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The American Urological Association (AUA) symptom score is a validated questionnaire that is used to objectively measure LUTS. Seven symptoms—incomplete emptying, frequency, inter- mittency, urgency, weak stream, straining, and nocturia—are scored from 0 to 5. Total scores from 0 to 7 are considered mild, scores from 7 to 19 are considered moderate, and scores of 20 to 35 are considered severe.

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Benign prostatic hyperplasia (BPH) occurs commonly in older men, with historical reports of up to 50% prevalence in men over the age of 50 years and increasing incidence with increasing age (1, 2). BPH anatomically compresses the urethra, causing increased bladder outlet resistance. This results in lower urinary tract symptoms (LUTS), which include symptoms of difficult void- ing (hesitancy, straining, weak stream, sensation of incomplete emptying) and irritable voiding (frequency, urgency, urge incontinence)

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Flashcard 5731199618316

Question
med: abbrev: LUTS
Answer
lower urinary tract symptoms (LUTS)


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That old men suffer with their prostates is a cliché as old as time. According to Hippocrates: ‘Diseases about the kidneys and bladder are cured with difficulty in old men’. 1 In the book of Ecclesiastes, in the Bible, old age is discussed, while writing in 1811, Sir Everard Home (1756–1832) was convinced that the phrase, ‘or the pitcher be broken at the fountain, or the wheel broken at the cistern’, represented the two principal effects of prostatic disease, the involuntary leakage of urine (the spilled water of the pitcher) and retention (the broken wheel preventing the drawing of water from the well). 2 However, at the time of the Ancient Greeks and Hebrews, the prostate was not recognised as a potential source of men’s urinary distress

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The other controversy regarding prostatic enlargement was whether this was pathological or an inevitable consequence of old age.

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Sir Benjamin Brodie, however, regarded the enlarged prostate as an almost invariable accompaniment of advanced age, elegantly stating that ‘when the hair becomes grey and scanty, when specks of earthy matter begin to be deposited in the tunics of the arteries, and when a white zone is formed at the margin of the cornea, at this same period the prostate gland usually – I might, perhaps, say invariably becomes increased in size’. 10

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However, Mr Wilson, in his lectures at the College of Surgeons, in 1821, must have caused considerable confusion. After having first stated that ‘that it appears to occur most frequently in those persons, who, either from living a life of strict celibacy, have not used the genital organs so much as nature seems to have intended, or who have injured both the genital and the urinary organs by a life of excess’ then unhelpfully adds, ‘many persons have suffered much from the enlargement of the prostate gland, who have lived a moderate and quiet life, without approaching to either of the above-mentioned extremes’. 12

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Flashcard 5731271183628

Question
Sir Benjamin Brodie, however, regarded the enlarged [...] as an almost invariable accompaniment of advanced age, elegantly stating that ‘when the hair becomes grey and scanty, when specks of earthy matter begin to be deposited in the tunics of the arteries, and when a white zone is formed at the margin of the cornea, at this same period the prostate gland usually – I might, perhaps, say invariably becomes increased in size’. 10
Answer
prostate


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Sir Benjamin Brodie, however, regarded the enlarged prostate as an almost invariable accompaniment of advanced age, elegantly stating that ‘when the hair becomes grey and scanty, when specks of earthy matter begin to be deposited in the tunics of

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Flashcard 5731274591500

Question
That old men suffer with their prostates is a cliché as old as [...]. According to Hippocrates: ‘Diseases about the kidneys and bladder are cured with difficulty in old men’. 1 In the book of Ecclesiastes, in the Bible, old age is discussed, while writing in 1811, Sir Everard Home (1756–1832) was convinced that the phrase, ‘or the pitcher be broken at the fountain, or the wheel broken at the cistern’, represented the two principal effects of prostatic disease, the involuntary leakage of urine (the spilled water of the pitcher) and retention (the broken wheel preventing the drawing of water from the well). 2 However, at the time of the Ancient Greeks and Hebrews, the prostate was not recognised as a potential source of men’s urinary distress
Answer
time


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That old men suffer with their prostates is a cliché as old as time. According to Hippocrates: ‘Diseases about the kidneys and bladder are cured with difficulty in old men’. 1 In the book of Ecclesiastes, in the Bible, old age is discussed, while writin

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Watchful Waiting As treatment of BPH-related LUTS is aimed at improvement of quality of life, watchful waiting is recommended by both the AUA and the European Association of Urology for patients with mild symptoms (AUA symptom score <8) or moderate-to-severe symptoms with minimal impairment in quality of life (47, 48). Watchful waiting should include education, modification of lifestyle factors (e.g., weight loss, increase in physical activity, and reduction of caffeine and alcohol intake), and yearly re-evaluation. Patients should be counseled appropriately on their risk of AUR, which is increased for those with moderate-to-severe symptoms, diminished urinary flow rates, larger prostate volumes, increasing serum PSA, and older age (33, 49, 50). Watchful waiting is inappropriate for patients with complications of bladder outlet obstruction related to BPH—such as renal insufficiency due to obstructive uropathy, recurrent UTI, bladder stones, and refractory urinary retention (failed at least one voiding trial after catheter removal).

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Phosphodiesterase type 5 inhibitors. Phosphodiesterase type 5 inhibitors (PDE5-I) are well known to be effective in the treatment of erectile dysfunction. PDE5-I increase nitric oxide sig- naling in genitourinary tract tissues, which causes calcium-dependent relaxation of endothelial smooth muscle and increased blood flow. By the same mechanism of action, PDE5-I have been found to improve BPH based on preclinical studies (13, 14). Five randomized placebo-controlled trials have demonstrated the efficacy of PDE5-I (sildenafil, tadalafil, and vardenafil) on LUTS related to BPH, with significant improvements in AUA symptom scores, but no effect on urinary flow rate or PVR (55–59). Currently, daily tadalafil is the only PDE5-I that is approved by the FDA for treatment of BPH. Although these medications are expensive, patients with concomitant erectile dysfunction may derive the most benefit from PDE5-I treatment for their LUTS related to BPH

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Flashcard 5731282193676

Question

patients with [...] erectile dysfunction

:towarzyszący

Answer
concomitant /kənˈkämədənt/


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res, but no effect on urinary flow rate or PVR (55–59). Currently, daily tadalafil is the only PDE5-I that is approved by the FDA for treatment of BPH. Although these medications are expensive, <span>patients with concomitant erectile dysfunction may derive the most benefit from PDE5-I treatment for their LUTS related to BPH <span>

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Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum

\({\displaystyle \ln n!=\sum _{j=1}^{n}\ln j}\)

with an integral:

\({\displaystyle \sum _{j=1}^{n}\ln j\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1.}\)

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Stirling's approximation - Wikipedia
Stirling's formula 7 Versions suitable for calculators 8 Estimating central effect in the binomial distribution 9 History 10 See also 11 Notes 12 References 13 External links Derivation[edit ] <span>Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum ln ⁡ n ! = ∑ j = 1 n ln ⁡ j {\displaystyle \ln n!=\sum _{j=1}^{n}\ln j} with an integral : ∑ j = 1 n ln ⁡ j ≈ ∫ 1 n ln ⁡ x d x = n ln ⁡ n − n + 1. {\displaystyle \sum _{j=1}^{n}\ln j\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1.} The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n!, one considers its natural logarithm , as this is a slowly varying




Preface

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in real analysis, a branch of mathematics, a slowly varying function is a function of a real variable whose behaviour at infinity is in some sense similar to the behaviour of a function converging at infinity. Similarly, a regularly varying function is a function of a real variable whose behaviour at infinity is similar to the behaviour of a power law function (like a polynomial) near infinity.

...

this makes sense now: ln(x) "converges" at infinity, i.e. if x = infinity, the function doesn't grow anymore! This is bc to increase value of ln(x) by constant, you'd e.g. need to double x - in this sense it has converged.

regularly varying functions behave like power law functions because if you scale input argument, the function grows by a function of the scale (as opposed to e.g. exponentials, where scaling input scales output by the output - hence exponential: growth depends on itself)

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Slowly varying function - Wikipedia
Wikipedia Join the WPWP Campaign to help improve Wikipedia articles with photos and win a prize Slowly varying function From Wikipedia, the free encyclopedia Jump to navigation Jump to search I<span>n real analysis, a branch of mathematics, a slowly varying function is a function of a real variable whose behaviour at infinity is in some sense similar to the behaviour of a function converging at infinity. Similarly, a regularly varying function is a function of a real variable whose behaviour at infinity is similar to the behaviour of a power law function (like a polynomial) near infinity. These classes of functions were both introduced by Jovan Karamata,[1][2] and have found several important applications, for example in probability theory. Contents 1 Basic definitions 2




Definition 1 . A measurable function L : (0,+∞) → (0,+∞) is called slowly varying (at infinity) if for all a > 0 ,

\(\lim _{{x\to \infty }}{\frac {L(ax)}{L(x)}}=1.\)

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Slowly varying function - Wikipedia
roperties 2.1 Uniformity of the limiting behaviour 2.2 Karamata's characterization theorem 2.3 Karamata representation theorem 3 Examples 4 See also 5 Notes 6 References Basic definitions[edit] <span>Definition 1. A measurable function L : (0,+∞) → (0,+∞) is called slowly varying (at infinity) if for all a > 0, lim x → ∞ L ( a x ) L ( x ) = 1. {\displaystyle \lim _{x\to \infty }{\frac {L(ax)}{L(x)}}=1.} Definition 2. A function L : (0,+∞) → (0,+∞) for which the limit g ( a ) = lim x → ∞ L ( a x ) L ( x ) {\displaystyle g(a)=\lim _{x\to \infty }{\frac {L(ax)}{L(x)}}} is finite but nonze




Definition 2 . A function L : (0,+∞) → (0,+∞) for which the limit

\(g(a)=\lim _{{x\to \infty }}{\frac {L(ax)}{L(x)}}\)

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Slowly varying function - Wikipedia
measurable function L : (0,+∞) → (0,+∞) is called slowly varying (at infinity) if for all a > 0, lim x → ∞ L ( a x ) L ( x ) = 1. {\displaystyle \lim _{x\to \infty }{\frac {L(ax)}{L(x)}}=1.} <span>Definition 2. A function L : (0,+∞) → (0,+∞) for which the limit g ( a ) = lim x → ∞ L ( a x ) L ( x ) {\displaystyle g(a)=\lim _{x\to \infty }{\frac {L(ax)}{L(x)}}} is finite but nonzero for every a > 0, is called a regularly varying function. These definitions are due to Jovan Karamata.[1][2] Note. In the regularly varying case, the sum of two




Definition 2 . A function L : (0,+∞) → (0,+∞) for which the limit

\(g(a)=\lim _{{x\to \infty }}{\frac {L(ax)}{L(x)}}\)

is finite but nonzero for every a > 0 , is called a regularly varying function.

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Slowly varying function - Wikipedia
measurable function L : (0,+∞) → (0,+∞) is called slowly varying (at infinity) if for all a > 0, lim x → ∞ L ( a x ) L ( x ) = 1. {\displaystyle \lim _{x\to \infty }{\frac {L(ax)}{L(x)}}=1.} <span>Definition 2. A function L : (0,+∞) → (0,+∞) for which the limit g ( a ) = lim x → ∞ L ( a x ) L ( x ) {\displaystyle g(a)=\lim _{x\to \infty }{\frac {L(ax)}{L(x)}}} is finite but nonzero for every a > 0, is called a regularly varying function. These definitions are due to Jovan Karamata.[1][2] Note. In the regularly varying case, the sum of two slowly varying functions is again slowly varying function. Basic properties[edit]




Note. In the regularly varying case, the sum of two slowly varying functions is again slowly varying function.

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Slowly varying function - Wikipedia
displaystyle g(a)=\lim _{x\to \infty }{\frac {L(ax)}{L(x)}}} is finite but nonzero for every a > 0, is called a regularly varying function. These definitions are due to Jovan Karamata.[1][2] <span>Note. In the regularly varying case, the sum of two slowly varying functions is again slowly varying function. Basic properties[edit] Regularly varying functions have some important properties:[1] a partial list of them is reported below. More extensive analyses of the properties characterizing




Described in an informal way, if \(f_{n}\) converges to \(f\) uniformly, then the rate at which \(f_{n}(x)\) approaches \(f(x)\) is "uniform" throughout its domain in the following sense: in order to determine how large \(n\) needs to be to guarantee that \(f_{n}(x)\) falls within a certain distance \(\epsilon \) of \(f(x)\), we do not need to know the value of \(x\in E\) in question — there is a single value of \({\displaystyle N=N(\epsilon )}\) independent of \(x\), such that choosing \(n\) to be larger than \(N\) will suffice

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Uniform convergence - Wikipedia
f N + 2 , … {\displaystyle f_{N},f_{N+1},f_{N+2},\ldots } differ from f {\displaystyle f} by no more than ϵ {\displaystyle \epsilon } at every point x {\displaystyle x} in E {\displaystyle E} . <span>Described in an informal way, if f n {\displaystyle f_{n}} converges to f {\displaystyle f} uniformly, then the rate at which f n ( x ) {\displaystyle f_{n}(x)} approaches f ( x ) {\displaystyle f(x)} is "uniform" throughout its domain in the following sense: in order to determine how large n {\displaystyle n} needs to be to guarantee that f n ( x ) {\displaystyle f_{n}(x)} falls within a certain distance ϵ {\displaystyle \epsilon } of f ( x ) {\displaystyle f(x)} , we do not need to know the value of x ∈ E {\displaystyle x\in E} in question — there is a single value of N = N ( ϵ ) {\displaystyle N=N(\epsilon )} independent of x {\displaystyle x} , such that choosing n {\displaystyle n} to be larger than N {\displaystyle N} will suffice. The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept,




In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions \((f_{n})\) converges uniformly to a limiting function \(f\) on a set \(E\) if, given any arbitrarily small positive number \(\epsilon \), a number \(N\) can be found such that each of the functions \({\displaystyle f_{N},f_{N+1},f_{N+2},\ldots }\) differ from \(f\) by no more than \(\epsilon \) at every point \(x\) in \(E\).

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Uniform convergence - Wikipedia
Uniform convergence - Wikipedia Uniform convergence From Wikipedia, the free encyclopedia Jump to navigation Jump to search In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ( f n ) {\displaystyle (f_{n})} converges uniformly to a limiting function f {\displaystyle f} on a set E {\displaystyle E} if, given any arbitrarily small positive number ϵ {\displaystyle \epsilon } , a number N {\displaystyle N} can be found such that each of the functions f N , f N + 1 , f N + 2 , … {\displaystyle f_{N},f_{N+1},f_{N+2},\ldots } differ from f {\displaystyle f} by no more than ϵ {\displaystyle \epsilon } at every point x {\displaystyle x} in E {\displaystyle E} . Described in an informal way, if f n {\displaystyle f_{n}} converges to f {\displaystyle f} uniformly, then the rate at which f n ( x ) {\displaystyle f_{n}(x)} approaches f ( x ) {\dis




Flashcard 5731738062092



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In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.[

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Power law - Wikipedia
her uses, see Power. An example power-law graph that demonstrates ranking of popularity. To the right is the long tail, and to the left are the few that dominate (also known as the 80–20 rule). <span>In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.[1] Contents 1 Empirical examples 2 Properties 2.1 Scale invariance 2.2 Lack of well-defined average value 2.3 Universality 3 Power-law functions 3.1 Examples 3.1.1 Astronomy 3.1.2 Crimin




One attribute of power laws is their scale invariance. Given a relation \(f(x)=ax^{-k}\), scaling the argument \(x\) by a constant factor \(c\) causes only a proportionate scaling of the function itself. That is,

\({\displaystyle f(cx)=a(cx)^{-k}=c^{-k}f(x)\propto f(x),\!}\)

where \(\propto \) denotes direct proportionality. That is, scaling by a constant \(c\) simply multiplies the original power-law relation by the constant \(c^{{-k}}\). Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both \(f(x)\) and \(x\), and the straight-line on the log–log plot is often called the signature of a power law.

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Power law - Wikipedia
ds for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature. Properties[edit] Scale invariance[edit] <span>One attribute of power laws is their scale invariance. Given a relation f ( x ) = a x − k {\displaystyle f(x)=ax^{-k}} , scaling the argument x {\displaystyle x} by a constant factor c {\displaystyle c} causes only a proportionate scaling of the function itself. That is, f ( c x ) = a ( c x ) − k = c − k f ( x ) ∝ f ( x ) , {\displaystyle f(cx)=a(cx)^{-k}=c^{-k}f(x)\propto f(x),\!} where ∝ {\displaystyle \propto } denotes direct proportionality. That is, scaling by a constant c {\displaystyle c} simply multiplies the original power-law relation by the constant c − k {\displaystyle c^{-k}} . Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both f ( x ) {\displaystyle f(x)} and x {\displaystyle x} , and the straight-line on the log–log plot is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of d




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plot the exponentials on a log-log scale
 figure loglog(x, y1, '.k' , x, y2, '.b' , x, y3, '.r' ); title ( 'Three exponentials on a loglog scale' ) xlabel( 'x' ); ylabel( 'y' ); legend({ 'y = e^x' , 'y = e^{2x}' , 'y = e^{3x}' }, 'Location' , 'Northwest' )

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EXPERIMENT 2: Working with logarithmic scales
logx(x, y1, '.k', x, y2, '.b', x, y3, '.r'); title ('Three exponentials on a semilog x scale') xlabel('x'); ylabel('y'); legend({'y = e^x', 'y = e^{2x}', 'y = e^{3x}'}, 'Location', 'Northwest') <span>plot the exponentials on a log-log scale figure loglog(x, y1, '.k', x, y2, '.b', x, y3, '.r'); title ('Three exponentials on a loglog scale') xlabel('x'); ylabel('y'); legend({'y = e^x', 'y = e^{2x}', 'y = e^{3x}'}, 'Location', 'Northwest') Experiment 2c: Behavior of logarithms when plotted using different scales Create data for 3 exponential functions for points in the interval [0.1, 10] x = 0.1:0.1:10; y1 = log(x); % Eva




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plot the polynomials on a log-log scale
 figure loglog(x, y1, '.k' , x, y2, '.b' , x, y3, '.r' ); title ( 'Three polynomials on a loglog scale' ) xlabel( 'x' ); ylabel( 'y' ); legend({ 'y = x' , 'y = x^2' , 'y = x^3' }, 'Location' , 'Northwest' )

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EXPERIMENT 2: Working with logarithmic scales
gure semilogx(x, y1, '.k', x, y2, '.b', x, y3, '.r'); title ('Three polynomials on a semilog x scale') xlabel('x'); ylabel('y'); legend({'y = x', 'y = x^2', 'y = x^3'}, 'Location', 'Northwest') <span>plot the polynomials on a log-log scale figure loglog(x, y1, '.k', x, y2, '.b', x, y3, '.r'); title ('Three polynomials on a loglog scale') xlabel('x'); ylabel('y'); legend({'y = x', 'y = x^2', 'y = x^3'}, 'Location', 'Northwest') Experiment 2b: Behavior of exponentials when plotted using different scales Create data for 3 exponential functions for points in the interval [0.1, 10] x = 0.1:0.1:10; y1 = exp(x); % E




A power-law \({\displaystyle x^{-k}}\) has a well-defined mean over \({\displaystyle x\in [1,\infty )}\) only if \({\displaystyle k>2}\), and it has a finite variance only if \({\displaystyle k>3}\); most identified power laws in nature have exponents such that the mean is well-defined but the variance is not, implying they are capable of black swan behavior

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Power law - Wikipedia
-normal distribution).[citation needed] Thus, accurately fitting and validating power-law models is an active area of research in statistics; see below. Lack of well-defined average value[edit] <span>A power-law x − k {\displaystyle x^{-k}} has a well-defined mean over x ∈ [ 1 , ∞ ) {\displaystyle x\in [1,\infty )} only if k > 2 {\displaystyle k>2} , and it has a finite variance only if k > 3 {\displaystyle k>3} ; most identified power laws in nature have exponents such that the mean is well-defined but the variance is not, implying they are capable of black swan behavior.[2] This can be seen in the following thought experiment:[10] imagine a room with your friends and estimate the average monthly income in the room. Now imagine the world's richest perso




The black swan theory or theory of black swan events is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalised after the fact with the benefit of hindsight.

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Black swan theory - Wikipedia
wan theory - Wikipedia Black swan theory From Wikipedia, the free encyclopedia Jump to navigation Jump to search Theory of response to surprise events A black swan (Cygnus atratus) in Australia <span>The black swan theory or theory of black swan events is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalised after the fact with the benefit of hindsight. The term is based on an ancient saying that presumed black swans did not exist – a saying that became reinterpreted to teach a different lesson after the first European encounter with t




This concept is often contrasted with uniform convergence. To say that

\(\lim_{n\rightarrow\infty}f_n=f\ \mbox{uniformly}\)

means that

\({\displaystyle \lim _{n\rightarrow \infty }\,\sup\{\,\left|f_{n}(x)-f(x)\right|:x\in A\,\}=0,}\)

where \(A\) is the common domain of \(f\) and \(f_{n}\). That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent. For example, if \({\displaystyle f_{n}:[0,1)\rightarrow [0,1)}\) is a sequence of functions defined by \(f_{n}(x)=x^{n}\), then \({\displaystyle \lim _{n\rightarrow \infty }f_{n}(x)=0}\) pointwise on the interval [0,1), but not uniformly.

...

i've missed some intuition first time around: in closed domain, pointwise = uniform convergence (because sup = max, and this max x needs to converge by pointwise criteria as well). For pointwise, every "point" x, the function family f_n approaches f. Can imagine this as one function slowly creeping towards the other one. The speed of convergence doesn't matter. For uniform it's somewhat different: For open intervals, there might be a supremum that doesn't converge, while all xs converge (i.e. we have pointwise, but not uniform convergence). This means for some given n, the functions have nearly approached each other, but towards boundary of domain, we can always find an x that still hasn't approached f. It refuses to approach, so to speak. It's probably called "uniform convergence" because if it holds, then all x's (even the ones at boundary) converge uniformly - as opposed to example before, where we could find f_n(x)'s at domain boundary that stayed far away from f.

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Pointwise convergence - Wikipedia
\lim _{n\rightarrow \infty }f_{n}(x)=f(x)} for every x in the domain. The function f {\displaystyle f} is said to be the pointwise limit function of f n {\displaystyle f_{n}} . Properties[edit] <span>This concept is often contrasted with uniform convergence. To say that lim n → ∞ f n = f uniformly {\displaystyle \lim _{n\rightarrow \infty }f_{n}=f\ {\mbox{uniformly}}} means that lim n → ∞ sup { | f n ( x ) − f ( x ) | : x ∈ A } = 0 , {\displaystyle \lim _{n\rightarrow \infty }\,\sup\{\,\left|f_{n}(x)-f(x)\right|:x\in A\,\}=0,} where A {\displaystyle A} is the common domain of f {\displaystyle f} and f n {\displaystyle f_{n}} . That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent. For example, if f n : [ 0 , 1 ) → [ 0 , 1 ) {\displaystyle f_{n}:[0,1)\rightarrow [0,1)} is a sequence of functions defined by f n ( x ) = x n {\displaystyle f_{n}(x)=x^{n}} , then lim n → ∞ f n ( x ) = 0 {\displaystyle \lim _{n\rightarrow \infty }f_{n}(x)=0} pointwise on the interval [0,1), but not uniformly. The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example, f ( x ) = lim n → ∞ cos ⁡ ( π x ) 2 n




The sequence \((f_{n})\) converges pointwise to the function \(f\), often written as

\(\lim_{n\rightarrow\infty}f_n=f\ \mbox{pointwise},\)

if and only if

\({\displaystyle \lim _{n\rightarrow \infty }f_{n}(x)=f(x)}\)

for every x in the domain. The function \(f\) is said to be the pointwise limit function of \(f_{n}\).

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Pointwise convergence - Wikipedia
n[edit] Suppose ( f n ) {\displaystyle (f_{n})} is a sequence of functions sharing the same domain and codomain. The codomain is most commonly the reals, but in general can be any metric space. <span>The sequence ( f n ) {\displaystyle (f_{n})} converges pointwise to the function f {\displaystyle f} , often written as lim n → ∞ f n = f pointwise , {\displaystyle \lim _{n\rightarrow \infty }f_{n}=f\ {\mbox{pointwise}},} if and only if lim n → ∞ f n ( x ) = f ( x ) {\displaystyle \lim _{n\rightarrow \infty }f_{n}(x)=f(x)} for every x in the domain. The function f {\displaystyle f} is said to be the pointwise limit function of f n {\displaystyle f_{n}} . Properties[edit] This concept is often contrasted with uniform convergence. To say that lim n → ∞ f n = f uniformly {\displaystyle \lim _{n\rightarrow \infty }f_{n}=f\ {\mbox{uniformly}




Regularly varying functions have some important properties:[1] a partial list of them is reported below. More extensive analyses of the properties characterizing regular variation are presented in the monograph by Bingham, Goldie & Teugels (1987).

Uniformity of the limiting behaviour [ edit ]

Theorem 1 . The limit in definitions 1 and 2 is uniform if a is restricted to a compact interval.

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Slowly varying function - Wikipedia
unction. These definitions are due to Jovan Karamata.[1][2] Note. In the regularly varying case, the sum of two slowly varying functions is again slowly varying function. Basic properties[edit] <span>Regularly varying functions have some important properties:[1] a partial list of them is reported below. More extensive analyses of the properties characterizing regular variation are presented in the monograph by Bingham, Goldie & Teugels (1987). Uniformity of the limiting behaviour[edit] Theorem 1. The limit in definitions 1 and 2 is uniform if a is restricted to a compact interval. Karamata's characterization theorem[edit] Theorem 2. Every regularly varying function f : (0,+∞) → (0,+∞) is of the form f ( x ) = x β L ( x ) {\displaystyle f(x)=x^{\beta }L(x)} where




In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral.

\({\displaystyle \int _{a}^{b}f(x)\,dx}\).

The trapezoidal rule works by approximating the region under the graph of the function \(f(x)\) as a trapezoid and calculating its area. It follows that

\({\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\cdot {\tfrac {f(a)+f(b)}{2}}}\).

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Trapezoidal rule - Wikipedia
ule (differential equations). For the explicit trapezoidal rule for solving initial value problems, see Heun's method. The function f(x) (in blue) is approximated by a linear function (in red). <span>In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral. ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} . The trapezoidal rule works by approximating the region under the graph of the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area. It follows that ∫ a b f ( x ) d x ≈ ( b − a ) ⋅ f ( a ) + f ( b ) 2 {\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\cdot {\tfrac {f(a)+f(b)}{2}}} . The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated b




Flashcard 5731771616524

Question
trapezoidal rule
Answer

\({\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\cdot {\tfrac {f(a)+f(b)}{2}}}\).


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Open it
\({\displaystyle \int _{a}^{b}f(x)\,dx}\). The trapezoidal rule works by approximating the region under the graph of the function \(f(x)\) as a trapezoid and calculating its area. It follows th<span>at \({\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\cdot {\tfrac {f(a)+f(b)}{2}}}\). <span>

Original toplevel document

Trapezoidal rule - Wikipedia
ule (differential equations). For the explicit trapezoidal rule for solving initial value problems, see Heun's method. The function f(x) (in blue) is approximated by a linear function (in red). <span>In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral. ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} . The trapezoidal rule works by approximating the region under the graph of the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area. It follows that ∫ a b f ( x ) d x ≈ ( b − a ) ⋅ f ( a ) + f ( b ) 2 {\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\cdot {\tfrac {f(a)+f(b)}{2}}} . The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated b







Flashcard 5731776335116

Tags
#has-images
Question
graphical intuition: trapezoidal rule
Answer


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#mental_models

Inversion technique for solving problems:

Let’s take a look at some examples. Say you want to improve innovation in your organization. Thinking forward, you’d think about all of the things you could do to foster innovation. If you look at the problem by inversion, however, you’d think about all the things you could do that would discourage innovation. Ideally, you’d avoid those things. Sounds simple right? I bet your organization does some of those ‘stupid’ things today.

...

this is similar to easy ELO gaining: Look at mistakes, then don't do them (e.g. no blunders in chess)

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Inversion: The Power of Avoiding Stupidity
about them forwards and backward. Inversion often forces you to uncover hidden beliefs about the problem you are trying to solve. “Indeed,” says Munger, “many problems can’t be solved forward.” <span>Let’s take a look at some examples. Say you want to improve innovation in your organization. Thinking forward, you’d think about all of the things you could do to foster innovation. If you look at the problem by inversion, however, you’d think about all the things you could do that would discourage innovation. Ideally, you’d avoid those things. Sounds simple right? I bet your organization does some of those ‘stupid’ things today. Another example, rather than think about what makes a good life, you can think about what prescriptions would ensure misery. Avoiding stupidity is easier than seeking brilliance. While




All I want to know is where I’m going to die, so I’ll never go there

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Inversion: The Power of Avoiding Stupidity
ty Inversion and The Power of Avoiding Stupidity Reading Time: 2 minutes Charlie Munger, the business partner of Warren Buffett and Vice Chairman of Berkshire Hathaway, is famous for his quote “<span>All I want to know is where I’m going to die, so I’ll never go there.” That thinking was inspired by the German mathematician Carl Gustav Jacob Jacobi, famous for some work on elliptic functions that I’ll never understand. Jacobi often solved difficult p




Flashcard 5731788655884

Question
Thinking: Quote that distills inversion technique
Answer
All I want to know is where I’m going to die, so I’ll never go there


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All I want to know is where I’m going to die, so I’ll never go there

Original toplevel document

Inversion: The Power of Avoiding Stupidity
ty Inversion and The Power of Avoiding Stupidity Reading Time: 2 minutes Charlie Munger, the business partner of Warren Buffett and Vice Chairman of Berkshire Hathaway, is famous for his quote “<span>All I want to know is where I’m going to die, so I’ll never go there.” That thinking was inspired by the German mathematician Carl Gustav Jacob Jacobi, famous for some work on elliptic functions that I’ll never understand. Jacobi often solved difficult p