let $\alpha,\beta$ be the solutions of $t^2=xt+1$.since $$\alpha+\beta=x,\quad \alpha\beta=-1$$ we have $$p(n+1)=(\alpha+\beta)p(n)-\alpha\beta p(n-1)$$so, we get $$p(n+1)-\alpha p(n)=\beta (p(n)-\alpha p(n-1))=\cdots =\beta^{n-1}(p(2)-\alpha p(1))$$ and $$p(n+1)-\beta p(n)=\alpha (p(n)-\beta p(n-1))=\cdots =\alpha^{n-1}(p(2)-\beta p(1))$$subtracting the latter from the former gives $$(\beta-\alpha)p(n)=\beta^{n-1}(x-\alpha)-\alpha^{n-1}(x-\beta),$$ i.e. $$p(n)=\frac{\beta^{n-1}(x-\alpha)-\alpha^{n-1}(x-\beta)}{\beta-\alpha}$$ where $$\alpha=\frac{x-\sqrt{x^2+4}}{2},\quad\beta=\frac{x+\sqrt{x^2+4}}{2}$$

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we lose nothing by starting with the equation $$ y'' + 2by' + cy = 0, $$ since the coefficient of $y''$ must be nonzero. we can proceed in a couple of different ways:reduction of orderthe simplest way is to suppose that $\alpha$ is a root of the characteristic equation $k^2+2bk+c=0$, and substitute $ y=ue^{\alpha x} $. this gives $$ 0 = y'' + 2by' + cy = e^{\alpha x} \left( (\alpha^2+2b\alpha+c)u + u''+ 2(\alpha+b) u' \right). $$ the first term in the bracket vanishes since $\alpha$ is a root, so we end up having to solve $$ u'' + 2(\alpha+b)u' = 0, $$ a first-order equation. now, suppose that $\beta$ is the other root of the characteristic equation. we have $$ k^2+2bk+c = (k-\alpha)(k-\beta) = k^2 - (\alpha+\beta)k+\alpha\beta, $$ so ...

yes.the number $r(k)$ is an element of the splitting field $f$ of the irreducible polynomial $f(x)=x^{n}-2$ where $n=\operatorname{lcm}(1,2,\ldots,k)$). indeed, if $\alpha$ is the unique positive root of $f$, then we have $$ r(k)=\sum_{n=2}^k\alpha^{n/n}$$ let $p$ be a prime with $\frac k2<p\le k$ (such a prime exists by bertrand's postulate), and let $\beta=\alpha\cdot e^{2\pi i/p}$. then $p\mid n$ and so $f(\beta)=0$. hence there exists an automorphism $\phi$ of $f$ that maps $\alpha\mapsto \beta$. note that $\beta^{n/n}=\alpha^{n/n}$ for most $n$. indeed, we have $\beta^{n/n}\ne\alpha^{n/n}$ if and only if $n/n$ is not a multiple of $p$. as $p^2\nmid n$, this is the case precisely for the case $n=p$. it follows that $$\phi(r(k))=r(k)+\beta^{n/p}-\alpha^{n/p}\ne r(k)$$ and therefore $

following cherng-tiao perng's proof for the other inequality, here is what i find. i don't have immediate access to folkmar bornemann's references, so i'm not sure the ideas here might be similar.begin by writing $t_k=\frac1{\sqrt{\alpha+\beta k^2}}t_k\sqrt{\alpha+\beta k^2}$, for $\alpha, \beta>0$ to be specified later. then, we apply the cauchy-schwarz inequality as follows \begin{align}\left(\sum_kt_k\right)^2&=\left(\sum_k\frac1{\sqrt{\alpha+\beta k^2}}t_k\sqrt{\alpha+\beta k^2}\right)^2 \leq\left(\sum_{k=1}^{\infty}\frac1{\alpha+\beta k^2}\right)\left(\sum_{k=1}^{\infty}(\alpha+\beta k^2)t_k^2\right)\\ &<\left(\int_0^{\infty}\frac{dx}{\alpha+\beta x^2}\right)\left(\alpha\sum_{k=1}^{\infty}t_k^2+\beta\sum_{k=1}^{\infty}k^2t_k^2\right) =\frac{\pi}{2\sqrt{\alpha\beta}}\left

i have a doubt about convolution.i have found this definition :$$(f*g)(t)=\int_{-\infty}^{+\infty} f(t-\alpha) \ g(\alpha) \ d\alpha$$this integral does not converge:$$\cos(t)*t=\int_{-\infty}^{+\infty} \cos(t-\alpha) \ \alpha \ d\alpha$$contrariwise: $$ \mathscr{l} \{ \cos(t) * t \} =\mathscr{l} \{ \cos(t) \} \ \mathscr{l} \{t \}=\frac{1}{s^3+s}$$partial fraction decomposition:$$\frac{1}{s^3+s}=\frac{a}{s}+\frac{b}{s-i}+\frac{c}{s+i}$$$$a=\lim_{s\rightarrow 0} \ \frac{1}{s^2+1}=1$$ $$b=\lim_{s\rightarrow i} \ \frac{1}{s^2+is}=-\frac{1}{2}$$ $$c=\lim_{s\rightarrow -i} \ \frac{1}{s^2-is}=\frac{1}{2}$$$$\frac{1}{s^3+s}=\frac{1}{s}+\frac{-\frac{1}{2}}{s-i}+\frac{\frac{1} {2}}{s+i}$$$$\mathscr{l}^{-1} \{ \frac{1}{s}+\frac{-\frac{1}{2}}{s-i}+\frac{\frac{1}{2}}{s+i} \}=1-\frac{1}{2} \ e^{it}+\fr

when the "null hypothesis" includes more than one state of nature, the actual false positive rate (fpr) may vary with that state. all we can do is guarantee a limit on the fpr no matter what that state of nature might be--but we cannot always guarantee the fpr actually equals $\alpha$.(there are other reasons why the fpr might not actually equal its targeted value $\alpha$, such as when the test statistic is discrete. these situations usually can be cured by using randomized decision procedures. as such they do not provide any fundamental insight into the question.)consider the cl ical one-tailed test where the statistic $x$ is umed to have a normal distribution of unknown mean $\mu$ and (for simplicity) known standard deviation $\sigma$. $\mu$ is to be compared to $0$. the null hypothesis

to add to rob's answer, i wanted to expand on where this naming convention comes from.the international astronomical union (iau) is the organization which generally sets conventions and definitions. they're the ones who demoted pluto to being a dwarf planet in 2006. anyway, before any exoplanets were found, there existed a convention for naming multiple-star systems. the rule was that you gave the system a name, for example alpha centauri, and then the brightest star in that system is designated by "a". any other objects are then given the letters "b", "c", etc. the alpha centauri system has three stars in it and so they're named alpha centauri a (the brightest of the three), alpha centauri b, and alpha centauri c (a.k.a. proxima centauri).this convention then continued on to nami...

all the numbers you list are algebraic.lemma.if $\alpha$ and $\beta$ are algebraic, then $\alpha+\beta$ and $\alpha\beta$ are algebraic.proof. since $\beta$ is algebraic (over $\mathbb{q}$), it is also algebraic over $\mathbb{q}(\alpha)$. hence $\mathbb{q}(\alpha,\beta)$ is a finite extension of $\mathbb{q}(\alpha)$, which in turn is finite over $\mathbb{q}$. therefore $\mathbb{q}(\alpha,\beta)$ is finite by the dimension formula: $$ [\mathbb{q}(\alpha,\beta):\mathbb{q}]= [\mathbb{q}(\alpha,\beta):\mathbb{q}(\alpha)] [\mathbb{q}(\alpha):\mathbb{q}] $$ any finite extension of $\mathbb{q}$, say $n$-dimensional, consists of algebraic elements, because if $\gamma$ is an element, $\{1,\gamma,\gamma^2,\dots,\gamma^n\}$ is linearly dependent, so we find a polynomial with coefficients in $\mathbb{

when a nucleus decays the reaction is characterised by the release of a fixed amount of energy called the q-value of the reaction.this diagram shows what happens when am-241 decays to np-237 with the emission of an alpha particle.the energy levels in both nuclei and well defined and so the energies of the alpha particles are well defined. $\rm energy_{\rm decay} =energy_{excited \,daughter} + energy_{alpha}$ so in the example shown the energies of the emitted alpha particles will be 5.48, 5.54, 5.58, 5.61 and 5.64 mev.the excited daughter nucleus then gets rid of the surplus energy with the emission of a gamma.however for beta decay the quantum jumps as characterised by those shown below for alpha particles are accompanied by the emission of two particles which together carry away a fix...

yes, this holds in general. recall that an extension $f/\bbb q$ is algebraic iff for each $\alpha\in f$ we have$$[\bbb q(\alpha):\bbb q]=\dim_{\bbb q} \bbb q(\alpha)< \infty$$that is, if every element in it generates a finite extension. similarly any finite extension is algebraic since if $\beta\in f$ then $\bbb q(\beta)\subseteq f\implies [\bbb q(\beta):\bbb q]\le [f:\bbb q]$. for a nested radical expression, this is easy then, take the most nested radical, adjoin the needed roots and proceed by induction.in your first example $\alpha=\sqrt{11+2\sqrt 7}$, then we see that the field $\bbb q(\sqrt 7)$ has a polynomial for this which is ${1\over 2}(x^2-11)-\sqrt 7 = 0$ so the splitting field of this polynomial, which has degree at most $2$ over $\bbb q(\sqrt 7)$, and therefore degree at m

tgg_overlord|43m ago|news|0|award-winning veteran games developer and international games label team17 has today announced its partnership with freiburg, germany-based independent developer radiation blue to publish genesis alpha one, a stunning new sci-fi game powered by unreal engine 4 and combining a unique mix of roguelike, fps, base building and survival elements on pc and console. game developergenesis alpha onepcps4team 17xbox onethegg.netread full story >>thegg.net

#1: no. norms which are induced by inner products are exactly those satisfying the "parallelogram law": $2 \| x \|^2 + 2 \| y \|^2 = \| x+y \|^2 + \| x-y \|^2$. in this case you have an inner product defined by the "polarization identity" $\langle x,y \rangle = \frac{1}{4} \left ( \| x+y \|^2 - \| x - y \|^2 \right )$ (with some small change in the complex case). two norms on $\mathbb{r}^n$ for $n>1$ that do not have this property are the $1$ norm, $\| x \|_1=\sum_{i=1}^n |x_i|$, and the $\infty$ norm, $\| x \|_\infty = \max_i |x_i|$.#2: no, there are many metric spaces which are not normed spaces. many of these are not even vector spaces, but we can even equip vector spaces with metrics which are not compatible with norms (in the sense that there is no norm such that $\| x \|=d(x,0)$).

let $$\textbf{sphd}(\alpha;x)=\int_{0}^xt^{\alpha t}dt$$i want make a g hic like this by using mathematicai tried:plot[table[nintegrate[t^(a t), {t, 0, x}], {a, -1, 10, 1}], {x, 1, 4}] but it is very slow..

the alpha particle is a quantum mechanical system, and it is not clear what we might mean by drawing pictures of billiard balls arranged according to cl ical polyhedra.in particular, the alpha has quantum numbers $j^\pi=0^+$, so it has complete spherical symmetry. in a shell model picture, which provides a simple guide to the exact 4-body wave function, the alpha is a state in which all four particle (a neutron with spin up/down, and a proton with spin up/down) occupy the same 1s (spherically symmetric) orbital. this implies that the alpha should be drawn as a blob, with smeared out protons and neutrons. the shell model wave function is not exact, and there are short range correlations, that means if i detect a spin up proton at the origin, then there is a slightly enhanced/reduced probabi

wolfyseyes|1h ago|article|0|the mmo alpha and beta list is the most complete list of pre-launch events in the mmo industry. find your next game and catch all the latest news! devindustrypcmmogames.comread full story >>mmogames.com

wolfyseyes|2m ago|news|0|developer citadel studios announces plans for legends of aria alpha 2, future testing goals, and a shift in the game's steam release. legends of ariapcmmogames.comread full story >>mmogames.com

john2|9m ago|news|0|swdtech games has released a pre-alpha demo for pixel noir. pcpixel noirdsogaming.comread full story >>dsogaming.com

from the theory of linear mappings, we know linear maps over a vector space isfy two properties:additivity: $$f(v+w)=f(v)+f(w)$$homogeneity: $$f(\alpha v)=\alpha f(v)$$which $\alpha\in \mathbb{f}$ is a scalar in the field which the vector space is defined on, and neither of these conditions implies the other one. if $f$ is defined over the complex numbers, $f:\mathbb{c}\longrightarrow \mathbb{c}$, then finding a mapping which is additive but not homogenous is simple; for example, $f(c)=c^*$. but can any one present an example on the reals, $f:\mathbb{r}\longrightarrow \mathbb{r}$, which is additive but not homogenous?

roll_dmg|52m ago|news|3|independent developer ninja theory has updated their progress on the playstation blog that their ps4 exclusive, hellblade: senua’s sacrifice has reached the alpha stage in development. game developerhellblade: senua’s sacrificeninja theoryps4hardcoregamer.comread full story >>hardcoregamer.com

closeskip in skipxembedxshare a clip from community reporter jerold macdonald-evoy's tour of the homeless encampment near mesa known as "camp alpha" as they work to comply with adot's order for them to relocate. hannah gaber/azcentral.com “12-string” & dog penny live at camp alpha near the 202 and mckellips drive wednesday, dec. 21, 2016 in mesa, ariz. camp alpha is a homeless encampment that focuses on housing veterans, but won't turn anyone away.(photo: david kadlubowski /the republic) 168 connect 1 linkedinemailmorecorrections & clarifications: an upcoming tempe community action agency fundraiser aims to raise $25,000. an earlier version listed the wrong amount. at any given time in 2015, there were 564,708 homeless people across th...

closeskip in skipxembedxshare a clip from community reporter jerold macdonald-evoy's tour of the homeless encampment near mesa known as "camp alpha" as they work to comply with adot's order for them to relocate. hannah gaber/azcentral.com “12-string” & dog penny live at camp alpha near the 202 and mckellips drive wednesday, dec. 21, 2016 in mesa, ariz. camp alpha is a homeless encampment that focuses on housing veterans, but won't turn anyone away.(photo: david kadlubowski /the republic) 170 connect 1 linkedinemailmorecorrections & clarifications: an upcoming tempe community action agency fundraiser aims to raise $25,000. an earlier version listed the wrong amount. at any given time in 2015, there were 564,708 homeless people across th...

closeskip in skipxembedxshare a clip from community reporter jerold macdonald-evoy's tour of the homeless encampment near mesa known as "camp alpha" as they work to comply with adot's order for them to relocate. hannah gaber/azcentral.com “12-string” & dog penny live at camp alpha near the 202 and mckellips drive wednesday, dec. 21, 2016 in mesa, ariz. camp alpha is a homeless encampment that focuses on housing veterans, but won't turn anyone away.(photo: david kadlubowski /the republic) 169 connect 1 linkedinemailmorecorrections & clarifications: an upcoming tempe community action agency fundraiser aims to raise $25,000. an earlier version listed the wrong amount. at any given time in 2015, there were 564,708 homeless people across th...

closeskip in skipxembedxshare a clip from community reporter jerold macdonald-evoy's tour of the homeless encampment near mesa known as "camp alpha" as they work to comply with adot's order for them to relocate. hannah gaber/azcentral.com “12-string” & dog penny live at camp alpha near the 202 and mckellips drive wednesday, dec. 21, 2016 in mesa, ariz. camp alpha is a homeless encampment that focuses on housing veterans, but won't turn anyone away.(photo: david kadlubowski /the republic) 174 connect 1 linkedinemailmorecorrections & clarifications: an upcoming tempe community action agency fundraiser aims to raise $25,000. an earlier version listed the wrong amount. at any given time in 2015, there were 564,708 homeless people across th...

closeskip in skipxembedxshare a clip from community reporter jerold macdonald-evoy's tour of the homeless encampment near mesa known as "camp alpha" as they work to comply with adot's order for them to relocate. hannah gaber/azcentral.com “12-string” & dog penny live at camp alpha near the 202 and mckellips drive wednesday, dec. 21, 2016 in mesa, ariz. camp alpha is a homeless encampment that focuses on housing veterans, but won't turn anyone away.(photo: david kadlubowski /the republic) 173 connect 1 linkedinemailmorecorrections & clarifications: an upcoming tempe community action agency fundraiser aims to raise $25,000. an earlier version listed the wrong amount. at any given time in 2015, there were 564,708 homeless people across th...

closeskip in skipxembedxshare a clip from community reporter jerold macdonald-evoy's tour of the homeless encampment near mesa known as "camp alpha" as they work to comply with adot's order for them to relocate. hannah gaber/azcentral.com “12-string” & dog penny live at camp alpha near the 202 and mckellips drive wednesday, dec. 21, 2016 in mesa, ariz. camp alpha is a homeless encampment that focuses on housing veterans, but won't turn anyone away.(photo: david kadlubowski /the republic) 171 connect 1 linkedinemailmorecorrections & clarifications: an upcoming tempe community action agency fundraiser aims to raise $25,000. an earlier version listed the wrong amount. at any given time in 2015, there were 564,708 homeless people across th...

closeskip in skipxembedxshare a clip from community reporter jerold macdonald-evoy's tour of the homeless encampment near mesa known as "camp alpha" as they work to comply with adot's order for them to relocate. hannah gaber/azcentral.com “12-string” & dog penny live at camp alpha near the 202 and mckellips drive wednesday, dec. 21, 2016 in mesa, ariz. camp alpha is a homeless encampment that focuses on housing veterans, but won't turn anyone away.(photo: david kadlubowski /the republic) 166 connect 1 linkedinemailmorecorrections & clarifications: an upcoming tempe community action agency fundraiser aims to raise $25,000. an earlier version listed the wrong amount. at any given time in 2015, there were 564,708 homeless people across th...