on 28-Aug-2020 (Fri)

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.

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)

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

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

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

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

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).

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

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:

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

Preface

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.

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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)

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

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

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

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

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.

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

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

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 \(E\).

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.^[

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.

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' )

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' )

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

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.

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.

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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.

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 \(f_{n}\).

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.

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

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

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.

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

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