac {n}{d}}\right)^{\mu (d)},\forall n\geq 1.}$ ## Series relations Let ${\displaystyle a_{n}=\sum _{d\mid n}b_{d}}$ so that ${\displaystyle b_{n}=\sum _{d\mid n}\mu \left({\frac {n}{d}}\right)a_{d}}$ is its transform. The transforms are related by means of series: the Lambert series ${\displaystyle \sum _{n=1}^{\infty }a_{n}x^{n}=\sum _{n=1}^{\infty }b_{n}{\frac {x^{n}}{1-x^{n}}}}$ and the Dirichlet series: ${\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}=\zeta (s)\sum _{n=1}^{\infty }{\frac {b_{n}}{n^{s}}}}$ where ζ(s) is the Riemann zeta function. ## Repeated transformations Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation. For example, if one starts with Euler's totient function φ, and repeatedly applies the transformation process, one obtains: 1. φ the totient function 2. φ1 = I, where I(n) = n is the identity function 3. I1 = σ1 = σ, the divisor function If the starting function is the Möbius function itself, the list of functions is: 1. μ, the Möbius function 2. μ1 = ε where ${\displaystyle \varepsilon (n)={\begin{cases}1,&{\text{if }}n=1\\0,&{\text{if }}n>1\end{cases}}}$ is the unit function 3. ε1 = 1, the constant function 4. 11 = σ0 = d = τ, where d = τ is the number of divisors of n, (see divisor function). Both of these lists of functions extend infinitely in both directions. The Möbius inversion formula enables these lists to be traversed backwards. As an example the sequence starting with φ is: ${\displaystyle f_{n}={\begin{cases}\underbrace {\mu \ldots \mu } _{-n{\text{ factors}}}\varphi &{\text{if }}n<0\\[8px]\varphi &{\text{if }}n=0\\[8px]\varphi \underbrace {{\mathit {1}}\ldots {\mathit {1}}} _{n{\text{ factors}}}&{\text{if }}n>0\end{cases}}}$ The generated sequences can perhaps be more easily understood by considering the corresponding Dirichlet series: each repeated application of the transform corresponds to multiplication by the Riemann zeta function. ## Generalizations A related inversion formula more useful in combinatorics is as follows: suppose F(x) and G(x) are complex-valued functions defined on the interval [1, ∞) such that ${\displaystyle G(x)=\sum _{1\leq n\leq x}F\left({\frac {x}{n}}\right)\quad {\mbox{ for all }}x\geq 1}$ then ${\displaystyle F(x)=\sum _{1\leq n\leq x}\mu (n)G\left({\frac {x}{n}}\right)\quad {\mbox{ for all }}x\geq 1.}$ Here the sums extend over all positive integers n which are less than or equal to x. This in turn is a special case of a more general form. If α(n) is an arithmetic function possessing a Dirichlet inverse α−1(n), then if one defines ${\displaystyle G(x)=\sum _{1\leq n\leq x}\alpha (n)F\left({\frac {x}{n}}\right)\quad {\mbox{ for all }}x\geq 1}$ then ${\displaystyle F(x)=\sum _{1\leq n\leq x}\alpha ^{-1}(n)G\left({\frac {x}{n}}\right)\quad {\mbox{ for all }}x\geq 1.}$ The previous formula arises in the special case of the constant function α(n) = 1, whose Dirichlet inverse is α−1(n) = μ(n). A particular application of the first of these extensions arises if we have (complex-valued) functions f(n) and g(n) defined on the positive integers, with ${\displaystyle g(n)=\sum _{1\leq m\leq n}f\left(\left\lfloor {\frac {n}{m}}\right\rfloor \right)\quad {\mbox{ for all }}n\geq 1.}$ By defining F(x) = f(⌊x⌋) and G(x) = g(⌊x⌋), we deduce that ${\displaystyle f(n)=\sum _{1\leq m\leq n}\mu (m)g\left(\left\lfloor {\frac {n}{m}}\right\rfloor \right)\quad {\mbox{ for all }}n\geq 1.}$ A simple example of the use of this formula is counting the number of reduced fractions 0 < a/b < 1, where a and b are coprime and bn. If we let f(n) be this number, then g(n) is the total number of fractions 0 < a/b < 1 with bn, where a and b are not necessarily coprime. (This is because every fraction a/b with gcd(a,b) = d and bn can be reduced to the fraction a/d/b/d with b/dn/d, and vice versa.) Here it is straightforward to determine g(n) = n(n − 1)/2, but f(n) is harder to compute. Another inversion formula is (where we assume that the series involved are absolutely convergent): ${\displaystyle g(x)=\sum _{m=1}^{\infty }{\frac {f(mx)}{m^{s}}}\quad {\mbox{ for all }}x\geq 1\quad \Longleftrightarrow \quad f(x)=\sum _{m=1}^{\infty }\mu (m){\frac {g(mx)}{m^{s}}}\quad {\mbox{ for all }}x\geq 1.}$ As above, this generalises to the case where α(n) is an arithmetic function possessing a Dirichlet inverse α−1(n): ${\displaystyle g(x)=\sum _{m=1}^{\infty }\alpha (m){\frac {f(mx)}{m^{s}}}\quad {\mbox{ for all }}x\geq 1\quad \Longleftrightarrow \quad f(x)=\sum _{m=1}^{\infty }\alpha ^{-1}(m){\frac {g(mx)}{m^{s}}}\quad {\mbox{ for all }}x\geq 1.}$ For example, there is a well known proof relating the Riemann zeta function to the prime zeta function that uses the series-based form of Möbius inversion in the previous equation when ${\displaystyle s=1}$. Namely, by the Euler product representation of ${\displaystyle \zeta (s)}$ for ${\displaystyle \Re (s)>1}$ ${\displaystyle \log \zeta (s)=-\sum _{p\mathrm {\ prime} }\log \left(1-{\frac {1}{p^{s}}}\right)=\sum _{k\geq 1}{\frac {P(ks)}{k}}\iff P(s)=\sum _{k\geq 1}{\frac {\mu (k)}{k}}\log \zeta (ks),\Re (s)>1.}$ These identities for alternate forms of Möbius inversion are found in.[2] A more general theory of Möbius inversion formulas partially cited in the next section on incidence algebras is constructed by Rota in.[3] ## Multiplicative notation As Möbius inversion applies to any abelian group, it makes no difference whether the group operation is written as addition or as multiplication. This gives rise to the following notational variant of the inversion formula: ${\displaystyle {\mbox{if }}F(n)=\prod _{d\|n}f(d),{\mbox{ then }}f(n)=\prod _{d\|n}F\left({\frac {n}{d}}\right)^{\mu (d)}.}$ ## Proofs of generalizations The first generalization can be proved as follows. We use Iverson's convention that [condition] is the indicator function of the condition, being 1 if the condition is true and 0 if false. We use the result that ${\displaystyle \sum _{d\|n}\mu (d)=\varepsilon (n),}$ that is, ${\displaystyle 1*\mu =\varepsilon }$, where ${\displaystyle \varepsilon }$ is the unit function. We have the following: {\displaystyle {\begin{aligned}\sum _{1\leq n\leq x}\mu (n)g\left({\frac {x}{n}}\right)&=\sum _{1\leq n\leq x}\mu (n)\sum _{1\leq m\leq {\frac {x}{n}}}f\left({\frac {x}{mn}}\right)\\&=\sum _{1\leq n\leq x}\mu (n)\sum _{1\leq m\leq {\frac {x}{n}}}\sum _{1\leq r\leq x}[r=mn]f\left({\frac {x}{r}}\right)\\&=\sum _{1\leq r\leq x}f\left({\frac {x}{r}}\right)\sum _{1\leq n\leq x}\mu (n)\sum _{1\leq m\leq {\frac {x}{n}}}\left[m={\frac {r}{n}}\right]\qquad {\text{rearranging the summation order}}\\&=\sum _{1\leq r\leq x}f\left({\frac {x}{r}}\right)\sum _{n\|r}\mu (n)\\&=\sum _{1\leq r\leq x}f\left({\frac {x}{r}}\right)\varepsilon (r)\\&=f(x)\qquad {\text{since }}\varepsilon (r)=0{\text{ except when }}r=1\end{aligned}}} The proof in the more general case where α(n) replaces 1 is essentially identical, as is the second generalisation. ## On posets For a poset P, a set endowed with a partial order relation ${\displaystyle \leq }$, define the Möbius function ${\displaystyle \mu }$ of P recursively by ${\displaystyle \mu (s,s)=1{\text{ for }}s\in P,\qquad \mu (s,u)=-\sum _{s\leq t (Here one assumes the summations are finite.) Then for ${\displaystyle f,g:P\to K}$, where K is a commutative ring, we have ${\displaystyle g(t)=\sum _{s\leq t}f(s)\qquad {\text{ for all }}t\in P}$ if and only if ${\displaystyle f(t)=\sum _{s\leq t}g(s)\mu (s,t)\qquad {\text{ for all }}t\in P.}$ (See Stanley's Enumerative Combinatorics, Vol 1, Section 3.7.) ## Contributions of Weisner, Hall, and Rota The statement of the general Möbius inversion formula [for partially ordered sets] was first given independently by Weisner (1935) and Philip Hall (1936); both authors were motivated by group theory problems. Neither author seems to have been aware of the combinatorial implications of his work and neither developed the theory of Möbius functions. In a fundamental paper on Möbius functions, Rota showed the importance of this theory in combinatorial mathematics and gave a deep treatment of it. He noted the relation between such topics as inclusion-exclusion, classical number theoretic Möbius inversion, coloring problems and flows in networks. Since then, under the strong influence of Rota, the theory of Möbius inversion and related topics has become an active area of combinatorics.[4] ## Notes 1. ^ Möbius 1832, pp. 105–123 2. ^ NIST Handbook of Mathematical Functions, Section 27.5. 3. ^ [On the foundations of combinatorial theory, I. Theory of Möbius Functions\|https://link.springer.com/content/pdf/10.1007/BF00531932.pdf] 4. ^ Bender & Goldman 1975, pp. 789–803 ## References • Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 • Bender, Edward A.; Goldman, J. R. (1975), "On the applications of Möbius inversion in combinatorial analysis", Amer. Math. Monthly, 82 (8): 789–803, doi:10.2307/2319793, JSTOR 2319793 • Ireland, K.; Rosen, M. (2010), A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics (Book 84) (2nd ed.), Springer-Verlag, ISBN 978-1-4419-3094-1 • Kung, Joseph P.S. (2001) [1994], "Möbius inversion", Encyclopedia of Mathematics, EMS Press • Möbius, A. F. (1832), "Über eine besondere Art von Umkehrung der Reihen.", Journal für die reine und angewandte Mathematik, 9: 105–123 • Stanley, Richard P. (1997), Enumerative Combinatorics, vol. 1, Cambridge University Press, ISBN 0-521-55309-1 • Stanley, Richard P. (1999), Enumerative Combinatorics, vol. 2, Cambridge University Press, ISBN 0-521-56069-1	2022-12-05 22:33:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 36, "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": 0.959649384021759, "perplexity": 1163.783652448591}, "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-2022-49/segments/1669446711045.18/warc/CC-MAIN-20221205200634-20221205230634-00473.warc.gz"}
https://proofwiki.org/wiki/Mathematician:Mathematicians/Sorted_By_Nation/Netherlands	Mathematician:Mathematicians/Sorted By Nation/Netherlands For more comprehensive information on the lives and works of mathematicians through the ages, see the MacTutor History of Mathematics archive, created by John J. O'Connor and Edmund F. Robertson. The army of those who have made at least one definite contribution to mathematics as we know it soon becomes a mob as we look back over history; 6,000 or 8,000 names press forward for some word from us to preserve them from oblivion, and once the bolder leaders have been recognised it becomes largely a matter of arbitrary, illogical legislation to judge who of the clamouring multitude shall be permitted to survive and who be condemned to be forgotten.' -- Eric Temple Bell: Men of Mathematics, 1937, Victor Gollancz, London Holland Adriaan Metius (1571 – 1635) Dutch geometer and astronomer. Best known now for his approximation $\dfrac {355} {113}$ for $\pi$ (pi), known to the Chinese and Arabic mathematical traditions centuries earlier. show full page Willebrord van Royen Snell (1580 – 1626) Dutch applied mathematician and astronomer who founded the modern science of geodesy, by pioneering the technique of triangulation. Developed an improved method for determining the value of $\pi$ (pi) using polygons. Discovered the Sine Law. Known today for rediscovering the Snell-Descartes Law in 1621, governing the refraction of light. He did not publish himself. It first appeared in 1703 when it was published in Christiaan Huygens' Dioptrica. show full page Dutch Republic Johannes van Waveren Hudde (1628 – 1704) Johannes Hudde, also rendered Johann Hudde or Jan Hudde, was Dutch mathematician, who was also at one time the mayor of Amsterdam and governor of the Dutch East India Company. Organised the regulation of the waterways of Amsterdam, in the process making major steps towards improvements in sanitation. Collaborated on a translation into Latin of La Géométrie by René Descartes. show full page Christiaan Huygens (1629 – 1695) Dutch mathematician, astronomer, physicist and horologist. Studied the rings of Saturn and discovered its moon Titan. Invented the pendulum clock. Believed that light travels in waves, hence the Huygens-Fresnel Principle. show full page Daniel Bernoulli (1700 – 1782) Dutch / Swiss mathematician who worked mostly on fluid dynamics, probability theory and statistics. Considered by many to be the first mathematical physicist. Son of Johann Bernoulli and the brother of Nicolaus II Bernoulli and Johann II Bernoulli. Famously suffered from the jealousy and bad temper of his father Johann Bernoulli who, among other unpleasantnesses, tried to steal his Hydrodynamica and pass it off as his own, naming it Hydraulica. show full page Pieter Nieuwland (1764 – 1794) Dutch nautical scientist, chemist, mathematician and poet. Has been called the Dutch Isaac Newton. Known for finding the largest cube that can pass through a hole in a unit cube. show full page Netherlands Thomas Joannes Stieltjes (1856 – 1894) Thomas Joannes Stieltjes (whose name is also rendered Thomas Jan Stieltjes) was a Dutch mathematician whose main fields of study included continued fractions and measure theory. show full page Jan Cornelis Kluyver (1860 – 1932) Dutch mathematician who made important contributions to analysis, number theory and geometry. Professor at Leiden University between 1892 and 1930. show full page Willem Abraham Wythoff (1865 – 1939) Dutch mathematician known for his work in in combinatorial game theory and number theory. Also known for his work in geometry, in particular for the Wythoff construction of uniform tilings and uniform polyhedra. show full page Luitzen Egbertus Jan Brouwer (1881 – 1966) Known to his friends as Bertus. Dutch mathematician working in topology, set theory, measure theory and complex analysis. Founded the mathematical philosophy of intuitionism. show full page Hendrik Anthony Kramers (1894 – 1952) Dutch physicist who worked with Niels Bohr to understand how electromagnetic waves interact with matter. show full page Dirk Jan Struik (1894 – 2000) Dutch mathematician, historian of mathematics and Marxian theoretician who spent most of his life in the United States. show full page Arend Heyting (1898 – 1980) Dutch mathematician and logician of the Intuitionist school. show full page Bartel Leendert van der Waerden (1903 – 1996) Dutch mathematician and historian of mathematics. show full page Tjalling Charles Koopmans (1910 – 1985) Dutch American mathematician and economist. show full page Nicolaas Govert de Bruijn (1918 – 2012) Dutch mathematician known for his contributions to analysis, number theory, combinatorics and logic. show full page Cornelis Gerrit Lekkerkerker (1922 – 1999) Dutch mathematician who worked on analytic and geometric number theory. Later he worked on topics in functional analysis. show full page Adrianus Johannes Wilhelmus Duijvestijn (1927 – 1998) Dutch computer scientist and mathematician best known for finding the Smallest Perfect Square Dissection. show full page Edsger Wybe Dijkstra (1930 – 2002) Hugely influential Dutch pioneer of computer science. show full page Dirk van Dalen (b. 1932 ) Dutch mathematician and historian of science. show full page Hermanus Johannes Joseph te Riele (b. 1947 ) Dutch mathematician specializing in computational number theory. Proving the correctness of the Riemann Hypothesis for the first $1.5$ billion non-trivial zeros of the Riemann zeta function, with Jan van de Lune and Dik Winter. Disproved the Mertens Conjecture, with Andrew Michael Odlyzko. Known for factoring large numbers of world record size. Found a new upper bound for $\pi \left({x}\right) - Li \left({x}\right)$. show full page Hendrik Willem Lenstra Jr. (b. 1949 ) Dutch mathematician working principally in computational number theory. Well known as the discoverer of the elliptic curve factorization method. Co-discoverer of the Lenstra-Lenstra-Lovász Lattice Basis Reduction Algorithm. The Cohen-Lenstra Heuristics, a set of precise conjectures about the structure of class groups of quadratic fields, is named after him. show full page Frits Beukers (b. 1953 ) Dutch mathematician, who works on number theory and hypergeometric functions. show full page Arjen Klaas Lenstra (b. 1956 ) Dutch mathematician active in cryptography and computational number theory, especially in areas such as integer factorization. Co-discoverer of the Lenstra-Lenstra-Lovász Lattice Basis Reduction Algorithm. show full page Eric Eleterius Coralie van Damme (b. 1956 ) Dutch economist known for his contributions to game theory. show full page	2018-08-21 02:26:58	{"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": 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": 0.7300397157669067, "perplexity": 7252.3031822453795}, "config": {"markdown_headings": false, "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-2018-34/segments/1534221217909.77/warc/CC-MAIN-20180821014427-20180821034427-00120.warc.gz"}
https://allthingsstatistics.com/miscellaneous/partial-product-multiplication/	# Partial Product Multiplication – Explained with Examples - The partial product method of multiplication is a method that allows us to multiply two numbers quickly in our heads by decomposing each number as a sum of two numbers. This method is most useful when we are dealing with small numbers having two or three digits. The disadvantage of this method is that it is not very suitable when we wish to multiply two large numbers. ## Steps for Partial Product Multiplication: 1. Write each of the two numbers as a sum of units and tens. For example, 23 = 20 + 3. 2. Use the distributive property of multiplication to expand the brackets. 3. Calculate each product and add the resulting numbers to get the final answer. Let us try to understand this method by looking at some examples. ### Example 1: Suppose you want to calculate: 33 \times 11. Step 1: We write each number as a sum of two simpler numbers as follows: 33 =30 + 3. 11 =10 + 1. Step 2: Apply the distributive property and expand the brackets: \begin{align}33 \times 11 &= (30 + 3)(10 + 1) \\ &= 30 \times (10 + 1) + 3 \times (10 + 1) \\ &= (30 \times 10) + (30 \times 1) + (3 \times 10) + (3 \times 1)\end{align} Step 3: Add up the resulting numbers: \begin{align}33 \times 11 &= (30 \times 10) + (30 \times 1) + (3 \times 10) + (3 \times 1) \\ &= 300 + 30 +30 + 3 \\ &= 363. \end{align} So the final answer is 33 \times 11 = 363 . ### Example 2: Calculate the product 47 \times 86 . \begin{align}47 \times 86 &= (40 + 7)(80 + 6) \\ &= 40 \times (80 + 6) + 7 \times (80 + 6) \\ &= (40 \times 80) + (40 \times 6) + (7 \times 80) + (7 \times 6) \\ &= 3200 + 240 + 560 + 42 \\ &= 4042. \end{align} ### Example 3: Calculate the product 327 \times 16 using the partial multiplication method. \begin{align}327 \times 16 &= (300 + 20 + 7)(10 + 6) \\ &= 300 \times (10 + 6) + 20 \times (10 + 6) + 7 \times (10 + 6) \\ &= (300 \times 10) + (300 \times 6) + (20 \times 10) + (20 \times 6) + (7 \times 10) + (7 \times 6) \\ &= 3000 + 1800 + 200 + 120 + 70 + 42 \\ &= 5232. \end{align} So the final answer is 327 \times 16 = 5232 . Summary Article Name Partial Product Multiplication - Explained with Examples Description Steps for Partial Product Multiplication: 1) Write each of the two numbers as a sum of units and tens. For example, 23 = 20 + 3. 2) Use the distributive property of multiplication to expand the brackets. 3) Calculate each product and add the resulting numbers to get the final answer. Publisher Name allthingsstatistics.com Hey 👋 I'm currently pursuing a Ph.D. in Maths. Prior to this, I completed my master's in Maths & bachelors in Statistics. I created this website for explaining maths and statistics concepts in the simplest possible manner. If you've found value from reading my content, feel free to support me in even the smallest way you can.	2023-01-31 00:51:51	{"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": 1.0000100135803223, "perplexity": 414.03864184306144}, "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/1674764499831.97/warc/CC-MAIN-20230130232547-20230131022547-00555.warc.gz"}
http://mathhelpforum.com/differential-equations/187356-integrating-factor-problem-print.html	# Integrating Factor Problem • Sep 5th 2011, 05:08 PM VitaX Integrating Factor Problem Find the solution of the initial value problem. $ty' + (t+1)y = t$ ; $y(ln2) = 1$ ; $t > 0$ $\frac{dy}{dx} + p(x)y = Q(x)$ $y' + \left(1 + \frac{1}{t} \right)y = 1$ $p(t) = 1 + \frac{1}{t}$ ; $Q(t) = 1$ $\int p(x) = \int 1 + \frac{1}{t} dt = t + ln(t)$ Integrating Factor: $e^{t + ln(t)} = e^{t}e^{ln(t)} = te^t$ $te^t \left[y' + \left(1 + \frac{1}{t} \right)y \right] = te^t$ $te^t y' + \left(1 + \frac{1}{t} \right)y = te^t$ $u'v + v'u = \frac{d}{dx} (uv)$ ; $u = y$ ; $v = te^t$ $\frac{d}{dt}[yte^t] = te^t$ $\int d[yte^t] = \int te^t dt$ $yte^t = \int te^t dt$ Integrate by Parts: $\int te^t dt = te^t - e^t$ $yte^t = te^t - e^t$ $y(t) = 1 - \frac{1}{t} + C$ Initial Condition: $y(ln2) = 1$ $y(ln2) = 1 = 1 - \frac{1}{ln2} + C$ $C = \frac{1}{ln2}$ My Solution: $y(t) = 1 - \frac{1}{t} + \frac{1}{ln2}$ Book Solution: $y(t) = \frac{t - 1 + 2e^{-t}}{t}$ Where exactly did I go wrong? • Sep 5th 2011, 05:37 PM Chris L T521 Re: Integrating Factor Problem Quote: Originally Posted by VitaX Find the solution of the initial value problem. $ty' + (t+1)y = t$ ; $y(ln2) = 1$ ; $t > 0$ $\frac{dy}{dx} + p(x)y = Q(x)$ $y' + \left(1 + \frac{1}{t} \right)y = 1$ $p(t) = 1 + \frac{1}{t}$ ; $Q(t) = 1$ $\int p(x) = \int 1 + \frac{1}{t} dt = t + ln(t)$ Integrating Factor: $e^{t + ln(t)} = e^{t}e^{ln(t)} = te^t$ $te^t \left[y' + \left(1 + \frac{1}{t} \right)y \right] = te^t$ $te^t y' + \left(1 + \frac{1}{t} \right)y = te^t$ $u'v + v'u = \frac{d}{dx} (uv)$ ; $u = y$ ; $v = te^t$ $\frac{d}{dt}[yte^t] = te^t$ $\int d[yte^t] = \int te^t dt$ $yte^t = \int te^t dt$ Integrate by Parts: $\int te^t dt = te^t - e^t$ $yte^t = te^t - e^t$ This is where you made your mistake. $y(t) = 1 - \frac{1}{t} + C$ Initial Condition: $y(ln2) = 1$ $y(ln2) = 1 = 1 - \frac{1}{ln2} + C$ $C = \frac{1}{ln2}$ My Solution: $y(t) = 1 - \frac{1}{t} + \frac{1}{ln2}$ Book Solution: $y(t) = \frac{t - 1 + 2e^{-t}}{t}$ Where exactly did I go wrong? You made your mistake when you said $yte^t = te^t-e^t$. It should be $yte^t=te^t-e^t{\color{red} + C}$. That way, when you divide through by $te^t$, you get $y=1-\frac{1}{t}+\frac{1}{t}Ce^{-t}$ Now apply your initial condition to get the book's solution. I hope this helps. • Sep 5th 2011, 05:41 PM VitaX Re: Integrating Factor Problem Ah, OK. Thanks for clearing that up.	2017-11-19 07:24:33	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 54, "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": 0.9937414526939392, "perplexity": 4146.601606455433}, "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/1510934805417.47/warc/CC-MAIN-20171119061756-20171119081756-00021.warc.gz"}
https://andrewpwheeler.wordpress.com/tag/racial-bias/	# New preprint: Allocating police resources while limiting racial inequality I have a new working paper out, Allocating police resources while limiting racial inequality. In this work I tackle the problem that a hot spots policing strategy likely exacerbates disproportionate minority contact (DMC). This is because of the pretty simple fact that hot spots of crime tend to be in disadvantaged/minority neighborhoods. Here is a graph illustrating the problem. X axis is the proportion of minorities stopped by the police in 500 by 500 meter grid cells (NYPD data). Y axis is the number of violent crimes over along time period (12 years). So a typical hot spots strategy would choose the top N areas to target (here I do top 20). These are all very high proportion minority areas. So the inevitable extra police contact in those hot spots (in the form of either stops or arrests) will increase DMC. I’d note that the majority of critiques of predictive policing focus on whether reported crime data is biased or not. I think that is a bit of a red herring though, you could use totally objective crime data (say swap out acoustic gun shot sensors with reported crime) and you still have the same problem. The proportion of stops by the NYPD of minorities has consistently hovered around 90%, so doing a bunch of extra stuff in those hot spots will increase DMC, as those 20 hot spots tend to have 95%+ stops of minorities (with the exception of one location). Also note this 90% has not changed even with the dramatic decrease in stops overall by the NYPD. So to illustrate my suggested solution here is a simple example. Consider you have a hot spot with predicted 30 crimes vs a hot spot with predicted 28 crimes. Also imagine that the 30 crime hot spot results in around 90% stops of minorities, whereas the 28 crime hot spot only results in around 50% stops of minorities. If you agree reducing DMC is a reasonable goal for the police in-and-of-itself, you may say choosing the 28 crime area is a good idea, even though it is a less efficient choice than the 30 crime hot spot. I show in the paper how to codify this trade-off into a linear program that says choose X hot spots, but has a constraint based on the expected number of minorities likely to be stopped. Here is an example graph that shows it doesn’t always choose the highest crime areas to meet that racial equity constraint. This results in a trade-off of efficiency though. Going back to the original hypothetical, trading off a 28 crime vs 30 crime area is not a big deal. But if the trade off was 3 crimes vs 30 that is a bigger deal. In this example I show that getting to 80% stops of minorities (NYC is around 70% minorities) results in hot spots with around 55% of the crime compared to the no constraint hot spots. So in the hypothetical it would go from 30 crimes to 17 crimes. There won’t be a uniform formula to calculate the expected decrease in efficiency, but I think getting to perfect equality with the residential pop. will typically result in similar large decreases in many scenarios. A recent paper by George Mohler and company showed similar fairly steep declines. (That uses a totally different method, but I think will be pretty similar outputs in practice — can tune the penalty factor in a similar way to changing the linear program constraint I think.) So basically the trade-off to get perfect equity will be steep, but I think the best case scenario is that a PD can say "this predictive policing strategy will not make current levels of DMC worse" by applying this algorithm on-top-of your predictive policing forecasts. I will be presenting this work at ASC, so stop on by! Feedback always appreciated. # Quantifying racial bias in peremptory challenges A question came up recently on cross validated about putting some numbers on the amount of bias in jury selection. I had a previous question of a similar nature, so it had been on my mind previously. The original poster did not say this was specifically for a Batson challenge, but that is simply my presumption. It is both amazing and maddening that given the same question four different potential analyses were suggested. Although it is a bit out of the norm for what I talk about, I figured it would be worth a post. # Some background on Batson For some background, Batson challenges are specifically in the context of selecting jurors for a trial. (Everything that follows is specific to what I know about law in the US.) To select a jury first the court selects potential jurors for the venire from the general public. Then both the prosecution and defence counsels have the opportunity to question individuals in the venire. A typical flow seems to be that a panel of the venire is selected (say 10), then the court has a set of standardized questions they ask every individual potential juror. This part is referred to as voir dire. If the individual states they can not be impartial, or there is some other characteristic that indicates they cannot be impartial that potential juror can be eliminated for cause. Without intervention of counsel the standard questions by the court typically weeds out any obvious cases. After the standard questions both counsels have the opportunity to ask their own questions and further identify challenges for cause. There are no limits on who can be eliminated for cause. The wrinkle specific to Batson though is that each counsel is given a fixed number of peremptory challenges. The number is dictated by the severity of the case (in more serious cases each side has a higher number of challenges). The wikipedia page says in some circumstances the defense gets more than the prosecution, but the total number is always fixed in advance. The logic behind peremptory challenges is that either counsel can use personal discretion to eliminate potential jurors without needing a justification. Basically it is a fail-safe of the court to allow gut feelings of either counsel to eliminate jurors they believe will be partial to the opposing side. But based on the equal protection clause it was decided in Batson vs. Kentucky that one can not use the challenges solely based on race. As a side effect of allowing so many peremptory challenges, one can easily eliminate a particular minority group, as being a minority group they will only have a few representatives in the venire. During the voir dire if the opposing counsel believes the opposition is using the peremptory challenges in a racially discriminatory manner, they can object with a Batson challenge. The supreme court decided on three steps to evaluate the challenge. 1. The party that objected has the burden to prove a prima facie case that the challenges were used in a discriminatory manner. This includes an argument that the group is discriminated against is cognizable, and that there is additional numerical evidence of discrimination. 2. Then the burden shifts on the party being challenged to justify the use of the peremptory challenges based on race neutral reasons. 3. The burden then shifts back to the original challenging party. This is to dispute whether the reasons proferred for the use of the peremptory challenges are purely pretextual. Witnessing the proceedings for this particular case in the New York court of appeals case is what prompted my interest, and I recommend reading their decision as a good general background on Batson challenges (the wikipedia page is lacking quite a bit). What follows is some number crunching specific to the first part, establishing a prima facie case of discrimination. # Now some numbers Batson challenges are made in situ during voir dire. All the cases I am familiar with simply use fractions to establish that the peremptory challenges are being used in a discriminatory manner. The fact that the numbers are changing during voir dire makes the calculations of statistics more difficult. But I will address the ex post facto assessment of the first step given the final counts of the number of peremptory challenges and the total number on the venire with their racial distribution. This presumption I will later discuss how it might impact on the findings in a more realistic setting. Consider the case of People vs. Hecker (linked to above). It happened that the peremptory challenges by the defence to exclude two Asian’s from the jury panel is what prompted the Batson challenge. Later on one other Asian juror was seated to the jury. The appeals court considered in this case whether step 1 was justified, so it is not a totally academic question to attempt to quantify the chances of two out of three Asian’s being challenged. First, I will specify how we might put a number of this chance occurrence. If a person randomly selected 13 names out of a hat with 39 people, and of those names 3 individuals were Asian, what is the probability that 2 of those selected would be Asian? This probability is dictated by the hypergeometric distribution. More generally, the set up is: • n equals the total number of eligible cases that are subject to be challenged • p equals the total number of the race in question that are subject to be challenged • k equals the total number of peremptory challenges used on the racial group in question • d equals the total number of peremptory challenges And the hypergeometric distribution is calculated using binomial coefficients as: $\frac{{p \choose k} {n-p \choose d-k} }{{n \choose d}}$ So plugging the numbers listed above into the formula, we get the probability of two out of three Asian jurors being challenged if the challenges were made randomly would be equal to below according to Wolfram Alpha: $\frac{{p \choose k} {n-p \choose d-k} }{{n \choose d}} = \frac{{3 \choose 2} {39-3 \choose 13-2} }{{39 \choose 13}} = \frac{156}{703} \approx 0.22$ So the probability of a chance occurrence given this particular set of circumstances is 22%, not terribly small, although may be sufficient given other circumstances to justify the first step (it seems the first step is intended that the burden is rather light). Where did my numbers come from though exactly? Given the circumstances of the case in the appeal decision the only number that would be uncontroversial would be p = 2, that two Asian jurors were excluded based on peremptory challenges. p = 3 comes from after the case, in which one other Asian juror was actually seated. It appears in the appeals case I linked to they consider the jury composition after the Batson challenge was initially brought, but obviously the initial trial judge can’t use that future information. I choose to use d = 13 because the defence only used 13 of their 15 peremptory challenges by the end of the seating. The total number n = 39 is the most difficult to come by. The appeal case states that in the first pool 18 jurors were brought for questioning, 5 were eliminated for cause, and that both the defence and prosecution used 5 peremptory challenges. I chose to count the total number as 13 for this round, 18 minus the 5 eliminated for cause. In our pulling names out of the hat experiment though you may consider the number to be only 8, so not count the cases the prosecution used their peremptory challenges on. The second round brought another 18 potential jurors, of which 4 were eliminated for challenge. In this round the two Asian jurors were challenged by the defence when the judge asked to evaluate the first 9 of this panel (given the language it appears defence used 3 peremptory challenges during this evaluation of the first 9). At the end of the second round both parties used another 5 peremptory challenges. So to get the the total number of 39, I use 13 for the first round plus 14 for the second, although I could reasonably use 8 for the first and 9 for the second. I end up at 39 by using 13 + 14 + 12 – the last Asian juror (Kazuko) was the the 13th to be questioned on the third panel. One prior juror had been challenged for cause, so I count this as 12 towards the total of 39. There were a total of 26 panellists in this round, and the defence used a total of another 3 peremptory challenges, and there is no other information on whether the prosecution used any more peremptory challenges. (I’m unsure the total number of jurors seated for the case, so I can’t make many other guesses – the total number of jurors selected in the prior two rounds were 7). I don’t worry about the selection of the alternate juror for this analysis. So lets try to apply this same analysis at the exact time the Batson challenge was raised, when the second Asian juror was eliminated. I prefer to make the calculations at the time the challenge was raised, as the opposing counsel may later alter their behavior in light of a prior Batson challenge. In doing this, now we have p = 2 and k = 2 (there was no other mention of any Asian’s being challenged for cause). We have a bit of uncertainty about d and n though. d at a minimum for the defence has to be 7 at this point (five in the first round plus the two Asians in the second round), but could be as many as 10 (5 in the first and 5 in the second). n could be as mentioned before 8 + 9 = 17 (excluding cases the prosecution challenged) or 13 + 14 = 27 (including all cases not challenged for cause). In the middle you may consider n to be 13 + 9 = 22, the exact point when the defence was asked to bring forth challenges for the first 9 seated on that round of the panel. So what difference does this make on the estimated probabilities? Well lets just graph the estimates for all values of d between 7 and 10, and n between 17 and 27. Here the lines are for different values of d (with labels at the beginning left part of the line), and the x axis is the different values of n. We can see the probabilities follow the pattern that as n increases and d decreases the probability of that combination goes down. Even over all these values the probability never goes below 5%. I do not know if a 5% probability is sufficient for the numerical justification of the prima facie case of discrimination. 22% seems too low a threshold to me (by chance about 1 in 5 times) but 5% may be good enough (by chance 1 in 20 times). So lets try this same sensitivity analysis for the entire case. For the evaluation of everything after the fact I think p = 3 and k = 2 are largely uncontroversial, but lets vary d between 7 and 15 and n between 23 (chosen to be at the lower end of cases that were legitimately evaluated in the pool by the end I believe) to 45. Some of these situations are not commensurate with the limited information we have (e.g. d = 7 and n = 45 is not possible) but I think the graph will be informative anyway. My labelling could use more work, but the line on top is d = 15 and they increment until the line on bottom where d = 7. So here we can see that the probability after the fact never gets much below 10%, and that is for lower values of d and higher values of n that are not likely possible given the data. So basically in the case that looks the worst for the defence here the probability of selecting two out of three Asian’s by random (giving varying numbers of peremptory challenges and varying the pool from which to draw them) is never below 10%. Not a terribly strong case that the numerical portion of step 1 has been satisfied. Basically, no matter what reasonable values you put in for d or n in this circumstance the probability of choosing 2 out of three Asian jurors randomly is not going to be much below 10%. # On some of the other suggested analyses On the original question on CV I mentioned previously, besides my own suggested analysis here there were 3 other suggested analysis: • Using a regression model to predict the probability of a racial group being challenged • Analysis of Contingency tables • Calculating all potential permutations, and then counting the percentage of those permutations that meet some criteria. All three I do not think are completely unreasonable, but I prefer the approach I listed above. I will attempt to articulate those reasons. So first I will talk about the regression approach. This is generically a model of the form predicting the probability of a peremptory challenge based on the race of the potential juror: $\text{Prob}(\text{Challenge}) = f(\beta_0 + \beta_1(\text{Racial Group}_i))$ The anonymous function is typically a logit (for a logistic model) or a probit function. This generalized linear model is then estimated via maximum likelihood, and one formulates hypothesis tests for the B_1 coefficient. The easy critique of this is that the test is not likely to be very powerful with the small samples – as the estimates are based on maximum likelihood and are only guaranteed to unbiased asymptotically. This could be a fairly simple exercise to attempt to see the behavior of this bias in the small samples, but I suspect it reduces the power of the test greatly. Also note that in the case where the subgroup of interest is always challenged, such as in the two out of two Asian’s in the Hecker case mentioned, the equation is not identified due to perfect separation. There are alternative ways to estimate the equation in the case of perfect separation, but this does not mitigate the small sample problem. More generally, my original formulation of the data generating mechanism being the hypergeometric distribution, drawing names out of hat, is quite different than this. This is a model of the probability of anyone being peremptory challenged. One then estimates the model to see if the probability is increased among the racial group of interest. This is arguably not the question of interest. For instance, say the model estimated the probability of an Asian being challenged to be only 6%, and the probability of anyone else to be 4%. In one sense, this establishes the prima facie case of discrimination of Asian’s compared to everyone else, but does only a probability of 6% of using a peremptory challenge warrant a Batson challenge? I don’t think so. If you think that a challenge will never come with such low probabilities, you are right in that the expected probabilities for the racial group of question will not be that low when a Batson challenge is made, but once you consider the uncertainty in the estimates (e.g. 95% confidence intervals) they could easily be that low. On the flip side if the racial group is struck 96% of the time, but everyone else is struck 94% of the time, does that establish the numerical evidence of discrimination? I’m not sure, it may if this prevents any of the particular racial group being seated. The analysis of contingency tables, in particular Fisher’s Exact Test, is exactly the same as my hypergeometric approach if one only considers the racial group of interest against all other parties. Fisher’s exact test is a reasonable approach over the more typical chi-square because, 1) the cells will be quite small, and 2) this is one of the unusual cases where the marginals are fixed. So making a 2 by 2 contingency table based on the very first example I gave (which resulted in a probability of around 22%) would be a table: Challenge No Challenge Total Asian 2 1 3 Other 11 25 36 Total 13 26 39 For the formula the 2 by 2 table is referred to as: Challenge No Challenge Total Asian a b a + b Other c d c + d Total a + c b + d n Which Fisher’s Exact Test can be formulated by the binomial coefficients: $\frac{{a + b \choose a} {c+d \choose c} }{{n \choose a+c}} = \frac{{2 + 1 \choose 2} {11+25 \choose 11} }{{39 \choose 13}}$ Which if you look closely is exactly the same set of binomial coefficients for the hypergeometric test I listed previously. So, as long as one only tests the one racial group against all others Fisher’s Exact test of a 2 by 2 contingency table is exactly the same as my recommendation. I don’t particularly think the historical p-value <= 0.05 standard is necessary, but it is the same information. What bothers me more about this approach is when people start adding other cells in the contingency table. In the first step you need to establish a pattern of discrimination against one particular cognizable group. The treatment of other groups is non sequitur to this question in the first step. (In People vs Black evaluations of how unemployment was treated for non-black jurors was considered is steps 2 and 3, but not in the first step.) Including other groups into the table though will change the outcome. Such ad hoc decisions on what racial groups to consider should not have any effect on the evidence of discrimination against the specific racial group of interest. This problem of what groups is similarly applicable to the regression approach mentioned above. Hypothesis tests of the coefficients will be dependent on what particular contrasts you wish to draw and will change the estimates if certain groups are specified in the equation. The final approach, counting up particular permutations that meet a particular threshold is intuitive, but again has an ad hoc element, the same problem with choosing which racial groups will impact the test statistic. All of the approaches (including my own) need to be explicit about the groups being tested beforehand. Only monitoring one group as in the hypergeometric test I presented earlier is much simpler to justify ex post facto, but it still would be best to establish the cognizable groups before voir dire takes place. For the tests that use other racial or ethnic groups in the calculations are much more suspect to justification, as the picking and choosing of the other groups will impact the calculations. # Some recommendations A typical question I get asked as an academic is, So what would you recommend to improve the situation? Totally reasonable question that I often don’t have a good answer to. It is easy to throw out recommendations without considering the entirety of the situation, and the complexities of the criminal justice system are no exception. With full awareness that no one with any authority will likely read my recommendations, my suggestions follow none-the-less. The first is, only slightly in jest, is to only allow 1 peremptory challenge. There is no bright line rule on numerical evidence presented that is necessary to establish discrimination, but the NY State court of appeals case I mentioned did indicate that it takes more than 1 challenge to establish a pattern of discrimination. This may seem extreme, but the logic applies to the same to allowing fewer peremptory challenges. The fewer the challenges, the less capability either counsel has to entirely eliminate a particular racial group from the jury. It simultaneously makes counsels use of the challenges more precious, so they should be more hesitant to use them based on gut feelings predicated solely by racial stereotyping. As another side effect (good or bad depending on how you look at it) it also makes the evaluation of whether one is using the challenges in a racially discriminatory manner much clearer. As I shown above, when d decreased the probability of the outcome generally decreased. For example with the Hecker case, pretend there were only a total of 5 peremptory challenges. So in this hypothetical situation have n = 20, d = 5, and p and k = 2 (if the number of n was much higher with the number of peremptory challenges limited to 5 for each side the jury would have to be close to set already by the time 20 individuals were questioned). The probability of this is 5%, whereas if d = 7 the probability is 11%. To be very generic, having fewer challenges makes particular racial patterns less likely by chance. I understand the motivation for peremptory challenges, but it is unclear to me why such a large number are currently afforded for most cases. Also to the extent that timeliness is a priority for the court, allowing fewer challenges would certainly decrease the necessary time needed for voir dire. (Which was a concern in the Hecker case, as the court only allowed a very short time for questioning the panels.) The second is that case-law should be established for cognizable groups, particularly given the racial make up of the defendent(s) and victim(s). Or, conversely, counsels should be required a priori to voir dire to establish the cognizable groups. This avoids cherry picking any group for a Batson challenge, as one could always specify a group based on the ex post facto characteristics of the groups used for peremptory challenges. To put a probability to whether a certain number of a particular group could be chosen at at random it is necessary to supply the hypothesis before looking at data. Ad hoc selections of a group could always occur, and with some of the other statistical tests ad hoc inclusion of cells in a contingency table or to include in a test statistic could impact the analysis. Making such a case a priori should prevent any nefarious manipulation of the numbers after the fact. Neither of these appear to be too onerous to me to be reasonable suggestions. I doubt any lawyer or judge is going to be typing binomial coefficients into Wolfram Alpha during voir dire anytime soon though. Maybe I should make a look up table or nomogram for hypergeometric probabilities for typical values that would come up during voir dire. It would be pretty easy for a lawyer to keep a tally and then do a look up, or keep in mind before hand at what point a set of challenges is unlikely due to chance. With many peremptory challenges and few uses of a particular group, I suspect the probabilities of that happening by chance are much larger than people expect. The two out of two Asian’s in the Hecker case is a good example where the numerical evidence of discrimination is very weak no matter how you plug in the numbers.	2019-01-18 07:39:04	{"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": 4, "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": 0.5460852384567261, "perplexity": 825.7143538859674}, "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/1547583659944.3/warc/CC-MAIN-20190118070121-20190118092121-00570.warc.gz"}
https://docs.scipy.org/doc/scipy-1.2.1/reference/generated/scipy.odr.ODR.html	# scipy.odr.ODR¶ class scipy.odr.ODR(data, model, beta0=None, delta0=None, ifixb=None, ifixx=None, job=None, iprint=None, errfile=None, rptfile=None, ndigit=None, taufac=None, sstol=None, partol=None, maxit=None, stpb=None, stpd=None, sclb=None, scld=None, work=None, iwork=None)[source] The ODR class gathers all information and coordinates the running of the main fitting routine. Members of instances of the ODR class have the same names as the arguments to the initialization routine. Parameters: Other Parameters: data : Data class instance instance of the Data class model : Model class instance instance of the Model class beta0 : array_like of rank-1 a rank-1 sequence of initial parameter values. Optional if model provides an “estimate” function to estimate these values. delta0 : array_like of floats of rank-1, optional a (double-precision) float array to hold the initial values of the errors in the input variables. Must be same shape as data.x ifixb : array_like of ints of rank-1, optional sequence of integers with the same length as beta0 that determines which parameters are held fixed. A value of 0 fixes the parameter, a value > 0 makes the parameter free. ifixx : array_like of ints with same shape as data.x, optional an array of integers with the same shape as data.x that determines which input observations are treated as fixed. One can use a sequence of length m (the dimensionality of the input observations) to fix some dimensions for all observations. A value of 0 fixes the observation, a value > 0 makes it free. job : int, optional an integer telling ODRPACK what tasks to perform. See p. 31 of the ODRPACK User’s Guide if you absolutely must set the value here. Use the method set_job post-initialization for a more readable interface. iprint : int, optional an integer telling ODRPACK what to print. See pp. 33-34 of the ODRPACK User’s Guide if you absolutely must set the value here. Use the method set_iprint post-initialization for a more readable interface. errfile : str, optional string with the filename to print ODRPACK errors to. Do Not Open This File Yourself! rptfile : str, optional string with the filename to print ODRPACK summaries to. Do Not Open This File Yourself! ndigit : int, optional integer specifying the number of reliable digits in the computation of the function. taufac : float, optional float specifying the initial trust region. The default value is 1. The initial trust region is equal to taufac times the length of the first computed Gauss-Newton step. taufac must be less than 1. sstol : float, optional float specifying the tolerance for convergence based on the relative change in the sum-of-squares. The default value is eps(1/2) where eps is the smallest value such that 1 + eps > 1 for double precision computation on the machine. sstol must be less than 1. partol : float, optional float specifying the tolerance for convergence based on the relative change in the estimated parameters. The default value is eps(2/3) for explicit models and eps**(1/3) for implicit models. partol must be less than 1. maxit : int, optional integer specifying the maximum number of iterations to perform. For first runs, maxit is the total number of iterations performed and defaults to 50. For restarts, maxit is the number of additional iterations to perform and defaults to 10. stpb : array_like, optional sequence (len(stpb) == len(beta0)) of relative step sizes to compute finite difference derivatives wrt the parameters. stpd : optional array (stpd.shape == data.x.shape or stpd.shape == (m,)) of relative step sizes to compute finite difference derivatives wrt the input variable errors. If stpd is a rank-1 array with length m (the dimensionality of the input variable), then the values are broadcast to all observations. sclb : array_like, optional sequence (len(stpb) == len(beta0)) of scaling factors for the parameters. The purpose of these scaling factors are to scale all of the parameters to around unity. Normally appropriate scaling factors are computed if this argument is not specified. Specify them yourself if the automatic procedure goes awry. scld : array_like, optional array (scld.shape == data.x.shape or scld.shape == (m,)) of scaling factors for the errors in the input variables. Again, these factors are automatically computed if you do not provide them. If scld.shape == (m,), then the scaling factors are broadcast to all observations. work : ndarray, optional array to hold the double-valued working data for ODRPACK. When restarting, takes the value of self.output.work. iwork : ndarray, optional array to hold the integer-valued working data for ODRPACK. When restarting, takes the value of self.output.iwork. data : Data The data for this fit model : Model The model used in fit output : Output An instance if the Output class containing all of the returned data from an invocation of ODR.run() or ODR.restart() Methods restart([iter]) Restarts the run with iter more iterations. run() Run the fitting routine with all of the information given and with full_output=1. set_iprint([init, so_init, iter, so_iter, …]) Set the iprint parameter for the printing of computation reports. set_job([fit_type, deriv, var_calc, …]) Sets the “job” parameter is a hopefully comprehensible way. #### Previous topic scipy.odr.Model.set_meta #### Next topic scipy.odr.ODR.restart	2021-05-06 01:32:01	{"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": 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": 0.22188825905323029, "perplexity": 3511.1661928857247}, "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/1620243988724.75/warc/CC-MAIN-20210505234449-20210506024449-00464.warc.gz"}
http://giantbattlingrobots.blogspot.com/2009/01/random-walk-down-monopoly-lane.html	## 09 January 2009 ### A Random Walk Down Monopoly Lane Since my Candyland post seemed to get a good response, I thought I'd take a stab at a few other games everybody knows. "How about Monopoly?", I thought. [Let's do this right at least once: it's Monopoly®.] A small amount of research revealed a wealth of material on probabilities of landing on each square, expected incomes, expected payback times, probability of landing in jail and much much more. Most of those links take you to the superb Probabilities in the Game of Monopoly page by Truman Collins. Since Truman and others have already done such a good job on this topic, and I already discussed the idea of a Markov chain earlier, I decided I needed a different angle on this game. Describing the probabilities of landing on each square as a Markov process is useful information, and you might easily use that to improve your play, but it doesn't describe the actual progress of the game. Unlike Candyland, Monopoly isn't about what square you are on and getting to the end, it's about accumulating wealth. ALL the wealth. A full description of how this works in Monopoly would be quite long, but we might learn a bit by constructing a much simpler sort of game which just keeps track of wealth. Consider a game where two players, call them P1 and P2, start with equal amounts of money, and each turn they trade some random amount of money to the other player. If we think about the difference in wealth between Players 1 and 2 (P1 $minus P2$), and plot this value over a series of turns, it might look something like this (to right). Here I set up a spreadsheet where each player randomly gives the other player between $0 and$19 each turn. This sort of series is known as a Random Walk. Here is another plot with ten similar random walks. In these instances P1 is behind P2 in wealth at turn 200 (the difference is negative) more often than not. This just random variation, and if we looked at many such plots, we expect should expect that P1 will be winning (wealthier than P2) 50% of the time. In this respect the game is fair; it is completely random and no player has any advantage. "Now what?", you say. I created a really boring game that no one would ever play. Let's make the game, which I will now refer to as "Monopoly-Walk", a little more complicated by adding or changing some rules that make it a little bit more like the original Monopoly: 1) Let each player start with $2000. 2) Let each player gain$20 per turn. 3) On each turn both players randomly give each other a randomly determined amount of money between 0% and 6% of the other player's wealth. 4) When one player controls all the wealth, the game is over. Rules #1 and #2 are similar to Monopoly; each player starts with equal wealth, and gain wealth over time. My version has a fixed income every turn for simplicity. Rule #3 is our random walk, but this time it's different, because if one player gains greater wealth the other will probably end up paying them more money. Neither player starts with any advantage, but sooner or later one player gains a big advantage. This is a big simplification from the original game, but it still captures the essential element of Monopoly: THE RICH GET RICHER. Rule #4 is identical to our original Monopoly game. So here are two plots of a game of Monopoly-Walk. These are identical plots, just a different time scales to better show what is going on. This "walk" (red line) looks much smoother than the random walks above, but the scale on the y-axis is much greater and so the difference do not look as jagged. The blue "cone" line represents the total wealth in the game (or maximum difference in wealth) as both positive and negative values. When the red line touches one of the blue, the game is over and one player has won (P1 is the difference is positive, P2 if it is negative). In this example the players are nearly equal in wealth for the first 60 turns (difference is ~0), but then player 2 gains a small advantage which eventually leads to a win about turn 125. Here is the spreadsheet I used to create the plots: Monopoly2.xls. You can try adjusting the starting values and see how that changes the game (See my comments for more explanation). If you don't want to try that, here are some more plots you can look at to get a better idea what is going on. You can see here that the game usually remains somewhat "flat" at first, with no big swings in wealth. Later one, when one player has more than a tiny advantage, that advantage quickly translates into a win. Let's think a bit more about the remaining differences between this simple game and Monopoly. Monopoly has properties which must be purchased, thus reducing a player's wealth. The properties give small rent payments at first, but later on are developed (also reducing wealth) to allow greater rents, earning the owner a greater share of the other player's money on future turns. These things are a delaying effects between first acquiring wealth and turning that wealth into higher rents, and so the game remains essentially random for a longer time before one player gains a decisive advantage. Monopoly also has "taxes" that remove wealth from a player and the total wealth available in the game. This serves as another sort of delay to a player gaining that decisive advantage. These taxes also add additional randomness, making the game even less predictable. Add in three or more players and gentle random walk turns into a wild roller-coaster ride governed by luck and player moxie. So that's my angle on the game Monopoly. I doubt it will help anyone play better, but it may allow the opportunity to appreciate a different sort of mathematical process, one that also happens to be a lot of fun. PS: If one reader would try my spreadsheet and report success or failure, that would be appreciated.	2017-08-21 14:03:17	{"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": 0.48701679706573486, "perplexity": 704.0229268329772}, "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-2017-34/segments/1502886108709.89/warc/CC-MAIN-20170821133645-20170821153645-00670.warc.gz"}
http://stackoverflow.com/questions/2840712/optimization-math-computation-multiplication-and-summing	# optimization math computation (multiplication and summing) Suppose you want to compute the sum of the square of the differences of items: $\sum_{i=1}^{N-1} (x_i - x_{i+1})^2$ the simplest code (the input is std::vector<double> xs, the ouput sum2) is: double sum2 = 0.; double prev = xs[0]; for (vector::const_iterator i = xs.begin() + 1; i != xs.end(); ++i) { sum2 += (prev - (i)) (prev - (i)); // only 1 - with compiler optimization prev = (i); } I hope that the compiler do the optimization in the comment above. If N is the length of xs you have N-1 multiplications and 2N-3 sums (sums means + or -). Now suppose you know this variable: $x_1^2 + x_N^2 + 2 \sum_{i=2}^{N-1} x_i^2$ and call it sum. Expanding the binomial square: $sum_i^{N-1} (x_i-x_{i+1})^2 = sum - 2\sum_{i=1}^{N-1} x_i x_{i+1}$ so the code becomes: double sum2 = 0.; double prev = xs[0]; for (vector::const_iterator i = xs.begin() + 1; i != xs.end(); ++i) { sum2 += (i) prev; prev = (i); } sum2 = -sum2 2. + sum; Here I have N multiplications and N-1 additions. In my case N is about 100. Well, compiling with g++ -O2 I got no speed up (I try calling the inlined function 2M times), why? - Please try to format LaTex stuff properly. SO supports it. – Hamish Grubijan May 15 '10 at 15:55 Somehow I suspect that a matrix operation might help. Try using GPU or vectorize this :) – Hamish Grubijan May 15 '10 at 15:58 @Hamish1: are you sure? How? @Hamish2: a vectorization of the operation should help, but now I'm thinking only about a semplification of the mathematical computation. – Ruggero Turra May 15 '10 at 16:18 You might try looking at the assembly code gcc emits to see what optimizations it makes. (There is a compiler switch for that). Speculating wildly, perhaps the additions are not your program's performance bottleneck, e.g. because multiplications are a lot more expensive? – meriton May 15 '10 at 16:47 @Hamish: This is proper LaTex, but stackoverflow doesn't support it. I think mathoverflow does... – Lucas May 15 '10 at 16:49 The multiplications are much more costly than additions in term of execution time. Also, depending on the processor additions and multiplications will be done in parallel. Ie. it will start the next multiplication while it's doing the addition (see http://en.wikipedia.org/wiki/Out-of-order_execution). So reducing the number of additions will not help much for the performance. What you can do is make it easier for the compiler to vectorize your code, or vectorize by yourself. To make it easier for the compiler to vectorize, I would use a regular array of doubles, use subscripts and not pointers. EDIT: N = 100 might also be a small number to see the difference in execution time. Try a N bigger. Dirty code but shows the perf improvement. Output: 1e+06 59031558 1e+06 18710703 The speedup you get is ~3x. #include <vector> #include <iostream> using namespace std; unsigned long long int rdtsc(void) { unsigned long long int x; unsigned a, d; __asm__ volatile("rdtsc" : "=a" (a), "=d" (d)); return ((unsigned long long)a) \| (((unsigned long long)d) << 32);; } double f(std::vector<double>& xs) { double sum2 = 0.; double prev = xs[0]; vector<double>::const_iterator iend = xs.end(); for (vector<double>::const_iterator i = xs.begin() + 1; i != iend; ++i) { sum2 += (prev - (i)) (prev - (i)); // only 1 - with compiler optimization prev = (i); } return sum2; } double f2(double xs, int N) { double sum2 = 0; for(int i = 0; i < N - 1; i+=1) { sum2 += (xs[i+1] - xs[i])(xs[i+1] - xs[i]); } return sum2; } int main(int argc, char* argv[]) { int N = 1000001; std::vector<double> xs; for(int i=0; i<N; i++) { xs.push_back(i); } unsigned long long int a, b; a = rdtsc(); std::cout << f(xs) << endl; b = rdtsc(); cout << b - a << endl; a = rdtsc(); std::cout << f2(&xs[0], N) << endl; b = rdtsc(); cout << b - a << endl; } - N is fixed by the problem – Ruggero Turra May 15 '10 at 17:28 Right. But if you can, it's easier to have N bigger to check speedups. – Kamchatka May 15 '10 at 17:31 a) why do you use i+=1 instead of ++i? b) the speed up is because you are using plain array instead of std::vector or because you are not using the prev variable? – Ruggero Turra May 15 '10 at 17:43 a) because I tried loop unrolling and i+=4. it didn't give much more speedup. b) the speedup is because I use plain array mostly (you can try by removing the prev in your example). Also, compilers are doing a best job if you use standard constructs (though it's not a general rule). If this is possible, you should try ICC (Intel C Compiler). There is a free version for personal use and it is really good a loop analysis. – Kamchatka May 16 '10 at 4:53 with g++ -O2 I don't get your improvement: 1e+06->7422819, 1e+06->7186527 – Ruggero Turra May 16 '10 at 19:34 Addition can be free when done as x+=ab. The compiler should be able to figure that out in the first version, if the architecture supports it. The math is probably happening in parallel with i which could be slower. Do not call xs.end() at every loop iteration unless you expect the return value to change. If the compiler cannot optimize it out it will dwarf the rest of the loop. -	2015-07-06 07:30:06	{"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": 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": 0.6049026846885681, "perplexity": 6110.409888991646}, "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/1435375098071.98/warc/CC-MAIN-20150627031818-00020-ip-10-179-60-89.ec2.internal.warc.gz"}
https://mattxgalloway.com/SCPME/articles/Details.html	## Introduction Consider the case where we observe $$n$$ independent, identically distributed copies of the random variable ($$X_{i}$$) where $$X_{i} \in \mathbb{R}^{p}$$ is normally distributed with some mean, $$\mu$$, and some variance, $$\Sigma$$. That is, $$X_{i} \sim N_{p}\left( \mu, \Sigma \right)$$. Because we assume independence, we know that the probability of observing these specific observations $$X_{1}, ..., X_{n}$$ is equal to \begin{align} f(X_{1}, ..., X_{n}; \mu, \Sigma) &= \prod_{i = 1}^{n}(2\pi)^{-p/2}\left\| \Sigma \right\|^{-1/2}\exp\left[ -\frac{1}{2}\left( X_{i} - \mu \right)^{T}\Sigma^{-1}\left( X_{i} - \mu \right) \right] \\ &= (2\pi)^{-nr/2}\left\| \Sigma \right\|^{-n/2}\mbox{etr}\left[ -\frac{1}{2}\sum_{i = 1}^{n}\left( X_{i} - \mu \right)\left( X_{i} - \mu \right)^{T}\Sigma^{-1} \right] \end{align} where $$\mbox{etr}\left( \cdot \right)$$ denotes the exponential trace operator. It follows that the log-likelihood for $$\mu$$ and $$\Sigma$$ is equal to the following: $l(\mu, \Sigma \| X) = const. - \frac{n}{2}\log\left\| \Sigma \right\| - tr\left[ \frac{1}{2}\sum_{i = 1}^{n}\left(X_{i} - \mu \right)\left(X_{i} - \mu \right)^{T}\Sigma^{-1} \right]$ If we are interested in estimating $$\mu$$, it is relatively straight forward to show that the maximum likelihood estimator (MLE) for $$\mu$$ is $$\hat{\mu}_{MLE} = \sum_{i = 1}^{n}X_{i}/n$$ which we typically denote as $$\bar{X}$$. However, in addition to $$\mu$$, many applications require the estimation of $$\Sigma$$ as well. We can also find a maximum likelihood estimator: \begin{align} &\hat{\Sigma}_{MLE} = \arg\max_{\Sigma \in \mathbb{S}_{+}^{p}}\left\{ const. - \frac{n}{2}\log\left\| \Sigma \right\| - tr\left[ \frac{1}{2}\sum_{i = 1}^{n}\left(X_{i} - \mu \right)\left(X_{i} - \mu \right)^{T}\Sigma^{-1} \right] \right\} \\ &\nabla_{\Sigma}l(\mu, \Sigma \| X) = -\frac{n}{2}\Sigma^{-1} + \frac{1}{2}\sum_{i = 1}^{n}\left(X_{i} - \mu \right)\left(X_{i} - \mu \right)^{T}\Sigma^{-2} \\ \Rightarrow &\hat{\Sigma}_{MLE} = \left[ \frac{1}{n}\sum_{i = 1}^{n}\left(X_{i} - \bar{X} \right)\left(X_{i} - \bar{X} \right)^{T} \right] \end{align} By setting the gradient equal to zero and plugging in the MLE for $$\mu$$, we find that the MLE for $$\Sigma$$ is our usual sample estimator often denoted as $$S$$. It turns out that we could have just as easily computed the maximum likelihood estimator for the precision matrix $$\Omega \equiv \Sigma^{-1}$$ and taken its inverse: $\hat{\Omega}_{MLE} = \arg\min_{\Omega \in S_{+}^{p}}\left\{ tr\left(S\Omega\right) - \log\left\|\Omega\right\| \right\}$ so that $$\hat{\Omega}_{MLE} = S^{-1}$$. Beyond the formatting convenience, computing estimates for $$\Omega$$ as opposed to $$\Sigma$$ often poses less computational challenges – and accordingly, the literature has placed more emphasis on efficiently solving for $$\Omega$$ instead of $$\Sigma$$. As in regression settings, we can construct a penalized log-likelihood estimator by adding a penalty term, $$P\left(\Omega\right)$$, to the likelihood: $\hat{\Omega} = \arg\min_{\Omega \in S_{+}^{p}}\left\{ tr\left(S\Omega\right) - \log\left\|\Omega \right\| + P\left( \Omega \right) \right\}$ $$P\left( \Omega \right)$$ is often of the form $$P\left(\Omega \right) = \lambda\\|\Omega \\|_{F}^{2}/2$$ or $$P\left(\Omega \right) = \\|\Omega\\|_{1}$$ where $$\lambda > 0$$, $$\left\\|\cdot \right\\|_{F}^{2}$$ is the Frobenius norm and we define $$\left\\|A \right\\|_{1} = \sum_{i, j} \left\| A_{ij} \right\|$$. These penalties are the ridge and lasso, respectively. The penalty proposed by Molstad and Rothman (2017) is one of the following form: $P\left(\Omega\right) = \lambda\left\\| A\Omega B - C \right\\|_{1}$ where $$A \in \mathbb{R}^{m \times p}, B \in \mathbb{R}^{p \times q}, \mbox{ and } C \in \mathbb{R}^{m \times q}$$ are matrices assumed to be known and specified by the user. Solving the full penalized log-likelihood for $$\Omega$$ results in solving $\hat{\Omega} = \arg\min_{\Omega \in S_{+}^{p}}\left\{ tr\left(S\Omega\right) - \log\left\|\Omega \right\| + \lambda\left\\| A\Omega B - C \right\\|_{1} \right\}$ This form of penalty is particularly useful because matrices $$A, B, \mbox{ and } C$$ can be constructed so that we penalize the sum, absolute value of a characteristic of the precision matrix $$\Omega$$. This type of penalty leads to many new, interesting, and novel estimators for $$\Omega$$. An example of one such estimator (suppose we observe $$n$$ samples of $$Y_{i} \in \mathbb{R}^{r}$$) would be one where we set $$A = I_{p}, B = \Sigma_{xy}, \mbox{ and } C = 0$$ where $$\Sigma_{xy}$$ is the covariance matrix of $$X$$ and $$Y$$. This penalty has the effect of assuming sparsity in the forward regression coefficient $$\beta \equiv \Omega\Sigma_{xy}$$. Of course, in practice we do not know the true covariance matrix $$\Sigma_{xy}$$ but we might consider using the sample estimate $$\hat{\Sigma}_{xy} = \sum_{i = 1}^{n}\left(X_{i} - \bar{X}\right)\left(Y_{i} - \bar{Y}\right)^{T}/n$$ We will explore how to solve for $$\hat{\Omega}$$ in the next section. This section requires general knowledge of the alternating direction method of multipliers (ADMM) algorithm. I would recommend reading this overview I have written here before proceeding. The ADMM algorithm - thanks to it’s flexibility - is particularly well-suited to solve penalized-likelihood optimization problems that arise naturally in several statistics and machine learning applications. Within the context of Molstad and Rothman (2017), this algorithm would consist of iterating over the following three steps: \begin{align} \Omega^{k + 1} &= \arg\min_{\Omega \in \mathbb{S}_{+}^{p}}L_{\rho}(\Omega, Z^{k}, \Lambda^{k}) \\ Z^{k + 1} &= \arg\min_{Z \in \mathbb{R}^{n \times r}}L_{\rho}(\Omega^{k + 1}, Z, \Lambda^{k}) \\ \Lambda^{k + 1} &= \Lambda^{k} + \rho\left(A\Omega^{k + 1}B - Z^{k + 1} - C \right) \end{align} where $$L_{p}(\cdot)$$ is the augmented lagrangian defined as $L_{\rho}(\Omega, Z, \Lambda) = f\left(\Omega\right) + g\left(Z\right) + tr\left[\Lambda^{T}\left(A\Omega B - Z - C\right)\right] + \frac{\rho}{2}\left\\|A\Omega B - Z - C\right\\|_{F}^{2}$ with $$f\left(\Omega\right) = tr\left(S\Omega\right) - \log\left\|\Omega\right\|$$ and $$g\left(Z\right) = \lambda\left\\|Z\right\\|_{1}$$. However, instead of solving the first step exactly, the authors propose an alternative, approximating objective function ($$\tilde{L}$$) based on the majorize-minimize principle – the purpose of which is to find a solution that can be solved in closed form. The approximating function is defined as \begin{align} \tilde{L}_{\rho}\left(\Omega, Z^{k}, \Lambda^{k}\right) = f\left(\Omega\right) &+ tr\left[(\Lambda^{k})^{T}(A\Omega B - Z^{k} - C) \right] + \frac{\rho}{2}\left\\|A\Omega B - Z^{k} - C \right\\|_{F}^{2} \\ &+ \frac{\rho}{2}vec\left(\Omega - \Omega^{k}\right)^{T}Q\left(\Omega - \Omega^{k}\right) \end{align} where $$Q = \tau I_{p} - \left(A^{T}A \otimes BB^{T}\right)$$, $$\otimes$$ is the Kronecker product, and $$\tau$$ is chosen such that $$Q$$ is positive definite. Note that if $$Q$$ is positive definite (p.d.), then $\frac{\rho}{2}vec\left(\Omega - \Omega^{k} \right)^{T}Q\left(\Omega - \Omega^{k} \right) > 0$ since $$\rho > 0$$ and $$vec\left(\Omega - \Omega^{k}\right)$$ is always nonzero whenever $$\Omega \neq \Omega^{k}$$. Thus $$L_{\rho}\left(\cdot\right) \leq \tilde{L}\left(\cdot\right)$$ for all $$\Omega$$ and $$\tilde{L}$$ is a majorizing function. To see why this particular function was used, consider the Taylor’s expansion of $$\rho\left\\|A\Omega B - Z^{k} - C\right\\|_{F}^{2}/2$$: \begin{align} \frac{\rho}{2}\left\\| A\Omega B - Z^{k} - C \right\\|_{F}^{2} &\approx \frac{\rho}{2}\left\\| A\Omega^{k} B - Z^{k} - C \right\\|_{F}^{2} \\ &+ \frac{\rho}{2}vec\left( \Omega - \Omega^{k}\right)^{T}\left(A^{T}A \otimes BB^{T}\right)vec\left(\Omega - \Omega^{k}\right) \\ &+ \rho vec\left(\Omega - \Omega^{k}\right)^{T}vec\left(BB^{T}\Omega^{k}A^{T}A - B(Z^{k})^{T}A - BC^{T}A \right) \end{align} Note: \begin{align} &\nabla_{\Omega}\left\{ \frac{\rho}{2}\left\\|A\Omega B - Z - C\right\\|_{F}^{2} \right\} = \rho BB^{T}\Omega A^{T}A - \rho BZ^{T}A - \rho BC^{T}A \\ &\nabla_{\Omega}^{2}\left\{ \frac{\rho}{2}\left\\|A\Omega B - Z - C \right\\|_{F}^{2} \right\} = \rho\left(A^{T}A \otimes BB^{T} \right) \end{align} This implies that \begin{align} \frac{\rho}{2}\left\\| A\Omega B - Z^{k} - C \right\\|_{F}^{2} &+ \frac{\rho}{2}vec\left(\Omega - \Omega^{k} \right)^{T}Q\left(\Omega - \Omega^{k} \right) \\ &\approx \frac{\rho}{2}\left\\| A\Omega^{k} B - Z^{k} - C \right\\|_{F}^{2} + \frac{\rho}{2}vec\left(\Omega - \Omega^{k} \right)^{T}Q\left(\Omega - \Omega^{k} \right) \\ &+ \frac{\rho}{2}vec\left( \Omega - \Omega^{k}\right)^{T}\left(A^{T}A \otimes BB^{T}\right)vec\left(\Omega - \Omega^{k}\right) \\ &+ \rho vec\left(\Omega - \Omega^{k}\right)^{T}vec\left(BB^{T}\Omega^{k}A^{T}A - B(Z^{k})^{T}A - BC^{T}A \right) \\ &= \frac{\rho}{2}\left\\| A\Omega^{k} B - Z^{k} - C \right\\|_{F}^{2} + \frac{\rho\tau}{2}\left\\|\Omega - \Omega^{k}\right\\|_{F}^{2} \\ &+ \rho tr\left[\left(\Omega - \Omega^{k}\right)\left(BB^{T}\Omega^{k}A^{T}A - B(Z^{k})^{T}A - BC^{T}A \right)\right] \end{align} Let us now plug in this equality into our optimization problem in step one: \begin{align} \Omega^{k + 1} &:= \arg\min_{\Omega \in \mathbb{S}_{+}^{p}}\tilde{L}_{\rho}(\Omega, Z^{k}, \Lambda^{k}) \\ &= \arg\min_{\Omega \in \mathbb{S}_{+}^{p}}\left\{\begin{matrix} tr\left(S\Omega\right) - \log\left\|\Omega\right\| + tr\left[(\Lambda^{k})^{T}(A\Omega B - Z^{k} - C) \right] + \frac{\rho}{2}\left\\|A\Omega B - Z^{k} - C \right\\|_{F}^{2} \end{matrix}\right. \\ &+ \left.\begin{matrix} \frac{\rho}{2}vec\left(\Omega - \Omega^{k}\right)^{T}Q\left(\Omega - \Omega^{k}\right) \end{matrix}\right\} \\ &= \arg\min_{\Omega \in \mathbb{S}_{+}^{p}}\left\{\begin{matrix} tr\left(S\Omega\right) - \log\left\|\Omega\right\| + tr\left[(\Lambda^{k})^{T}(A\Omega B - Z^{k} - C) \right] + \frac{\rho}{2}\left\\|A\Omega^{k} B - Z^{k} - C \right\\|_{F}^{2} \end{matrix}\right. \\ &+ \left.\begin{matrix} \frac{\rho\tau}{2}\left\\|\Omega - \Omega^{k}\right\\|_{F}^{2} + \rho tr\left[\left(\Omega - \Omega^{k}\right)\left(BB^{T}\Omega^{k}A^{T}A - B(Z^{k})^{T}A - BC^{T}A \right)\right] \end{matrix}\right\} \\ &= \arg\min_{\Omega \in \mathbb{S}_{+}^{p}}\left\{\begin{matrix} tr\left[\left(S + \rho A^{T}(A\Omega^{k}B - Z^{k} - C + \Lambda^{k}/\rho)B^{T} \right)\Omega\right] \end{matrix}\right. \\ &- \left.\begin{matrix} \log\left\|\Omega\right\| + \frac{\rho\tau}{2}\left\\|\Omega - \Omega^{k}\right\\|_{F}^{2} \end{matrix}\right\} \\ &= \arg\min_{\Omega \in \mathbb{S}_{+}^{p}}\left\{ tr\left[\left(S + G^{k} \right)\Omega\right] - \log\left\|\Omega\right\| + \frac{\rho\tau}{2}\left\\|\Omega - \Omega^{k}\right\\|_{F}^{2} \right\} \\ \end{align} where $$G^{k} = \rho A^{T}(A\Omega^{k}B - Z^{k} - C + \Lambda^{k}/\rho)B^{T}$$. The augmented ADMM algorithm is the following: \begin{align} \Omega^{k + 1} &= \arg\min_{\Omega \in \mathbb{S}_{+}^{p}}\left\{tr\left[\left(S + G^{k}\right)\Omega\right] - \log\left\|\Omega\right\| + \frac{\rho\tau}{2}\left\\|\Omega - \Omega^{k}\right\\|_{F}^{2} \right\} \\ Z^{k + 1} &= \arg\min_{Z \in \mathbb{R}^{n \times r}}\left\{\lambda\left\\|Z\right\\|_{1} + tr\left[(\Lambda^{k})^{T}(A\Omega B - Z^{k} - C) \right] + \frac{\rho}{2}\left\\|A\Omega B - Z^{k} - C \right\\|_{F}^{2} \right\} \\ \Lambda^{k + 1} &= \Lambda^{k} + \rho\left(A\Omega^{k + 1}B - Z^{k + 1} - C \right) \end{align} ### Algorithm Set $$k = 0$$ and repeat steps 1-6 until convergence. 1. Compute $$G^{k} = \rho A^{T}\left( A\Omega^{k} B - Z^{k} - C + \rho^{-1}Y^{k} \right)B^{T}$$ 2. Decompose $$S + \left( G^{k} + (G^{k})^{T} \right)/2 - \rho\tau\Omega^{k} = VQV^{T}$$ (via the spectral decomposition). 3. Set $$\Omega^{k + 1} = V\left( -Q + (Q^{2} + 4\rho\tau I_{p})^{1/2} \right)V^{T}/(2\rho\tau)$$ 4. Set $$Z^{k + 1} = \mbox{soft}\left( A\Omega^{k + 1}B - C + \rho^{-1}Y^{k}, \rho^{-1}\lambda \right)$$ 5. Set $$Y^{k + 1} = \rho\left( A\Omega^{k + 1} B - Z^{k + 1} - C \right)$$ 6. Replace $$k$$ with $$k + 1$$. where $$\mbox{soft}(a, b) = \mbox{sign}(a)(\left\| a \right\| - b)_{+}$$. ### Proof of (2-3): $\Omega^{k + 1} = \arg\min_{\Omega \in \mathbb{S}_{+}^{p}}\left\{tr\left[\left(S + G^{k}\right)\Omega\right] - \log\left\|\Omega\right\| + \frac{\rho\tau}{2}\left\\|\Omega - \Omega^{k}\right\\|_{F}^{2} \right\}$ \begin{align} &\nabla_{\Omega}\left\{tr\left[\left(S + G^{k}\right)\Omega\right] - \log\left\|\Omega\right\| + \frac{\rho\tau}{2}\left\\|\Omega - \Omega^{k}\right\\|_{F}^{2} \right\} \\ &= 2S - S\circ I_{p} + G^{k} + (G^{k})^{T} - G^{k}\circ I_{p} - 2\Omega^{-1} + \Omega^{-1}\circ I_{p} \\ &+ \frac{\rho\tau}{2}\left[2\Omega - 2(\Omega^{k})^{T} + 2\Omega^{T} - 2\Omega^{k} - 2(\Omega - \Omega^{k})^{T}\circ I_{p} \right] \end{align} Note that we need to honor the symmetric constraint given by $$\Omega$$. By setting the gradient equal to zero and multiplying all off-diagonal elements by $$1/2$$, this simplifies to $S + \frac{1}{2}\left(G^{k} + (G^{k})^{T}\right) - \rho\tau\Omega^{k} = (\Omega^{k + 1})^{-1} - \rho\tau\Omega^{k + 1}$ We can then decompose $$\Omega^{k + 1} = VDV^{T}$$ where $$D$$ is a diagonal matrix with diagonal elements equal to the eigen values of $$\Omega^{k + 1}$$ and $$V$$ is the matrix with corresponding eigen vectors as columns. $S + \frac{1}{2}\left(G^{k} + (G^{k})^{T}\right) - \rho\tau\Omega^{k} = VD^{-1}V^{T} - \rho\tau VDV^{T} = V\left(D^{-1} - \rho\tau D\right)V^{T}$ This equivalence implies that $\phi_{j}\left( D^{k} \right) = \frac{1}{\phi_{j}(\Omega^{k + 1})} - \rho\tau\phi_{j}(\Omega^{k + 1})$ where $$\phi_{j}(\cdot)$$ is the $$j$$th eigen value and $$D^{k} = S + \left(G^{k} + (G^{k})^{T}\right)/2 - \rho\tau\Omega^{k}$$. Therefore \begin{align} &\Rightarrow \rho\tau\phi_{j}^{2}(\Omega^{k + 1}) + \phi_{j}\left( D^{k} \right)\phi_{j}(\Omega^{k + 1}) - 1 = 0 \\ &\Rightarrow \phi_{j}(\Omega^{k + 1}) = \frac{-\phi_{j}(D^{k}) \pm \sqrt{\phi_{j}^{2}(D^{k}) + 4\rho\tau}}{2\rho\tau} \end{align} In summary, if we decompose $$S + \left(G^{k} + (G^{k})^{T}\right)/2 - \rho\tau\Omega^{k} = VQV^{T}$$ then $\Omega^{k + 1} = \frac{1}{2\rho\tau}V\left[ -Q + (Q^{2} + 4\rho\tau I_{p})^{1/2}\right] V^{T}$ ### Proof of (4) $Z^{k + 1} = \arg\min_{Z \in \mathbb{R}^{n \times r}}\left\{ \lambda\left\\| Z \right\\|_{1} + tr\left[(\Lambda^{k})^{T}\left(A\Omega^{k + 1}B - Z - C\right)\right] + \frac{\rho}{2}\left\\| A\Omega^{k + 1}B - Z - C \right\\|_{F}^{2} \right\}$ \begin{align} \partial&\left\{ \lambda\left\\| Z \right\\|_{1} + tr\left[(\Lambda^{k})^{T}\left(A\Omega^{k + 1}B - Z - C\right)\right] + \frac{\rho}{2}\left\\| A\Omega^{k + 1}B - Z - C \right\\|_{F}^{2} \right\} \\ &= \partial\left\{ \lambda\left\\| Z \right\\|_{1} \right\} + \nabla_{\Omega}\left\{ tr\left[(\Lambda^{k})^{T}\left(A\Omega^{k + 1}B - Z - C\right)\right] + \frac{\rho}{2}\left\\| A\Omega^{k + 1}B - Z - C \right\\|_{F}^{2} \right\} \\ &= \mbox{sign}(Z)\lambda - \Lambda^{k} - \rho\left( A\Omega^{k + 1}B - Z - C \right) \end{align} where $$\mbox{sign(Z)}$$ is the elementwise sign operator. By setting the gradient/sub-differential equal to zero, we arrive at the following equivalence: $Z_{ij}^{k + 1} = \frac{1}{\rho}\left( \rho(A\Omega_{ij}^{k + 1}B - C) + \Lambda_{ij}^{k} - Sign(Z_{ij}^{k + 1})\lambda \right)$ for all $$i = 1,..., p$$ and $$j = 1,..., p$$. We observe two scenarios: • If $$Z_{ij}^{k + 1} > 0$$ then $\rho\left(A\Omega_{ij}^{k + 1}B - C\right) + \Lambda_{ij}^{k} > \lambda\alpha$ • If $$Z_{ij}^{k + 1} < 0$$ then $\rho\left(A\Omega_{ij}^{k + 1}B - C\right) + \Lambda_{ij}^{k} < -\lambda\alpha$ This implies that $$\mbox{sign}(Z_{ij}^{k + 1}) = \mbox{sign}\left(\rho(A\Omega_{ij}^{k + 1}B - C) + \Lambda_{ij}^{k}\right)$$. Putting all the pieces together, we arrive at \begin{align} Z_{ij}^{k + 1} &= \frac{1}{\rho}\mbox{sign}\left(\rho(A\Omega_{ij}^{k + 1}B - C) + \Lambda_{ij}^{k}\right)\left( \left\| \rho(A\Omega_{ij}^{k + 1}B - C) + \Lambda_{ij}^{k} \right\| - \lambda \right)_{+} \\ &= \frac{1}{\rho}\mbox{soft}\left(\rho(A\Omega_{ij}^{k + 1}B - C) + \Lambda_{ij}^{k}, \lambda\right) \end{align} where soft is the soft-thresholding function. ## Stopping Criterion In discussing the optimality conditions and stopping criterion, we will follow the steps outlined in Boyd et al. (2011) and cater them to the SCPME method. Below we have three optimality conditions: 1. Primal: $A\Omega^{k + 1}B - Z^{k + 1} - C = 0$ 1. Dual: \begin{align} 0 &\in \partial f\left(\Omega^{k + 1}\right) + \frac{1}{2}\left(B(\Lambda^{k + 1})^{T}A + A^{T}\Lambda^{k + 1}B^{T} \right) \\ 0 &\in \partial g\left(Z^{k + 1}\right) - \Lambda^{k + 1} \end{align} The first dual optimality condition is a result of taking the sub-differential of the lagrangian (non-augmented) with respect to $$\Omega^{k + 1}$$ (note that we must honor the symmetric constraint of $$\Omega^{k + 1}$$) and the second is a result of taking the sub-differential of the lagrangian with respect to $$Z^{k + 1}$$ (no symmetric constraint). We will define the left-hand side of the primal optimality condition as the primal residual $$r^{k + 1} = A\Omega^{k + 1}B - Z^{k + 1} - C$$. At convergence, the optimality conditions require that $$r^{k + 1} \approx 0$$. The second residual we will define is the dual residual: $s^{k + 1} = \frac{\rho}{2}\left( B(Z^{k + 1} - Z^{k})^{T}A + A^{T}(Z^{k + 1} - Z^{k})B^{T} \right)$ This residual is derived from the following: Because $$\Omega^{k + 1}$$ is the argument that minimizes $$L_{p}\left( \Omega, Z^{k}, \Lambda^{k} \right)$$, \begin{align} 0 &\in \partial \left\{ f\left(\Omega^{k + 1}\right) + tr\left[ \Lambda^{k}\left( A\Omega^{k + 1}B - Z^{k} - C \right) \right] + \frac{\rho}{2}\left\\| A\Omega^{k + 1}B - Z^{k} - C \right\\|_{F}^{2} \right\} \\ &= \partial f\left(\Omega^{k + 1} \right) + \frac{1}{2}\left(B(\Lambda^{k})^{T}A + A^{T}\Lambda^{k}B^{T} \right) + \frac{\rho}{2}\left( BB^{T}\Omega^{k + 1}A^{T}A + A^{T}A\Omega^{k + 1}BB^{T} \right) \\ &- \frac{\rho}{2}\left( A^{T}(Z^{k} + C)B^{T} + B(Z^{k} + C)^{T}A \right) \\ &= \partial f\left(\Omega^{k + 1} \right) + \frac{1}{2}\left(B(\Lambda^{k})^{T}A + A^{T}\Lambda^{k}B^{T} \right) \\ &+ \frac{\rho}{2}\left( B(B^{T}\Omega^{k + 1}A^{T} - (Z^{k})^{T} - C^{T})A + A^{T}(A\Omega^{k + 1}B - Z^{k} - C)B^{T} \right) \\ &= \partial f\left(\Omega^{k + 1} \right) + \frac{1}{2}\left( B(\Lambda^{k})^{T}A + A^{T}\Lambda^{k}B^{T} \right) + \frac{\rho}{2}\left(A^{T}(A\Omega^{k + 1}B - Z^{k + 1} + Z^{k + 1} - Z^{k} - C)B^{T} \right) \\ &+ \frac{\rho}{2}\left(B(B^{T}\Omega^{k + 1}A^{T} - (Z^{k + 1})^{T} + (Z^{k + 1})^{T} - (Z^{k})^{T} - C^{T})A \right) \\ &= \partial f\left(\Omega^{k + 1} \right) + \frac{1}{2}\left[ B\left((\Lambda^{k})^{T} + \rho(B^{T}\Omega^{k + 1}A^{T} - (Z^{k + 1})^{T} - C^{T}) \right)A \right] \\ &+ \frac{1}{2}\left[ A^{T}\left(\Lambda^{k} + \rho(A\Omega^{k + 1}B - Z^{k + 1} - c)B \right)B^{T} \right] + \frac{\rho}{2}\left(B(Z^{k + 1} - Z^{k})^{T}A + A^{T}(Z^{k + 1} - Z^{k})B^{T} \right) \\ &= \partial f\left(\Omega^{k + 1} \right) + \frac{1}{2}\left(B(\Lambda^{k + 1})^{T}A + A^{T}\Lambda^{k + 1}B^{T} \right) + \frac{\rho}{2}\left(B(Z^{k + 1} - Z^{k})^{T}A + A^{T}(Z^{k + 1} - Z^{k})B^{T} \right) \\ \Rightarrow 0 &\in \frac{\rho}{2}\left( B(Z^{k + 1} - Z^{k})^{T}A + A^{T}(Z^{k + 1} - Z^{k})B^{T} \right) \end{align} Like the primal residual, at convergence the optimality conditions require that $$s^{k + 1} \approx 0$$. Note that the second dual optimality condition is always satisfied: \begin{align} 0 &\in \partial \left\{ g\left(Z^{k + 1}\right) + tr\left[ \Lambda^{k}\left( A\Omega^{k + 1}B - Z^{k + 1} - C \right) \right] + \rho\left\\| A\Omega^{k + 1}B - Z^{k + 1} - C \right\\|_{F}^{2} \right\} \\ &= \partial g\left(Z^{k + 1}\right) - \Lambda^{k} - \rho\left(A\Omega^{k + 1}B - Z^{k + 1} - C \right) \\ &= \partial g\left(Z^{k + 1}\right) - \Lambda^{k + 1} \\ \end{align} One possible stopping criterion is to set $$\epsilon^{rel} = \epsilon^{abs} = 10^{-3}$$ and stop the algorithm when $$\epsilon^{pri} \leq \left\\| r^{k + 1} \right\\|_{F}$$ and $$\epsilon^{dual} \leq \left\\| s^{k + 1} \right\\|_{F}$$ where \begin{align} \epsilon^{pri} &= \sqrt{nr}\epsilon^{abs} + \epsilon^{rel}\max\left\{ \left\\| A\Omega^{k + 1}B \right\\|_{F}, \left\\| Z^{k + 1} \right\\|_{F}, \left\\| C \right\\|_{F} \right\} \\ \epsilon^{dual} &= p\epsilon^{abs} + \epsilon^{rel}\left\\| \left( B(\Lambda^{k + 1})^{T}A + A^{T}\Lambda^{k + 1}B^{T} \right)/2 \right\\|_{F} \end{align} ## References Boyd, Stephen, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, and others. 2011. “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers.” Foundations and Trends in Machine Learning 3 (1). Now Publishers, Inc.: 1–122. Molstad, Aaron J, and Adam J Rothman. 2017. “Shrinking Characteristics of Precision Matrix Estimators.” Biometrika.	2021-10-19 12:47:10	{"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": 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": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000098943710327, "perplexity": 2724.892397400201}, "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/1634323585265.67/warc/CC-MAIN-20211019105138-20211019135138-00004.warc.gz"}
https://www.shaalaa.com/concept-notes/properties-of-whole-numbers-commutativity-property-of-whole-number_14144	Tamil Nadu Board of Secondary EducationSSLC (English Medium) Class 6th Commutative Property of Whole Numbers: In both cases, we reach 8. 5 + 3 is the same as 3 + 5. You can add two whole numbers in any order. We say that addition is commutative for whole numbers. This property is known as the Commutativity of addition. 2. Commutativity of multiplication: You will observe that 3 × 4 = 4 × 3. You can multiply two whole numbers in any order. We say multiplication is commutative for whole numbers. 3. Commutativity of Subtraction: a - b ≠ b - a a = 3, b = 5 a - b = b - a 3 - 5 = 5 - 3 - 2 ≠ 2 Thus, Subtraction is not commutative for whole numbers. 4. Commutativity of Division: a ÷ b ≠ b ÷ a a = 9 and b = 3 9/3 = 3/9 3 ≠ 0.333. Thus, Division is not commutative for whole numbers. If you would like to contribute notes or other learning material, please submit them using the button below. Shaalaa.com Commutative Property of Whole Numbers [00:08:59] S 0%	2022-05-25 19:28:58	{"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": 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": 0.4239414930343628, "perplexity": 1026.0299963430632}, "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/1652662593428.63/warc/CC-MAIN-20220525182604-20220525212604-00042.warc.gz"}
https://math.stackexchange.com/questions/1076667/in-the-monty-hall-problem-is-it-correct-to-think-the-probabilities-of-the-two-u	# In the Monty Hall problem, is it correct to think the probabilities of the two unchosen doors as shifting? I was explaining the Monty Hall problem to someone thus: You have three doors, and you pick one, giving you a $1/3$ chance of being right. The presenter opens one of the other two doors, knowing which door knows has a goat behind it. Then: 1. you would not pick the opened door as it contains a goat, and not the expensive prize. So the probability that the opened door is a winner, is now reduced from $1/3$ to $0$. 2. the remaining closed door must now have a $2/3$ chance of winning, an increase from the original $1/3$. The person insisted that I was incorrect in stating that the probabilities shift in 1, 2 above. I do not understand if the opposer is correct or not. In some sense, I understand what they were saying: that the outcome does not change the probability. But at the same time, if we label the doors as "door chosen, door opened by presenter, and door remaining", then the probabilities will always be, respectively, $1/3$, $0$, $2/3$, right? The car will be behind the door that is untouched 2/3 times, no matter what. Was I wrong to suggest that as the game is played, the probabilities of the two unchosen doors shift from $1/3$ and $1/3$ to $0$ and $2/3$? • You could rather explain the solution like this: imagine, from the get-go, that you're intent upon switching when the game show host asks you. It's easy to see that if the first door you picked was a goat, them you end up with a car. Thus the probability of winning is $2/3$ if you decide to swap. Dec 21 '14 at 15:18 • @Arthur But isn't what I asked distinctly different? I wasn't asking the probability that we win. I was asking if I was correct in saying that the probability that the car is behind the remaining door is 2/3. The person insisted that I was incorrect, and that my reassigning the probability of this door from 1/3 to 2/3 is actually illogical... Dec 21 '14 at 15:53 • That is why I wrote it as a comment, not an answer. It's a suggestion for a different explanation that your friend might accept, not a solution to your question. Dec 21 '14 at 16:02 • @Arthur I agree totally with your suggestion. Usually when I am explaining this problem to people, I try many lines of attack until one gets through. But I'm scared that I don't understand the problem correctly because I'm doubting the validity of my original explanation. Dec 21 '14 at 16:03 • @Arthur Yes, but the person said that your two choices are essentially: stay with the current door (1/3 probability), or switch to the other two doors (2/3 probability) because you aren't really choosing the only door remaining. You are in some sense choosing both doors at once, and each one has a 1/3 probability of winning, so together they have a 2/3 chance of winning. Is that what is really going on here? Dec 21 '14 at 16:15 $$P\left({\mbox{Prize is behind chosen door}}\right){}\neq{}P\left({\mbox{Prize is behind chosen door}\,\|\,\mbox{Prize is not behind door with goat}}\right)\,,$$ in general. • Such a brief response. Dec 21 '14 at 15:59 • @user46944 : Over the course of the game there are two closely related but, in general, different experiments: 1) choosing the door with the prize Without knowing where the prize is and 2) choosing the door with the prize while knowing one of the doors that does Not have the prize. The relevant question, therefore, is to know which stage of the game you are in, and then apply the appropriate conditional probability distributions. – ki3i Dec 21 '14 at 16:16 You might want to nail down what probabilities you're talking about. In the first case, it's "the probability that the prize is behind that door, GIVEN what you know now." In the second, it's the same thing, given what you know AFTER the revelation. The numbers don't change -- they are different numbers all along. Consider a simpler experiment: I have two pennies -- one an ordinary one, the other a 1909 SVDB penny worth 300. I place them behind me and grab one in each hand behind my back. I don't know which hand has which penny. I ask you "what's the probability that the valuable one is in my left hand?" You say "50%". Now I do the same thing, but this time, having grabbed the coins, I peek in my left hand and say to you that the valuable coin is there. I then ask you "What's the probability that the valuable coin is in my left hand?" Assuming you trust me, the answer is surely 100%. But as you'll observe, these are different probabilities. The tricky thing in the Monty Hall problem is that \begin{align} Pr[\text{Doorn$contains car} \| \text{door$kdoes not contain car}] \end{align} always has the value2/3$, but the value of that probability to you is nothing until you're informed that door$k$does not have the car. • So basically I wasn't wrong with my explanation? The person was absolutely convinced beyond a doubt that my explanation of the probability that the car is behind the last remaining door is 2/3 was incorrect, and that probabilities don't shift. They actually said that to believe such a thing is illogical. But I don't think so, and it doesn't seem like you think so, either... Dec 21 '14 at 15:52 • Also, I think your scenario with the coins is a tiny bit different. With the Monty Hall problem, a decision is made in the very beginning, and then one of the doors is revealed. But in your scenario, you give away information before any decision is made. Dec 21 '14 at 15:55 • Yes, you were wrong in your explanation. You said that a probability changed. What actually happened was that you were looking at two different probabilities, and calling them both by the same name ("the prob. that the car is behind door 1"). My coin-in-hand example was meant to illustrate this in a case with no ambiguity, but where "the probability that the good coin is in the left hand" clearly refers to two different probabilities, whose values are completely evident. Dec 21 '14 at 16:19 • I'm still confused. The two scenarios in the Monty Hall problem are linked by the initial choice you make. That is, the remaining probabilities in both scenarios are dependent on the first choice that you made. So why is it wrong to suggest that the remaining probabilities shift? What harm can this do? Is there some counterexample or concrete reason why this is an incorrect explanation? Dec 21 '14 at 16:23 • Done. (And perhaps the edit wasn't insignificant, either!) Glad that I was of some use, if only in pointing to ki3i's answer. :) Dec 22 '14 at 17:06 I think your explanation is interesting. I think the person who is complaining about your explanation is confused about the effect of the reveal upon the probabilities. That the initial chance you chose correctly is $$1/3$$, and the open door -- post reveal -- has 0 likelihood of being the car, I think the confusion is where did the other $$1/3$$ go? You assert, correctly, that it belongs to the third door which was not initially selected or revealed. But you are arguing with someone who may think that that 1/3 is now evenly shared between your original door and the third door. I think others have already explained the correct way to solve the game above. But, I will attempt to just point out a few things you might do to improve your explanation and convince your friend. • Approach the question from their point of view -- get them to talk about it. Do they think that there is an equal likelihood of winning the prize behind the chosen and remaining doors? • Use a visual. Sketch out a tree diagram. The visual will help your friend to see your point of view. Finally, I suppose I would point out that when you ask Was I wrong to suggest that as the game is played, the probabilities of the two unchosen doors shift from 1/3 and 1/3 to 0 and 2/3? We're all saying your result is correct. But, maybe we should also point out that there are two components to what you are saying. There is a resulting probability $$2/3$$ chance of winning if you change, which you correctly state. Then there is an interpretation of why this is, which you explain by "shifting probabilities." Maybe trying out a different way to explain the result will help convince your friend. ### EDIT: Let's be clear. Chosen the remaining door caries with it a $$2/3$$ chance of winning the car. When you say that this person is adamant about the probabilities remaining 1/3, 1/3, 1/3 at the end of the game and that you are not sure who to believe - it seems like part of your question is not just how to convince this commenter, but also how to convince yourself. ## Tree diagram Explanation Rob gave a similar/equivalent/better explanation of this above, maybe mine is slightly different, but probably not significantly. You have 1/3, 1/3, 1/3 chance of selecting each of the doors represented at the top of the diagram. You obviously can't know which door you've chosen. 2/3 of the time you'll pick one of the two incorrect doors, 1/3 of the time you'll pick the correct door. Now, the winning strategy if you picked the incorrect door is to choose the remaining door -- this is easy to see. However it's not perfect, 1/3 of the time you'll pick the correct door to begin with and changing means you lose. But the way I would explain the game is simply that since $$2/3$$ of the time you will select the wrong door to begin with, the best move is to choose the remaining door. We'd expect to win 2/3 of the time. ### Edit 2: Equivalent Statements In a comment you stated Or is the probability that the remaining door has a car still 1/3? The person suggested that even though it is more advantageous to switch because the probability that you lose is 2/3 if you stay, that doesn't imply that the remaining door has 2/3 probability of winning. The probability that the car is in one of the remaining two doors is still 2/3 (obvious), and that's why switching is better. But it doesn't mean the specific door untouched has a 2/3 probability of winning. This is nonsense. If the probability that you stay and lose is $$2/3$$, then the probability that the car is behind the only remaining door with the chance of concealing the car must be $$2/3$$. Your statement and your friend's statement are identical. To the person playing the game, you can't know which door the car is behind, and the car can't switch as the game is played - so it might seem logical to think that since the car can't move there must always be a 1/3 chance that the car is behind any door. Before we start the game this is exactly the situation -- there is a 1/3 chance that the car is behind any of the doors. After the game when we know the location of the car, we could I suppose say that the probabilities are $$\{1, 0, 0\}$$ -- two doors do not have a car, one does. Further you said that "the probability that the car is still behind one of the remaining two doors is 2/3" -- no, the probability that the car is still behind one of the remaining two doors is 1 (unless you have some reason to doubt the construct of the game to begin with). It can't be the case that the probability of the remaining door has a car is still 1/3 and that there is a 2/3 chance of winning if you change. Since (winning by changing) and (the car is behind the remaining door) are two ways to say the same thing. • Thanks for your answer! The truth is, I made a YouTube video about the Monty Hall problem, and one of the users that seemed to understand the problem very well took issue with my explanation of the probabilities changing. Specifically, I said that once the presenter opened the door, the probability of that door having a car went from 1/3 to 0/3, and that forced the probability of the remaining door to go from 1/3 to 2/3 because the probability of the initial choice being correct doesn't change. Dec 22 '14 at 3:00 • The user said that actually the probability of each door is still 1/3 because an outcome doesn't change the probability of the door. I still don't understand why my explanation would be wrong, but he was adamant that probabilities don't change based on outcomes. Since the doors each started with 1/3 probability of winning, that is how they stay even after the false door is revealed. I don't really know whom to believe anymore. Dec 22 '14 at 3:02 • I saw your edits. I agree with everything you said. I just don't know why that user disagreed with me to the point that they shook the foundations of my understanding (as it seems you also say, what the person said doesn't make any sense!). A lot of people answered this question trying to give me tips on how to explain the problem better, but that wasn't what I was asking at all. Thank you for answering my real question. Dec 22 '14 at 12:52 • By the way, when I said "the probability that the car is behind one of the two remaining doors is 2/3", I meant that the two remaining doors represents the two doors not chosen by the player at first. Dec 22 '14 at 13:11 • @user46944 ok, we'll again that's logically equivalent to saying that the probability that the car is behind the single unopened remaining door is$2/3$. Dec 22 '14 at 16:40 I think it is a little confusing to talk of the probabilities changing. However, at the point that the presenter opens the door and reveals a goat, the probabilities of your chosen door, the open door and the other door being the one with the car are 1/3, 0 and 2/3 as you say. It is more usual to talk about conditional probabilities: what is the probability of one event given that some other event has occurred? You can avoid talking about conditional probabilities altogether by explaining the Monty Hall problem like this: if you picked the car initially, then switching makes you lose, while if you picked a goat initially, then switching makes you win. The probability of picking the car initially is$\frac{1}{3}$and the probability of picking a goat initially is$\frac{2}{3}$, but switching interchanges the outcomes and hence interchanges these probabilities. In a nutshell, interchanging success and failure in a situation where failure is originally more likely than success is a winning strategy. • Thanks for the alternate explanation, but the point of my question is actually whether or not my talk of probabilities changing is incorrect (illogical), or if there is nothing wrong with the validity of the explanation, regardless of how helpful it might be to someone. Dec 21 '14 at 17:45 • As I said, I think it is confusing to talk about the probabilities changing. However, at the point that the presenter opens the door and reveals a goat, the probabilities of your chosen door, the open door and the other door being the one with the car are$1/3$,$0$and$2/3$as you say. Dec 21 '14 at 20:06 • Confusing or not, I'm questioning the validity of the statement. Is it logical to say that the probabilities shift? If not, why not? Dec 21 '14 at 20:37 • The validity of what statement? Dec 21 '14 at 23:20 You can make sense of probabilities "shifting". What you are doing here is assigning two probability measures; the first one$P_0$might assign probabilities such as •$P_0(\text{car behind door #2}) = 1/3$•$P_0(\text{car behind door #1 and Monty opens door #3}) = 1/6$and the second probability measure$P_1$assigns •$P_1(\text{car behind door #2}) = 2/3$•$P_1(\text{car behind door #1 and Monty opens door #3}) = 0$While it is reasonable to say that the probabilities of events change when switching from$P_0$to$P_1$, do take care to note that$P_0(\text{car behind door #2})$does not change. Now, this is not normally how people use probability theory to account for new information; instead they they use conditional probabilities. In fact, your$P_1$can be expressed in terms of$P_0$: $$P_1(\text{some event}) = P_0(\text{some event} \mid \text{Monty opens door #2})$$ So the typical way to mathematically model the situation only ever uses a single way to assign probabilities to events; they work with$P_0\$ throughout the entire problem. What changes with new information is which probabilities we are interested in. In particular, we expect the actual outcomes to be governed by the conditional probabilities. First off, it is very wrong to say that probability "shifts." What really happens should be a three-step solution that, in general, works like this: 1. You start with a set of events {E1,E2,E3,E4,...EN} whose probabilities add up to 1. That is, P1+P2+P3+P4+...+PN=1. 2. Some of these events get eliminated by information you learn. The probabilities of what remains add up to less than 1. For example, eliminating E2 and E3, we get P1+P4+...+PN=X<1. 3. Since the updated probabilities need to add up to 1 again, we divide each of them by X: P1'=P1/X, P4'=P4/X, etc. This can be called "renormalization." In the classic explanation for the Monty Hall problem, we have: • E1 = "Car behind your door"; P1=1/3 • E2 = "Car behind the next higher door, wrapping to #1 if necessary"; P2=1/3 • E3 = "Car behind the next lower door, wrapping to #3 if necessary"; P3=1/3 Say E3 is eliminated as a possibility when Monty Hall opens a door. Then: • P1' = P1/(P1+P2) = (1/3)/[(1/3)+(1/3)] = 1/2 • P2' = P1/(P1+P2) = (1/3)/[(1/3)+(1/3)] = 1/2 Wait a minute, that's wrong. But then, so is the classic explanation for the correct set of answers, P1'=1/3 and P2'=2/3. It says that P1'=P1 and P2'=P2+P3 (that is, that probability "shifts"). Any true mathematician should be embarrassed to offer such an explanation. Yes, it gets the right answers, but for entirely wrong reasons. And using the classic explanation is one of the main reasons the Monty Hall Problems is still controversial. It fails in two very obvious ways: it doesn't explain what is wrong with the solution I just gave; and it is less mathematically sound since it implies the improper concept of "shifting" probability. Why should a mathematical novice accept an explanation that has these obvious faults? The correct solution, following the trivial pattern I stated above, needs four events. It is clearer if I list them backwards: 1. E4 = "Car behind the next lower door. Monty Hall opens the next higher door."; P4=1/3. 2. E3 = "Car behind the next higher door. Monty Hall opens the next lower door." P3=1/3. 3. E2 = "Car is behind your door, and Monty Hall opens the next higher door." P2=1/6. 4. E1 = "Car is behind your door, and Monty Hall opens the next lower door." P1=1/6. Then the solution is that Monty Hall opened only one door - say it is the next higher one. This eliminates E2 and E4. • P1' = P1/(P1+P3) = (1/3)/[(1/3)+(1/6)] = 2/3. • P3' = P3/(P1+P3) = (1/6)/[(1/3)+(1/6)] = 1/3. This gets the same answer as the "probability shifts" explanation, but with no shifting, and it shows what is wrong with the previous solution. The information you learn does not "eliminate" every game where there is a goat behind a certain door, it eliminates only those where that certain door was opened. Since another door can get opened even when that certain door has a goat, the two events are not the same. The big question that may convince your friend: Are you counting the possibility that the host will open a door to show you a car? If he doesn't know what's behind the doors, this is a real possibility. (Also, are you allowed to switch doors if he shows you a car? It's not in the rules. Does it even matter?) In probability, a "sample space" is (roughly speaking) the set of all possible outcomes that you are considering. A fair coin has 1/2 chance of showing heads and 1/2 chance of showing tails. You ignore all other possibilities (the coin lands on its edge; the coin falls into a storm drain and cannot be read; etc.). If the host knows what's behind the doors and avoids ever showing you a car behind a door you have not chosen, then he has given you information, and the odds change accordingly. You get the same result of change in odds (for a statistical analysis after the fact) if the host knows nothing, but you discard all occasions when he "accidentally" shows you a car.	2022-01-16 11:17:04	{"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": 8, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7338494658470154, "perplexity": 739.834516180859}, "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/1642320299852.23/warc/CC-MAIN-20220116093137-20220116123137-00418.warc.gz"}
https://scriptinghelpers.org/questions/12465/how-do-i-instance-a-intvalue	-2 # How do i instance a intvalue? [closed] im trying to make an rpg and for the exp i am using and intvalue and i cant seem to get it to go into the player game.Players.PlayerAdded:connect(function() local expe = Instance.new("IntValue") expe.Parent = game.Players.LocalPlayer expe.Value = 0 end) 0 its actually more organized than that i dont know what happened reloce 0 — 7y ### Locked by TheeDeathCaster, Redbullusa, and BlueTaslem This question has been locked to preserve its current state and prevent spam and unwanted comments and answers. 1 Remember, LocalPlayer can only be accessed by a LocalScript, and you have another error; Other then the LocalPlayer, you left out something in your PlayerAdded event; the Player, you forgot to specify the Player in the function/event, let's fix up your script; game.Players.PlayerAdded:connect(function(plr) --'plr' here will specify the Player that has joined the server/game local expe = Instance.new("IntValue") --Correct; this will create a new Instance [in this case, it will create the IntValue] expe.Parent = plr --This will set the Parent of 'expe' to 'plr' [the Player] expe.Value = 0 --This will set 'expe''s Value to 0 end) --This ends the code block for the event/function Hope this helped! 0 game.Players.PlayerAdded:connect(function() local expe = Instance.new("IntValue") expe.Parent = game.Players.LocalPlayer expe.Value = 0 end) To organize a Script, you need to put Code Blocks in, or press the Lua button before answering or making a post, the one in a blue bubble. As for your question.. it looks right to me, but I can't be sure. game.Workspace.Players.PlayerAdded:connect(function(plr)	2022-08-08 01:27:51	{"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": 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": 0.398630291223526, "perplexity": 5034.008537175763}, "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/1659882570741.21/warc/CC-MAIN-20220808001418-20220808031418-00279.warc.gz"}
http://mathhelpforum.com/calculus/140777-properties-summations.html	# Math Help - Properties of Summations 1. ## Properties of Summations Hello, I'm working through a proof of the coefficient of linear regression (r) from its verbose form to its concise one. I realize the concept is statistics, but the the proof seems more algebra and possibly calculus-based. Verbose Form: $ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{n\sum x^2 - (\sum x)^2} \sqrt{n\sum y^2 - (\sum y)^2}} $ Concise Form: $ r = \frac{\sum (z_{x} z_{y})}{n-1} $ It seems easiest to work backward from the Concise Form to the Verbose one, and to do so I'm using the following definitions: $ z_{x} = \frac{x - \bar{x}}{s_{x}} $ $ \bar{x} = \frac{\sum x}{n} $ $ s_{x} = \sqrt{\frac{n\sum x^2 - (\sum x)^2}{n(n-1)}} $ Doing basic substitution in the Verbose Form I get (didn't substitute for $s_{x}$ or $s_{y}$ to keep some semblance of readability): $ r = \frac{\sum (\frac{x - \frac{\sum x}{n}}{s_{x}} * \frac{y - \frac{\sum y}{n}}{s_{y}})}{n-1} $ I'm kind of stuck on what to do with the numerator, which seems to result in distributing a summation to other summations. Since x and y are two "paired" sets n will be the same for all summations and also a constant. To simplify my question: Am I able to distribute the $\sum$ like so?: $\sum (\frac{x - \frac{\sum x}{n}}{s_{x}})$ -> $\sum(\frac{\frac{nx - \sum x}{n}}{s_{x}})$ -> $\frac{\frac {n\sum x - \sum \sum x}{n}}{\sum s_{x}}$ If so, what would the term $\sum \sum x$ resolve to? Note: I don't want anyone to solve the proof here, I'm just trying to understand how I might be able to resolve the summations. I'd like to work through the proof myself to understand how it works. Whew, okay first time using LaTeX that took a lot out of me... 2. Originally Posted by jstandard Hello, I'm working through a proof of the coefficient of linear regression (r) from its verbose form to its concise one. I realize the concept is statistics, but the the proof seems more algebra and possibly calculus-based. Verbose Form: $ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{n\sum x^2 - (\sum x)^2} \sqrt{n\sum y^2 - (\sum y)^2}} $ Concise Form: $ r = \frac{\sum (z_{x} z_{y})}{n-1} $ It seems easiest to work backward from the Concise Form to the Verbose one, and to do so I'm using the following definitions: $ z_{x} = \frac{x - \bar{x}}{s_{x}} $ $ \bar{x} = \frac{\sum x}{n} $ $ s_{x} = \sqrt{\frac{n\sum x^2 - (\sum x)^2}{n(n-1)}} $ Doing basic substitution in the Verbose Form I get (didn't substitute for $s_{x}$ or $s_{y}$ to keep some semblance of readability): $ r = \frac{\sum (\frac{x - \frac{\sum x}{n}}{s_{x}} * \frac{y - \frac{\sum y}{n}}{s_{y}})}{n-1} $ I'm kind of stuck on what to do with the numerator, which seems to result in distributing a summation to other summations. Since x and y are two "paired" sets n will be the same for all summations and also a constant. To simplify my question: Am I able to distribute the $\sum$ like so?: $\sum (\frac{x - \frac{\sum x}{n}}{s_{x}})$ -> $\sum(\frac{\frac{nx - \sum x}{n}}{s_{x}})$ -> $\frac{\frac {n\sum x - \sum \sum x}{n}}{\sum s_{x}}$ If so, what would the term $\sum \sum x$ resolve to? Note: I don't want anyone to solve the proof here, I'm just trying to understand how I might be able to resolve the summations. I'd like to work through the proof myself to understand how it works. Whew, okay first time using LaTeX that took a lot out of me... For your double sum, what is the indexing? 3. It actually doesn't list the indexing in the formula definition (at least in the text I'm using), just $\sum$ I would guess it would have to be: $\sum_{i=1}^{n} \sum_{i=1}^{n} x_{i}$ since all of the calculations are done for 2 sets of values, both sets having n number of values (since they're paired). I should mention, I'm not even certain if my formula progression is correct and I'm able to "distribute" the summation in that manner. 4. The sum $\sum_{i=1}^nx_i$ has only n as a free variable. Therefore the sum $\sum_1^n\sum_{i=1}^n x_i$ is just $n \sum_{i=1}^n x_i$. But I don't think that's what this sum actually is. Note that you may NOT (!!) distibute a sum across a quotient vis. $\sum \frac ab \neq {\sum a \over \sum b}$, which it looks like you've done. I'm confused about what the summation is over in $s_x$. There do not appear to be any free variables under the summation sign. Thanks, that makes sense about the not being able to distribute the sum across a fraction. For the summation in the denom, $s_{x} = \sqrt{\frac{n\sum_{i=1}^n x_i^2 - (\sum_{i=1}^n x_i)^2}{n(n-1)}}$ 6. Ah, I see. What text are you working out of? 7. The text is: Elementary Statistics 11th ed, Mario F. Triola. All of the summations in the book are listed simply as $\sum$ without any indexing. The original formulas are the ones I have above in the "Concise" and "Verbose" forms, where I'm trying to track the proof from "Concise" to "Verbose" using whatever manipulations possible. I should mention I spoke with a tutor at school who also came to the conclusion that the double summation term resolves to $n \sum_{i=1}^n x_i$, but that presents a problem because then you'd have: $n\sum_{i=1}^n x_i - \sum _{i=1}^n \sum _{i=1}^n x_i$ -> $n \sum_{i=1}^n x_i - n \sum_{i=1}^n x_i$ -> = 0. Eliminating the terms you need in the numerator of the "Verbose" form of the equation. Ha, sorry, I realize the "medium" of communication for this isn't necessarily the greatest, so let me know if I'm not doing a good job explaining things. I'm somewhat in new territory here.	2016-05-30 14:37:29	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 38, "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": 0.9033923745155334, "perplexity": 459.9324052074774}, "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/1464051035374.76/warc/CC-MAIN-20160524005035-00096-ip-10-185-217-139.ec2.internal.warc.gz"}
https://physics.stackexchange.com/questions/288798/mixed-states-and-matrix-representation	# Mixed states and matrix representation On page 47 of Quantum Machine Learning you can read that: Interference terms-the off-diagonal elements-are present in the density matrix of a pure state (Equation 3.10), but they are absent in a mixed state (Equation 3.11) Where 3.10 is a pure state on $\|\psi> = a\|0>+b\|1>$ $\rho = \|\psi><\psi\|= \begin{bmatrix}\|a\|^2&ab^\\a^b&\|b\|^2\end{bmatrix}$ and 3.11, described as a sum of projectors: $\rho_{mixed} = \|a\|^2 \|0><0\| + \|b\|^2 \|1><1\| = \begin{bmatrix}\|a\|^2&0\\0&\|b\|^2\end{bmatrix}$ He is saying that the matrices for mixed states are purely diagonal. I do not agree with what he said, especially because looking at the definition of mixed state (on all the books I found), to build one, I can use any old state that I want, and thus on off diagonal elements I can have non zero values. Can anyone clarify better what is written there, and why? If I’m not wrong, that quote is true only if you put in the density matrix of the mixed states only states that are basis for that space (so you have only diagonal matrices in the summation, as in the example). Thank you! • could you also include equations 3.10 and 3.11? – glS Oct 25 '16 at 16:35 • Please make your post self-contained. Don't expect anyone to go that other site to get your whole query. – user36790 Oct 25 '16 at 16:35 • The statement refers to a very specific example for which it is correct (indeed, if the density matrix of a pure state is diagonal, then it cannot have more than one nonzero element on the diagonal). However, I find this section very convoluted and not very enlightening. – Martin Oct 25 '16 at 16:42 • The way to express bra-ket notation in latex (in the absence of the "physics" package) is \langle ... \| ... \rangle. – Adomas Baliuka Oct 25 '16 at 20:14	2019-07-19 21:13:02	{"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": 0.7243043780326843, "perplexity": 396.7295256917885}, "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/1563195526359.16/warc/CC-MAIN-20190719202605-20190719224605-00123.warc.gz"}
https://cs184-firesim.github.io/final-report/	# Fire Simulation and Rendering ## Authors: Eric Ge, Debbie Liang, Ziyao Zhang In this project, we achieved real-time simulation and rendering of fire effects in both 2D and 3D. Our implementation utilized the Unity engine and adopted popular techniques in noise generation, physical simulation, and volumetric rendering. Our 2D fire is implemented with panning Voronoi noise and Perlin noise in Unity's built-in shader graph, while our 3D fire relies on custom fluid-simulation and ray-marching HLSL shaders. We measured our rendering effects and performance on an NVIDIA 2060 graphics card and obtained real-time results. # Technical Approach ## 2D Fire We multiplied Voronoi noise and Perlin noise textures and interpolated their product with a standard uv texture. We then used this modified uv to sample a single Gaussian blob to create fire-like variations. Lastly, we applied a color gradient to give an orange color to the result. ## 3D Fire ### Simulation We decided to use 3D textures to hold different attributes that describe a fire. In our simulation, we calculated the corresponding values for velocity, smoke density, temperature, and fuel level in each cell of a 3D voxelized grid. We based our simulation off the Navier Stokes equation for fluid dynamics: $$\frac{\partial u}{\partial t} = -(u \cdot \nabla) u - \frac{1}{\rho} \nabla p + f$$ In the formula, $$u$$ is the velocity, $$p$$ is the pressure, $$\rho$$ is the density of the molecule mass, and $$f$$ is the external force on molecules. The first term represents the advection, which is the velocity of a fluid that causes the fluid to transport objects, densities, and other quantities along with the flow. The term inside the parenthesis is the divergence, which represents the rate at which density exits the region. The second term simulates how pressure gathers and generates force and provides accelerations to the surrounding molecules. For each time step, we compute the advection and divergence based on the interaction between neighboring voxels, and propagate the changes in temperature, pressure, and velocity correspondingly. We then update the values in each 3D texture and render the resulting volume to the screen with the rendering pipeline. For the external force field of the Navier Stokes equation, we simulated both buoyant force and vorticity force using the following equations: $$f_{buoyancy} = \frac{Pmg}{R} (\frac{1}{T_0} - \frac{1}{T})z$$ $$f_{vc} = \epsilon(\Psi \times \omega)\delta x$$ $$\Psi = \frac{\eta}{\|\eta\|}$$ $$\eta = \nabla \|\omega\|$$ The buoyant force is influenced by temperature and density, and it changes the velocity of molecules to make the simulation more realistic. With a higher temperature, the molecules will rise with a larger velocity. The other force that we applied to our molecules is vorticity force. This force helps us restore some of the curling behavior of smoke that was lost due to the discrete nature of the simulation. Below is the simulation pipeline that we used. It outputs the values in a set of 3D Textures. ### Rendering To render the generated fire, we start by injecting a shader into the post-processing stage of the main camera. Then, we perform ray marching to sample the simulation result passed in as 3D textures. Ray marching is a GPU-friendly technique for sampling a 3D volume along a ray. Specifically, we generate rays starting at the center of the camera in the direction of each pixel. Then, we calculate whether the ray hits our target volume. If it doesn't, then the value we are trying to sample must be 0. Otherwise, we march through the volume in small steps and sample the volume at each step. The average of the samples weighted by the length of the ray inside of the target volume is the value we will use to color that pixel. Without accounting for embers, we sample two textures from the simulation result: density and reaction coordinate (fuel). The density value is used to determine the thickness of the smoke, while the reaction coordinate is used to determine the color of the flame. We map the value of the reaction coordiante to a gradient specified by three colors: the core, the border, and the smoke. The end result is a realistic-looking fire: Our smoke rendering pipeline borrows exsiting literature [8] on how to render volumetric clouds. Since clouds are comparably brighter and more misty than smoke, we multiply the end result by the smoke color for a more realistic look. In the equation below, $$d$$ is the density, $$p$$ is the precipitation constant (not used here), $$\theta$$ is the angle between the camera ray and the sun, and $$g$$ is the eccentricity that defaults to $$0.2$$. ### Special Effects We also added an additional ember effect to the top of the flames. Ember effect are distinct spurts of burning particles that are cause by incompletely combusted fuel that are advected into the air. This special effect is integrated into our physical model by simulating a additional Voxel space of ember particles that are randomly generated at the bottom of the fire with Perlin noise and propelled upwards through advection. Our raymarching procedure detects these particles and add a small glowing aura around them to make them appear realistic. # Results We experimented our simulation and rendering procedure on an NVIDIA RTX 2060 graphics card and achieved real-time (60+ fps) results. We set our voxel grid size to 128128128. For our GPU kernels, we used 161616 GPU blocks, each containing 512 threads. For a view of our full simulation and rendering results, we have put together a demo video below: # Problems & Lessons Learned The main difficulty of our implementation came from debugging the propagation of the attributes across timestamps. HLSL shaders typically don't accommodate print debugging so we created a script in Python to cross-validate our formulae. It turned out that the values propagated as expected, which proved that our formulae was correct. We then created a debugging shader, which fills positions in a 3D texture when the value that we are testing passes a certain threshold. We also experimented with debuggers such as RenderDoc and eventually managed to peak inside of our shaders by setting breakpoints. An interesting quirk of HLSL compute shaders is that a global variables cannot be assigned within the shader file. However, using an assignment operator to declare such variables does not result in a syntax error. We encountered a problem where the values of velocity and pressure remained zero despite our advection kernel changing them at every time step. Using a debug shader, we eventually located the problem as being caused by a zero time-step variable seemingly initialized to a non-zero value at declaration. Manually assigning this variable in our C# driver script before each invocation of the compute kernels solved this problem. # References 2. Real-Time Simulation and Rendering of 3D Fluids, Keenan Crane, Ignacio Llamas, Sarah Tariq. [link] 3. Fast Fluid Dynamics Simulation on the GPU, Mark J. Harris. GPU Gems, Chapter 38. University of North Carolina at Chapel Hill. [link] 8. Real-time Volumetric Cloudscapes. Andrew Schneider. GPU Pro 7. [link]	2020-07-13 12:15:54	{"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": 2, "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": 0.3722319006919861, "perplexity": 1380.5667041818926}, "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/1593657143365.88/warc/CC-MAIN-20200713100145-20200713130145-00377.warc.gz"}
http://en.wikisource.org/wiki/Elements_of_the_Differential_and_Integral_Calculus/Chapter_V_part_3	# Elements of the Differential and Integral Calculus/Chapter V part 3 Elements of the Differential and Integral Calculus by William Anthony GranvilleChapter V, § 54–61 54. Differentiation of $\operatorname{vers} \ v$. Let $\ y$ $= \operatorname{vers}\ v\$. By Trigonometry this may be written $\ y$ $= 1 - \cos v\$. Differentiating, $\frac{dy}{dx}$ $= \sin v \frac{dv}{dx}$. XVII ∴ $\frac{d}{dx}(\operatorname{vers} v)$ $= \sin v \frac{dv}{dx}$. In the derivation of our formulas so far it has been necessary to apply the General Rule, p. 29 [§ 31] (i.e. the four steps), only for the following: III $\frac{d}{dx}(u + v - w)$ $= \frac{du}{dx} + \frac{dv}{dx} - \frac{dw}{dx}$ Algebraic sum. V $\frac{d}{dx}(uv)$ $= u \frac{dv}{dx} + v \frac{du}{dx}$. Product. VII $\frac{d}{dx} \left ( \frac{u}{v} \right )$ $= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$. Quotient. VIII $\frac{d}{dx}(\log_a v)$ $= \log_a e \frac{\frac{dv}{dx}}{v}$. Logarithm. XI $\frac{d}{dx}(\sin v)$ $= \cos v \frac{dv}{dx}$ Sine. XXV $\frac{dy}{dx}$ $= \frac{dy}{dv} \cdot \frac{dv}{dx}$. Function of a function. XXVI $\frac{dy}{dx}$ $= \frac{1}{\frac{dx}{dy}}$. Inverse functions. Not only do all the other formulas we have deduced depend on these, but all we shall deduce hereafter depend on them as well. Hence it follows that the derivation of the fundamental formulas for differentiation involves the calculation of only two limits of any difficulty, viz., $\lim_{v \to 0} \frac{\sin v}{1}$ $= 1\$ by § 22, p. 21 and $\lim_{v \to 0}(1 + v)^{\frac{1}{v}}$ $= e\$. By § 23, p. 22 EXAMPLES Differentiate the following: 1. $y = sin ax^2\$. $\frac{dy}{dx}$ $= \cos ax^2 \frac{d}{dx}(ax^2)$ by XI [$v = ax^2$.] 2. $y = \tan \sqrt{1 - x}$. $\frac{dy}{dx}$ $= \sec^2 \sqrt{1 - x} \frac{d}{dx}(1 - x)^{\frac{1}{2}}$ by XIII [$v = \sqrt{1 - x}$.] $= \sec^2 \sqrt{1 - x} \cdot \frac{1}{2} (1 - x)^{-\frac{1}{2}}(-1)$. $= -\frac{\sec^2 \sqrt{1 - x}}{2\sqrt{1 - x}}$. 3. $y = \cos^3 x$. This may also be written, $y$ $= (\cos x)^3$. $\frac{dy}{dx}$ $= 3(\cos x)^2 \frac{d}{dx}(\cos x)$ by VI [$v = \cos x$ and $n = 3$.] $= 3 \cos^2 x (-\sin x)$ by XII $= -3 \sin x \cos^2 x$ 4. $y = \sin nx \sin^n x$. $\frac{dy}{dx}$ $= \sin nx \frac{d}{dx}(\sin x)^n + \sin^n x \frac{d}{dx}(\sin nx)$ by V [$v = \sin nx$ and $v = \sin^n x$.] $= \sin nx \cdot n(\sin x)^{n - 1} \frac{d}{dx}(\sin x) + \sin^n x \cos nx \frac{d}{dx}(nx)$ by VI and XI $= n \sin nx \cdot \sin^{n - 1} x \cos x + n \sin^n x \cos nx$ $= n \sin^{n - 1} x(\sin nx \cos x + \cos nx \sin x)$ $= n \sin^{n - 1} x \sin(n + 1) x$. 5. $y = \sec ax$. Ans. $\frac{dy}{dx}$ $a \sec ax \tan ax$. 6. $y = \tan(ax + b)$. $\frac{dy}{dx}$ $= a \sec^2 (ax + b)$. 7. $s = \cos 3 ax$. $\frac{ds}{dx}$ $= -3a \sin 3 ax$. 8. $s = \cot(2t^2 + 3)$. $\frac{ds}{dt}$ $= -4 t \csc^2 (2t^2 + 3)$. 9. $f(y) = \sin 2y \cos y$. $f'(y)$ $= 2 \cos 2y \cos y - \sin 2y \sin y$. 10. $F(x) = \cot^2 5x$ $F'(x)$ $= -10 \cot 5x \csc^2 5x$. 11. $F(\theta) = \tan \theta - \theta$. $F'(\theta)$ $= \tan^2 \theta$. 12. $f(\phi) = \phi \sin \phi + \cos \phi$ $f'(\phi)$ $= \phi \cos \phi$. 13. $f(t) = \sin^3 t \cos t$ $f'(t)$ $= \sin^2 t(3\cos^t - \sin^2 t)$. 14. $r = a \cos 2\theta$. $\frac{dr}{d\theta}$ $= -2a \sin 2\theta$. 15. $\frac{d}{dx} \sin^2 x = \sin 2x$. 16. $\frac{d}{dx} \cos^3 x^2 = -6x \cos^2 x^2 \sin x^2$. 17. $\frac{d}{dt} \csc \frac{t^2}{2} = -t \csc \frac{t^2}{2} \cot \frac{t^2}{2}$. 18. $\frac{d}{ds} a \sqrt{\cos 2s} = -\frac{a \sin 2s}{\sqrt{\cos 2s}}$. 19. $\frac{d}{d\theta} a(1 - \cos \theta) = a sin \theta$. 20. $\frac{d}{dx}(\log \cos x) = -\tan x$. 21. $\frac{d}{dx}(\log \tan x) = \frac{2}{\sin 2x}$. 22. $\frac{d}{dx}(\log \sin^2 x) = 2 \cot x$. 23. $\frac{d}{dt} \cos \frac{a}{t} = \frac{a}{t^2} \sin \frac{a}{t}$. 24. $\frac{d}{d\theta} \sin \frac{1}{\theta^2} = -\frac{2}{\theta^3} \cos \frac{1}{\theta^2}$. 25. $\frac{d}{dx} e^{\sin x} = e^{\sin x} \cos x$. 26. $\frac{d}{dx} \sin(\log x) = \frac{\cos(\log x)}{x}$. 27. $\frac{d}{dx} \tan(\log x) = \frac{\sec^2(\log x)}{x}$. 28. $\frac{d}{dx} a \sin^3 \frac{\theta}{3} = a \sin^2 \frac{\theta}{3} \cos \frac{\theta}{3}$. 29. $\frac{d}{d\alpha} \sin(\cos \alpha) = -\sin \alpha \cos(\cos \alpha)$. 30. $\frac{d}{dx} \frac{\tan x - 1}{\sec x} = \sin x + \cos x$. 31. $y = \log \sqrt{ \frac{1 + \sin x}{1 - \sin x} }$. $\frac{dy}{dx}$ $= \frac{1}{\cos x}$. 32. $y = \log \tan \left ( \frac{\pi}{4} + \frac{x}{2} \right )$. $\frac{dy}{dx}$ $= \frac{1}{\cos x}$. 33. $f(x) = \sin(x + a) \cos(x - a)$ $f'(x)$ $= \cos 2x$. 34. $y = a^{\tan nx}$. $y'$ $= na^{\tan nx} \sec^2 nx \log a$. 35. $y = e^{\cos x} \sin x$. $\ y'$ $= e^{\cos x} (\cos x - \sin^2 x)\$. 36. $y = e^x \log \sin x$. $y'$ $= e^x(\cot x + \log \sin x)$. 37. Differentiate the following functions: (a) $\frac{d}{dx} \sin 5x^2$. (f) $\frac{d}{dx} \csc(\log x)$. (k) $\frac{d}{dt} e^{a - b\cos t}$. (b) $\frac{d}{dx} \cos(a - bx)$. (g) $\frac{d}{dx} \sin^3 2x$ (l) $\frac{d}{dt} \sin \frac{t}{3} \cos^2 \frac{t}{3}$. (c) $\frac{d}{dx} \tan \frac{ax}{b}$. (h) $\frac{d}{dx} \cos^2(\log x)$. (m) $\frac{d}{d\theta} \cot \frac{b}{\theta^2}$. (d) $\frac{d}{dx} \cot \sqrt{ax}$. (i) $\frac{d}{dx} \tan^2 \sqrt{1 - x^2}$. (n) $\frac{d}{d\phi} \sqrt{1 + \cos^2 \phi}$. (e) $\frac{d}{dx} \sec e^{3x}$. (j) $\frac{d}{dx} \log(\sin^2 ax)$. (o) $\frac{d}{ds} \log \sqrt{1 - 2\sin^2 s}$. 38. $\frac{d}{dx}(x^n e^{\sin x}) = x^{n - 1} e^{\sin x} (n + x\cos x)$. 39. $\frac{d}{dx} (e^{ax} \cos mx) = e^{ax}(a \cos mx - m \sin mx)$. 40. $f(\theta) = \frac{1 + \cos \theta}{1 - \cos \theta}$. $\ f'(\theta)$ $=\ -\frac{2 \sin \theta}{(1 - \cos \theta)^2}$. 41. $f(\phi) = \frac{e^{a\phi}(a \sin \phi - \cos \phi)}{a^2 + 1}$. $\ f'(\phi)$ $=\ e^{a\phi} \sin \phi$. 42. $\ f(s) = (s \cot s)^2$. $\ f'(s)$ $=\ 2s \cot s (\cot s - s \csc^2 s)$. 43. $r = \frac{1}{3} \tan^3 \theta - \tan \theta + \theta$. $\frac{dr}{d\theta}$ $= \tan^4 \theta$. 44. $y = x^{\sin x}\$. $\frac{dy}{dx}$ $=\ x^{\sin x} \left ( \frac{\sin x}{x} + \log x \cos x \right )$. 45. $y = (\sin x)^x\$. $\ y'$ $=\ (\sin x)^x [ \log \sin x + x \cot x]$. 46. $y = (\sin x)^{\tan x}\$. $\ y'$ $=\ (\sin x)^{\tan x} (1 + \sec^2 x \log \sin x)$. 47. Prove $\frac{d}{dx} \cos v = -\sin v \frac{dv}{dx}$, using the General Rule. 48. Prove $\frac{d}{dx} \cot v = -\csc^2 v \frac{dv}{dx}$ by replacing $\cot v$ by $\frac{\cos c}{\sin v}$. 55. Differentiation of $\arcsin v$. Let $\ y$ $= \arcsin\ v$;[1] then $\ v$ $= \sin\ y$. Differentiating with respect to $y$ by XI, $\frac{dv}{dy}$ $= \cos\ y$; therefore $\frac{dy}{dv}$ $= \frac{1}{\cos y}$. By (C), p. 46 [§ 43] But since $v$ is a function of $x$, this may be substituted in $\frac{dy}{dx}$ $= \frac{dy}{dv} \cdot \frac{dv}{dx}$ (A), p. 45 [§ 42] giving $\frac{dy}{dx}$ $= \frac{1}{\cos y} \cdot \frac{dv}{dx}$. $= \frac{1}{\sqrt{1 - v^2}} \frac{dv}{dx}$. $[ \cos y = \sqrt{1 - \sin^2 y} = \sqrt{1 - v^2}$, the positive sign of the radical being taken, since $\cos y$ is positive for all values of $y$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ inclusive.] XVIII ∴ $\frac{d}{dx}(\arcsin v)$ $= \frac{\frac{dv}{dx}}{\sqrt{1 - v^2}}$. 56. Differentiation of $\arccos v$. Let $\ y$ $= \arccos\ v$;[2] then $\ y$ $= \cos\ y$. Differentiating with respect to $y$ by XII, $\frac{dv}{dy}$ $= -\sin\ y$. therefore $\frac{dy}{dv}$ $= -\frac{1}{\sin y}$. By (C), p. 46 [§ 43] But since $v$ is a function of $x$, this may be substituted in the formula $\frac{dy}{dx}$ $= \frac{dy}{dv} \cdot \frac{dv}{dx}$, (A), p. 45 [§ 42] giving $\frac{dy}{dx}$ $= -\frac{1}{\sin y} \cdot \frac{dv}{dx}$ $= - \frac{1}{\sqrt{1 - v^2}} \frac{dv}{dx}$. [ $\sin y = \sqrt{1 - \cos^2 y} = \sqrt{1 - v^2}$, the plus sign of the radical being taken, since $\sin y$ is positive for all values of y between 0 and π inclusive.] XIX ∴ $\frac{d}{dx}(\arccos v)$ $= -\frac{\frac{dv}{dx}}{\sqrt{1 - v^2}}$. 57. Differentiation of $\arctan v$. Let $\ y$ $=\ \arctan v$;[3] then $\ y$ $=\ \tan y$. Differentiating with respect to $y$ by XIV, $\frac{dv}{dy}$ $=\ \sec^2 y$; therefore $\frac{dy}{dv}$ $= \frac{1}{\sec^2 y}$. By (C), p. 46 [§ 43] But since $v$ is a function of $x$, this may be substituted in the formula $\frac{dy}{dx}$ $= \frac{dy}{dv} \cdot \frac{dv}{dx}$, (A), p. 45 [§ 42] giving $\frac{dy}{dx}$ $= \frac{1}{\sec^2 y} \cdot \frac{dv}{dx}$ $= \frac{1}{1 + v^2} \frac{dv}{dx}$. [$\sec^2 y = 1 + \tan^2 y = 1 + v^2$] XX ∴ $\frac{d}{dx} (\arctan v)$ $= \frac{\frac{dv}{dx}}{1 + v^2}$ 58. Differentiation of $\arccot u$.[4] Following the method of the last section, we get XXI $\frac{d}{dx}(\arccot v) = -\frac{\frac{dv}{dx}}{1 + v^2}$. 59. Differentiation of $\arcsec u$. Let $\ y$ $=\ \arcsec v$;[5] then $\ v$ $=\ \sec y$. Differentiating with respect to $y$ by IV, $\frac{dv}{dy}$ $=\ \sec y \tan y$; therefore $\frac{dy}{dv}$ $= \frac{1}{\sec y \tan y}$ By (C), p. 46 [§ 43] But since $v$ is a function of $x$, this may be substituted in the formula $\frac{dy}{dx}$ $= \frac{dy}{dv} \cdot \frac{dv}{dx}$, (A), p. 45 [§ 42] giving $\frac{dy}{dx}$ $= \frac{1}{\sec y \tan y} \frac{dv}{dx}$ $= \frac{1}{v \sqrt{v^2 - 1}} \frac{dv}{dx}$. [$\sec y = v$, and $\tan y = \sqrt{\sec y - 1} = \sqrt{v^2 - 1}$, the plus sign of the radical being taken, since $\tan y$ is positive for an values of $y$ between 0 and $\frac{\pi}{2}$ and between $-\pi$ and $-\frac{\pi}{2}$, including 0 and $-\pi$]. XXII ∴ $\frac{d}{dx} (\arcsec v)$ $= \frac{\frac{dv}{dx}}{v \sqrt{v^2 - 1}}$. 60. Differentiation of $\arccsc v$.[6] Let $\ y$ $=\ \arccsc v$; then $\ v$ $=\ \csc y$. Differentiating with respect to $y$ by XVI and following the method of the last section, we get XXIII $\frac{d}{dx}(\arccsc v)$ $= -\frac{\frac{dv}{dx}}{v\sqrt{v^2 - 1}}$. 61. Differentiation of $\operatorname{arc vers} v$. Let $\ y$ $= \operatorname{arcvers} v$;[7] then $\ v$ $= \operatorname{vers} y$. Differentiating with respect to $y$ by XVII, $\frac{dv}{dy}$ $=\ \sin y$; therefore $\frac{dy}{dv}$ $= \frac{1}{\sin y}$ By (C), p. 46 [§ 43] But since $v$ is a function of $x$, this may be substituted in the formula $\frac{dy}{dx}$ $= \frac{dy}{dv} \cdot \frac{dv}{dx}$ (A), p. 45 [§ 42] giving $\frac{dy}{dx}$ $= \frac{1}{\sin y} \cdot \frac{dv}{dx}$ $= \frac{1}{\sqrt{2v - v^2}} \frac{dv}{dx}$ [$\sin y = \sqrt{1 - \cos^2 y} = \sqrt{1 - (1 - \operatorname{vers} y)^2} = \sqrt{2v - v^2}$, the plus sign of the radical being taken, since $\sin y$ is positive for all values of $y$ between 0 and $\pi$ inclusive.] XXIV ∴ $\frac{d}{dx} (\operatorname{arcvers} v)$ $= \frac{\frac{dv}{dx}}{\sqrt{2v - v^2}}$. 1. It should be remembered that this function is defined only for values of $v$ between -1 and +1 inclusive and that $y$ (the function) is many-valued, there being infinitely many arcs whose sines will equal $v$. Thus, in the figure (the locus of $y = \arcsin v$), when $v = OM, y = MP_1, MP_2, MP_3, \cdots, MQ_1 MQ_2, \cdots$. In the above discussion, in order to make the function single-valued; only values of $y$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ inclusive (points on arc $QOP$) are considered; that is, the arc of smallest numerical value whose sine is $v$. 2. This function is defined only for values of $v$ between -1 and +1 inclusive, and is many-valued. In the figure (the locus of $y = \arccos v$), when $v=OM, y=MP_1, MP_2, \cdots, MQ_1 MQ_2, \cdots$. In order to make the function single-valued, only values of $y$ between 0 and π inclusive are considered; that is, $y$ the smallest positive arc whose cosine is $v$. Hence we confine ourselves to arc QP of the graph. 3. This function is defined for all values of $v$, and is many-valued, as is clearly shown by its graph. In order to make it single-valued, only values of $y$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ are considered; that is, the arc of smallest numerical value whose tangent is $v$ (branch $AOE$). 4. This function is defined for all values of $v$, and is many-valued, as is seen from its graph (Fig. a). In order to make it single-valued, only values of $y$ between 0 and $\pi$ are considered; that is, the smallest positive arc whose cotangent is $v$. Hence we confine ourselves to branch AB. 5. This function is defined for all values of $v$ except those lying between -1 and +1, and is seen to be many-valued. To make the function single-valued, $y$ is taken as the arc of smallest numerical value whose secant is $v$. This means that if $v$ is positive, we confine ourselves to points on arc AB (Fig. b), $y$ taking on values between 0 and $\frac{\pi}{2}$ (0 may be included); and if $v$ is negative, we confine ourselves to points on arc DC, $y$ taking on values between $-\pi$ and $-\frac{\pi}{2}$ ($-\pi$ may be included). 6. This function is defined for all values of $v$ except those lying between -1 and +1, and is seen to be many-valued. To make the function single-valued, $y$ is taken as the arc of smallest numerical value whose cosecant is $v$. This means that if $v$ is positive, we confine ourselves to points on the arc AB (Fig. a), $y$ taking on values between 0 and $\frac{\pi}{2}$ ($\frac{\pi}{2}$ may be included); and if $v$ is negative, we confine ourselves to points on the arc CD, $y$ taking on values between $-\pi$ and $-\frac{\pi}{2}$ ($-\frac{\pi}{2}$ may be included). 7. Defined only for values of $v$ between 0 and 2 inclusive, and is many-valued. To make the function continuous, $y$ is taken as the smallest positive arc whose versed sine is $v$; that is, $y$ lies between 0 and $\pi$ inclusive. Hence we confine ourselves to arc $OP$ of the graph (Fig. a).	2013-12-10 15:37:26	{"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": 0, "img_math": 341, "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": 0.9640704393386841, "perplexity": 556.4945705755025}, "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-2013-48/segments/1386164021066/warc/CC-MAIN-20131204133341-00093-ip-10-33-133-15.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/41848/diagram-like-perturbation-theory-in-quantum-mechanics	# Diagram-like perturbation theory in quantum mechanics There seems to be a formalism of quantum mechanics perturbation that involve something like Feynman diagrams. The advantage is that contrary to the complicated formulas in standard texts, this formalism is intuitive and takes almost zero effort to remember (to arbitrary orders). For example, consider a two level atom $\{\|g\rangle, \|e\rangle\}$ coupled to an external ac electric field of frequency $\omega$. Denote the perturbation by $\hat V$, with nonzero matrix element $\langle e\|\hat V \|g\rangle$. Then the second order energy correction reads $$E^{(2)} = \langle e\|\hat V \|g\rangle\frac{1}{\omega_g - \omega_e +\omega} \langle g\|\hat V \|e\rangle + \langle e\|\hat V \|g\rangle\frac{1}{\omega_g - \omega_e -\omega} \langle g\|\hat V \|e\rangle$$ where the first term corresponds to the process absorb a photon then emit a photon while the second process is emit a photon then absorb a photon. Does anybody know the name of this formalism? And why it is equivalent to the formalism found in standard texts? - add comment ## 2 Answers There is an exposition of a diagrammatic representation of the terms in the quantum mechanical perturbation expansion here. Basically the diagrams are just used to represent the combinatorial properties resulting from eigenstate degeneracy. (Just for fun there is also a diagrammatic approach to perturbation expansions in classical mechanics here). - Thank you! But the first reference is sophisticated and too mathematical. The formalism I'm looking for is quite intuitive and physical. – ChenChao Oct 28 '12 at 5:59 add comment Try Shankar Intro to Quantum Mechanincs page 489, he discusses the mathematical connection and the relation to feynman diagrams, or rather all the possible paths of interaction: If you find something better, I'd be curious to know too. - add comment	2014-04-24 21:53:46	{"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": 0.8779613971710205, "perplexity": 567.0763210112635}, "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-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00080-ip-10-147-4-33.ec2.internal.warc.gz"}
https://bookstore.ams.org/view?ProductCode=TRANS2/163	An error was encountered while trying to add the item to the cart. Please try again. The following link can be shared to navigate to this page. You can select the link to copy or click the 'Copy To Clipboard' button below. Copy To Clipboard Successfully Copied! Third Siberian School: Algebra and Analysis Edited by: L. A. Bokut′ Russian Academy of Sciences M. Hazewinkel CWI Yu. G. Reshetnyak Russian Academy of Sciences Available Formats: Electronic ISBN: 978-1-4704-3374-1 Product Code: TRANS2/163.E 188 pp List Price: $122.00 MAA Member Price:$109.80 AMS Member Price: $97.60 Click above image for expanded view Third Siberian School: Algebra and Analysis Edited by: L. A. Bokut′ Russian Academy of Sciences M. Hazewinkel CWI Yu. G. Reshetnyak Russian Academy of Sciences Available Formats: Electronic ISBN: 978-1-4704-3374-1 Product Code: TRANS2/163.E 188 pp List Price:$122.00 MAA Member Price: $109.80 AMS Member Price:$97.60 • Book Details American Mathematical Society Translations - Series 2 Volume: 1631995 MSC: Primary 16; 17; 18; 19; 28; 46; 52; 53; 57; This book contains papers presented at the Third Siberian School: Algebra and Analysis, held in Irkutsk in the summer of 1989. Drawing 130 participants from all over the former Soviet Union, the school sought to acquaint Siberian and other mathematicians with the latest achievements in a wide variety of mathematical areas and to give young researchers an opportunity to present their work. The papers presented here range over topics in algebra, analysis, geometry, and topology. Research mathematicians. • Cover • Title page • Contents • On the III Siberian School “Algebra and analysis”, Irkutsk, August 30–September 4, 1989 • Quasiderivations in diagonal matrix algebras • 𝐿(²) Atiyah-Bott-Lefschetz theorem • Topological structure of 𝑘-saddle surfaces • On uniqueness of reconstruction of the form of convex and visible bodies from their projections • On subalgebras of maximal rank of semisimple Lie algebras • Algebraic principles of building mathematical structures • On the absence of Sullivan’s cusp finiteness theorem in higher dimensions • The variety of all rings has Higman’s property • Boolean-valued introduction to the theory of vector lattices • The Whitehead groups of algebraic groups and applications to some problems of algebraic group theory • Diffeomorphicity criteria for simply connected manifolds • On the 𝐾-theory of generalized fibre bundles and some of their twisted forms • On mappings preserving convexity • Affine crystallographic groups • Integrals with respect to vector-valued measures: Theoretical problems and applications • Generalized derivations of algebras • Back Cover • Request Review Copy • Get Permissions Volume: 1631995 MSC: Primary 16; 17; 18; 19; 28; 46; 52; 53; 57; This book contains papers presented at the Third Siberian School: Algebra and Analysis, held in Irkutsk in the summer of 1989. Drawing 130 participants from all over the former Soviet Union, the school sought to acquaint Siberian and other mathematicians with the latest achievements in a wide variety of mathematical areas and to give young researchers an opportunity to present their work. The papers presented here range over topics in algebra, analysis, geometry, and topology. Research mathematicians. • Cover • Title page • Contents • On the III Siberian School “Algebra and analysis”, Irkutsk, August 30–September 4, 1989 • Quasiderivations in diagonal matrix algebras • 𝐿(²) Atiyah-Bott-Lefschetz theorem • Topological structure of 𝑘-saddle surfaces • On uniqueness of reconstruction of the form of convex and visible bodies from their projections • On subalgebras of maximal rank of semisimple Lie algebras • Algebraic principles of building mathematical structures • On the absence of Sullivan’s cusp finiteness theorem in higher dimensions • The variety of all rings has Higman’s property • Boolean-valued introduction to the theory of vector lattices • The Whitehead groups of algebraic groups and applications to some problems of algebraic group theory • Diffeomorphicity criteria for simply connected manifolds • On the 𝐾-theory of generalized fibre bundles and some of their twisted forms • On mappings preserving convexity • Affine crystallographic groups • Integrals with respect to vector-valued measures: Theoretical problems and applications • Generalized derivations of algebras • Back Cover Please select which format for which you are requesting permissions.	2023-01-28 12:50:14	{"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": 0.34939613938331604, "perplexity": 4325.041090203786}, "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/1674764499634.11/warc/CC-MAIN-20230128121809-20230128151809-00732.warc.gz"}
https://en.wikipedia.org/wiki/CRC-based_framing	CRC-based framing CRC-based framing is a kind of frame synchronization used in Asynchronous Transfer Mode (ATM) and other similar protocols. The concept of CRC-based framing was developed by StrataCom, Inc. in order to improve the efficiency of a pre-standard Asynchronous Transfer Mode (ATM) link protocol. This technology was ultimately used in the principal link protocols of ATM itself and was one of the most significant developments of StrataCom. An advanced version of CRC-based framing was used in the ITU-T SG15 G.7041 Generic Framing Procedure (GFP), which itself is used in several packet link protocols. Overview of CRC-based framing The method of CRC-Based framing re-uses the header cyclic redundancy check (CRC), which is present in ATM and other similar protocols, to provide framing on the link with no additional overhead. In ATM, this field is known as the Header Error Control/Check (HEC) field. It consists of the remainder of the division of the 32 bits of the header (taken as the coefficients of a polynomial over the field with two elements) by the polynomial ${\displaystyle x^{8}+x^{2}+x+1}$. The pattern 01010101 is XORed with the 8-bit remainder before being inserted in the last octet of the header.[1] Constantly checked as data is transmitted, this scheme is able to correct single-bit errors and detect many multiple-bit errors.[clarification needed] For a tutorial and an example of computing the CRC see mathematics of cyclic redundancy checks. The header CRC/HEC is needed for another purpose within an ATM system, to improve the robustness in cell delivery. Using this same CRC/HEC field for the second purpose of link framing provided a significant improvement in link efficiency over what other methods of framing, because no additional bits were required for this second purpose. A receiver utilizing CRC-based framing bit-shifts along the received bit stream until it finds a bit position where the header CRC is correct for a number of times. The receiver then declares that it has found the frame. A hysteresis function is applied to keep the receiver in lock in the presence of a moderate error rate. In links where there is already a byte lock mechanism present such as within an E-carrier or SDH frame, the receiver need only byte-shift (rather than of bit-shifting) along the receive data stream to find lock. Length/HEC-Based Framing An advanced, variable frame size version of CRC-Based framing is used in ITU-T SG15 G.7041 GFP links where it is known as Length/HEC-based framing. An offset to the next valid header is present in a fixed position relative to the CRC/HEC. The receiver looks for a position in the receive data stream following the rules that the header CRC/HEC is correct and the byte offset correctly points to the next valid header CRC/HEC. Invention of CRC-based framing StrataCom produced the first (pre-standard) ATM commercial product, the IPX. The IPX used 24 byte cells instead of ATM's 53 byte cells, and the field definitions were slightly different, but the basic idea of using short, fixed length cells was identical. StrataCom's first product had T1 (1.544 Mbit/s) based links which included a 5 bit header CRC, similar to ATM's 8 bit header CRC. T1 is a time-division multiplexing (TDM) protocol with 24 byte payloads carried in a 193 bit frame. The first bit of each frame carries one bit out of a special pattern. A receiver finds this special pattern by sequentially looking for the bit position in the receive data where a bit from this pattern shows up every 193rd byte. It was convenient for StrataCom to make the length of one cell equal to the length of one T1 frame[2] because a useful T1 framer Integrated Circuit from Rockwell was on the market. This device found the 193 bit long TDM frame and put out the 24 bytes in a form that could be used effectively. When it came time to produce a European product, the benefit of using 24 byte frames became a liability. The European E-carrier (E1) format has a 32 byte frame of which 30 bytes could carry data. The development team's first proposal used the HDLC protocol to encapsulate a sequence of 24 byte cells into a byte stream collected from the 30 byte E1 payloads. This was highly inefficient because HDLC has a heavy and data-dependent overhead. The project team subsequently realized they could base the framing on the CRC.[3] A circuit was designed which examined the incoming byte stream emerging from the E1 framer device and found a byte position for which the header CRC value was consistently correct. This team also went on to create a more error tolerant form of the technique.[4] A related technique was patented in 1984. That technique uses the CRC to find the start of 50 bit frames composed of a 36 bit data payload, a 13 bit CRC, and a single 1 bit start-of-frame indicator.[5] Notes and references 1. ^ "ATM User-Network Interface Specification V3.0". Retrieved 2007-09-17. 2. ^ Previous Stratacom patent using DS-1's framing 3. ^ Original Stratacom CRC-based framing patent 4. ^ More error tolerant Stratacom CRC-based framing patent 5. ^	2018-11-15 08:53:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.23689116537570953, "perplexity": 2833.9487027759606}, "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-2018-47/segments/1542039742569.45/warc/CC-MAIN-20181115075207-20181115100641-00029.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-7-exponential-and-logarithmic-functions-7-2-graph-exponential-decay-functions-7-2-exercises-problem-solving-page-490/30a	## Algebra 2 (1st Edition) $\approx119.651$ Plugging in $I=200, t=1.5$ into the given function we get: $A=200(0.71)^{1.5}\approx119.651$	2021-04-14 13:29:39	{"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": 0.885347306728363, "perplexity": 1758.4642207290563}, "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-2021-17/segments/1618038077818.23/warc/CC-MAIN-20210414125133-20210414155133-00422.warc.gz"}
https://math.stackexchange.com/questions/988619/does-the-definition-of-derivative-exclude-the-possibility-for-discontinuous-rate	# Does the definition of derivative exclude the possibility for discontinuous rate of change? Is it possible to have a function whose instantaneous rate at every X are different to each other such that there are no pattern of gradual change between them but the definition of derivative fails to acquired the rate at every X? such that the derivative at X1=1 =7 X2=1.0000001 =-987 X3=1.0000002 =0.0089 The definition of derivative would fail since it expects that the neighbor instantaneous rate at X2 assumes the it is infinitely close to it which means it gradually change to X2 but it's not the case.A function who has an instantaneous rate at every point but its instantaneous rate to every point are connected such that no gradual change appears not even a single interval.Would a function exist? and why not? • ... come again? – Clarinetist Oct 24 '14 at 4:28 • It looks to me as if the question is whether, if $f$ is a differentiable function, the derivative $f\,'$ also has to be continuous. – Lubin Oct 24 '14 at 4:32 • @Lubin: it depends on whether you read the question title or the question body. In any case whether the question is about a continuous function whose change is so irregular that it nowhere has a derivative, or about a function that does have a derivative in each point but which is itself not continuous, such things are possible (and in fact common!). However you need to sharpen your idea of what a function and being differentiable are to appreciate these examples; an explanation at the vague "rate of change" level of precision seems very hard to give. – Marc van Leeuwen Oct 24 '14 at 4:45 • You seem to think that $10^{-7}$ is automatically small. It is not. Functions and their derivatives can do anything in that interval. It is not hard to find a nice smooth function that meets your specs (aside from $1=7$ and the like). We just need to make sure the $h$ in the definition of derivative is smaller than $10^{-7}$, but that is OK. – Ross Millikan Oct 24 '14 at 4:45 The derivative of a function can exist, but be discontinuous at some points. The standard example is $x^2\sin(1/x)$ when $x\ne 0$, and $f(0)=0$. This is differentiable everywhere, but the derivative is not continuous at $x=0$.	2020-10-29 10:45:18	{"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": 0.826999843120575, "perplexity": 247.51358089586049}, "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-2020-45/segments/1603107904039.84/warc/CC-MAIN-20201029095029-20201029125029-00120.warc.gz"}
https://search.datacite.org/repositories/laam.ptb-oar?resource-type-id=dataset	### Datasets of high spatial resolution scans of the airborne ultrasound field of an ultrasonic welding machine either with or without an artificial head at a worker’s sedentary position Robert Schöneweiß, Christoph Kling & Christian Koch This dataset contains measuring data which are the result of investigations of the influence of a person on an airborne ultrasound field. These investigations have been conducted within the scope of the EMPIR project 15HLT03: “Ears II - Metrology for modern hearing assessment and protecting public health from emerging noise sources”. In the context of “Ears II”, they served gaining knowledge for occupational safety and health. Here, the aim was to investigate the influence of... ### Round robin comparison on quantitative nanometer scale magnetic field measurements by magnetic force microscopy Xiukun Hu As a standard tool for nano-scale investigation of magnetic domain structures, magnetic force microscopy (MFM) measures the local stray magnetic field landscape of the measured sample, however, generally providing only qualitative data. To quantify the stray magnetic fields, the MFM system could be calibrated by a so-called transfer function (TF) approach which fully considers the finite extent of the MFM tip. However, albeit being comprehensive, the TF approach is not yet well established, mainly due... ### Sound intensity and sound pressure values after measurements in various surrounding environments. Spyros Brezas, Volker Wittstock & Fabian Heisterkamp The files contain the values of acoustical and correction quantities used for the analysis presented in the corresponding report. For scanning measurements, the sound pressure and sound intensity values for each scan path are given. For measurements at discrete points, the sound pressure and sound intensity values at each measurement point are given. ### Additional data for the publication \"Optical frequency ratio of a 171Yb+ single-ion clock and a 87Sr lattice clock\" Sören Dörscher, Nils Huntermann, Roman Schwarz, Richard Lange, Erik Benkler, Burghard Lipphardt, Uwe Sterr, Ekkehard Peik & Christian Lisdat The enclosed data set provides additional data on the measurements presented in the publication entitled "Optical frequency ratio of a 171Yb+ single-ion clock and a 87Sr lattice clock" in Metrologia (DOI: 10.1088/1681-7575/abc86f). ### Measurement and simulation data for \"Identification of aliasing effects in measurements of unknown MTFs\" Markus Schake & Michael Schulz The provided data set and simulation software enables the immediate replication of the results presented in "Identification of aliasing effects in measurements of unknown MTFs." The research data is structured with respect to the figures presented in the contribution. Each subfolder contains all necessary measurement or simulation data to receive the figure from the contribution by running a MATLAB file. Therefore, it is possible to compare the results with full context to those presented in... ### Datasets for Multidimensional model to correct PV device performance measurements taken under diffuse irradiation to reference conditions Fabian Plag, Stefan Riechelmann, Ingo Kröger & Stefan Winter The datasets provided here contain spectral- and angular-dependent responsivities of three different reference solar cells measured in Plag , et al., 2017, Angular dependent spectral responsivity — Traceable measurements on optical losses in PV devices. Prog Photovolt Res Appl.;1–14. and used in Plag, et al., 2018, Multidimensional model to correct PV device performance measurements taken under diffuse irradiation to reference conditions. The additionally provided uvspec input file allow radiative transfer simulations that are similar to... ### Dataset for the calibration of torque measurement under constant rotation in a wind turbine test bench Paula Weidinger, Gisa Foyer, Stefan Kock, Jonas Gnauert & Rolf Kumme This dataset is the basis for the publication "Calibration of torque measurement under constant rotation in a wind turbine test bench" accepted for publication in the "Journal of Sensors and Sensor Systems". ### Good Practice Guide on Making Rectangular Waveguide Connections at Frequencies above 100 GHz - Measurement data Karsten Kuhlmann & Thorsten Probst This dataset contains the results from test measurements used for the document "Good Practice Guide on Making Rectangular Waveguide Connections at Frequencies above 100 GHz" (DOI: 10.7795/530.20190805). ### Additional data for the publication ‘Simple and compact diode laser system stabilized to Doppler-broadened iodine-lines at 633 nm’ Florian Krause, Erik Benkler, Christian Nölleke, Patrick Leisching & Uwe Sterr The data sets provide additional information on the experiments and the physical background discussed in the publication entitled ‘Simple and compact diode laser system stabilized to Doppler-broadened iodine-lines at 633 nm’ in Applied Optics (DOI: 10.1364/AO.409308). Description of the individual files: The file 'Trans_spectrum.dat' provides the normalized iodine transmission data as a function of the laser diode temperature (shown in fig. 2). The file 'nu_R(74)_8-4.dat' provide a time series of the absolute frequency calculated of... ### Additional data for the publication \"The blackbody radiation shift in strontium lattice clocks revisited\" Christian Lisdat, Sören Dorscher, Ingo Nosske & Uwe Sterr The data sets provide additional information on the manuscript including data shown in graphs. ### Measurement and simulation data for \"Optical form measurement employing a tiltable line scanning low coherence interferometer for annular subaperture stitching interferometry\" Markus Schake, Jörg Riebeling, Peter Lehmann & Gerd Ehret The provided data set and simulation software enables the immediate replication of the results presented in "Optical form measurement employing a tiltable line scanning low coherence interferometer for annular subaperture stitching interferometry." The research data is structured with respect to the figures presented in the contribution. Each subfolder contains all necessary measurement or simulation data to receive the figure from the contribution by running a MATLAB file. Therefore, it is possible to compare the results... ### Fractional frequency ratio difference time series data sets (Optical frequency comb comparison) Erik Benkler, Uwe Sterr, Burghard Lipphardt, Rafal Wilk, Thomas Puppe & Felix Rohde ### Simulated simultaneous 18F-NaF PET/MR with physiological motion and ground truth information Johannes Mayer & Christoph Kolbitsch This dataset is the basis for the publication "Evaluation of synergistic image registration for motion-corrected coronary NaF-PET-MR" Dataset containing simulated cardiac and respiratory motion of simultaneous PET/MR data with 18F-NaF uptake in the right coronary artery. Supplied are PET and MR images, as well as underlying ground truth motion and regions of interests (ROI) in which the uptake and background signal are located. The dataset is based on the XCAT phantom and a custom simulation... ### Dataset for the publication \"Coherent Excitation of the Highly Forbidden Electric Octupole Transition in 172Yb+\" Henning Fürst, Chih-Han Yeh, Dimitri Kalincev, André Kulosa, Laura Dreissen, Richard Lange, Erik Benkler, Nils Huntermann, Ekkehard Peik & Tanja Mehlstäubler Measurement data for the publication "Coherent Excitation of the Highly Forbidden Electric Octupole Transition in 172Yb+" by H.A. Fürst et al, Phys. Rev. Lett. 125, 163001 – Published 16 October 2020. ### Dataset for Combining Harmonic Laser Beams by Fiber Components for Refractivity–Compensating Two-Color Interferometry Xiukun Hu, Anni Röse, Günther Prellinger, Paul Köchert, Jigui Zhu & Florian Pollinger A perfect spatial overlap with multiple beams of different wavelengths is a prerequisite for multi–wavelength interferometry. Beam combination with the help of fibers seems to be an interesting method for this. We investigated three different types of fiber components, multi–mode wavelength division multiplexers (MM–WDMs), single–mode wavelength division multiplexers (SM–WDMs), and endlessly single–mode polarization maintaining photonic crystal fibers (PM–PCFs). All three seem potential candidates for a perfect spatial overlap of laser beams separated by an octave,... ### Catalogue of unfiltered x-ray spectra from tungsten-, molybdenum-, and rhodium-anode-based x-ray tubes with generating voltages from 10kV to 50kV in steps of 1kV Steffen Ketelhut & Ludwig Büermann Set of experimental unfiltered x-ray spectra in the range of mammography qualities. The spectra can be computationally filtered to generate a wide range of x-ray spectra in this energy range. ### Dataset for Dynamic characterization of multi-component sensors for force and moment Jan Nitsche, Rolf Kumme & Rainer Tutsch This Dataset is the basis for the publication "Dynamic characterization of multi-component sensors for force and moment" accepted for publication in "Journal of Sensors and Sensor Systems". It contains acceleration measurement data of a scanning laser vibrometer, displacement measurement of a photogrammetric measurement setup and force measurement of a multi-component force and moment sensor. ### Modulation-based long-range interferometry as basis for an optical two-color temperature sensor [Dataset] Anni Röse, Yang Liu, Paul Köchert, Günther Prellinger, Eberhard Manske & Florian Pollinger Investigations of an 2f/3f detection scheme for phase detection. The laser frequency of the light source is sinusoidally modulated, and the interferometric phase derived from the harmonic signals. In this article we discuss measurement scheme, approach-included challenges and present first verification experiments over a range of 20m. ### Additional data for the publication \"Coherent Suppression of Tensor Frequency Shifts through Magnetic Field Rotation\" Robert Lange, Nils Huntermann, Christian Sanner, Hu Shao, Burghard Lipphardt, Christian Tamm & Ekkehard Peik Data on simulations of the magnetic field and on the frequency shift measurement of the E3 and E2 transition used for a determination of the quadrupole moments. ### Dataset for the publication \"Umfrage zum Forschungsdatenmanagement in der Physik\" Holger Israel, Esther Tobschall & Frank Tristram Der Umgang mit Forschungsdaten gewinnt mit fortschreitender Digitalisierung an gesellschaftlicher Aufmerksamkeit. Während ein weitgehender Konsens zu den in den FAIR-Prinzipien dargestellten Anforderungen an gutes Forschungsdatenmanagement besteht, ist ihre Umsetzung eine längerfristige Aufgabe. Das im Rahmen der Nationalen Forschungsdateninfrastruktur (NFDI) gegründete Konsortium NFDI4Phys führte im Frühjahr 2020 eine Umfrage zur Ermittlung des status quo und der vordringlichen Bedarfe an das Forschungsdatenmanagement in der Physik durch. In diesem Dokument stellen wir die Methodik und detaillierte Auswertung der... ### Collection of spectra of direct photons recorded in ISO neutron fields Harald Dombrowski & Ralf Nolte Compilation of prompt photon spectra recorded in ISO neutron fields produced at PIAF (PTB accelerator facility) by using a 1.5'' x 1.5'' CeBr3 detector. This data set is an appendix to the article “Quantitative investigation of gamma radiation in accelerator produced ISO neutron fields” by H. Dombrowski and R. Nolte. The spectra are depicted in this article. The time-of-flight method was used to detect direct photons. By performing shadow cone measurements, a background subtraction was... ### Dataset for DC measurement of the dressed states in a coupled 100 GHz resonator system using single quasiparticle transistor as a sensitive microwave detector Sergey Lotkhov, Marat Khabipov & Ralf Dolata This data set is the basis for the publication "DC measurement of the dressed states in a coupled 100 GHz resonator system using single quasiparticle transistor as a sensitive microwave detector". It contains the following data: Lotkhov_APL19_fig_2a.dat contains a 2D matrix with autonomous signal current values I_sset as function of bias and normalized gate voltages. The measurement was done on a test sample without qubit with deviating parameters Lotkhov_APL19_fig_2b.dat contains a 2D matrix with signal... ### Long Slender Piezo-Resistive Silicon Microprobes for Fast Measurements of Roughness and Mechanical Properties inside Micro-Holes with Diameters below 100 µm Uwe Brand, Min Xu, Lutz Doering, Jannick Langfahl-Klabes, Heinrich Behle, Sebastian Büteﬁsch, Thomas Ahbe, Bodo Mickan, Erwin Peiner, Stefan Völlmeke, Thomas Frank, Ilia Kiselev, Michael Drexel & Michael Hauptmannl During the past decade, piezo-resistive cantilever type silicon microprobes for high-speed roughness measurements inside high-aspect-ratio microstructures, like injection nozzles or critical gas nozzles have been developed. This article summarizes their metrological properties for fast roughness and shape measurements including noise, damping, tip form, tip wear, and probing forces and presents the ﬁrst results on the measurement of mechanical surface parameters. Due to the small mass of the cantilever microprobes, roughness measurements at very high traverse... ### Additional data for the publication \"Long term measurement of the 87Sr clock frequency at the limit of primary Cs clocks\" Roman Schwarz, Sören Dörscher, Ali Al-Masoudi, Erik Benkler, Thomas Legero, Uwe Sterr, Stefan Weyers, Johannes Rahm, Burghard Lipphardt & Christian Lisdat The data sets provide additional information on the experiments and the physical background discussed in the publication entitled "Long term measurement of the $^{87}$Sr clock frequency at the limit of primary Cs clocks" in Physical Review Research (DOI: 10.1103/PhysRevResearch.2.033242). • 2021 8 • 2020 10 • 2019 4 • 2018 2 • Dataset 24	2021-10-16 17:15:09	{"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": 0.3790053129196167, "perplexity": 11353.748172553645}, "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/1634323584913.24/warc/CC-MAIN-20211016170013-20211016200013-00352.warc.gz"}
https://electronics.stackexchange.com/questions/367052/replace-bss138-with-ao3400a-in-level-shifter-circuit	# Replace BSS138 with AO3400A in level shifter circuit I have to use level shifter circuit to interface MAX31855 with Arduino running on 5V. I found easy to build level shifter circuit at SparkFun website. It is using BSS138 (datasheet) which I don't have. I am thinking to build same circuit with AO3400A (datasheet), which I already ordered last week for a pack of 100, so I hope to receive it by next few days. My question is whether it will be completely OK to replace BSS138 with AO3400A in this particular circuit? As per both datasheet, I think AO3400A is much much better in terms of $$\R_{DS(ON)}\$$. $$\V_{GS(th)}\$$ is also good for AO3400A. $$\Q_g\$$ is just slightly higher on AO3400A. (These are the parameters I usually compare to choose MOSFETs). I am asking this because I don't have yet AO3400A to test. If AO3400A is wrong here, then I can order BSS138 soon. However, I don't think I will find any other use for BSS138, particularly because of its large $$\R_{DS(ON)}\$$ (datasheet says it is extremely low $$\R_{DS(ON)}\$$ !!) and low drain current capability. • @TomCarpenter Thanks. In fact, I don't know exactly how this circuit works! – Junaid Apr 7 '18 at 15:13 • I've added a bit on how the circuit works. – Tom Carpenter Apr 7 '18 at 15:44 TL;DR; It will work, but it will be slow. Probably no more than 200kHz data rate. $$\R_{ds(on)}\$$ is irrelevant in this circuit to be honest. The channel resistance essentially acts as a potential divider with R4 or R3 when the output is pulled low. So even if it were 100Ω the output low voltage would still be no more than 100mV. A larger on-resistance will slow down the falling edge slightly, but given the rising edge is entirely driven by the pull-up resistor anyway, the level shifter is no good for high-speed circuits (a few MHz tops depending on trace length and capacitive load). Any MOSFET with a $$\V_{gs(th)}\$$ of less than $$\V_{LV} - V_{sd}\$$ (low voltage supply minus body diode forward voltage) should suffice for the circuit to work at some frequency, AO3400A meets that requirement. However, as @SpehroPefhany points out, this is not the full story. As already mentioned, these circuits are slow, in part due to the pull-up resistors, but also due to the capacitance of the MOSFET, its $$\C_{iss}\$$ and $$\C_{oss}\$$ ("input" and "output" capacitances). MOSFETs have capacitances between each of the terminals, which can be considered as capacitors between gate and source ($$\C_{iss}\$$), and between drain and source ($$\C_{oss}\$$). These capacitors will also act to slow the circuit down. The higher the capacitance, the slower the circuit. The capacitance has the biggest impact in down-shifting on both rising and falling edges. For up-shifting the effect is minimised on the rising edge as C_{iss} helps give a boost, however the falling edge is affected just as much as the down-shifting mode. Your chosen MOSFET has very high capacitance, which will limit the speed of the circuit. As a very quick approximate simulation, this shows the circuit in operation at 2MHz for both the BSS138 and the AO3400A. A screenshot of the results is shown below in case the URL breaks. As expected, the AO3400A performs badly at this frequency. In fact slowing down the frequency from the simulation we can see for down-shifting about a 800ns rise time (10% to 80% Vdd) and about 50ns fall time. This compares to only a 40ns rise time for the BSS138. You can probably get ~200kHz using the proposed transistor vs. ~4MHz with the BSS138. ### How it works I'll add a bit about how the circuit works for completeness. There are two modes of operation, up-shifting (LV1 is input, HV1 is output), and down-shifting (HV1 is input, LV1 is output). For simplicity of the explanation, I'll assume that LV is 3.3V, and HV is 5V. The list numbers correspond to the number on the diagram at the bottom of the answer. Up-Shifting (LV1 = Input, HV1 = Output) Up-shifting is the easiest to understand. 1. When the input pin is high, then the $$\V_{gs}\$$ of the MOSFET will be zero - both the gate and source are at 3.3V. As such the MOSFET is turned off, and the output pin will be pulled up to 5V by R4. 2. When the input is pulled low, the source will be at 0V, but the gate will remain at 3.3V. As such $$\V_{gs}\$$ is now 3.3V and the MOSFET turns on. The MOSFET will pull the drain down to the source voltage (0V), which means the output pin will now be low. (The input pin is sinking current from both R3 and R4). Down-Shifting (LV1 = Output, HV1 = Input) Down shifting is a little more complicated. 1. When the input pin is high, there is nothing in the circuit pulling the output down. As such it will be pulled up to 3.3V by R3. This will make the $$\V_{gs}\$$ of the MOSFET zero, preventing any current flowing from input to output. As such the output voltage cannot exceed 3.3V, even though the input is 5V. 2. When the input is pulled low, the body diode of the MOSFET which goes from source to drain will start conducting and pull the output down. As the output is pulled down, the source voltage moves towards $$\V_{sd}\$$ (body diode forward voltage). 3. As this happens, $$\V_{gs}\$$ will now be $$\3.3V - V_{sd}\$$, which must be sufficient to turn the MOSFET on. Once the MOSFET turns on, the output voltage will then drop towards zero as the current through R3 flows through the channel rather than the body diode. • Thanks for detailed explanation! As you guessed it was easy to understand up-shifting. Down-shifting with input high is also not hard to understand. – Junaid Apr 7 '18 at 16:15 • As I mentioned it is to interface with MAX31855 to communicate over SPI. Up to what speed it will be safe to use? – Junaid Apr 7 '18 at 16:20 • @Junaid I've gotten 4MHz out of this type of level shifter before by using 2.2k pull-up resistors. That was over a very short run (~2cm wire). It was also for a device that was very sensitive to rounding of the signal edges (it was an SPI flash device). – Tom Carpenter Apr 7 '18 at 22:26 • However that was with a BS170, which has a much lower Ciss and Coss than the AO3400A, so will be faster. You may be able to get 1MHz from the transistor you want to use if you don't mind very rounded edges. – Tom Carpenter Apr 7 '18 at 22:42 AO3400A is a power MOSFET and has much higher Cgs (20x higher) and gate charge and Coss. If you don't care at all about speed it will work. Speed is already pretty terrible because of the 10K pullup and other capacitance. If you reduce the pullup resistor you can gain some of the speed back but whatever is driving it will have to sink more current (and have an appropriately low output voltage to yield enough noise immunity) and the power consumption will be higher. Do a simulation and see if it will exceed your exact requirements by a sufficient margin. • Need to learn how to simulate circuits :) – Junaid Apr 7 '18 at 16:21 • As I mentioned, it is to interface with MAX31855 to communicate over SPI. Up to what speed it will be safe to use? – Junaid Apr 7 '18 at 16:21 • You can simulate it or build it and test it. My crystal ball is a bit cloudy this afternoon. – Spehro Pefhany Apr 7 '18 at 17:02 Well, this question is rather old but I guess I chould chip in: For SPI, sometimes you don't need a level shifter for connecting 3.3 to 5V devices. ## MISO: Often the 3.3V on the MISO line is enough to drive the input pins of your microcontroller to HIGH. You need to read the electrical characteristics on your device. I'm going to assume you use an Arduino Uno, which uses a ATMEGA328P, let's see the datasheet: It's specified as: 2. “Min” means the lowest value where the pin is guaranteed to be read as high. So at 5V, your Atmega328P is GUARANTEED to read as 1 as long as the pin is 0.7 times Vcc. So 3.5V at 5V. Your device runs at 3.3V. So it "may or may not work". In my experience, it works at normal room temperature and low speeds. But It WILL vary with VCC and, temperature, and specific devices. ## MOSI, SCLK: Trivial to set up a voltage divider to drive 5V down to about 3.3V. May not work at very high MHz but I've gotten it to work with a MAX31856 at 5MHz or so. Sometimes you can also get away with just a high value resistor: the device most likely has clamping diodes to VDD, so the voltage will be safely clamped and the series resistor will prevent it from burning. This may or may not work with all devices. For an atmega328 (and most microcontrollers) an input looks like this: The diodes are, of course, internal to the microcontroller. Cpin is the parasitic capacitance, and Rpu is the weak pull up. I couldn't find wether the MAX31855 has this kind of input, but chances are, it does (it's really easy to test: use a 100K resistor and feed 5V into the pin while supplying 3.3V at VDD. Measure the voltage across the resistor, and it should be near 3.3V). A 10K series resistor may be enough to limit the current without damaging the pin. You may need to go lower for higher speeds, but too low will make too much current flow, and mess with, in this particular case, the MAX31855's internal temperature sensor for cold junction). ## CS: CS is special. It can be driven with an open drain output. You can use a resistor from 3.3V to your CS line, and set your microcontroller to OUTPUT LOW to write a 0, or INPUT (High-Z) to let it float back to 1. For peace of mind you can use a reverse diode (Cathode to microcontroller pin), which would block an accidental 5V from flowing into the device. It would need to be a Schottky diode, because a regular 1N4148 silicon diode may keep the anode at 0.7V and prevent the device from triggering. In my experience the 1N4148 I was using worked fine, though.	2020-02-24 10:02:36	{"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": 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": 18, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45053863525390625, "perplexity": 1742.4839059385856}, "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-2020-10/segments/1581875145910.53/warc/CC-MAIN-20200224071540-20200224101540-00424.warc.gz"}
https://math.stackexchange.com/questions/2913602/prove-that-an-m-by-n-matrix-is-invertible-iff-its-columns-form-a-basis-in	# Prove that an $m$ by $n$ matrix is invertible iff its columns form a basis in $\mathbb F^n$. In the linear algebra book I'm working through, we've already proved two related theorems, namely: 1. $A: V\mapsto{W}$ is an isomorphism iff $Av_k = w_k$ where $v_1,...,v_n$ and $w_1,...,w_n$ are bases in $V$ and $W$ respectively. 2. Let $A: V\mapsto{W}$ be a linear transformation. $A$ is invertible iff for any right side $b\in{Y}$ the equation $Ax = b$ has a unique solution $x\in{X}$. The corollary: an $m$ by $n$ matrix is invertible iff its columns form a basis in $\mathbb F^n$ is then given without proof. How can we show that the corollary is true from the two theorems above? I'm assuming it should be fairly intuitive since the proof is omitted, but I'm struggling to put the pieces together in a rigorous way. There are many ways to think about it. Since the given theorems verse about linear transformations while your corollary about matrices, you should first keep in mind that any linear transformation $T$ has an associated matrix $A_T$ such that $[T (x)]_B = A_T [x]_B$ for a given basis $B$, from where it follows that the matrix $A_T$ is invertible iff $T$ is (or equivalently, iff $T$ is an isomorphism.) Also, isomorphisms can only be established between spaces of equal dimensions, so necessarily $m = n$ in your statement. Consider the second theorem. If the columns of matrix $A$ do not form a basis of $\mathbb {R}^n$, then the columns of $A$: 1) are not linearly independent; and/or 2) do not span all $\mathbb{R}^n$ If 1) is the case, the solutions for $A x = b$ are not unique; if 2) is the case, there are vectors $b$ for which the equation has no solutions. In both cases, A is not invertible (cause its associated linear transformation is not invertible.) The main issue is "can every $b$ be written as a linear combination of the columns of $A$." The answer is "yes" by your second theorem. Which linear combination is it? The unique solution $x$. The set $B = \{e_i\}_{1 \leq i \leq n}$ with $(e_i)_j = \delta_{ij}$ is a basis for $\mathbb{F}^n$. That is, the vector $e_i$ has zeros in every coordinate but the $i$-th one, where it has a one. By $(1)$, $A$ will be an isomorphism iff $Ae_i$ forms a basis of $\mathbb{F}^n$. Note (or prove) that these are precisely the columns of $A$.	2019-11-17 00:41:15	{"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": 0.9699994325637817, "perplexity": 88.11314867885686}, "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-2019-47/segments/1573496668772.53/warc/CC-MAIN-20191116231644-20191117015644-00291.warc.gz"}