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Well, I have the following recursive formula (where $\text{n}$ gives the position in the sequence):
$$\text{P}_\text{n}=\alpha\cdot\text{P}_{\text{n}-1}+\text{P}_{\text{n}-2}\tag1$$
For arbitrary $\alpha\in\mathbb{N}^+$.
And I know that $\text{P}_1=\beta$ and $\text{P}_2=\gamma$, where $\beta\space\wedge\space\gamma\in\mathbb{N}^+$.
How can I write a program that gives me the value of the nth position in the sequence?
Example, find the value of the 5th position in the sequence when we know that $\beta=1$ and $\alpha=\gamma=2$. Now it has to give the value $\text{P}_5=29$.
So, I think that the code has to start with:
\[Alpha] =2;\[Beta] =1;\[Gamma] =2;n =5;
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Yeah, this software cannot be too easy to install, my installer is very professional looking, currently not tied into that code, but directs the user how to search for their MikTeX and or install it and does a test LaTeX rendering
Some body like Zeta (on codereview) might be able to help a lot... I'm not sure if he does a lot of category theory, but he does a lot of Haskell (not that I'm trying to conflate the two)... so he would probably be one of the better bets for asking for revision of code.
he is usually on the 2nd monitor chat room. There are a lot of people on those chat rooms that help each other with projects.
i'm not sure how many of them are adept at category theory though... still, this chat tends to emphasize a lot of small problems and occasionally goes off tangent.
you're project is probably too large for an actual question on codereview, but there is a lot of github activity in the chat rooms. gl.
In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966). It was also written down as the Fourier transform off^(z)=∏m=1∞(cos...
Defined as the probability that $\sum_{n=1}^\infty2^{-n}\zeta_n$ will be less than $x$, where the $\zeta_n$ are chosen randomly and independently from the unit interval
@AkivaWeinberger are you familiar with the theory behind Fourier series?
anyway here's a food for thought
for $f : S^1 \to \Bbb C$ square-integrable, let $c_n := \displaystyle \int_{S^1} f(\theta) \exp(-i n \theta) (\mathrm d\theta/2\pi)$, and $f^\ast := \displaystyle \sum_{n \in \Bbb Z} c_n \exp(in\theta)$. It is known that $f^\ast = f$ almost surely.
(a) is $-^\ast$ idempotent? i.e. is it true that $f^{\ast \ast} = f^\ast$?
@AkivaWeinberger You need to use the definition of $F$ as the cumulative function of the random variables. $C^\infty$ was a simple step, but I don't have access to the paper right now so I don't recall it.
> In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.
I am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (and maybe this doesnt't help my intuition), but I am having trouble seeing where in the setup they d...
Usually it is a great advantage to consult the notes, as they tell you exactly what has been done. A book will teach you the field, but not necessarily help you understand the style that the prof. (who creates the exam) creates questions.
@AkivaWeinberger having thought about it a little, I think the best way to approach the geometry problem is to argue that the relevant condition (centroid is on the incircle) is preserved by similarity transformations
hence you're free to rescale the sides, and therefore the (semi)perimeter as well
so one may (for instance) choose $s=(a+b+c)/2=1$ without loss of generality
that makes a lot of the formulas simpler, e.g. the inradius is identical to the area
It is asking how many terms of the Euler Maclaurin formula do we need in order to compute the Riemann zeta function in the complex plane?
$q$ is the upper summation index in the sum with the Bernoulli numbers.
This appears to answer it in the positive: "By repeating the above argument we see that we have analytically continued the Riemann zeta-function to the right-half plane σ > 1 − k, for all k = 1, 2, 3, . . .."
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Here we want to give an easy mathematical bootstrap argument why solutions to the time independent 1D Schrödinger equation (TISE) tend to be rather nice. First formally rewrite the differential form$$-\frac{\hbar^2}{2m} \psi^{\prime\prime}(x) + V(x) \psi(x) ~=~ E \psi(x) \tag{1}$$into the int...
[Some time travel comments] Since in the previous paragraph, we have explained how travelling to the future will not necessary result in you to arrive in the future that is resulted as if you have never time travelled (via twin paradox), what is the reason that the past you travelled back, has to be the past you learnt from historical records :?
@0ßelö7 Well, I'd omit the explanation of the notation on the slide itself, and since there seems to be two pairs of formulae, I'd just put one of the two and then say that there's another one with suitable substitutions.
I mean, "Hey, I bet you've always wondered how to prove X - here it is" is interesting. "Hey, you know that statement everyone knows how to prove but doesn't bother to write down? Here is the proof written down" significantly less so
Sorry I have a quick question: For questions like this physics.stackexchange.com/questions/356260/… where the accepted answer clearly does not answer the original question what is the best thing to do; downvote, flag or just leave it?
So this question says express $u^0$ in terms of $u^j$ where $u$ is the four-velocity and I get what $u^0$ and $u^j$ are but I'm a bit confused how to go about this one? I thought maybe using the space-time interval and evaluating for $\frac{dt}{d\tau}$ but it's not workin out for me... :/ Anyone give me a quickie starter please? :p
Although a physics question, this is still important to chemistry. The delocalized electric field is related to the force (and therefore the repulsive potential) between two electrons. This in turn is what we need to solve the Schrödinger Equation to describe molecules. Short answer: You can calculate the expectation value of the corresponding operator, which comes close to the mentioned superposition. — Feodoran13 hours ago
If we take an electron that's delocalised w.r.t position, how can one evaluate the electric field over some space? Is it some superposition or a sort of field with all the charge at the expectation value of the position?
@0ßelö7 I just looked back at chat and noticed Phase's question, I wasn't purposefully ignoring you - do you want me to look over it? Because I don't think I'll gain much personally from reading the slides.
Maybe it's just me having not really done much with Eigenbases but I don't recognise where I "put it in terms of M's eigenbasis". I just wrote it down for some vector v, rather than a space that contains all of the vectors v
If we take an electron that's delocalised w.r.t position, how can one evaluate the electric field over some space? Is it some superposition or a sort of field with all the charge at the expectation value of the position?
Honey, I Shrunk the Kids is a 1989 American comic science fiction film. The directorial debut of Joe Johnston and produced by Walt Disney Pictures, it tells the story of an inventor who accidentally shrinks his and his neighbor's kids to a quarter of an inch with his electromagnetic shrinking machine and throws them out into the backyard with the trash, where they must venture into their backyard to return home while fending off insects and other obstacles.Rick Moranis stars as Wayne Szalinski, the inventor who accidentally shrinks his children, Amy (Amy O'Neill) and Nick (Robert Oliveri). Marcia...
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Speaker
Sakshin Bunthawin (Biotechnology of Electromechanics Research Unit, Science of Physics, Faculty of Technology and Environment, Prince of Songkla University, Kathu, Phuket 83120, Thailand)
Description
The present study employs exponential decay pulse-electric field inductions to enhance sex reversal of Nile tilapia eggs in suspensions for monosex-male. The novel technique was the pilot study to initiate the usage of natural androgen hormone of $\textit{Butea superba}$ (Red Kwao Krua) extract instead of our previous electroporation medium of 17$\alpha$ - methyltestosterone (MT). The eggs of Nile tilapia obtained from our parent breeding stocks were carefully selected for inducement in pulse electric field strengths of 0.25- 87.50 kV.m$^{-1}$ generated between narrow plate electrodes. Electric fields of 3-5 exponential decay pulses of 5-100 $\mu$s durations were applied to optimize egg’s electroporation. The prototype of the electrode equipped with the sequential signal pulse-generator (SPG) for on-site inductions was fabricated which could induce $\approx$ 50 eggs/SPG unit at one time. The electroporation medium (EPM) was prepared using HEPES buffer with a minimized concentration of the androgen hormone, $\textit{Butea superba}$. Roxb Root (Red Kwao Krua, RKK) extract. Only minimal volumes (50 ml) of buffered MT medium were required. The electroporation of induced membrane pores were observed by determination of pore sizes and pore densities through SEM micrographs. After the process of sex reversal, induced eggs were grown up to frysizes big enough (2 months old) to verify sex reversal. This novel technique reduced the RKK dose down to a minimized value of 1,500 $\mu$g.l$^{-1}$ and achieved a 69.14% male: female sex ratio.
Primary author
Sakshin Bunthawin (Biotechnology of Electromechanics Research Unit, Science of Physics, Faculty of Technology and Environment, Prince of Songkla University, Kathu, Phuket 83120, Thailand)
Co-authors
Mr Kata Jaruwongrungsee (Nanoelectronics and MEMS Laboratory, National Electronics and Computer Technology Center (NECTEC), National Science and Technology Development Agency (NSTDA), Ministry of Science and Technology (MOST), Pathumthani 12120, Thailand) Mrs Thanyada Sornsilpa (Biotechnology of Electromechanics Research Unit, Science of Physics, Faculty of Technology and Environment, Prince of Songkla University, Kathu, Phuket 83120, Thailand)
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I want to find a two term expansion of the form $x\sim x_0 + \epsilon^\alpha x_1 + \epsilon^\beta x_2 + \ldots$, with $\alpha < \beta < \ldots$, for small $\epsilon$, of each solution $x$ of the following equation: $$ x^2 - 2x + (1 - \epsilon^2)^{25} = 0 $$ I've substituted $x\sim x_0 + \epsilon^\alpha x_1 + \epsilon^\beta x_2 + \ldots$ into the equation to arrive at: $$ (x_0 + \epsilon^\alpha x_1 + \epsilon^\beta x_2 + \ldots)^2 -2(x_0 + \epsilon^\alpha x_1 + \epsilon^\beta x_2 + \ldots) + (1 - \epsilon^2)^{25} = 0 $$ Since I want to find a two term expansion for the equation I want to find $x_0, x_1$ and $\alpha$. I will try to do so by inspecting the terms in the above equation.
$\mathcal{O}(1)$: If I look at the $\mathcal{O}(1)$ terms I find that $x_0$ needs to be chosen such that it satisfies $$ x_0^2 - 2x_0 + 1 = 0 $$ The $1$ in this equation comes from the fact that $(1 - \epsilon^2)^{25} = 1+ \text{terms involving $\epsilon$}$. Solving this equation gives that $x_0 = 1$. Also since the $(1 - \epsilon^2)^{25}$ involves a term that is of order $\epsilon^2$, I know that I need $\alpha = 2$ for balance.
$\mathcal{O}(\epsilon^2)$: If I look at the $\mathcal{O}(\epsilon^2)$ terms I find that I need to choose $x_1$ such that it satisfies $$ 2x_0x_1 - 2x_1 - 1 = 0 $$ Where the $-1$ comes from the fact that $(1 - \epsilon^2)^{25}$ expansion contains $-\epsilon^2$. Since $x_0 = 1$ this means that I need to find $x_1$ such that $$ 2x_1 - 2x_1 -1 = 0 $$ But this is not possible..
Question: How do I find a two term expansion for the given equation? What am I doing wrong in my approach? Edit: As Maxim pointed out, I did not consider the possibility that $\alpha = 1$. If $\alpha = 1$ I get $2x_0x_1 - 2x_1 = 0$ which is fine since $x_0= 1$. If $\alpha = 1$ then $\beta = 2$ because I need a term to balance the $-\epsilon^2$ in $(1 - \epsilon^2)^{25}$. With $\alpha = 1$ and $\beta = 2$ I get the following equation:$$2x_0x_2 -2x_2 + x_1 -1 = 0\Leftrightarrow x_1 = 1$$Since I now know $x_0$ and $x_1$ I think I have a two term asymptotic expansion that looks like this:$$x = 1 + \epsilon + \mathcal{O}(\epsilon^3)$$I'm not sure that this expansion is the result that I want though since I know that the equation$$x^2 - 2x + (1 - \epsilon^2)^{25} = 0$$has two solutions. As $\epsilon$ goes to zero, the first solution approaches one from below while the second solution approaches one from above. It seems that the expansion that I found is only an approximation of the second solution. Is this correct?
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, the Haar wavelets method [ 3 ], Legendre wavelets method [ 4 ], Rationalized haar wavelet [ 5 ], Hermite cubic splines [ 6 ], Coifman wavelet scaling functions [ 7 ], CAS wavelets [ 8 ], Bernoulli wavelets [ 9 ], wavelet preconditioned techniques [ 25 , 26 , 27 , 28 ]. Some of the papers are found for solving Abel′s integral equations using the wavelet based methods, such as Legendre wavelets [ 10 ] and Chebyshev wavelets [ 11 ].Abel′s integral equations have applications in various fields of science and engineering. Such as microscopy,seismology
720 728[25] Bulut, H., Yel, G. and Baskonus, H.M. 2016. An Application Of Improved Bernoulli Sub-Equation Function Method To The Nonlinear Time-Fractional Burgers Equation, Turkish Journal of Mathematics and Computer Science, 5, 1-17. Bulut H. Yel G. Baskonus H.M. 2016 An Application Of Improved Bernoulli Sub-Equation Function Method To The Nonlinear Time-Fractional Burgers Equation Turkish Journal of Mathematics and Computer Science 5 1 17[26] Dusunceli, F. 2018. "Solutions for the Drinfeld-Sokolov Equation Using an IBSEFM Method" MSU Journal Of Science. 6
Newton–Raphson method for nonlinear equations by modified Adomian decomposition method Applied Mathematics and Computation 145 887 893 10.1016/s0096-3003(03)00282-0[22] S.S. Ganji, D.D. Ganji, A.G. Davodi, S. Karimpour, (2009), Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using max–min approach , Applied Mathematical Modelling 34 2676-2684. 10.1016/j.apm.2009.12.002 Ganji S.S. Ganji D.D. Davodi A.G. Karimpour S. 2009 Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using
Sk. Sarif Hassan, Moole Parameswar Reddy and Ranjeet Kumar Rout
.V., 1993. Circuit implementation of synchronized chaos with applications to communications. Physical review letters , 71(1), p.65. Cuomo K.M. Oppenheim A.V. 1993 Circuit implementation of synchronized chaos with applications to communications Physical review letters 71 1 65[4] Lü, J. and Chen, G., 2002. A new chaotic attractor coined. International Journal of Bifurcation and chaos , 12(03), pp.659-661. Lü J. Chen G. 2002 A new chaotic attractor coined International Journal of Bifurcation and chaos 12 03 659 661[5] Pehlivan, I. and Uyaroğlu, Y., 2010. A new
Dushko Josheski, Elena Karamazova and Mico Apostolov
interval ( a , b ), then this function is concave on ( a , b ) if: ∀ x ∈ ( a , b ), f ″( x ) < 0. Or a C 2 function: g : A → R n on the open and convex set A ⊂ R n is concave if and only if ∂ 2 f ( x ) < 0 and is semidefinite for all x , then f is strictly concave. In the literature of this king very important term is marginal cost pricing equilibrium which is a family of consumption, production plans, lump sum taxes and prices such that such that households are maiming their utility subject to their budget constraints and firms production plans
, 3). For example, the 3D and 2D plots of the bell-shaped solitary wave solution (41) are displayed in Figure 1 with ε = 1, k 1 = 1, k 2 = 1.2, k 3 = 2.2, c 2 = 1.5, δ = 2.5 when α = 0.95. Figure 2 shows the 3D and 2D plots of the kink-shaped solitary wave solution (45) for ε = −1, k 1 = 0.2, k 2 = 1.5, k 3 = 0.25, c 2 = 0.5, δ = −1.2 when α = 0.9. In Figure 3 , the 3D and 2D plots of the singular soliton solution (50) are depicted for ε = 1, k 1 = 0.5, k 2 = 1.5, k 3 = 0.3, c 2 = 1.3, δ = −1 when α = 0
Juan Pedro Ruiz-Fernández, Javier Benlloch, Miguel A. López and Nelia Valverde-Gascueña
calendars bring to light in those countries where laws protect the right of holiday choice. In these cases, the calendar factor is no longer deterministic.On the other hand, other authors (e.g., Tucker & Rahilly, 1982 [ 19 ]; Koehn & Brown, 1985 [ 14 ]; Chan & Kumaraswamy, 1995 [ 6 ]; El-Rayes & Mosehli 2001 [ 9 ];Wiliams, 2008 [ 21 ]; Odabasi, 2009 [ 15 ]) have explored the climatic factor giving rise to some predictive models with varying success. Nevertheless, we have not found any studies measuring the influence of all the seasonal factors in the construction
Esin İnan Eskitaşçıoğlu, Muhammed Bahadırhan Aktaş and Haci Mehmet Baskonus
{array}{}\displaystyle4{u_{{\rm{xt}}}} + {u_{{\rm{xxxt}}}} + 8{u_x}{u_{{\rm{xy}}}} +4{u_{{\rm{xx}}}}{u_y} - {\rm{\gamma }}{{\rm{u}}_{{\rm{xx}}}} = 0,\end{array}$$ (1)where γ is a real constant with a non-zero value, by using the sine-Gordon expansion method (SGEM).2Fundamental Properties of the SGEMLet us consider the following sine-Gordon equation [ 24 , 25 , 26 ]:u x x − u t t = m 2 sin u ,$$\begin{array}{}\displaystyle{u_{xx}} - {u_{tt}} = {m^2}\sin \left( u \right),\end{array}$$ (2)where u = u ( x , t ) and m is a real constant. When we
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In the formulation, presumably on the right side what is intended are 3-dimensional non-degenerate quadratic spaces (up to isomorphism), with discriminant 1 (same as $4^3$ mod squares as John Ma notes). But to make this work also in characteristic 2, it is better to proceed with a different point of view: that of conformal isometry of quadratic spaces (i.e., isomorphisms $T:V \simeq V'$ such that $q' \circ T = \lambda q$ for some $\lambda \in k^{\times}$). More specifically, we claim that away from characteristic 2,
every 3-dimensional non-degenerate quadratic space is conformal to a unique one with discriminant 1. Thus, by working with conformal isometry classes we will be able to work in a fully characteristic-free manner.
To see what is going on, recall that the set of isomorphism classes of central simple algebras of dimension 4 is ${\rm{H}}^1(k, {\rm{PGL}}_2)$, and the set of conformal isometry classes of dimension 3 is ${\rm{H}}^1(k, {\rm{PGO}}_3)$. But ${\rm{GO}}_{2m+1} = {\rm{GL}}_1 \times {\rm{SO}}_{2m+1}$, so ${\rm{PGO}}_{2m+1} = {\rm{SO}}_{2m+1}$. Hence, ${\rm{PGO}}_3 = {\rm{SO}}_3$. Since ${\rm{SO}}_3 \simeq {\rm{PGL}}_2$ through the representation of ${\rm{PGL}}_2$ via conjugation on the 3-dimensional space of traceless $2 \times 2$ matrices equipped with the determinant as the standard split non-degenerate quadratic form $xy - z^2$ (preserved by that conjugation action!), that answers the entire question at the level of isomorphism classes of objects. (The link to ${\rm{SO}}_3$ encodes the link to discriminant 1.)
But we can do better than keep track of isomorphism classes: we can also keep track of isomorphisms, as explained below. This is a refinement of John Ma's answer, as well as that of Matthias Wendt (which appeared at almost exactly the same time as this answer first appeared, so I didn't see it until this one was done).
The following notation will permit considering finite fields on equal footing with all other fields. For a finite-dimensional central simple algebra $A$ over an arbitrary field $k$, let ${\rm{Trd}}:A \rightarrow k$ be its "reduced trace" and ${\rm{Nrd}}:A \rightarrow k$ be its "reduced norm". These are really most appropriately viewed as "polynomial maps" in the evident sense. That is, if $\underline{A}$ is the "ring scheme" over $k$ representing the functor $R \rightsquigarrow A \otimes_k R$ (i.e., an affine space over $k$ equipped with polynomial maps expressing the $k$-algebra structure relative to a choice of $k$-basis) then we have $k$-morphisms ${\rm{Trd}}:\underline{A} \rightarrow \mathbf{A}^1_k$ and ${\rm{Nrd}}:\underline{A} \rightarrow \mathbf{A}^1_k$.
For $A$ of dimension 4 we set $\underline{V}_A$ to be the kernel of ${\rm{Trd}}:\underline{A} \rightarrow \mathbf{A}^1_k$; speaking in terms of kernel of ${\rm{Trd}}$ is a bit nicer than speaking in terms of orthogonal complements so that one doesn't need to separately consider characteristic 2 (where the relationship between quadratic forms and symmetric bilinear forms breaks down). This $\underline{V}_A$ is an affine space of dimension 3 over $k$ on which ${\rm{Nrd}}$ is a non-degenerate quadratic form $q_A$ (i.e., zero locus is a
smooth conic in the projective plane $\mathbf{P}(V_A^{\ast})$, where $V_A := \underline{V}_A(k)$): indeed, these assertions are "geometric" in nature, so it suffices to check them over $k_s$, where $A$ becomes a matrix algebra and we can verify everything by inspection.
We will show that the natural map of affine varieties $$\underline{{\rm{Isom}}}(\underline{A}, \underline{A}') \simeq \underline{{\rm{CIsom}}}((\underline{V}_A, q_A), (\underline{V}_{A'}, q_{A'}))/\mathbf{G}_m$$from the "isomorphism variety" to the "variety of conformal isometries mod unit-scaling" is an isomorphism; once that is shown, by Hilbert 90 we could pass to $k$-points to conclude that isomorphisms among such $A$'s correspond exactly to conformal isometries among such $(V_A, q_A)$'s up to unit scaling. It suffices to check this isomorphism assertion for varieties over $k_s$, where it becomes the assertion that the natural map$$\underline{{\rm{Aut}}}_{{\rm{Mat}}_2/k} \rightarrow {\rm{CAut}}_{({\rm{Mat}}_2^{{\rm{Tr}}=0}, \det)/k}/\mathbf{G}_m$$from the Aut-scheme to the scheme of conformal isometries up to unit-scaling is an isomorphism. But this is precisely the natural map$${\rm{PGL}}_2 \rightarrow {\rm{PGO}}(-z^2-xy) = {\rm{PGO}}(xy+z^2) = {\rm{SO}}(xy+z^2) = {\rm{SO}}_3$$between smooth affine $k$-groups that is classically known to be an isomorphism over any field $k$ (can check bijectivity on geometric points and the isomorphism property on tangent spaces at the identity points).
Finally, we want to show that
every 3-dimensional non-degenerate quadratic space $(V, q)$ is conformal to $(V_A, q_A)$ for some $A$.Note that if such an $A$ exists then it is unique up to unique isomorphism in the sense that if $A$ and $A$ are two such equipped with conformal isometries $(V_A, q_A) \simeq (V, q) \simeq (V_{A'}, q_{A'})$ then this composite conformal isometry arises from a unique isomorphism $A \simeq A'$ of $k$-algebras. Hence, by Galois descent (!) it suffices to check existence over $k_s$! But over a separably closed field the smooth projective conic has a rational point, so $(V, q)_{k_s}$ contains a hyperbolic plane and thus is isometric to $xy + \lambda z^2$ on $k_s^3$ for some $\lambda \in k_s^{\times}$. This is conformal to $(-1/\lambda)q_{k_s}$, but $(1/\lambda)xy - z^2$ and $-x'y' - z^2$ are clearly isometric, and the latter is ${\rm{Mat}}_2^{{\rm{Tr}}=0}$ equipped with the restriction of det.
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Latex Support
LaTeX has been integrated into this site, making it easy to include mathematics in the pages you edit. This is not a standard feature of PmWiki, so it is not covered in the standard documentation on text formatting rules. This page explains how to include LaTeX code in the pages you edit and addresses some complications that arise.
You can find more explanations in Original Latex Support (thanks for the LaTeX engine to Joe Miller, and his now defunct Effective Randomness wiki). You can find additional features not necessary about LaTeX in Advanced Formatting.
The syntax used to produce LaTeX input is the same as what you would expect. For example, the formula is produced by the code
$A\in 2^\omega$. The usual markup is also provided for display mathematics.
Here some examples of formulas are presented. Press Edit button to see the code.
$\sqrt{2^\frac{1}{x^2+1}+1}$
$A=\left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right)$
For all :
$$\sum_{i=0}^n i=\frac{n(n+1)}{2}$$
Integral calculus:
$$\int_0^{+\infty} \frac{\sin t}{t}\, dt=\frac{\pi}{2}$$
Inline syntax:
$$a^2=b$$
\[c=d+d\]
Begin/end syntax:
\begin{align*} x_k&=Ax_{k-1}+Bu_k+w_k,\\ z_k&=Hx_k+v_k, \end{align*}
Important: The begin/end syntax can only be used for formulas that fit one line in the source file (although they may occupy more than one line after compilation).
Some formulas do not work:
$a^{(N)}$provides . But you can always use
$a^{\left(N\right)}$, which provides , or
$a^{\:(N)\:}$provides .
$[-Y,Y]$is not handled well. But you can always use
$ [-Y,Y] $, which provides .
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A Belyi-extender (or dessinflateur) is a rational function $q(t) = \frac{f(t)}{g(t)} \in \mathbb{Q}(t)$ that defines a map
\[ q : \mathbb{P}^1_{\mathbb{C}} \rightarrow \mathbb{P}^1_{\mathbb{C}} \] unramified outside $\{ 0,1,\infty \}$, and has the property that $q(\{ 0,1,\infty \}) \subseteq \{ 0,1,\infty \}$.
An example of such a Belyi-extender is the power map $q(t)=t^n$, which is totally ramified in $0$ and $\infty$ and we clearly have that $q(0)=0,~q(1)=1$ and $q(\infty)=\infty$.
The composition of two Belyi-extenders is again an extender, and we get a rather mysterious monoid $\mathcal{E}$ of all Belyi-extenders.
Very little seems to be known about this monoid. Its units form the symmetric group $S_3$ which is the automrphism group of $\mathbb{P}^1_{\mathbb{C}} – \{ 0,1,\infty \}$, and mapping an extender $q$ to its degree gives a monoid map $\mathcal{E} \rightarrow \mathbb{N}_+^{\times}$ to the multiplicative monoid of positive natural numbers.
If one relaxes the condition of $q(t) \in \mathbb{Q}(t)$ to being defined over its algebraic closure $\overline{\mathbb{Q}}$, then such maps/functions have been known for some time under the name of
dynamical Belyi-functions, for example in Zvonkin’s Belyi Functions: Examples, Properties, and Applications (section 6).
Here, one is interested in the complex dynamical system of iterations of $q$, that is, the limit-behaviour of the orbits
\[ \{ z,q(z),q^2(z),q^3(z),… \} \] for all complex numbers $z \in \mathbb{C}$.
In general, the 2-sphere $\mathbb{P}^1_{\mathbb{C}} = S^2$ has a finite number of open sets (the Fatou domains) where the limit behaviour of the series is similar, and the union of these open sets is dense in $S^2$. The complement of the Fatou domains is the Julia set of the function, of which we might expect a nice fractal picture.
Let’s take again the power map $q(t)=t^n$. For a complex number $z$ lying outside the unit disc, the series $\{ z,z^n,z^{2n},… \}$ has limit point $\infty$ and for those lying inside the unit circle, this limit is $0$. So, here we have two Fatou domains (interior and exterior of the unit circle) and the Julia set of the power map is the (boring?) unit circle.
Fortunately, there are indeed dynamical Belyi-maps having a more pleasant looking Julia set, such as this one
But then, many dynamical Belyi-maps (and Belyi-extenders) are systems of an entirely different nature, they are
completely chaotic, meaning that their Julia set is the whole $2$-sphere! Nowhere do we find an open region where points share the same limit behaviour… (the butterfly effect).
There’s a nice sufficient condition for chaotic behaviour, due to Dennis Sullivan, which is pretty easy to check for dynamical Belyi-maps.
A
periodic point for $q(t)$ is a point $p \in S^2 = \mathbb{P}^1_{\mathbb{C}}$ such that $p = q^m(p)$ for some $m > 1$. A critical point is one such that either $q(p) = \infty$ or $q'(p)=0$.
Sullivan’s result is that $q(t)$ is completely chaotic when all its critical points $p$ become eventually periodic, that is some $q^k(p)$ is periodic,
but $p$ itself is not periodic.
For a Belyi-map $q(t)$ the critical points are either comlex numbers mapping to $\infty$ or the inverse images of $0$ or $1$ (that is, the black or white dots in the dessin of $q(t)$) which are not leaf-vertices of the dessin.
Let’s do an example, already used by Sullivan himself:
\[ q(t) = (\frac{t-2}{t})^2 \] This is a Belyi-function, and in fact a Belyi-extender as it is defined over $\mathbb{Q}$ and we have that $q(0)=\infty$, $q(1)=1$ and $q(\infty)=1$. The corresponding dessin is (inverse images of $\infty$ are marked with an $\ast$)
The critical points $0$ and $2$ are not periodic, but they become eventually periodic:
\[
2 \rightarrow^q 0 \rightarrow^q \infty \rightarrow^q 1 \rightarrow^q 1 \] and $1$ is periodic.
For a general Belyi-extender $q$, we have that the image under $q$ of any critical point is among $\{ 0,1,\infty \}$ and because we demand that $q(\{ 0,1,\infty \}) \subseteq \{ 0,1,\infty \}$, every critical point of $q$ eventually becomes periodic.
If we want to avoid the corresponding dynamical system to be completely chaotic, we have to ensure that one of the periodic points among $\{ 0,1,\infty \}$ (and there is at least one of those) must be critical.
Let’s consider the very special Belyi-extenders $q$ having the additional property that $q(0)=0$, $q(1)=1$ and $q(\infty)=\infty$, then all three of them are periodic.
So, the system is always completely chaotic
unless the black dot at $0$ is not a leaf-vertex of the dessin, or the white dot at $1$ is not a leaf-vertex, or the degree of the region determined by the starred $\infty$ is at least two.
Going back to the mystery Manin-Marcolli sub-monoid of $\mathcal{E}$, it might explain why it is a good idea to restrict to very special Belyi-extenders having associated dessin a $2$-coloured tree, for then the periodic point $\infty$ is critical (the degree of the outside region is at least two), and therefore the conditions of Sullivan’s theorem are not satisfied. So, these Belyi-extenders do not necessarily have to be completely chaotic. (tbc)Leave a Comment
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In simple sense, hyperbola looks similar to to mirrored parabolas. The two halves are called the branches. When the plane intersect on the halves of a right circular cone angle of which will be parallel to the axis of the cone, a parabola is formed.
A hyperbola contains: two foci and two vertices. The foci of the hyperbola are away from the hyperbola’s center and vertices. Here is an illustration to make you understand:
The equation for hyperbola is,
\[\large \frac{(x-x_{0})^{2}}{a^{2}}-\frac{(y-y_{0})^{2}}{b^{2}}=1\]
Where,
$x_{0}, y_{0}$ are the center points. $a$ = semi-major axis. $b$ = semi-minor axis.
Let us learn the basic terminologies related to hyperbola formula:
MAJOR AXIS
The line that passes through the center, focus of the hyperbola and vertices is the Major Axis. Length of the major axis = 2a. The equation is given as:
\[\large y=y_{0}\]
MINOR AXIS
The line perpendicular to the major axis and passes by the middle of the hyperbola is the Minor Axis.
Length of the minor axis = 2b. The equation is given as:
\[\large x=x_{0}\]
ECCENTRICITY
The variation in the conic section being completely circular is eccentricity. It is usually greater than 1 for hyperbola. Eccentricity is $2\sqrt{2}$ for a regular hyperbola. The formula for eccentricity is:
\[\large \frac{\sqrt{a^{2}+b^{2}}}{a}\]
ASYMPTOTES
Two bisecting lines that is passing by the center of the hyperbola that doesn’t touch the curve is known as the Asymptotes. The equation is given as:
\[\large y=y_{0}+\frac{b}{a}x-\frac{b}{a}x_{0}\]
\[\large y=y_{0}-\frac{b}{a}x+\frac{b}{a}x_{0}\]
Directrix of a hyperbola
Directrix of a hyperbola is a straight line that is used in generating a curve. It can also be defined as the line from which the hyperbola curves away from. This line is perpendicular to the axis of symmetry. The equation of directrix is:
\[\large x=\frac{\pm a^{2}}{\sqrt{a^{2}+b^{2}}}\]
VERTEX
The point of the branch that is stretched and is closest to the center is the vertex. The vertex points are
\(\LARGE \left(a,y_{0}\right ) \ and \ \LARGE \left(-a,y_{0}\right )\)
Focus (foci)
On a hyperbola, focus (foci being plural) are the fixed points such that the difference between the distances are always found to be constant. The two focal points are:
\[\large\left(x_{0}+\sqrt{a^{2}+b^{2}},y_{0}\right)\]
\[\large \left(x_{0}-\sqrt{a^{2}+b^{2}},y_{0}\right)\]3
Solved examples Question: The equation of the hyperbola is given as: $\frac{(x-4)^{2}}{9^{2}}-\frac{(y-2)^{2}}{7^{2}}$
Find the following: Vertex, Asymptote, Major Axis, Minor Axis and Directrix?
Solution:
Given,
$x_{0}=4$ $y_{0}=2$ $a =9$ $b = 7$
The vertex point: $(a, y_{0})$ and $(-a, y_{0})$ are $(9, 2)$ and $(-9, 2)$
Asymptote
$y=2+\frac{7}{9}x-\frac{37}{9}$
$y=2-\frac{7}{9}x-\frac{23}{9}$
$y=2+0.77x+4.1=6.1+0.77x$
$y=2-0.77x+2.5=4.5+0.77x$
Major Axis
$y=y_{o}$
$y_{o}=2$
Minor Axis
$x=x_{o}$
$x_{o} =4$
Directrix
$x=\frac{\pm 9^{2}}{\sqrt{9^{2}+7^{2}}} = \pm \frac{81}{\sqrt{81+49}}=7.1$
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A linear-quadratic control problem of uncertain discrete-time switched systems
1.
School of Science, Nanjing Forestry University, Nanjing 210037, China
2.
School of Science, Nanjing University of Science & Technology, Nanjing 210094, China
This paper studies a linear-quadratic control problem for discrete-time switched systems with subsystems perturbed by uncertainty. Analytical expressions are derived for both the optimal objective function and the optimal switching strategy. A two-step pruning scheme is developed to efficiently solve such problem. The performance of this method is shown by two examples.
Keywords:Linear-quadratic model, uncertain switched system, optimal control, local pruning scheme, global pruning scheme. Mathematics Subject Classification:Primary: 49N10, 49L20; Secondary: 65K05. Citation:Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial & Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016
References:
[1]
A. Bemporad, F. Borrelli and M. Morari,
On the optimal control law for linear discrete time hybrid systems,
[2] [3]
F. Borrelli, M. Baotic, A. Bemporad and M. Morari,
Dynamic programming for contrained optimal control of discrete-time linear hybrid systems,
[4]
S. Boubakera, M. Djemaic, N. Manamannid and F. M'Sahlie,
Active modes and switching instants identification for linear switched systems based on discrete particle swarm optimization,
[5]
H. V. Esteban, C. Patrizio, M. Richard and B. Franco,
Discrete-time control for switched positive systems with application to mitigating viral escape,
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H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings,
Control parametrization enhancing technique for optimal discrete-valued control problems,
[10]
F. Li, P. Shi, L. Wu, M. V. Basin and C. C. Lim,
Quantized control design for cognitive radio networks modeled as nonlinear semi-Markovian jump systems,
[11] [12] [13]
B. Liu,
[14] [15] [16]
C. Liu, Z. Gong and E. Feng,
Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture,
[17]
R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling,
Optimal switching instants for a switched-capacitor DC/DC power converter,
[18]
K. L. Teo, C. J. Goh and K. H. Wong,
[19]
C. Tomlin, G. J. Pappas, J. Lygeros, D. N. Godbole and S. Sastry,
Hybrid control models of next generation air traffic management,
[20]
L. Y. Wang, A. Beydoun, J. Sun and I. Kolmanasovsky,
Optimal hybrid control with application to automotive powertrain systems,
[21] [22]
L. Wu, D. Ho and C. Li,
Sliding mode control of switched hybrid systems with stochastic perturbation,
[23]
X. Xu and P. Antsaklis,
Optimal control of switched systems based on parameterization of the switching instants,
[24] [25]
H. Yan and Y. Zhu,
Bang-bang control model with optimistic value criterion for uncertain switched systems,
[26]
W. Zhang, J. Hu and A. Abate,
On the value function of the discrete-time switched lqr problem,
[27] [28] [29]
Y. Zhu,
Uncertain optimal control with application to a portfolio selection model,
[30]
show all references
References:
[1]
A. Bemporad, F. Borrelli and M. Morari,
On the optimal control law for linear discrete time hybrid systems,
[2] [3]
F. Borrelli, M. Baotic, A. Bemporad and M. Morari,
Dynamic programming for contrained optimal control of discrete-time linear hybrid systems,
[4]
S. Boubakera, M. Djemaic, N. Manamannid and F. M'Sahlie,
Active modes and switching instants identification for linear switched systems based on discrete particle swarm optimization,
[5]
H. V. Esteban, C. Patrizio, M. Richard and B. Franco,
Discrete-time control for switched positive systems with application to mitigating viral escape,
[6] [7] [8] [9]
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings,
Control parametrization enhancing technique for optimal discrete-valued control problems,
[10]
F. Li, P. Shi, L. Wu, M. V. Basin and C. C. Lim,
Quantized control design for cognitive radio networks modeled as nonlinear semi-Markovian jump systems,
[11] [12] [13]
B. Liu,
[14] [15] [16]
C. Liu, Z. Gong and E. Feng,
Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture,
[17]
R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling,
Optimal switching instants for a switched-capacitor DC/DC power converter,
[18]
K. L. Teo, C. J. Goh and K. H. Wong,
[19]
C. Tomlin, G. J. Pappas, J. Lygeros, D. N. Godbole and S. Sastry,
Hybrid control models of next generation air traffic management,
[20]
L. Y. Wang, A. Beydoun, J. Sun and I. Kolmanasovsky,
Optimal hybrid control with application to automotive powertrain systems,
[21] [22]
L. Wu, D. Ho and C. Li,
Sliding mode control of switched hybrid systems with stochastic perturbation,
[23]
X. Xu and P. Antsaklis,
Optimal control of switched systems based on parameterization of the switching instants,
[24] [25]
H. Yan and Y. Zhu,
Bang-bang control model with optimistic value criterion for uncertain switched systems,
[26]
W. Zhang, J. Hu and A. Abate,
On the value function of the discrete-time switched lqr problem,
[27] [28] [29]
Y. Zhu,
Uncertain optimal control with application to a portfolio selection model,
[30]
Algorithm 1:(Two-step pruning scheme) 1: Set $\tilde{H}_{0}=\{(Q_{f}, 0)\}$; 2: for $k=0$ to $N-1$ do 3: for all $(P, \gamma)\in \tilde{H}_{k}$ do 4: $\Gamma_{k}(P, \gamma)=\emptyset$; 5: for i=1 to m do 6: $P^{(i)}=\rho_{i}(P)$, 7: $\gamma^{(i)}=\gamma+\frac{1}{3}\|\sigma_{N-k}\|^{2}_{P}$, 8: $\Gamma_{k}(P, \gamma)=\Gamma_{k}(P, \gamma)\bigcup\{(P^{(i)}, \gamma^{(i)})\}$; 9: end for 10: for i=1 to m do 11: if $(P^{(i)}, \gamma^{(i)})$ satisfies the condition in Lemma 5.3, then 12: $\Gamma_{k}(P, \gamma)=\Gamma_{k}(P, \gamma)\backslash\{(P^{(i)}, \gamma^{(i)})\}$; 13: end if 14: end for 15: end for 16: $\tilde{H}_{k+1}=\bigcup\limits_{(P, \gamma)\in \tilde{H}_{k}}\Gamma_{k}(P, \gamma)$; 17: $\hat{H}_{k+1}=\tilde{H}_{k+1}$; 18: for i=1 to $|\hat{H}_{k+1}|$ do 19: if $(\hat{P}^{(i)}, \hat{\gamma}^{(i)})$ satisfies the condition in Lemma 5.4, then 20: $\hat{H}_{k+1}=\hat{H}_{k+1}\backslash\{(\hat{P}^{(i)}, \hat{\gamma}^{(i)})\}$; 21: end if 22: end for 23: end for 24: $J(0, x_{0})=\min\limits_{(P, \gamma)\in \hat{H}_{N}}(\|x_{0}\|^{2}_{P}+\gamma).$
Algorithm 1:(Two-step pruning scheme) 1: Set $\tilde{H}_{0}=\{(Q_{f}, 0)\}$; 2: for $k=0$ to $N-1$ do 3: for all $(P, \gamma)\in \tilde{H}_{k}$ do 4: $\Gamma_{k}(P, \gamma)=\emptyset$; 5: for i=1 to m do 6: $P^{(i)}=\rho_{i}(P)$, 7: $\gamma^{(i)}=\gamma+\frac{1}{3}\|\sigma_{N-k}\|^{2}_{P}$, 8: $\Gamma_{k}(P, \gamma)=\Gamma_{k}(P, \gamma)\bigcup\{(P^{(i)}, \gamma^{(i)})\}$; 9: end for 10: for i=1 to m do 11: if $(P^{(i)}, \gamma^{(i)})$ satisfies the condition in Lemma 5.3, then 12: $\Gamma_{k}(P, \gamma)=\Gamma_{k}(P, \gamma)\backslash\{(P^{(i)}, \gamma^{(i)})\}$; 13: end if 14: end for 15: end for 16: $\tilde{H}_{k+1}=\bigcup\limits_{(P, \gamma)\in \tilde{H}_{k}}\Gamma_{k}(P, \gamma)$; 17: $\hat{H}_{k+1}=\tilde{H}_{k+1}$; 18: for i=1 to $|\hat{H}_{k+1}|$ do 19: if $(\hat{P}^{(i)}, \hat{\gamma}^{(i)})$ satisfies the condition in Lemma 5.4, then 20: $\hat{H}_{k+1}=\hat{H}_{k+1}\backslash\{(\hat{P}^{(i)}, \hat{\gamma}^{(i)})\}$; 21: end if 22: end for 23: end for 24: $J(0, x_{0})=\min\limits_{(P, \gamma)\in \hat{H}_{N}}(\|x_{0}\|^{2}_{P}+\gamma).$
1 2 3 4 5 6 7 8 9 10 2 5 4 4 7 7 4 7 7 7 2 2 2 3 3 2 3 3 3 3
1 2 3 4 5 6 7 8 9 10 2 5 4 4 7 7 4 7 7 7 2 2 2 3 3 2 3 3 3 3
0 2 - -0.7861 12.9774 1 2 0.6294 -0.2579 2.5122 2 2 0.8116 -0.1749 0.6456 3 2 -0.7460 0.0761 0.2084 4 2 0.8268 -0.1143 0.1582 5 1 0.2647 -0.0495 0.1137 6 2 -0.8049 0.1221 0.1113 7 2 -0.4430 0.0918 0.0765 8 2 0.0938 0.005 0.0443 9 2 0.9150 -0.1444 0.0678
0 2 - -0.7861 12.9774 1 2 0.6294 -0.2579 2.5122 2 2 0.8116 -0.1749 0.6456 3 2 -0.7460 0.0761 0.2084 4 2 0.8268 -0.1143 0.1582 5 1 0.2647 -0.0495 0.1137 6 2 -0.8049 0.1221 0.1113 7 2 -0.4430 0.0918 0.0765 8 2 0.0938 0.005 0.0443 9 2 0.9150 -0.1444 0.0678
1 2 3 4 5 6 7 8 9 10 2 5 12 9 9 9 9 9 9 9 2 4 3 3 3 3 3 3 3 3
1 2 3 4 5 6 7 8 9 10 2 5 12 9 9 9 9 9 9 9 2 4 3 3 3 3 3 3 3 3
0 5 - -0.7273 11.0263 1 1 0.6294 -0.1808 0.6251 2 2 0.8116 -0.0892 0.5116 3 1 -0.7460 0.1421 0.2063 4 2 0.8268 -0.0608 0.1428 5 2 0.2647 -0.0523 0.1121 6 2 -0.8049 0.1105 0.1113 7 2 -0.4430 0.0931 0.0690 8 1 0.0938 -0.0062 0.0426 9 2 0.9150 -0.1522 0.0615
0 5 - -0.7273 11.0263 1 1 0.6294 -0.1808 0.6251 2 2 0.8116 -0.0892 0.5116 3 1 -0.7460 0.1421 0.2063 4 2 0.8268 -0.0608 0.1428 5 2 0.2647 -0.0523 0.1121 6 2 -0.8049 0.1105 0.1113 7 2 -0.4430 0.0931 0.0690 8 1 0.0938 -0.0062 0.0426 9 2 0.9150 -0.1522 0.0615
[1]
Jin Feng He, Wei Xu, Zhi Guo Feng, Xinsong Yang.
On the global optimal solution for linear quadratic problems of switched system.
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Georg Vossen, Stefan Volkwein.
Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems.
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Decomposition of discrete linear-quadratic optimal control problems for switching systems.
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Solar Emission in the ISM band
The sun is 6.955e8 meters in diameter, and 1.496e11 meters away. Its effective black body temperature is 5778k.
The Planck black body spectrum is:
\Large { { 2 h {\nu}^3 } \over { c^2 ( e^{h \nu / k T } - 1 ) } } Watts / steradian m
2 Hz
For h \nu << kT , this approximates to:
\large { { 2 h {\nu}^3 } \over { c^2 ( 1 + h \nu / k T - 1 ) } }
\large { { 2 h {\nu}^3 } \over { c^2 h \nu / k T }}
\large { { 2 {\nu}^2 k T } \over { c^2 } }
{ 2 kT / {\lambda}^2 }
Multiplied by half the sky, 2 \pi steradians, and the bandwidth BW , the power per square meter is
4 \pi kT ~ BW / {\lambda}^2 W / m
2
Multiplied by the surface area of the sun 4 \pi {Rsun}^2 :
( 4 \pi Rsun / \lambda )^2 ~ kT ~ BW Watts
For the the ISM band, \lambda = 2.997e8 m / 5.8 GHz = 0.0517 meters and BW = 15 MHz. The Boltzmann constant k is 1.3806488 × 10-23 J/K so kT is 7.977E-20 Joules . 4 \pi Rsun / \lambda is 4 π 6.955e8 / 0.0517 = 1.69e11 (unitless). The power emission from the entire Sun in ISM is 2280 Watts / Hz or 34.2 GW for the entire ISM band.
Brighter than 1000 suns
If we tried to make 100 Terawatts of earth energy from space solar power (10kW / person, 10 billion people), and broadcast 120 Terawatts of ISM band power to do it, the brightness of the constellation would be 3500 times the entire sun radiating into all of space. If the energy was concentrated into a 1 Hz band, it could outshine the mostly-red-dwarf galaxy in that narrow band.
How much hits the earth? The earth's disk is π Re
2, Re =6370 km, and the sun is Ro = 1.496e8 km away, irradiating a sphere with an area of 4 π Ro 2. The fraction reaching earth is 0.25 (Re/Ro) 2 or 4.54E-10 of the sun's output, 15.5 Watts for the entire ISM band, or 1 μW / Hz . It is sobering to realize that a single wireless access point can generate more power in its channel than the Sun delivers to the whole planet.
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Define a multi-particle "breeding" random walk $\mathcal{W_p}$ in $d$ dimensions, for $p \in (0,1)$ as follows:
The state of $\mathcal{W_p}$ at integer time $t\geq 0$ consists of the pair $(k, x)$ where $k \in \Bbb{Z}^+ \cup 0$ and $x \in \left(\Bbb{Z}^d \right)^k$. Informally, the state at time $t$ consists of zero or more particles, each on a node of a $d$-dimensional rectilinear grid.
The state of $\mathcal{W_p}$ at $t=0$ is $(1, 0^d)$, that is, a single particle at the origin.
The transition rules from the state at time $t$ to that at time $t+1$ are described in two phases: the first phase is movement and breeding, in which for each particle (that is, for each of the $k$ projections of element of $x$, which in turn are elements of $\Bbb{Z}^d$), with probability $1-p$ that particle moves with equal probability to each of the $2\cdot d$ neighboring vertices in the grid (as in a usual unbiased random walk), and with probability $p$ splits into two particles, which move to the two neighbors along a uniformly randomly selected direction among the $d$ dimensions.
The second phase of the transition consists of dealing with cases in which multiple particles end up at the same node. Three separate models are of interest:
(a) The annihilating multiparticle walk, where the transition rule states that if two or more particles arrive at the same node at time $t$, all of them are removed from the state for time $t+1$. An annihilating multiparticle walk may reach a stopping position with zero particles, which we call "extinction."
(b) The limited multiparticle walk, where the transition rule states that if two or more particles arrive at the same node at time $t$, all but one of them are removed from the state for time $t+1$. The state in this model could equally well be described as a finite subset of $\Bbb{Z}^d$, rather than specifying the number of particles and listing the nodes of each particle.
(c) The exploding multiparticle walk, in which the second phase of the transition is trivial, and multiple particles can occupy the same node.
Have these models been studied? Of particular interest are two questions:
For a one-dimensional annihilating multiparticle walk, what is the probability of eventual extinction as a function of $p$?
Define a "return to the origin" as a time $t$ at which there is at least one particle at the origin (it is allowed that other particles may be elsewhere at that time). For a three-dimensional exploding random walk, for what (if any) values of $p$ does $\mathcal{W_p}$ almost surely return to the origin?
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Hi Reinhard,
Thanks for the detailed report.
I’m afraid this issue has been logged in the past (albeit relatively recently) and I haven’t had the time to look into it properly. It certainly needs to be addressed. See: https://github.com/wspr/unicode-math/issues/497
Regarding the loading order, I have even thought that unicode-math should load amsmath automatically, but I shy away from present unicode-math as a one-stop-shop. I’ll revisit whether I can force a more explicit warning or error, even if it’s via changes to amsmath.sty itself.
Thanks also for the invite to BachoTeX — I would truly love to go, I just need to line it up one year with some additional conference travel to make it worth the lengthy trip. (It looks like Adelaide to Bydgoszcz is 2 or 3 stops and 30+ hrs travel time…)
Regards,
Will
> On 28 Apr 2019, at 2:23 pm, Reinhard Kotucha <[log in to unmask]> wrote:
>
> Hello Will,
> when I load unicode-math after amsmath, the amsmath option intlimits
> has no effect any more. The options sumlimits and namelimits still
> work as expected.
>
> \documentclass{article}
> \usepackage[intlimits,sumlimits,namelimits]{amsmath}
> \usepackage{unicode-math}
>
> \begin{document}
> \[ \int_{0}^{1} \]
> \[ \sum_{i=0}^{1} \]
> \[ \lim_{x\to\infty} \]
> \end{document}
>
> Even worse, when I reverse the order (loading unicode-math first), the
> operator \lim is printed in italics. This is so strange that I can't
> believe it. Can anybody confirm this behavior?
>
> I tested with both, lualatex and xelatex and TeX Live 2012, 2013,..,2019.
> Same result, as expected.
>
> The above is a minimal example. When unicode-math is loaded first I
> only get a warning "Package amsmath Warning: Unable to redefine math
> accent \mathring.". But when I use other fonts like TeX Gyre Pagella
> with fontspec and texgyrepagella-math.otf with unicode-math, I get
> zillions of error messages about already defined control sequences.
>
> Thus I suppose that amsmath has to be loaded before unicode-math
> unconditionally. If this assumption is true it would be better if
> amsmath aborts with a meaningful error message if it detects an
> already loaded unicode-math.
>
> I'm aware that not all dependency issues can be resolved this way.
> ATM there are 180,000 files in TeX Live, three times more than ten
> years ago and the number is steadily growing. But if only the LaTeX
> core packages complain when loaded in an inappropriate order, that
> would be a great advantage.
>
> I'll meet Frank next week at the BachoTeX conference in Poland.
>
> Will, did you ever consider to attend this amazing conferece? Though
> it's [officially] the conference of the Polish TeX User Group, it's
> quite international. In the past it was attended by people from
> innumerable European countries, Japan, America, and Australia. The
> conference site (in a nature sanctuary) is an ideal place where
> families with [little] children can relax. BachoTeX even offers
> workshops which little kids can attend.
>
> See
> http://www.gust.org.pl/bachotex/bachotex-promo/bachotex-1920x1080.pdf
>
> If you didn't register already, I really hope to meet you there next
> year.
>
> Regards,
> Reinhard
>
> --
> ------------------------------------------------------------------
> Reinhard Kotucha Phone: +49-511-3373112
> Marschnerstr. 25
> D-30167 Hannover mailto:[log in to unmask]
> ------------------------------------------------------------------
|
Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping
Department of Mathematics, State University of Maringa, Maringa, 87020-900, Brazil
$ \begin{eqnarray*} i\psi_t + \Delta \psi + i \alpha b(x)(|\psi|^{2} + 1)\psi & = & \phi \psi \chi_{\omega} \; \hbox{in}\; \Omega \times (0, \infty), \; (\alpha >0)\ \\ \phi_{tt} - \Delta \phi + a(x) \phi_t & = & |\psi|^2 \chi_{\omega}\; \hbox{in}\; \Omega \times (0, \infty), \end{eqnarray*} $
$ \Omega $
$ \mathbb{R}^2 $
$ \Gamma $
$ \omega $
$ \partial \Omega $
$ \chi_{\omega} $
$ \omega $
$ a, b\in L^{\infty}(\Omega) $
$ a(x) \geq a_0 >0 $
$ \omega $
$ b(x) \geq b_{0} > 0 $
$ \omega $ Keywords:Klein-Gordon-Schrödinger, localized damping, exponential stability, asymptotic behavior, existence and uniqueness. Mathematics Subject Classification:Primary: 35L70, 35B40. Citation:Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations & Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041
References:
[1]
A. F. Almeida, M. M. Cavalcanti and J. P. Zanchetta,
Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping,
[2] [3] [4] [5]
J. B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Klein-Gordon-Schrödinger equations,
[6]
C. Banquet, L. C. F. Ferreira and E. J. Villamizar-Roa, On existence and scattering theory for the Klein-Gordon-Schrödinger system in an infinite L2-norm setting,
[7]
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,
[8] [9]
V. Bisognin, M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Soriano,
Uniform decay for the Klein-Gordon-Schrödinger equations with locally distributed damping,
[10]
C. A. Bortot and W. J. Corrêa,
Exponential stability for the defocusing semilinear Schrödinger equation with locally distributed damping on a bounded domain,
[11]
M. Cavalcanti and V. N. Domingos Cavalcanti,
Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations,
[12]
M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka,
Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction,
[13]
J. Colliander, J. Holmer and N. Tzirakis,
Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems,
[14] [15]
M. Daoulatli, I. Lasiecka and D. Toundykov,
Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions,
[16] [17] [18] [19] [20] [21] [22]
B. Guo and Y. Li,
Attractor for dissipative Klein-Gordon-Schrödinger equations in ${{\bf{R}}^{\bf{3}}}$,
[23] [24]
B. Guo and Y. Li,
Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations,
[25] [26] [27] [28]
H. Lange and B. Wang,
Regularity of the global attractor for the Klein-Gordon-Schrödinger equation,
[29] [30] [31]
I. Lasiecka and D. Toundykov,
Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,
[32]
G. Lebeau, Controle de l'equation de Schrödinger. (french) [control of the Schrödinger equation],
[33]
Y. Li, Q. Shi, C. Wang and S. Wang, Well-posedness for the nonlinear Klein-Gordon-Schrödinger equations with heterointeractions,
[34]
J. L. Lions,
[35]
J. L. Lions,
[36] [37] [38]
M. Ohta,
Stability of stationary states for the coupled Klein-Gordon-Schrödinger equations,
[39] [40] [41] [42]
M. N. Poulou and N. M. Stavrakakis, Uniform decay for a local dissipative Klein-Gordon-Schrödinger type system,
[43] [44] [45]
N. Tzirakis,
The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space,
[46]
B. Wang,
Classical global solutions for non-linear Klein-Gordon-Schrödinger equations,
[47]
show all references
References:
[1]
A. F. Almeida, M. M. Cavalcanti and J. P. Zanchetta,
Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping,
[2] [3] [4] [5]
J. B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Klein-Gordon-Schrödinger equations,
[6]
C. Banquet, L. C. F. Ferreira and E. J. Villamizar-Roa, On existence and scattering theory for the Klein-Gordon-Schrödinger system in an infinite L2-norm setting,
[7]
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,
[8] [9]
V. Bisognin, M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Soriano,
Uniform decay for the Klein-Gordon-Schrödinger equations with locally distributed damping,
[10]
C. A. Bortot and W. J. Corrêa,
Exponential stability for the defocusing semilinear Schrödinger equation with locally distributed damping on a bounded domain,
[11]
M. Cavalcanti and V. N. Domingos Cavalcanti,
Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations,
[12]
M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka,
Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction,
[13]
J. Colliander, J. Holmer and N. Tzirakis,
Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems,
[14] [15]
M. Daoulatli, I. Lasiecka and D. Toundykov,
Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions,
[16] [17] [18] [19] [20] [21] [22]
B. Guo and Y. Li,
Attractor for dissipative Klein-Gordon-Schrödinger equations in ${{\bf{R}}^{\bf{3}}}$,
[23] [24]
B. Guo and Y. Li,
Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations,
[25] [26] [27] [28]
H. Lange and B. Wang,
Regularity of the global attractor for the Klein-Gordon-Schrödinger equation,
[29] [30] [31]
I. Lasiecka and D. Toundykov,
Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,
[32]
G. Lebeau, Controle de l'equation de Schrödinger. (french) [control of the Schrödinger equation],
[33]
Y. Li, Q. Shi, C. Wang and S. Wang, Well-posedness for the nonlinear Klein-Gordon-Schrödinger equations with heterointeractions,
[34]
J. L. Lions,
[35]
J. L. Lions,
[36] [37] [38]
M. Ohta,
Stability of stationary states for the coupled Klein-Gordon-Schrödinger equations,
[39] [40] [41] [42]
M. N. Poulou and N. M. Stavrakakis, Uniform decay for a local dissipative Klein-Gordon-Schrödinger type system,
[43] [44] [45]
N. Tzirakis,
The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space,
[46]
B. Wang,
Classical global solutions for non-linear Klein-Gordon-Schrödinger equations,
[47]
[1] [2]
A. F. Almeida, M. M. Cavalcanti, J. P. Zanchetta.
Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping.
[3]
Fábio Natali, Ademir Pastor.
Orbital stability of periodic waves for the
Klein-Gordon-Schrödinger system.
[4]
Fábio Natali, Ademir Pastor.
Stability properties of periodic standing waves for the
Klein-Gordon-Schrödinger system.
[5]
Marilena N. Poulou, Nikolaos M. Stavrakakis.
Finite dimensionality of a Klein-Gordon-Schrödinger type system.
[6]
Ji Shu.
Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions.
[7]
Weizhu Bao, Chunmei Su.
Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes.
[8]
Salah Missaoui, Ezzeddine Zahrouni.
Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$.
[9]
Caidi Zhao, Gang Xue, Grzegorz Łukaszewicz.
Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations.
[10]
Pavlos Xanthopoulos, Georgios E. Zouraris.
A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves.
[11]
E. Compaan, N. Tzirakis.
Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $.
[12]
Tetsu Mizumachi, Dmitry Pelinovsky.
On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation.
[13]
Wen Feng, Milena Stanislavova, Atanas Stefanov.
On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion.
[14]
Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago.
Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis.
[15] [16] [17]
Boling Guo, Yan Lv, Wei Wang.
Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations.
[18]
Fathi Hassine.
Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping.
[19]
Xiaoyan Lin, Yubo He, Xianhua Tang.
Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential.
[20]
2018 Impact Factor: 1.048
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Prior to calculators, the standard way for determining a logarithm precisely was to use a look-up table for values. However, such tables were usually only printed for the common most bases, particularly 10 (log). To determine a logarithm for a different base, you needed to first convert to that base.
Luckily, it’s very simple to do so. The change of base formula is: \[\log_a b = \frac{\log_c b}{\log_c a}\]
This works for any base, but it’s most typical to use it for 10 and
e. For instance, if you want to find \(\log_6 29\) using a log (base 10) table, you find \(\log 6\) and \(\log 29\), then divide. That is, \[\log_6 29 = \frac{\log 29}{\log 6} \approx \frac{1.4624}{.7782} \approx 1.879\] Side note: You’ll notice that the log table I linked only goes up to 9.99. So how did I use it to find \(\log 29\)? I looked up \(\log 2.9\) and then added 1. This is why common log tables were most common for this purpose: They’re more flexible than natural log tables. You only need the values from 1.000 to 9.999 to find the log of any positive real number.
Since many calculators only have log and ln buttons, it’s useful to know this formula. Even for the calculator we use in class, you might find that it’s easier to use this technique instead of using
[2nd][Window/F1][5].
You’ll get the same answer either way because the calculator is performing a change of base anyway. Using
[2nd][Window/F1][5], find \(\log_0 0\) and \(\log_1 1\). You’ll get an error for each, but it’s a different error: In the first case, the error is Error: Domain because 0 is not a valid base. In the second case, the error is Error: Divide by 0. If the calculator were doing the logarithm directly, it should give another domain error (because 1 is not a valid base). By the change of base formula, though, you would calculate \(\log 1 \div \log 1 = 0 \div 0\). Proof
The change of base formula seems weird: Why does it work? How does making the base into its own log work? This is magic!
It’s not magic, it’s math. Logarithms do sometimes work in ways we might not immediately predict. The proof is remarkably short, given how unusual it looks.
First, let’s replace each part of the formula with a variable. That is, we’ll let \[ x = \log_a b \\ y = \log_c b \\ z = \log_c a\]
Let’s rewrite each of these into exponential form: \[ a^x = b \\ c^y = b \\ c^z = a\]
Replacing \(a\) in the first line with \(c^z\) from the third line, then using the transitive property on the first two lines, gives us: \[ (c^z)^x = c^y \]
Because of the properties of exponents, we can rewrite \((c^z)^x = c^{xz}\). Two exponential expressions with the same base (other than 1 and 0, which are excluded by definition) have to have the same exponent, that is: \[ xz = y \Rightarrow x = \frac{y}{z} \]
We can now replace the variables as we defined them in the first step, giving us: \[\log_a b = \frac{\log_c b}{\log_c a}\]
which is the change of base formula.
|
Our paper is on indeterminate strings, which are important for their applicability in bioinformatics. (They have been considered, for example, in Christodoulakis 2015 and Helling 2017.)
An interesting feature of indeterminate strings is the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimal alphabet size of indeterminates representing any given undirected graph corresponds to the size of the minimal clique cover of this graph. This paper first considers a related problem proposed in Helling 2017: characterize $late \Theta_n(m)$, which is the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph on vertices and edges. We provide improvements to the known upper bound for . Helling 2017 also presents an algorithm which finds clique covers in polynomial time. We build on this result with an original heuristic for vertex sorting which significantly improves their algorithm’s results, particularly in dense graphs.
This work was the result of building on Helling 2017 (see this post) and of a year of research undertaken by Ryan McIntyre under my (Michael Soltys) supervision at the California State University Channel Islands.
Very happy that our paper Computing covers from matchings with permutations, with Ariel Fernández and Ryszard Janicki, is going to appear in the International Journal of Computer Applications.
We present a matrix permutation algorithm for computing a minimal vertex cover from a maximal matching in a bipartite graph. Our algorithm is linear time and linear space, and provides an interesting perspective on a well known problem. Unlike most algorithms, it does not use the concept of alternating paths, and it is formulated entirely in terms of combinatorial operations on a binary matrix. The algorithm relies on permutations of rows and columns of a 0-1 matrix which encodes a bipartite graph together with its maximal matching. This problem has many important applications such as network switches which essentially compute maximal matchings between their incoming and outgoing ports.
Abstract: In this study, we provide mathematical and practice-driven justification for using [0, 1] normalization of inconsistency indicators in pairwise comparisons. The need for normalization, as well as problems with the lack of normalization, are presented. A new type of paradox of infinity is described.
Accepted for publication in the International Journal of Approximate Reasoning, April 2017.
A new paper:On normalization of inconsistency indicators in pairwise comparisons, by W.W. Koczkodaj, J.-P. Magnot, J. Mazurek, J.F. Peters, H. Rakhshani, M. Soltys, D. Strzałka, J. Szybowski and A. Tozzi.
Abstract: In this study, we provide mathematical and practice-driven justification for using [0,1] normalization of inconsistency indicators in pairwise comparisons. The need for normalization, as well as problems with the lack of normalization, are presented. A new type of paradox of infinity is described.
In a new paper, Square-free strings over alphabet lists, my PhD student Neerja Mhaskar and I, solve an open problem that was posed in A new approach to non repetitive sequences, by Jaroslaw Grytczuk, Jakub Kozik, and Pitor Micek, in arXiv:1103.3809, December 2010.
The problem can be stated as follows: Given an alphabet list $L=L_1,\ldots,L_n$, where $|L_i|=3$ and $0 \leq i \leq n$, can we always find a square-free string, $W=W_1W_2 \ldots W_n$, where $W_i\in L_i$? We show that this is indeed the case. We do so using an “offending suffix” characterization of forced repetitions, and a counting, non-constructive, technique. We discuss future directions related to finding a constructive solution, namely a polytime algorithm for generating square-free words over such lists.
I wrote a paper about finite games which I presented at Computability in Europe in Athens, 2009. Now it turns out, that the Vienna school of Economics, Wirtschaftsuniversität Wien, has been citing it repeatedly in the last few months, in particular Aurélien Fichet de Clairfontaine. It is very satisfying to see research being picked up by other areas!
Abstract: A word is non-repetitive if it does not contain a subword of the form vv. Given a list of alphabets L = L1, L2, . . . , Ln, we investigate the question of generating non-repetitive words w = w1w2 . . . wn, such that the symbol wi is a letter in the alphabet Li. This problem has been studied by several authors (e.g., [GKM10], [Sha09]), and it is a natural extension of the original problem posed and solved by A. Thue. While we do not solve the problem in its full generality, we show that such strings exist over many classes of lists. We also suggest techniques for tackling the problem, ranging from online algorithms, to combinatorics over 0-1 matrices, and to proof complexity. Finally, we show some properties of the extension of the problem to abelian squares.
Abstract: We show that shuffle, the problem of determining whether a string w can be composed from an order preserving shuffle of strings x and y, is not in AC0, but it is in AC1. The fact that shuffle is not in AC0 is shown by a reduction of parity to shuffle and invoking the seminal result of Furst et al., while the fact that it is in AC1 is implicit in the results of Mansfield. Together, the two results provide a lower and upper bound on the complexity of this combinatorial problem. We also explore an interesting relationship between graphs and the shuffle problem, namely what types of graphs can be represented with strings exhibiting the anti-Monge condition.
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External Direct Sum of Rings is Ring Theorem
Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be rings.
Then their (external) direct product: $\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$
is a ring.
Proof
Consider the structures $\left({R_1, +_1}\right), \left({R_2, +_2}\right), \ldots, \left({R_n, +_n}\right)$.
$\displaystyle \left({R, +}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k}\right)$
is a group.
Similarly, consider the structures $\left({R_1, \circ_1}\right), \left({R_2, \circ_2}\right), \ldots, \left({R_n, \circ_n}\right)$. $\displaystyle \left({R, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, \circ_k}\right)$
is a semigroup.
Hence the result, by definition of ring.
$\blacksquare$
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
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№ 9
All Issues Pelyukh G. P.
↓ Abstract
Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 129-138
We establish new properties of the solutions of a differential-functional equation with linearly transformed argument.
Fedorenko V. V., Ivanov А. F., Khusainov D. Ya., Kolyada S. F., Maistrenko Yu. L., Parasyuk I. O., Pelyukh G. P., Romanenko O. Yu., Samoilenko V. G., Shevchuk I. A., Sivak A. G., Tkachenko V. I., Trofimchuk S. I.
Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 257-260
On the asymptotic properties of continuous solutions of the systems of nonlinear functional equations
Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 119-125
For systems of nonlinear functional equations, we study asymptotic properties of their solutions continuously differentiable and bounded for $t \geq T > 0$ in a neighborhood of the singular point $t = +\infty$.
On properties of solutions of a limit problem for systems of nonlinear functional differential equations of neutral type
Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 217–224
For a class of systems of nonlinear differential-functional equations, we study asymptotic characteristics of their solutions continuously differentiable and bounded for
t > T > 0 (along with the first derivative). Investigation of the structure of the set of continuous solutions of systems of nonlinear difference equations with continuous argument
Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 99–108
We study the structure of the set of continuous solutions for one class of systems of nonlinear difference equations with continuous argument in the neighborhoods of equilibrium states.
On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points
Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1668–1676
We establish new properties of $C^1 (0, + ∞)$-solutions of the linear functional differential equation $\dot{x}(t) = ax(t) + bx(qt) + cx(qt)$ in the neighborhoods of the singular points $t = 0$$ and t = + ∞$.
Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1626-1633
We obtain new sufficient conditions for the existence and uniqueness of an
N-periodic solution ( N is a positive integer) of a nonlinear difference equation with continuous argument of the form x( t + 1) = f( x( t)) and investigate the properties of this solution. On Global Solutions of Systems of Nonlinear Functional Differential Equations with Deviating Argument Dependent on Unknown Functions
Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 402-407
For a system of nonlinear functional differential equations with nonlinear deviations of an argument, we obtain sufficient conditions for the existence of a continuously differentiable solution bounded for
t ∈ R.
Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 138-141
We establish conditions for the existence and uniqueness of continuous asymptotically periodic solutions of nonlinear difference equations with continuous argument.
On the Existence of Local Smooth Solutions of Systems of Nonlinear Functional Equations with Deviations Dependent on Unknown Functions
Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 64-77
We obtain conditions for the existence of a local differentiable solution of a system of nonlinear functional equations with nonlinear deviations of an argument.
Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 936-953
We investigate the structure of the general solution of a system of nonlinear difference equations with continuous argument in the neighborhood of an equilibrium state.
On the existence and uniqueness of solutions continuous and bounded on the real axis for nonlinear functional equations
Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 416-418
For one class of nonlinear functional equations, we establish conditions for the existence and uniqueness of solutions continuous and bounded on the real axis.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1368–1378
We investigate the structure of a general solution of systems of nonlinear difference equations with continuous argument in a neighborhood of the state of equilibrium.
Solutions of systems of nonlinear difference equations that are continuous and bounded on the entire real axis and their properties
Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1636–1645
For a system of nonlinear difference equations, we establish conditions for the existence and uniqueness of a solution bounded on the entire real axis and study its properties.
Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 304–308
We establish conditions for the existence and uniqueness of a periodic solution of one nonlinear difference equation.
Berezansky Yu. M., Boichuk О. A., Korneichuk N. P., Korolyuk V. S., Koshlyakov V. N., Kulik V. L., Luchka A. Y., Mitropolskiy Yu. A., Pelyukh G. P., Perestyuk N. A., Skorokhod A. V., Skrypnik I. V., Tkachenko V. I., Trofimchuk S. I.
Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 3–4
Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 140-145
We establish conditions for the existence of periodic solutions of systems of nonlinear difference equations with continuous argument.
Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1382–1387
We establish conditions for the existence of periodic solutions for a broad class of nonlinear difference equations with discrete argument.
Solutions of systems of nonlinear functional-differential equations bounded in the entire real axis and their properties
Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 737–747
For a system of nonlinear functional-differential equations with a linearly transformed argument, we establish the existence and uniqueness conditions for a solution bounded in the entire real axis and study the properties of this solution.
Ukr. Mat. Zh. - 1986. - 38, № 2. - pp. 261–265
Ukr. Mat. Zh. - 1983. - 35, № 4. - pp. 516—519
Ukr. Mat. Zh. - 1983. - 35, № 2. - pp. 173—181
Ukr. Mat. Zh. - 1978. - 30, № 1. - pp. 79–85
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Smoothing effects for some derivative nonlinear Schrödinger equations
1.
Department of Applied Mathematics, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162
2.
Instituto de Física y Matemáticas, Universidad Michoacana, AP 2-82, CP 58040, Morelia, Michoacana
3.
Department of Applied Mathematics, Science University of Tokyo,Tokyo 162-8601, Japan
$iu_t + u_{x x} = \mathcal N(u, \bar u, u_x, \bar u_x), \quad t \in \mathbf R,\ x\in \mathbf R;\quad u(0, x) = u_0(x),\ x\in \mathbf R,\qquad$ (A)
where $\mathcal N(u, \bar u, u_x, \bar u_x) = K_1|u|^2u+K_2|u|^2u_x +K_3u^2\bar u_x +K_4|u_x|^2u+K_5\bar u$ $u_x^2 +K_6|u_x|^2u_x$, the functions $K_j = K_j (|u|^2)$, $K_j(z)\in C^\infty ([0, \infty))$. If the nonlinear terms $\mathcal N =\frac{\bar{u} u_x^2}{1+|u|^2}$, then equation (A) appears in the classical pseudospin magnet model [16]. Our purpose in this paper is to consider the case when the nonlinearity $\mathcal N$ depends both on $u_x$ and $\bar u_x$. We prove that if the initial data $u_0\in H^{3, \infty}$ and the norms $||u_0||_{3, l}$ are sufficiently small for any $l\in N$, (when $\mathcal N$ depends on $\bar u_x$), then for some time $T > 0$ there exists a unique solution $u\in C^\infty ([-T, T]$\ $\{0\};\ C^\infty(\mathbb R))$ of the Cauchy problem (A). Here $H^{m, s} = \{\varphi \in \mathbf L^2;\ ||\varphi||_{m, s}<\infty \}$, $||\varphi||_{m, s}=||(1+x^2)^{s/2}(1-\partial_x^2)^{m/2}\varphi||_{\mathbf L^2}, \mathbf H^{m, \infty}=\cap_{s\geq 1} H^{m, s}.$
Mathematics Subject Classification:35Q5. Citation:Nakao Hayashi, Pavel I. Naumkin, Patrick-Nicolas Pipolo. Smoothing effects for some derivative nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 685-695. doi: 10.3934/dcds.1999.5.685
[1] [2] [3]
Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin.
Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation.
[4]
Hiroyuki Hirayama, Mamoru Okamoto.
Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity.
[5]
Nakao Hayashi, Pavel I. Naumkin.
Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited.
[6]
Kazumasa Fujiwara, Tohru Ozawa.
On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation.
[7]
Yuji Sagawa, Hideaki Sunagawa.
The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension.
[8]
Zihua Guo, Yifei Wu.
Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$.
[9]
Nakao Hayashi, Pavel Naumkin.
On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation.
[10] [11]
Hiroyuki Hirayama.
Well-posedness and scattering for a system of quadratic derivative
nonlinear Schrödinger equations with low regularity initial data.
[12]
Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani.
Representation formula
for traveling waves
to a derivative nonlinear Schrödinger equation
with the periodic boundary condition.
[13]
Brenton LeMesurier.
Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping.
[14]
Juan Belmonte-Beitia, Vladyslav Prytula.
Existence of solitary waves in nonlinear equations of Schrödinger type.
[15]
Van Duong Dinh, Binhua Feng.
On fractional nonlinear Schrödinger equation with combined power-type nonlinearities.
[16]
Hiroyuki Hirayama, Mamoru Okamoto.
Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity.
[17]
Changxing Miao, Bo Zhang.
Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations.
[18]
Binhua Feng.
On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities.
[19]
Jun-ichi Segata.
Initial value problem for the fourth order
nonlinear Schrödinger type equation
on torus and orbital stability of
standing waves.
[20]
Zhong Wang.
Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation.
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Definition:Scalar Ring Contents Definition
Let $\left({S, *_1, *_2, \ldots, *_n, \circ}\right)_R$ be an $R$-algebraic structure with $n$ operations, where:
$\left({R, +_R, \times_R}\right)$ is a ring $\circ: R \times S \to S$ is a binary operation. Then the ring $\left({R, +_R, \times_R}\right)$ is called the scalar ring of $\left({S, *_1, *_2, \ldots, *_n, \circ}\right)_R$. If the scalar ring is understood, then $\left({S, *_1, *_2, \ldots, *_n, \circ}\right)_R$ can be rendered $\left({S, *_1, *_2, \ldots, *_n, \circ}\right)$.
The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called
scalars.
The operation $\circ: R \times S \to S$ is called
scalar multiplication. Definition for Module
The same definition applies when $\left({S, *_1, *_2, \ldots, *_n}\right)$ is an abelian group $\left({G, +_G}\right)$.
In this case, $\left({G, +_G, \circ}\right)_R$ is a module.
Let $\left({G, +_G, \circ}\right)_K$ be a vector space, where:
$\left({K, +_K, \times_K}\right)$ is a field $\left({G, +_G}\right)$ is an abelian group $\left({G, +_G}\right)$ $\circ: K \times G \to G$ is a binary operation. Then the field $\left({K, +_K, \times_K}\right)$ is called the scalar field of $\left({G, +_G, \circ}\right)_K$. If the scalar field is understood, then $\left({G, +_G, \circ}\right)_K$ can be rendered $\left({G, +_G, \circ}\right)$.
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Definition:Quotient Ring Definition
Let $\struct {R, +, \circ}$ be a ring.
Let $J$ be an ideal of $R$.
Let $R / J$ be the (left) coset space of $R$ modulo $J$ with respect to $+$. Define an operation $+$ on $R / J$ by: $\forall x, y: \paren {x + J} + \paren {y + J} := \paren {x + y} + J$
Also, define the operation $\circ$ on $R / J$ by:
$\forall x, y: \paren {x + J} \circ \paren {y + J} := \paren {x \circ y} + J$ The algebraic structure $\struct {R / J, +, \circ}$ is called the quotient ring of $R$ by $J$. Also denoted as
While the inline form of the fraction notation $R / J$ is usually used for a
quotient ring, some presentations use the full $\dfrac R J$ form.
It is usual for the latter form to be used only when either of both of the expressions top and bottom are more complex than single symbols.
Also known as
This is also known as a
factor ring.
Some sources refer to this as a
residue class ring. Also see Quotient Ring Addition is Well-Defined Quotient Ring Product is Well-Defined Quotient Ring is Ring Congruence Relation and Ideal are Equivalent Results about Quotient Ringscan be found here. Sources 1965: Seth Warner: Modern Algebra... (previous) ... (next): $\S 22$ 1969: C.R.J. Clapham: Introduction to Abstract Algebra... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Theorem $42$ 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra... (previous) ... (next): $\S 2.2$: Homomorphisms 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra... (previous) ... (next): $\S 60$. Factor rings
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Upper and Lower Bounds of Integral Theorem
Let $\displaystyle \int_a^b \map f x \rd x$ be the definite integral of $\map f x$ over $\closedint a b$.
Then: $\displaystyle m \paren {b - a} \le \int_a^b \map f x \rd x \le M \paren {b - a}$
where:
on $\closedint a b$.
Suppose that $\forall t \in \left[{a \,.\,.\, b}\right]: \left|{f \left({t}\right)}\right| < \kappa$.
Then: $\displaystyle \forall \xi, x \in \left[{a \,.\,.\,b}\right]: \left|{\int_x^\xi f \left({t}\right) \rd t}\right| < \kappa \left|{x - \xi}\right|$ Proof
This follows directly from the definition of definite integral:
From Continuous Image of Closed Interval is Closed Interval it follows that $m$ and $M$ both exist.
The result follows.
$\blacksquare$
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I see a question quite a lot in past exam papers that goes like propose a quantum circuit that generates the state $|\psi \rangle$ given the initial state $|\phi\rangle$
Here's an example:
Given the initial state $|000 \rangle $ propose a quantum circuit that generates the state
$$|\psi \rangle=\tfrac{1}{\sqrt{2}} (|+++ \rangle - |--- \rangle)$$
Where $|\pm \rangle=(|0 \rangle \pm |1\rangle)/\sqrt{2}$
Now there's a square root of 2 involved so one would imagine a Hadamard gate is involved, but other than that I don't really see how you could just know the circuit apart from trial and error.
Are there any tips and tricks for making circuits that generate states given some initial state?
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Last time we revisited Robin’s theorem saying that 5040 being the largest counterexample to the bound
\[ \frac{\sigma(n)}{n~log(log(n))} < e^{\gamma} = 1.78107... \] is equivalent to the Riemann hypothesis.
\[
\Psi(n) = n \prod_{p | n}(1 + \frac{1}{p}) \]
where $p$ runs over the prime divisors of $n$. It is series A001615 in the online encyclopedia of integer sequences and it starts off with
1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, …
and here’s a plot of its first 1000 values
To understand this behaviour it is best to focus on the ‘slopes’ $\frac{\Psi(n)}{n}=\prod_{p|n}(1+\frac{1}{p})$.So, the red dots of minimal ‘slope’ $\approx 1$ correspond to the prime numbers, and the ‘outliers’ have a maximal number of distinct small prime divisors. Look at $210 = 2 \times 3 \times 5 \times 7$ and its multiples $420,630$ and $840$ in the picture. For this reason the primorial numbers, which are the products of the fist $k$ prime numbers, play a special role. This is series A002110 starting off with
1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870,…
In Patrick Solé and Michel Planat Extreme values of the Dedekind $\Psi$ function, it is shown that the primorials play a similar role for Dedekind’s Psi as the superabundant numbers play for the sum-of-divisors function $\sigma(n)$.That is, if $N_k$ is the $k$-th primorial, then for all $n < N_k$ we have that the 'slope' at $n$ is strictly below that of $N_k$ \[ \frac{\Psi(n)}{n} < \frac{\Psi(N_k)}{N_k} \] which follows immediately from the fact that any $n < N_k$ can have at most $k-1$ distinct prime factors and $p \mapsto 1 + \frac{1}{p}$ is a strictly decreasing function. Another easy, but nice, observation is that for all $n$ we have the inequalities
\[
n^2 > \phi(n) \times \psi(n) > \frac{n^2}{\zeta(2)} \]
where $\phi(n)$ is Euler’s totient function
\[
\phi(n) = n \prod_{p | n}(1 – \frac{1}{p}) \]
This follows as once from the definitions of $\phi(n)$ and $\Psi(n)$
\[
\phi(n) \times \Psi(n) = n^2 \prod_{p|n}(1 – \frac{1}{p^2}) < n^2 \prod_{p~\text{prime}} (1 - \frac{1}{p^2}) = \frac{n^2}{\zeta(2)} \] But now it starts getting interesting. In the proof of his theorem, Guy Robin used a result of his Ph.D. advisor Jean-Louis Nicolas
known as Nicolas’ criterion for the Riemann hypothesis: RH is true if and only if for all $k$ we have the inequality for the $k$-th primorial number $N_k$
\[ \frac{N_k}{\phi(N_k)~log(log(N_k))} > e^{\gamma} \] From the above lower bound on $\phi(n) \times \Psi(n)$ we have for $n=N_k$ that \[ \frac{\Psi(N_k)}{N_k} > \frac{N_k}{\phi(N_k) \zeta(2)} \] and combining this with Nicolas’ criterion we get \[ \frac{\Psi(N_k)}{N_k~log(log(N_k))} > \frac{N_k}{\phi(N_k)~log(log(N_k)) \zeta(2)} > \frac{e^{\gamma}}{\zeta(2)} \approx 1.08… \] In fact, Patrick Solé and Michel Planat prove in their paper Extreme values of the Dedekind $\Psi$ function that RH is equivalent to the lower bound \[ \frac{\Psi(N_k)}{N_k~log(log(N_k))} > \frac{e^{\gamma}}{\zeta(2)} \] holding for all $k \geq 3$.
In other words, it gives us the number of tiles needed in the Dedekind tessellation to describe the fundamental domain of the action of $\Gamma_0(n)$ on the upper half-plane by Moebius transformations.
When $n=6$ we have $\Psi(6)=12$ and we can view its fundamental domain via these Sage commands:
G=Gamma0(6) FareySymbol(G).fundamental_domain()
giving us the 24 back or white tiles (note that these tiles are each fundamental domains of the extended modular group, so we have twice as many of them as for subgroups of the modular group)
But, there are plenty of other, seemingly unrelated, topics where $\Psi(n)$ appears. To name just a few:
The number of points on the projective line $\mathbb{P}^1(\mathbb{Z}/n\mathbb{Z})$. The number of lattices at hyperdistance $n$ in Conway’s big picture. The number of admissible maximal commuting sets of operators in the Pauli group for the $n$ qudit.
and there are explicit natural one-to-one correspondences between all these manifestations of $\Psi(n)$, tbc.Leave a Comment
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Returns the group delay of the argument. Group delay is defined as:
\[ \frac{d}{dx}\left(\text{phase}\left(y\right)\right)\cdot \frac{1}{2\pi} \]
where ???MATH???y???MATH??? is the supplied vector and ???MATH???x???MATH??? is its reference. The GroupDelay function expects the result of an AC analysis where ???MATH???y???MATH??? is a voltage or current and its reference is frequency.
This function will yield an error if its argument is complex and has no reference.
Number
Type
Compulsory
Default
Description
1 real/complex array Yes Vector
Return type: real array
▲Function Summary▲ ◄ GraphLimits Groups ▶
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Lamé's Theorem - the Very First Application of Fibonacci Numbers
Among the unique properties of number five, Joe Roberts counts the appearance of five in one of the formulations of Lamé's theorem:
In carrying out the Euclidean algorithm to find the greatest common divisor of two positive integers \(a\) and \(b\), the number of steps needed will never exceed 5 times the number of base 10 digits in the smaller of the two integers \(a\) and \(b\).
Various aspects of this theorem, first proved by (Gabriel) Lamé in 1844, are quite regularly rediscovered. No doubt a "natural high" occurs each time this happens. (At least that was so in my case.)
In a 1992 publication, Roberts refers to a 1939 text on Number Theory, which tells me that Lamé's remarkable theorem is not very well known, at least among non-specialists. In his
Mathematical Gems II, Ross Honsberger refers to the 1924 edition of W. Sierpinski book Elementary Theory of Numbers; the book has been republished twice since, but by now became a bibliographic rarity. It is available online for download as djvu file.
Lamé's theorem and the Euclidean algorithm have been discussed at length in the 1969 D. Knuth's
Seminumerical Algorithms, now available in the third edition (1997) and lately in a most unusual book by V. H. Moll. So, we have three somewhat different formulations, each emphasizing a little different aspects of the theorem. Lamé's Theorem (Honsberger)
The number of steps (i.e., divisions) in an application of the Euclidean algorithm never exceeds 5 times the number of digits in the lesser.
Lamé's Theorem (Knuth)
For \(n\ge 1\), let integers \(u\) and \(v\), \(u\gt v\gt 0\), be such that processing \(u\) and \(v\) by the Euclidean algorithm takes exactly \(n\) division steps. Moreover, assume that \(u\) is the least possible number satisfying that requirement. Then \(u=F_{n+2}\) and \(v=F_{n+1}\), where \(\{F_k\}\) is the Fibonacci sequence.
Knuth also proves
Corollary
For \(0\lt u,v\lt N\), the number of the division steps needed by the Euclidean algorithm to process \(u\) and \(v\) does not exceed \(\big\lceil log_{\phi}(\sqrt{5}N)\big\rceil -2\).
Lamé's Theorem (Moll)
Let \(a,b\in\mathbb{N}\) with \(a\gt b\). The number of steps in the Euclidean algorithm is about \(\mbox{log}_{10}b/\mbox{log}_{10}\phi\). This is at most five times the number of decimal digits of \(b\).
\(\phi\) here is of course the Golden Ratio (\(\displaystyle\phi = \frac{1+\sqrt{5}}{2}\)) whose appearance may not be surprising after the Fibonacci sequence made its debut in Knuth's formulation. In all the proofs the Fibonacci numbers play a most fundamental role and, as Knuth observes, this was their first ever practical application; many more followed.
On this page it is convenient to define the Fibonacci numbers recursively
\(F_n= \begin{cases} 1 & \mbox{if } n = 0; \\ 1 & \mbox{if } n = 1; \\ F_{n-1}+F_{n-2} & \mbox{if } n > 1. \\ \end{cases} \)
Let \(p_n\) denote the number of digits in \(F_n\). As R. L. Duncan showed in 1966, the number of the division steps in the Euclidean algorithm required to compute \(\mbox{gcd}(F_{n+1}, F_{n})\) always satisfies the inequality
\(\displaystyle n\gt\frac{p_n}{\mbox{log}_{10}\phi} - 5\)
while Lamé's result could be reformulated as \(\displaystyle n\lt\frac{p_n}{\mbox{log}_{10}\phi} + 1\). In 1967, J. L. Brown has proved a stronger result:
There exist an infinite number of distinct positive integers \(n\) such that the determination of \(\mbox{gcd}(F_{n+1},F_{n})\) by the Euclidean algorithm requires exactly \(n\) divisions with \(n\) satisfying
\(\displaystyle n\gt\frac{p_n}{\mbox{log}_{10}\phi} - \frac{1}{2}\),
making the estimate in Lamé's theorem the best possible.
All proofs of the above-mentioned results use but basic properties of the Fibonacci numbers. The Honsberger/Sierpinski one is based on the following
Lemma
For all \(n\ge 1\), \(F_{n+5}\gt 10\cdot F_{n}\).
Proof
From the basic recurrence, \(F_{k+2}=F_{k+1}+F_{k}\), \(F_{k+2}=2\cdot F_{k}+F_{k-1}\), so we obtain successively
\( \begin{align} F_{n+5} &= F_{n+4} + F_{n+3} \\ &= 2\cdot F_{n+3} + F_{n+2} \\ &= 3\cdot F_{n+2} + 2\cdot F_{n+1} \\ &= 5\cdot F_{n+1} + 3\cdot F_{n} \\ &= 8\cdot F_{n} + 5\cdot F_{n-1} \\ &= 13\cdot F_{n-1} + 8\cdot F_{n-2} \\ &= 21\cdot F_{n-2} + 13\cdot F_{n-3} \\ &> 10\cdot (2F_{n-2} + F_{n-3}) \\ &= 10\cdot F_{n}. \end{align} \)
From this it is immediate that \(F_{n+5}\) has at least one more decimal digit than \(F_n\). It follows by induction that \(p_{n+5t}\gt 10^{t}p_{n}\). By direct inspection, numbers \(F_n\), for \(1\le n\le 5\) are single-digit. They
have at least \(2\) digits for \(5\lt n\le 10\),
have at least \(3\) digits for \(10\lt n\le 15\),
have at least \(4\) digits for \(15\lt n\le 20\),
and, in general, have at least \(k\) digits for \(5(k-1)\lt n\le 5k\). In other words, for \(5(k-1)\lt n\le 5k\), \(p_{n}\ge k\ge n/5\). Thus it is always the case that \(p_{n}\ge n/5\).
Let's remember that; this in an important step in proving Lamé's theorem. But in order to complete the proof, we need to bring in the Euclidean algorithm.
Given to integers, \(a\) and \(b\), \(a\gt b\gt 0\), we set \(a = r_0\), \(b=r_1\) and divide with remainder. Assuming Euclid's algorithm takes \(n\) steps,
\( \begin{align} r_{0} &= r_{1}q_{1} + r_{2}, 0\le r_{2}\lt r_{1} \\ r_{1} &= r_{2}q_{2} + r_{3}, 0\le r_{3}\lt r_{2} \\ r_{2} &= r_{3}q_{3} + r_{4}, 0\le r_{4}\lt r_{3} \\ &= \cdots \\ r_{n-2} &= r_{n-1}q_{n-1} + r_{n}, 0\le r_{n}\lt r_{n-1} \\ r_{n-1} &= r_{n}q_{n}. \\ \end{align} \)
Note that \(q_{n}\ge 2\), for, otherwise, we would have \(r_{n-1} = r_{n}\), in contradiction with the previous step of the algorithm (\(0\le r_{n}\lt r_{n-1}\).) We now proceed backwards, starting with \(r_{n}\ge 1\), which implies \(r_{n}\ge F_{1}.\) Next, \(r_{n-1}\ge 2r_{n} = F_{2}.\) And further,
\(r_{n-2} = r_{n-1}q_{n-1} + r_{n} \gt r_{n-1} + r_{n} \gt F_{2} + F_{1} = F_{3}.\)
In general, for \(1\le k\le n\),
\( \begin{align} r_{n-k} &= r_{n-k+1}q_{n-k+1} + r_{n-k+2} \\ & \gt r_{n-k+1} + r_{n-k+2} \\ & \gt F_{k} + F_{k-1} = F_{k+1}, \end{align} \)
such that at the end of the process(i.e., at the beginning of the algorithm) when \(k=n-1\) and \(k=n\), \(b=r_{1}\gt F_{n}\) and \(a=r_{0}\gt F_{n+1}\). Therefore, if integers \(a\) and \(b\), \(a\gt b\gt 0\), are such that the Euclidean algorithm takes exactly \(n\) steps, then necessarily \(b\gt F_{n}\) and has at least as many digits, which, as we found previously, is at least \(n/5\). And this proves the Sierpinski/Honsberger formulation.
Now, for Moll's formulation. Recollect that \(\phi ^{2}=\phi + 1\). We use that to prove by induction another
Lemma
For \(n\gt 1\), \(F_{n}\gt \phi ^{n-1}.\)
Proof
Although, the statement is to be proved for \(n\gt 1\), it saves time to observe that \(F_{1}=1\ge \phi ^ {0}\). Also \(F_{2}=2\gt \phi ^1\). Then
\(F_{3} = F_{2} + F_{1} \gt 1 + \phi = \phi ^{2}.\)
And, in general,
\(F_{k+2} = F_{k+1} + F_{k} \gt \phi ^{k} + \phi ^{k-1} = \phi ^{k-1} (1 + \phi) = \phi ^{k-1} \phi ^{2} = \phi ^{k+1}.\)
Now, as we already found, if, for \(a\) and \(b\), \(a\gt b\gt 0\), the Euclidean algorithm takes \(n\) steps, then \(b\gt F_{n}\) so that \(b\gt \phi ^{n-1}\), and \(n-1 \lt \frac{\mbox{log}_{10}b}{\mbox{log}_{10}\phi}\). If \(b\) is a \(k\)-digit integer, then \(10^{k-1}\le b\lt 10^k\), and since \(1/\mbox{log}_{10}(\phi)=4.78497...\), \(n-1\lt 5k\), or \(n\le 5k\). This exactly means that the number of the division steps in an application of Euclidean algorithm is at most five times the number of decimal digits of the smaller of the two numbers the algorithm has been applied to.
Almost obviously a pair of consecutive Fibonacci numbers provide a "worse case scenario" - the longest possible application of the Euclidean algorithm relative to the length of the numbers. This is because the quotients \(q_k\) in the application of the algorithm in this case are all 1 (meaning the least reduction in length), except of course the last one.
Finally, T. E. Moore has investigated the distribution of pairs \((a ,b)\) for a given number of steps (DC = "division count") in the Euclidean algorithm. The article includes a Basic programs that has been run on an Apple computer! Adding a 4-fold symmetry, Moore produced a sequence of mysterious diagrams:
References J. L. Brown, Jr., On Lamé's Theorem, Fibonacci Quarterly, v 5, n 2 (April 1967), 153-160 R. L. Duncan, Note on the Euclidean Algorithm, The Fibonacci Quarterly, v 4, n 4, (August 1966) 367-68. R. Honsberger, Mathematical Gems II, MAA, 1976 D. Knuth, The Art of Computer Programming, v2, Seminumerical Algorithms, Addison-Wesley, 1997 (3rd edition) V. H. Moll, Numbers and Functions: From a Classical-Experimental Mathematician's Point of View, AMS, 2012 T. E. Moore, Euclid's Algorithm and Lamé's Theorem on a Microcomputer, Fibonacci Quarterly, v 27, n 4 (August 1989), 290-295 J. Roberts, Lure of the Integers, MAA, 1992 W. Sierpinski, Elementary Theory of Numbers: Second English Edition, North Holland, 1988 J. V. Uspensky & M. A. Heaslet. Elementary Number Theory, McGraw-Hill, 1939. Modular Arithmetic Chinese Remainder Theorem Euclid's Algorithm Euclid's Algorithm (An Interactive Illustration) Euclid's Game Extension of Euclid's Game Binary Euclid's Algorithm gcd and the Fundamental Theorem of Arithmetic Extension of Euclid's Algorithm Lame's Theorem - First Application of Fibonacci Numbers Stern-Brocot Tree Farey series Pick's Theorem Fermat's Little Theorem Wilson's Theorem Euler's Function Divisibility Criteria Examples Equivalence relations A real life story
Copyright © 1996-2018 Alexander Bogomolny
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Difference between revisions of "Past Probability Seminars Spring 2019"
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[[Probability Seminar | Back to Current Probability Seminar Schedule ]]
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<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.
Latest revision as of 08:50, 23 May 2019 Contents 1 Past Seminars Spring 2019 1.1 January 31, Oanh Nguyen, Princeton 1.2 Wednesday, February 6 at 4:00pm in Van Vleck 911 , Li-Cheng Tsai, Columbia University 1.3 February 7, Yu Gu, CMU 1.4 February 14, Timo Seppäläinen, UW-Madison 1.5 February 21, Diane Holcomb, KTH 1.6 Probability related talk in PDE Geometric Analysis seminar: Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison 1.7 Wednesday, February 27 at 1:10pm Jon Peterson, Purdue 1.8 March 21, Spring Break, No seminar 1.9 March 28, Shamgar Gurevitch UW-Madison 1.10 April 4, Philip Matchett Wood, UW-Madison 1.11 April 11, Eviatar Procaccia, Texas A&M 1.12 April 18, Andrea Agazzi, Duke 1.13 April 25, Kavita Ramanan, Brown 1.14 Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, Kavita Ramanan, Brown 1.15 Tuesday, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) 1.16 Past Seminars Spring 2019 Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu
January 31, Oanh Nguyen, Princeton
Title:
Survival and extinction of epidemics on random graphs with general degrees
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$. Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.
Wednesday, February 6 at 4:00pm in Van Vleck 911 , Li-Cheng Tsai, Columbia University
Title:
When particle systems meet PDEs
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..
Title:
Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.
February 14, Timo Seppäläinen, UW-Madison
Title:
Geometry of the corner growth model
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).
February 21, Diane Holcomb, KTH
Title:
On the centered maximum of the Sine beta process Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.
Title: Quantitative homogenization in a balanced random environment
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).
Wednesday, February 27 at 1:10pm Jon Peterson, Purdue
Title:
Functional Limit Laws for Recurrent Excited Random Walks
Abstract:
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.
March 21, Spring Break, No seminar March 28, Shamgar Gurevitch UW-Madison
Title:
Harmonic Analysis on GLn over finite fields, and Random Walks
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the
character ratio:
$$ \text{trace}(\rho(g))/\text{dim}(\rho), $$
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant
rank. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM). April 4, Philip Matchett Wood, UW-Madison
Title:
Outliers in the spectrum for products of independent random matrices
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.
April 11, Eviatar Procaccia, Texas A&M Title: Stabilization of Diffusion Limited Aggregation in a Wedge.
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.
April 18, Andrea Agazzi, Duke
Title:
Large Deviations Theory for Chemical Reaction Networks
Abstract: The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.
April 25, Kavita Ramanan, Brown
Title:
Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.
Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, Kavita Ramanan, Brown
Title:
Tales of Random Projections
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.
Tuesday , May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto)
Title:
The directed landscape
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.
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Lower Semicontinuous with Lipschitz Coefficients
Lower Semicontinuous with Lipschitz Coefficients
Abstract
We are interested in integral functionals of the form
$$ \boldsymbol{J}(U, V) =\int_{\Omega }J\big(x, U(x), V(x)\big) dx,$$ where $J$ is Carath\'eodory positive integrand, satisfying some growth condition of order $p\in(1, \infty)$. We show that $\mathcal{A}(x, \partial)-$quasiconvexity of the integrand $J$ with respect to the third variable is a necessary and sufficient condition of lower semicontinuity of $\boldsymbol{J}$, where $\mathcal{A}(x, \partial)$ is a differential operator given by $$ \mathcal{A}(x, \partial)=\sum_{j=1}^{N}A^{(j)}(x)\partial_{x_{j}}, $$and the coefficients $A^{(j)}, j=1,...,N$ are only Lipschitzian, i.e. $A^{(j)}\in W^{1,\infty }\big(\Omega; \mathbb{M}^{l\times d}\big)$ and satisfy the condition of \textit{constant rank}. To this end, a framework of paradifferential calculus is needed to deal with the lower smoothness of the coefficients.
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This doesn't use the Peano-Baker series, but you can calculate the state transition matrix using NDSolve.The state transition matrix has the following properties:$\Phi(t_0,t_0) = I$, (where $I$ is the identity matrix)$\frac{d}{dt} \Phi(t,t_0) = A(t) \Phi(t,t_0) $For a time-varying matrix $A(t)$ of size $n \times n$, and a mathematica function fA[t] ...
The method used in Jason's answer, as noted, requires version 12. For those in earlier versions who are unable or unwilling to install cheminformatics toolboxes like RDKit, ChEMBL Beaker (see also the associated paper) provides an API that can be used to compute the ring counts required by the underlying formula, which is used in the code that follows:...
If HeavisideTheta[0] = 1, is HeavisideTheta in v11.3 equal to UnitStep in v5.0?For calculus purposes, sure.Can I replace UnitStep with HeavisideTheta?HeavisideTheta[] is what you should be using in version 6 and later versions, because it's the one now supported extensively by the calculus functions. In earlier functions, you have to settle for ...
The paper Fast Calculation of van der Waals Volume as a Sum of Atomic and Bond Contributions and Its Application to Drug Compounds, pointed out by theorist gives an approximate formula for the molecular volume as$$V_{\mathrm{vdW}} = \sum \mathrm{all\: atom\: contributions} - 5.92 N_\mathrm{B} - 14.7 R_\mathrm{A} - 3.8 R_\mathrm{NA}$$where $N_\mathrm{B}$...
You can use ConditionalExpression as in this answer linked by m_goldberg in comments:Manipulate[Plot[{f[x], ConditionalExpression[tangent[f, p, x], p - .5 <= x <= p + .5]},{x, -2 Pi, 2 Pi},Epilog -> {Red, PointSize[.015], Point@{p, Sin[p]}}], {p, -Pi, Pi}]You can replace ConditionalExpression[...] withPiecewise[{{tangent[f, p, x], ...
This is a kludge onto your code. I would have written the whole thing rather differently.Manipulate[Plot[{f[x], If[Abs[p - x] < 1, 100 tangent[f, p, x], 0]},{x, -2 Pi, 2 Pi},Epilog -> {Red, PointSize[.015], Point@{p, Sin[p]}}],{p, -Pi, Pi}]
The simple answer is a^0 == 1 for all a /; a != 0. Look at a plot:Plot3D[Evaluate@ReIm[a^(1/n)], {a, -2, 0.25}, {n, 1, 10000},AxesLabel -> Automatic,ScalingFunctions -> {None, "Log", None},ClippingStyle -> None,PlotLegends -> {Re, Im},PlotPoints -> 100]For a < 0 as n increases, the real part goes to 1 and the imaginary part ...
I can offer a solution by discretization to ElementMesh with external meshing software Gmsh. Currently it produces better quality mesh on curved surfaces that built-in methods. We need packages GmshLink, ImportMesh and MeshTools.Needs["GmshLink`"](*Set your path to Gmsh executable.*)$GmshDirectory = "my_path_to_current_release\\gmsh-4.4.1-Windows64";OP'...
It does not work , because MMA dosen't know a closed-form solution for the Sum.Workaround:Only for: -1 < x < 1, -1 < y < 1func = Sum[MellinTransform[(-1)^n*y^n*x^(a n^2), a, s] // PowerExpand , {n,0, \[Infinity]}, Assumptions -> {-1 < x < 1, -1 < y < 1, s > 0}]sol = Simplify[Series[func, {x, 0, 8}, {y, 0, 8}] // Normal] ...
Mathematica works very carefully and correct in this case. One additionally needs to say that the function is real.Integrate[DiracDelta[f[x]], {x, -∞, ∞}, Assumptions -> f[x] ∈ Reals && f[x]!= 0](* 0 *)
I managed to find an answer to my question and would like to share my experience with the community. It turns out that MA can do a lot with elliptic integrals, however, it does not know some identities, in particular Eq.8.126 from Table Of Integrals, Series And Products by Gradshteyn and Ryzhik. I retype them below in original notations$$K\!\left(\frac{2\...
You can get a list of the values using Map (i.e., shortcut /@)f[#] & /@ Range[4, 1, -1]{f[4], f[3], f[2], f[1]}To get the Dot product of the f's you can Apply (i.e., shortcut @@) the Dot function:Dot @@ (f[#] & /@ Range[4, 1, -1])
Not perfect, but it is a start.f = X \[Function] {2 Indexed[X, 1] +Sin[Indexed[X, 1] + Indexed[X, 2]],2 Indexed[X, 2] + Cos[Indexed[X, 1] + Indexed[X, 2]]};Df = X \[Function] Block[{x, y},D[f[{x, y}], {{x, y}, 1}] /. {x -> Indexed[X, 1], y -> Indexed[X, 2]}];g = Y \[Function] Block[{X}, X /. FindRoot[f[X] == Y, {X, 0. Y}]];...
Many times Mathematica gives enormous results to simple problemsIf Simplify still does not help reduce the antiderivative to what you like, you could always try Rubi<< Rubi`integrand = 1/Sqrt[1 + Sin[x]];sol = Int[integrand, x]D[sol, x] // SimplifyThere is a page here which compares different integrators with the size of antiderivatives ...
As I said, only a synthetic approach is possible. Please, look here (sec. 0.5) for a mathematical proof of some of the transitions.We have$$S=\int_0^1{\mathrm e^{\mathrm i\pi x}x^x(1-x)^{1-x}\mathrm dx} =\int_0^1 (1-x)\, \exp\left\{\left[\mathrm i\pi+\log x-\log(1-x)\right]x\right\}\, \mathrm dx$$First we verify numericallyp[z_] := (1-z) E^((I π + ...
The following does not answer the OP's question but does supply the answer to a few comments asked above:Rather than just a comment that might be missed, I am posting the links I've received from the Mathematics forum. I was interested in the integral and felt asking it's solution was more appropriate there: It's a beautiful example of using the residue ...
I am unable to Integrate it analytically (tried Integrate[E^(I Pi x) x^x (1 - x)^(1 - x), x, Assumptions -> 0 < x < 1] without success).I had to use NIntegrateNIntegrate[E^(I Pi x) x^x (1 - x)^(1 - x), {x, 0, 1}](* 1.73472*10^-17 + 0.355822 I *)and to get rid of the small Real portionChop[NIntegrate[E^(I Pi x) x^x (1 - x)^(1 - x), {x, 0, 1}]...
The Integral has no closed form solution, so use NIntegrate instead of Integrate:\[Mu] := 1;Eb := 0.040;\[CapitalGamma] := 1;(*Fitting parameter*)Eg := 2.354Ebj := Eg - Eb/j^2c := 1.4 (*fitting parameter*)andA[x_?NumericQ] := \[Mu]^2/x (Sum[(2 Eb/j^3 Sech[(x - Ebj)/\[CapitalGamma]]), {j, 1, 10}] +NIntegrate[Sech[(x - e)/\[CapitalGamma]] ...
I would likely write your function definition asopticalDepth[η_] := Integrate[ηe[u] σT a[u], {u, 0, η}]because a single letter variable like u is easy to type. To avoid problems with a previous top-level assignment to u, you can write it with a guard:opticalDepth[η_] := Block[{u},Integrate[ηe[u] σT a[u], {u, 0, η}]]However, using a formal symbol ...
A little dowdy but will do for the moment.v=0.75;f[x_]:=v ArcSin[x]+Cos[ArcSin[x]];F=InverseFunction[f];ifun=Interpolation[Table[{f[x],x},{x,v,1,0.001}]];inverse[y_]:=Piecewise[{{ifun[y],f[1]<y<f[v]},{Indeterminate,True}}]Plot[{f[x],F[x],inverse[x]},{x,-1.3,1.3},AxesOrigin->{0,0}]
If you do the integral first and then take the limit, you get an answer, but not the one you want.int = Integrate[x/E^(30/(a^16*Log[1/x])^(1/15)), {x, 0, 1}](* (Sqrt[15]*MeijerG[{{}, {}}, {{0, 1/15, 2/15, 1/5, 4/15, 1/3, 2/5,7/15, 8/15, 3/5, 2/3, 11/15, 4/5, 13/15, 14/15, 1}, {}},65536/a^16])/(256*Pi^7) *)In your preferred direction:...
This is a precision issue.$Version(* "12.0.0 for Mac OS X x86 (64-bit) (April 7, 2019)" *)Since you used a machine precision number (i.e., 0.5) in the integrand, the integration is done with only machine precision.Integrate[E^(-16/x^8^(-1) - (2*x)/(0.5)^9)/x^(9/8), {x, 0, Infinity}](* -0.0824979 *)Using exact numbersint = Integrate[E^(-16/x^8^(-...
Another case of using non-exact numbers with exact function.See the difference:integrand = x^(-9/8) Exp[-16 x^(-1/8) - 2 x/(1/2)^9];Integrate[integrand, {x, 0, Infinity}] // Nintegrand = x^(-9/8) Exp[-16 x^(-1/8) - 2 x/(0.5)^9];Integrate[integrand, {x, 0, Infinity}]Rule of thumb: use exact numbers when calling exact functions like DSolve, ...
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Conjugate Heat Transfer
In this blog post we will explain the concept of
conjugate heat transfer and show you some of its applications. Conjugate heat transfer corresponds with the combination of heat transfer in solids and heat transfer in fluids. In solids, conduction often dominates whereas in fluids, convection usually dominates. Conjugate heat transfer is observed in many situations. For example, heat sinks are optimized to combine heat transfer by conduction in the heat sink with the convection in the surrounding fluid.
Heat Transfer by Solids and Fluids Heat Transfer in a Solid
In most cases, heat transfer in solids, if only due to conduction, is described by Fourier’s law defining the conductive heat flux,
q, proportional to the temperature gradient: q=-k\nabla T.
For a time-dependent problem, the temperature field in an immobile solid verifies the following form of the heat equation:
Heat Transfer in a Fluid
Due to the fluid motion, three contributions to the heat equation are included:
The transport of fluid implies energy transport too, which appears in the heat equation as the convective contribution. Depending on the thermal properties on the fluid and on the flow regime, either the convective or the conductive heat transfer can dominate. The viscous effects of the fluid flow produce fluid heating. This term is often neglected, nevertheless, its contribution is noticeable for fast flow in viscous fluids. As soon as a fluid density is temperature-dependent, a pressure work term contributes to the heat equation. This accounts for the well-known effect that, for example, compressing air produces heat.
Accounting for these contributions, in addition to conduction, results in the following transient heat equation for the temperature field in a fluid:
Conjugate Heat Transfer Applications Effective Heat Transfer
Efficiently combining heat transfer in fluids and solids is the key to designing effective coolers, heaters, or heat exchangers.
The fluid usually plays the role of energy carrier on large distances. Forced convection is the most common way to achieve high heat transfer rate. In some applications, the performances are further improved by combining convection with phase change (for example liquid water to vapor phase change).
Even so, solids are also needed, in particular to separate fluids in a heat exchanger so that fluids exchange energy without being mixed.
Flow and temperature field in a shell-and-tube heat exchanger illustrating heat transfer between two fluids separated by the thin metallic wall.
Heat sinks are usually made of metal with high thermal conductivity (e.g. copper or aluminum). They dissipate heat by increasing the exchange area between the solid part and the surrounding fluid.
Temperature field in a power supply unit cooling due to an air flow generated by an extracting fan and a perforated grille. Two aluminum fins are used to increase the exchange area between the flow and the electronic components. Energy Savings
Heat transfer in fluids and solids can also be combined to minimize heat losses in various devices. Because most gases (especially at low pressure) have small thermal conductivities, they can be used as thermal insulators… provided they are not in motion. In many situations, gas is preferred to other material due to its low weight. In any case, it is important to limit the heat transfer by convection, in particular by reducing the natural convection effects. Judicious positioning of walls and use of small cavities helps to control the natural convection. Applied at the micro scale, the principle leads to the insulation foam concept where tiny cavities of air (bubbles) are trapped in the foam material (e.g. polyurethane), which combines high insulation performances with light weight.
Window cross section (left) and zoom-in on the window frame (right). Temperature profile in a window frame and glazing cross section from ISO 10077-2:2012 (thermal performance of windows). Fluid and Solid Interactions Fluid/Solid Interface
The temperature field and the heat flux are continuous at the fluid/solid interface. However, the temperature field can rapidly vary in a fluid in motion: close to the solid, the fluid temperature is close to the solid temperature, and far from the interface, the fluid temperature is close to the inlet or ambient fluid temperature. The distance where the fluid temperature varies from the solid temperature to the fluid bulk temperature is called the
thermal boundary layer. The thermal boundary layer size and the momentum boundary layer relative size is reflected by the Prandtl number (Pr=C_p \mu/k): for the Prandtl number to equal 1, thermal and momentum boundary layer thicknesses need to be the same. A thicker momentum layer would result in a Prandtl number larger than 1. Conversely, a Prandtl number smaller than 1 would indicate that the momentum boundary layer is thinner than the thermal boundary layer. The Prandtl number for air at atmospheric pressure and at 20°C is 0.7. That is because for air, the momentum and thermal boundary layer have similar size, while the momentum boundary layer is slightly thinner than the thermal boundary layer. For water at 20°C, the Prandtl number is about 7. So, in water, the temperature changes close to a wall are sharper than the velocity change. Normalized temperature (red) and velocity (blue) profile for natural convection of air close to a cold solid wall. Natural Convection
The natural convection regime corresponds to configurations where the flow is driven by buoyancy effects. Depending on the expected thermal performance, the natural convection can be beneficial (e.g. cooling application) or negative (e.g. natural convection in insulation layer).
The Rayleigh number, noted as Ra, is used to characterized the flow regime induced by the natural convection and the resulting heat transfer. The Rayleigh number is defined from fluid material properties, a typical cavity size, L, and the temperature difference,\Delta T, usually set by the solids surrounding the fluid:
The Grashof number is another flow regime indicator giving the ratio of buoyant to viscous forces:
The Rayleigh number can be expressed in terms of the Prandtl and the Grashof numbers through the relation Ra=Pr Gr.
When the Rayleigh number is small (typically <10
3), the convection is negligible and most of the heat transfer occurs by conduction in the fluid.
For a larger Rayleigh number, heat transfer by convection has to be considered. When buoyancy forces are large compared to viscous forces, the regime is turbulent, otherwise it is laminar. The transition between these two regimes is indicated by the critical order of the Grashof number, which is 10
9. The thermal boundary layer, giving the typical distance for temperature transition between the solid wall and the fluid bulk, can be approximated by \delta_\mathrm{T} \approx \frac{L}{\sqrt[4\,]{Ra}} when Pr is of order 1 or greater. Temperature profile induced by natural convection in a glass of cold water in contact with a hot surface . Forced Convection
The forced convection regime corresponds to configurations where the flow is driven by external phenomena (e.g. wind) or devices (e.g. fans, pumps) that dominate buoyancy effects.
In this case the flow regime can be characterized, similarly to isothermal flow, using the Reynolds number as an indicator,Re= \frac{\rho U L}{\mu}. The Reynolds number represents the ratio of inertial to viscous forces. At low Reynolds numbers, viscous forces dominate and laminar flow is observed. At high Reynolds numbers, the damping in the system is very low, giving small disturbances. If the Reynolds number is high enough, the flow field eventually ends up in turbulent regime.
The momentum boundary layer thickness can be evaluated, using the Reynolds number, by \delta_\mathrm{M} \approx \frac{L}{\sqrt{Re}}.
Streamlines and temperature profile around a heat sink cooling by forced convection. Radiative Heat Transfer
Radiative heat transfer can be combined with conductive and convective heat transfer described above.
In a majority of applications, the fluid is transparent to heat radiation and the solid is opaque. As a consequence, the heat transfer by radiation can be represented as surface-to-surface radiation transferring energy between the solid wall through transparent cavities. The radiative heat flux emitted by a diffuse gray surface is equal to \varepsilon n^2 \sigma T^4. When a surface is surrounded by bodies at a homogeneous T_\mathrm{amb}, the net radiative flux is q_\mathrm{r} = \varepsilon n^2 \sigma (T_\mathrm{amb}^4-T^4). When surrounding surfaces of different temperatures, each surface-to-surface exchange is determined by the surface’s view factors.
Nevertheless, both fluids and solids may be transparent or semitransparent. So radiation can occur in fluid and solids. In participating (or semitransparent) media, the radiation rays interact with the medium (solid or fluid) then absorb, emit, and scatter radiation.
Whereas radiative heat transfer can be neglected in applications with small temperature differences and lower emissivity, it plays a major role in applications with large temperature differences and large emissivities.
Comparison of temperature profiles for a heat sink with a surface emissivity \varepsilon = 0 (left) and \varepsilon = 0.9 (right). Conclusion
Heat transfer in solids and heat transfer in fluids are combined in the majority of applications. This is because fluids flow around solids or between solid walls, and because solids are usually immersed in a fluid. An accurate description of heat transfer modes, material properties, flow regimes, and geometrical configurations enables the analysis of temperature fields and heat transfer. Such a description is also the starting point for a numerical simulation that can be used to predict conjugate heat transfer effects or to test different configurations in order, for example, to improve thermal performances of a given application.
Notations
C_{p}: heat capacity at constant pressure (SI unit: J/kg/K)
g: gravity acceleration (SI unit: m/s
2)
Gr: Grashof number (dimensionless number)
k: thermal conductivity (SI unit: W/m/K)
L: characteristic dimension (SI unit: m)
n: refractive index (dimensionless number)
p_\mathrm{A}: absolute pressure (SI unit: Pa)
Pr: Prandtl number (dimensionless number)
q: heat flux (SI unit: W/m
2)
Q: heat source (SI unit: W/m
3)
Ra: Rayleigh number (dimensionless number)
S: strain rate tensor (SI unit: 1/s)
T: temperature field (SI unit:K)
T_\mathrm{amb}: ambient temperature (SI unit: K)
\bold{u}: velocity field (SI unit: m/s)
U: typical velocity magnitude (SI unit: m/s)
\alpha_{p}: thermal expansion coefficient (SI unit: 1/K)
\delta_\mathrm{M}: momentum boundary layer thickness (SI unit: m)
\delta_\mathrm{T}: thermal layer thickness (SI unit: m)
\Delta T: characteristic temperature difference (SI unit: K)
\varepsilon: surface emissivity (dimensionless number)
\rho: density (SI unit: kg/m
3)
\sigma: Stefan-Boltzmann constant (SI unit: W/m
2T 4)
\tau: viscous stress tensor (SI unit: N/m
2) Comments (27) CATEGORIES Chemical COMSOL Now Electrical Fluid General Interfacing Mechanical Today in Science TAGS CATEGORIES Chemical COMSOL Now Electrical Fluid General Interfacing Mechanical Today in Science
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Here we want to give an easy mathematical bootstrap argument why solutions to the time independent 1D Schrödinger equation (TISE) tend to be rather nice. First formally rewrite the differential form$$-\frac{\hbar^2}{2m} \psi^{\prime\prime}(x) + V(x) \psi(x) ~=~ E \psi(x) \tag{1}$$into the int...
[Some time travel comments] Since in the previous paragraph, we have explained how travelling to the future will not necessary result in you to arrive in the future that is resulted as if you have never time travelled (via twin paradox), what is the reason that the past you travelled back, has to be the past you learnt from historical records :?
@0ßelö7 Well, I'd omit the explanation of the notation on the slide itself, and since there seems to be two pairs of formulae, I'd just put one of the two and then say that there's another one with suitable substitutions.
I mean, "Hey, I bet you've always wondered how to prove X - here it is" is interesting. "Hey, you know that statement everyone knows how to prove but doesn't bother to write down? Here is the proof written down" significantly less so
Sorry I have a quick question: For questions like this physics.stackexchange.com/questions/356260/… where the accepted answer clearly does not answer the original question what is the best thing to do; downvote, flag or just leave it?
So this question says express $u^0$ in terms of $u^j$ where $u$ is the four-velocity and I get what $u^0$ and $u^j$ are but I'm a bit confused how to go about this one? I thought maybe using the space-time interval and evaluating for $\frac{dt}{d\tau}$ but it's not workin out for me... :/ Anyone give me a quickie starter please? :p
Although a physics question, this is still important to chemistry. The delocalized electric field is related to the force (and therefore the repulsive potential) between two electrons. This in turn is what we need to solve the Schrödinger Equation to describe molecules. Short answer: You can calculate the expectation value of the corresponding operator, which comes close to the mentioned superposition. — Feodoran13 hours ago
If we take an electron that's delocalised w.r.t position, how can one evaluate the electric field over some space? Is it some superposition or a sort of field with all the charge at the expectation value of the position?
@0ßelö7 I just looked back at chat and noticed Phase's question, I wasn't purposefully ignoring you - do you want me to look over it? Because I don't think I'll gain much personally from reading the slides.
Maybe it's just me having not really done much with Eigenbases but I don't recognise where I "put it in terms of M's eigenbasis". I just wrote it down for some vector v, rather than a space that contains all of the vectors v
If we take an electron that's delocalised w.r.t position, how can one evaluate the electric field over some space? Is it some superposition or a sort of field with all the charge at the expectation value of the position?
Honey, I Shrunk the Kids is a 1989 American comic science fiction film. The directorial debut of Joe Johnston and produced by Walt Disney Pictures, it tells the story of an inventor who accidentally shrinks his and his neighbor's kids to a quarter of an inch with his electromagnetic shrinking machine and throws them out into the backyard with the trash, where they must venture into their backyard to return home while fending off insects and other obstacles.Rick Moranis stars as Wayne Szalinski, the inventor who accidentally shrinks his children, Amy (Amy O'Neill) and Nick (Robert Oliveri). Marcia...
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Recently two nice papers appeared on the arXiv, the most recent by Galley and Masanes, and the oldest by López Grande et al.. They are both – although a bit indirectly – about the age old question of the equivalence between proper and improper mixtures.
A proper mixture is when you prepare the states $\ket{0}$ and $\ket{1}$ with probability $p$ and $1-p$, obtaining the density matrix
\[ \rho_\text{proper} = p\ket{0}\bra{0} + (1-p)\ket{1}\bra{1}.\] An improper mixture is when you prepare the entangled state $\sqrt{p}\ket{0}\ket{0} + \sqrt{1-p}\ket{1}\ket{1}$ and discard the second subsystem, obtaining the density matrix \[ \rho_\text{improper} = p\ket{0}\bra{0} + (1-p)\ket{1}\bra{1}.\] The question is then why do these different preparation procedures give rise to the same statistics (and therefore it is legitimate to represent them with the same density matrix).
Well, do they? I’m not so sure about that! The procedure to prepare the proper mixture is rather vague, so we can’t really answer whether is it appropriate to represent it via the density matrix $\rho_\text{proper}$. To remove the vagueness, I asked an experimentalist how she prepared the state $\frac12(\ket{0}\bra{0}+\ket{1}\bra{1})$ that was necessary for an experiment. “Easy”, she told me, “I prepared $n$ copies of $\ket{0}$, $n$ copies of $\ket{1}$, and then combined the statistics.
This sounds like preparing the state $\ket{0}^{\otimes n} \otimes \ket{1}^{\otimes n}$, not like preparing $\frac12(\ket{0}\bra{0}+\ket{1}\bra{1})$. Do they give the same statistics? Well, if I measure all states in the $Z$ basis, exactly $\frac12$ of the results will be $0$. But if I measure $\frac12(\ket{0}\bra{0}+\ket{1}\bra{1})$ in the $Z$ basis $2n$ times, the probability that $\frac12$ of the results are $0$ is
\[ \frac{1}{2^{2n}} {2n \choose n} \approx \frac{1}{\sqrt{n\pi}},\] so just by looking at this statistic I can guess with high probability which was the preparation. It is even easier to do that if I disregard her instructions and look at the order of the results: getting $n$ zeroes followed by $n$ ones is a dead giveaway.
Maybe one should prepare these states using a random number generator instead? If one uses the function
rand() from MATLAB to decide whether to prepare $\ket{0}$ or $\ket{1}$ at each round one can easily pass the two randomness tests I mentioned above. Maybe it can even pass all common randomness tests available in the literature, I don’t know how good
rand() is. But it cannot, however pass
all randomness tests, as
rand() is a deterministic algorithm using a finite seed, and is therefore restricted to outputting computable sequences of bits. One can, in fact, attack it, and this is the core of the paper of López Grande et al., showing how one can distinguish a sequence of bits that came from
rand() from a truly random one. More generally, even the best pseudorandom number generators we have are designed to be indistinguishable from truly random sources only by polynomial-time tests, and fail against exponential-time algorithms.
Clearly pseudorandomness is not enough to generate proper mixtures; how about true randomness instead? Just use a quantum random number generator to prepare bits with probabilities $p$ and $1-p$, and use these bits to prepare $\ket{0}$ or $\ket{1}$. Indeed, this is what people do when they are serious about preparing mixed states, and the statistics really are indistinguishable from those of improper mixtures. But why? To answer that, we need to model the quantum random number generator physically. We start by preparing a “quantum coin” in the state
\[ \sqrt{p}\ket{H}+\sqrt{1-p}\ket{T},\] which we should measure in the $\{\ket{H},\ket{V}\}$ basis to generate the random bits. Going to the Church of the Larger Hilbert Space, we model the measurement as \[ \sqrt{p}\ket{H}\ket{M_H}+\sqrt{1-p}\ket{T}\ket{M_T},\] and conditioned on the measurement we prepare $\ket{0}$ or $\ket{1}$, obtaining the state \[ \sqrt{p}\ket{H}\ket{M_H}\ket{0}+\sqrt{1-p}\ket{T}\ket{M_T}\ket{1}.\] We then discard the quantum coin and the measurement result, obtaining finally \[ p\ket{0}\bra{0} + (1-p)\ket{1}\bra{1},\] which is just the desired state, but now it is an improper mixture. So, at least in the Many-Worlds interpretation, there is no mystery about why proper and improper mixtures are equivalent: they are physically the same thing!
(A closely related question, which has a closely related answer, is why is it equivalent to prepare the states $\ket{0}$ or $\ket{1}$ with probability $\frac12$ each, or the states $\ket{+}$ or $\ket{-}$, again with probability $\frac12$? The equivalence fails for pseudorandomness, as shown by López Grande et al.; if we use true randomness instead, we are preparing the states
\[ \frac1{\sqrt{2}}(\ket{H}\ket{0}+\ket{T}\ket{1})\quad\text{or}\quad\frac1{\sqrt{2}}(\ket{H}\ket{+}+\ket{T}\ket{-})\] and discarding the coin. But note that if one applies a Hadamard to the coin of the first state one obtains the second, so the difference between then is just a unitary on a system that is discarded anyway; no wonder we can’t tell the difference! More generally, any two purifications of the same density matrix must be related by a unitary on the purifying system.)
Galley and Masanes want to invert the question, and ask for
which quantum-like theories proper and improper mixtures are equivalent. To be able to tackle this question, we need to define what improper mixtures even are in a quantum-like theory. They proceed by analogy with quantum mechanics: if one has a bipartite state $\ket{\psi}$, and are doing measurements $E_i$ only on the first system, the probabilities one obtains are given by \[ p(i) = \operatorname{tr}( (E_i \otimes \mathbb I) \ket{\psi}\bra{\psi} ),\] and the improper mixture is defined as the operator $\rho_\text{improper}$ for which \[ p(i) = \operatorname{tr}( E_i \rho_\text{improper})\] for all measurements $E_i$.
In their case, they are considering a quantum-like theory that is still based on quantum states, but whose probabilities are not given by the Born rule $p(i) = \operatorname{tr}(E_i \ket{\phi}\bra{\phi})$, but by some more general function $p(i) = F_i (\ket{\phi})$. One can then define the probabilities obtained by local measurements on a bipartite state as
\[ p(i) = F_i \star \mathbb I (\ket{\psi}),\] for some composition rule $\star$ and trivial measurement $\mathbb I$, and from that an improper mixture as the operator $\omega_\text{improper}$ such that \[ p(i) = F_i (\omega_\text{improper})\] for all measurements $F_i$.
Defining proper mixtures, on the other hand, is easy: if one can prepare the states $\ket{0}$ or $\ket{1}$ with probabilities $p$ and $1-p$, their proper mixture is the operator $\omega_\text{proper}$ such that for all measurements $F_i$
\[ p(i) = F_i(\omega_\text{proper}) = p F_i(\ket{0}) + (1-p) F_i(\ket{1}).\] That is, easy if one can generate true randomness that is not reducible to quantum-like randomness. I don’t think this makes sense, as one would have to consider a world where reductionism fails, or at least one where quantum-like mechanics is not the fundamental theory. Such non-reducible probabilities are uncritically assumed to exist anyway by people working on GPTs all the time1.
Now with both proper and improper mixtures properly defined, one can answer the question of whether they are equivalent: the answer is a surprising no, for any alternative probability rule that respects some basic consistency conditions. This has the intriguing consequence that if we were to modify the Born rule while keeping the rest of quantum mechanics intact, a wedge would be driven between the probabilities that come from the fundamental theory and some “external” probabilities coming from elsewhere. This would put the Many-Worlds interpretation under intolerable strain.
But such an abstract “no” result is not very interesting; I find it much more satisfactory to exhibit a concrete alternative to the Born rule where the equivalence fails. Galley and Masanes propose the function
\[ F_i(\ket{\psi}) = \operatorname{tr}(\hat F_i (\ket{\psi}\bra{\psi})^{\otimes 2})\] for some positive matrices $\hat F_i$ restricted by their consistency conditions. It is easy to see that the proper mixture of $\ket{0}$ and $\ket{1}$ described above is given by2 \[ \omega_\text{proper} = p \ket{00}\bra{00} + (1-p)\ket{11}\bra{11}.\] In quantum mechanics one would try to make it by discarding half of the state $\sqrt{p}\ket{0}\ket{0} + \sqrt{1-p}\ket{1}\ket{1}$. Here it doesn’t work, as nothing does, but I want to know what it gives us anyway. It is not easy to see that the improper mixture is given by the weirdo \begin{multline} \omega_\text{improper} = (p^2 + \frac{p(1-p)}{3})\ket{00}\bra{00} + \\ \frac{2p(1-p)}{3} (\ket{01}+\ket{10})(\bra{01}+\bra{10}) + ((1-p)^2 + \frac{p(1-p)}{3})\ket{11}\bra{11}.\end{multline}
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After developing and optimizing a method, the next step is to determine how well it works in the hands of a single analyst. Three steps make up this process: determining single-operator characteristics, completing a blind analysis of standards, and determining the method’s ruggedness. If another standard method is available, then we can analyze the same sample using both the standard method and the new method, and compare the results. If the result for any single test is unacceptable, then the method is not a suitable standard method.
14.2.1 Single Operator Characteristics
The first step in verifying a method is to determine the precision, accuracy, and detection limit when a single analyst uses the method to analyze a standard sample. The detection limit is determined by analyzing an appropriate reagent blank. Precision is determined by analyzing replicate portions of the sample, preferably more than ten. Accuracy is evaluated using a
t-test comparing the experimental results to the known amount of analyte in the standard. Precision and accuracy are evaluated for several different concentrations of analyte, including at least one concentration near the detection limit, and for each different sample matrix. Including different concentrations of analyte helps identify constant sources of determinate error and establishes the range of concentrations for which the method is applicable.
Note
See Chapter 4.8 for a discussion of detection limits. Pay particular attention to the difference between a detection limit, a limit of identification, and a limit of quantitation.
See Section 4.6.1 for a review of the
t-test.
See Chapter 4.2 for a review of constant determinate errors. Figure 4.3 illustrates how we can detect a constant determinate error by analyzing samples containing different amounts of analyte.
14.2.2 Blind Analysis of Standard Samples
Single-operator characteristics are determined by analyzing a standard sample that has a concentration of analyte known to the analyst. The second step in verifying a method is a
blind analysis of standard samples. Although the concentration of analyte in the standard is known to a supervisor, the information is withheld from the analyst. After analyzing the standard sample several times, the analyte’s average concentration is reported to the test’s supervisor. To be accepted, the experimental mean should be within three standard deviations—as determined from the single-operator characteristics—of the analyte’s known concentration.
Note
An even more stringent requirement is to require that the experimental mean be within two standard deviations of the analyte’s known concentration.
14.2.3 Ruggedness Testing
An optimized method may produce excellent results in the laboratory that develops a method, but poor results in other laboratories. This is not particularly surprising because a method typically is optimized by a single analyst using the same reagents, equipment, and instrumentation for each trial. Any variability introduced by the analysts, the reagents, the equipment, and the instrumentation is not included in the single-operator characteristics. Other less obvious factors may affect an analysis, including environmental factors, such as the temperature or relative humidity in the laboratory. If the procedure does not require their control, then they may contribute to variability. Finally, the analyst optimizing usually takes particular care to perform the analysis in exactly the same way during every trial, which may minimize the run to run variability.
An important step in developing a standard method is to determine which factors have a pronounced effect on the quality of the results. Once we identify these factors, we can write into the procedure instructions that specify how these factors must be controlled. A procedure that, when carefully followed, produces results of high quality in different laboratories is considered rugged. The method by which the critical factors are discovered is called
ruggedness testing. 6
Note
For example, if temperature is a concern, we might specify that it be held at 25 ± 2
oC.
Ruggedness testing usually is performed by the laboratory developing the standard method. After identifying potential factors, their effects are evaluated by performing the analysis at two levels for each factor. Normally one level is that specified in the procedure, and the other is a level likely to be encountered when the procedure is used by other laboratories.
This approach to ruggedness testing can be time consuming. If there are seven potential factors, for example, a 2
7 factorial design can evaluate each factor’s first-order effect. Unfortunately, this requires a total of 128 trials—too many trials to be a practical solution. A simpler experimental design is shown in Table 14.5, in which the two factor levels are identified by upper case and lower case letters. This design, which is similar to a 2 3 factorial design, is called a fractional factorial design. Because it includes only eight runs, the design provides information about only the eight first-order factor effects. It does not provide sufficient information to evaluate higher-order effects or interactions between factors, both of which are probably less important than the first-order effects.
Note
Why does this model estimate the seven first-order factor effects and not seven of the 20 possible first-order interactions? With eight experiments, we can only choose to calculate seven parameters (plus the average response). The calculation of
E D, for example, also gives the value for E AB. You can convince yourself of this by replacing each upper case letter with a +1 and each lower case letter with a –1 and noting that A × B = D. We choose to report the first-order factor effects because they are likely to be more important than interactions between factors.
factors
run
A
B
C
D
E
F
G
response
1
2
3
4
5
6
7
8
The experimental design in Table 14.5 is balanced in that each of a factor’s two levels is paired an equal number of times with the upper case and lower case levels for every other factor. To determine the effect,
E f, of changing a factor’s level, we subtract the average response when the factor is at its upper case level from the average value when it is at its lower case level.
\[E_\ce{f}=\dfrac{(\sum R_i)_{\textrm{upper case}}}{4}-\dfrac{(\sum R_i)_{\textrm{lower case}}}{4}\tag{14.16}\]
Because the design is balanced, the levels for the remaining factors appear an equal number of times in both summation terms, canceling their effect on
E f. For example, to determine the effect of factor A, E A, we subtract the average response for runs 5–8 from the average response for runs 1–4. Factor B does not affect E A because its upper case levels in runs 1 and 2 are canceled by the upper case levels in runs 5 and 6, and its lower case levels in runs 3 and 4 are canceled by the lower case levels in runs 7 and 8. After calculating each of the factor effects we rank them from largest to smallest without regard to sign, identifying those factors whose effects are substantially larger than the other factors.
Note
To see that this is design is balanced, look closely at the last four runs. Factor A is present at its level
a for all four of these runs. For each of the remaining factors, two levels are upper case and two levels are lower case. Runs 5–8 provide information about the effect of a on the response, but do not provide information about the effect of any other factor. Runs 1, 2, 5, and 6 provide information about the effect of B, but not of the remaining factors. Try a few other examples to convince yourself that this relationship is general.
We also can use this experimental design to estimate the method’s expected standard deviation due to the effects of small changes in uncontrolled or poorly controlled factors.
7
\[s=\sqrt {\frac{2}{7}\sum E_\ce{f}^2}\tag{14.17}\]
If this standard deviation is unacceptably large, then the procedure is modified to bring under greater control those factors having the greatest effect on the response.
Example 14.5
The concentration of trace metals in sediment samples collected from rivers and lakes can be determined by extracting with acid and analyzing the extract by atomic absorption spectrophotometry. One procedure calls for an overnight extraction using dilute HCl or HNO
3. The samples are placed in plastic bottles with 25 mL of acid and placed on a shaker operated at a moderate speed and at ambient temperature. To determine the method’s ruggedness, the effect of the following factors was studied using the experimental design in Table 14.5.
Factor A: extraction time
Factor B: shaking speed
Factor C: acid type
Factor D: acid concentration
Factor E: volume of acid
Factor F: type of container
Factor G: temperature
Eight replicates of a standard sample containing a known amount of analyte were carried through the procedure. The analyte’s recovery in the samples, given as a percentage, are shown here.
Determine which factors appear to have a significant effect on the response and estimate the method’s expected standard deviation.
Solution
To calculate the effect of changing each factor’s level we use equation 14.16 and substitute in appropriate values. For example,
E A is
\[E_\ce{A} = \dfrac{98.9 + 99.0 + 97.5 + 97.7}{4} - \dfrac{97.4 + 97.3 + 98.6 + 98.6}{4} = 0.30\]
and
E G is
\[E_\ce{A} = \dfrac{98.9 + 97.7 + 97.3 + 98.6}{4} - \dfrac{99.0 + 97.5 + 97.4 + 98.6}{4} = 0.00\]
Completing the remaining calculations and ordering the factors by the absolute values of their effects
Factor D
1.30
Factor A
0.35
Factor E
-0.10
Factor B
0.05
Factor C
-0.05
Factor F
0.05
Factor G
0.00
shows us that the concentration of acid (Factor D) has a substantial effect on the response, with a concentration of 0.05 M providing a much lower percent recovery. The extraction time (Factor A) also appears significant, but its effect is not as important as the acid’s concentration. All other factors appear insignificant. The method’s estimated standard deviation, from equation 14.27, is
\[s = \sqrt{\dfrac{2}{7}\left\{(1.30)^2 + (0.35)^2 + (-0.10)^2 + (0.05)^2 + (-0.05)^2 + (0.05)^2 + (0.00)^2\right\}} = 0.72\]
which, for an average recovery of 98.1% gives a relative standard deviation of approximately 0.7%. If we control the acid’s concentration so that its effect approaches that for factors B, C, and F, then the relative standard deviation becomes 0.18, or approximately 0.2%.
14.2.4 Equivalency Testing
If an approved standard method is available, then the new method should be evaluated by comparing results to those obtained with the standard method. Normally this comparison is made at a minimum of three concentrations of analyte to evaluate the new method over a wide dynamic range. Alternatively, we can plot the results using the new method against the results using the approved standard method. A slope of 1.00 and a
y-intercept of 0.0 provides evidence that the two methods are equivalent.
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I'm new to Mathematica and I'm trying to integrate this function:
K = Function[{x,theta}, ((b - x^3 (d/(Cos[theta])^2 - b/x)^3) (e - 2 b/x)^3/(((b - x^3 (e - 2 b/x)^3) (d/(Cos[theta])^2 - b/x)^3))) Sin[theta]]
from $\theta = 0$ to $\theta = \pi$ and from $x = c$ to $x = \frac{f \cos^2(\theta)}{a + \cos^2(\theta)}$ by executing this command:
Integrate[K, {theta, 0, Pi}, {x, c, f((Cos[theta])^2)/(a + (Cos[theta])^2)}].
However, when I do this, the output is $\pi \left(-\left(\left(\sqrt{\frac{a}{a+1}}-1\right) f+c\right)\right) Function\left(\{x,\text{theta}\},\frac{\sin (\text{theta}) \left(\left(e-\frac{2 b}{x}\right)^3 \left(b-x^3 \left(\frac{d}{\cos ^2(\text{theta})}-\frac{b}{x}\right)^3\right)\right)}{\left(b-x^3 \left(e-\frac{2 b}{x}\right)^3\right) \left(\frac{d}{\cos ^2(\text{theta})}-\frac{b}{x}\right)^3}\right)$
What is the meaning of the
Function part of this result?
The result seems like it's a product of the constants and the function itself, but this doesn't make sense because I clearly specified integration limits.
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Hello, TriKri!
I too got a fourth-degree equation, too.
But there is a way around it . . .
[quote]A 10-m ladder is leaned against a box (1 x 1 x 1 m) placed against a wall.
But the ground is so slippery that the ladder falls over the box and leans against the wall.
How high up on the wall does the ladder reach?
Code:
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*
* |y
10 * - - - *
* | 1 |
* 1| |1
* | |
* - - - - - - - * - - - *
x 1
From the large right triangle: .$\displaystyle (y+1)^2 + (1+x)^2 \:=\:10^2$ .[1]
The two smaller right triangles are similar. . . So we have: .$\displaystyle \frac{1}{x} \:=\:\frac{y}{1} \quad\Rightarrow\quad x \:=\:\frac{1}{y}$ .[2]
Substitute [2] into [1]: .$\displaystyle (y+1)^2 + \left(1 + \frac{1}{y}\right)^2 \:=\:100$
and we have: .$\displaystyle y^2 + 2y + 1 + 1 + \frac{2}{y} + \frac{1}{y^2} \;=\;100$ . . $\displaystyle \left(y^2 + 2 + \frac{1}{y^2}\right) + \left(2y + \frac{2}{y}\right) - 100 \:=\:0 $ . . $\displaystyle \left(y + \frac{1}{y}\right)^2 + 2\left(y + \frac{1}{y}\right) - 100 \:=\:0$
Let $\displaystyle u \:=\:y+\frac{1}{y}$ . . and we have the quadratic: .$\displaystyle u^2 + 2u - 100 \:=\:100$
Use the Quadratic Formula to solve for $\displaystyle u$, . . then back-substitute and solve for $\displaystyle y.$
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A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than be $v(t) = 3t^2-12t+9$ how could I find the intervals
Fix $c\in\{0,1,\dots\}$, let $K\geq c$ be an integer, and define $z_K=K^{-\alpha}$ for some $\alpha\in(0,2)$.I believe I have numerically discovered that$$\sum_{n=0}^{K-c}\binom{K}{n}\binom{K}{n+c}z_K^{n+c/2} \sim \sum_{n=0}^K \binom{K}{n}^2 z_K^n \quad \text{ as } K\to\infty$$but cannot ...
So, the whole discussion is about some polynomial $p(A)$, for $A$ an $n\times n$ matrix with entries in $\mathbf{C}$, and eigenvalues $\lambda_1,\ldots, \lambda_k$.
Anyways, part (a) is talking about proving that $p(\lambda_1),\ldots, p(\lambda_k)$ are eigenvalues of $p(A)$. That's basically routine computation. No problem there. The next bit is to compute the dimension of the eigenspaces $E(p(A), p(\lambda_i))$.
Seems like this bit follows from the same argument. An eigenvector for $A$ is an eigenvector for $p(A)$, so the rest seems to follow.
Finally, the last part is to find the characteristic polynomial of $p(A)$. I guess this means in terms of the characteristic polynomial of $A$.
Well, we do know what the eigenvalues are...
The so-called Spectral Mapping Theorem tells us that the eigenvalues of $p(A)$ are exactly the $p(\lambda_i)$.
Usually, by the time you start talking about complex numbers you consider the real numbers as a subset of them, since a and b are real in a + bi. But you could define it that way and call it a "standard form" like ax + by = c for linear equations :-) @Riker
"a + bi where a and b are integers" Complex numbers a + bi where a and b are integers are called Gaussian integers.
I was wondering If it is easier to factor in a non-ufd then it is to factor in a ufd.I can come up with arguments for that , but I also have arguments in the opposite direction.For instance : It should be easier to factor When there are more possibilities ( multiple factorizations in a non-ufd...
Does anyone know if $T: V \to R^n$ is an inner product space isomorphism if $T(v) = (v)_S$, where $S$ is a basis for $V$? My book isn't saying so explicitly, but there was a theorem saying that an inner product isomorphism exists, and another theorem kind of suggesting that it should work.
@TobiasKildetoft Sorry, I meant that they should be equal (accidently sent this before writing my answer. Writing it now)
Isn't there this theorem saying that if $v,w \in V$ ($V$ being an inner product space), then $||v|| = ||(v)_S||$? (where the left norm is defined as the norm in $V$ and the right norm is the euclidean norm) I thought that this would somehow result from isomorphism
@AlessandroCodenotti Actually, such a $f$ in fact needs to be surjective. Take any $y \in Y$; the maximal ideal of $k[Y]$ corresponding to that is $(Y_1 - y_1, \cdots, Y_n - y_n)$. The ideal corresponding to the subvariety $f^{-1}(y) \subset X$ in $k[X]$ is then nothing but $(f^* Y_1 - y_1, \cdots, f^* Y_n - y_n)$. If this is empty, weak Nullstellensatz kicks in to say that there are $g_1, \cdots, g_n \in k[X]$ such that $\sum_i (f^* Y_i - y_i)g_i = 1$.
Well, better to say that $(f^* Y_1 - y_1, \cdots, f^* Y_n - y_n)$ is the trivial ideal I guess. Hmm, I'm stuck again
O(n) acts transitively on S^(n-1) with stabilizer at a point O(n-1)
For any transitive G action on a set X with stabilizer H, G/H $\cong$ X set theoretically. In this case, as the action is a smooth action by a Lie group, you can prove this set-theoretic bijection gives a diffeomorphism
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Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension
1.
Department of Mathematics, Wayne State University, Detroit, MI 48202, United States
2.
Department of Mathematics, Information School, Renmin University of China, Beijing 100872, China
$\lambda_p(\Omega)=$i n f$_u\in H_0^1(\Omega),$u≠0(for some u)$\|\|\nabla u\|\|_2^2/\|\|u\|\|_p^2,$
where $|\|\cdot\||_p$ denotes $L^p$ norm. We derive in this paper a sharp form of the following improved Moser-Trudinger inequality involving the $L^p$-norm using the method of blow-up analysis:
$s u p_{u\in H_0^1(\Omega),\|\|\nabla u\|\|_2=1}\int_{\Omega} e^{4\pi (1+\alpha\|\|u\|\|_p^2)u^2}dx<+\infty$
for $0\leq \alpha <\lambda_p(\Omega)$, and the supremum is infinity for all $\alpha\geq \lambda_p(\Omega)$. We also prove the existence of the extremal functions for this inequality when $\alpha$ is sufficiently small.
Keywords:blow-up analysis, Sharp constant, Euler-Lagrange equation, extremal function, Moser-Trudinger inequality, concentration-compactness.. Mathematics Subject Classification:Primary: 35J20; Secondary: 35J6. Citation:Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963
[1]
Changliang Zhou, Chunqin Zhou.
Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian.
[2]
Prosenjit Roy.
On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions.
[3] [4] [5]
Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli.
On the validity of the Euler-Lagrange system.
[6]
Yutian Lei, Zhongxue Lü.
Axisymmetry of locally bounded solutions to an Euler-Lagrange
system of the weighted Hardy-Littlewood-Sobolev inequality.
[7] [8]
Jian Zhang, Shihui Zhu, Xiaoguang Li.
Rate of $L^2$-concentration of the blow-up solution
for critical nonlinear Schrödinger equation with potential.
[9]
Agnieszka B. Malinowska, Delfim F. M. Torres.
Euler-Lagrange equations
for composition functionals in calculus of variations on time
scales.
[10] [11] [12] [13]
Hiroyuki Takamura, Hiroshi Uesaka, Kyouhei Wakasa.
Sharp blow-up for semilinear wave equations with non-compactly supported data.
[14]
Jong-Shenq Guo.
Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition.
[15]
Helin Guo, Yimin Zhang, Huansong Zhou.
Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential.
[16] [17] [18]
Yohei Fujishima.
Blow-up set for a superlinear heat equation and pointedness of the initial data.
[19] [20]
István Győri, Yukihiko Nakata, Gergely Röst.
Unbounded and blow-up solutions for a delay logistic equation with positive feedback.
2018 Impact Factor: 1.143
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Yes, you read that right.
I've recently been embroiled in a lovely debate on Numberphile's video, "Infinity is bigger than you think", in which Dr. James Grime starts off: "We're going to break a rule. We're breaking one of the rules of Numberphile. We're talking about something that isn't a number. We're going to talk about infinity."
I, too, was a longtime believer of what high school students all over are told: "Infinity is not a number; infinity is a concept." As my studies of mathematics progressed, however, I began to see that perhaps the things I had always taken for granted were not as black-and-white as they had seemed. There was a lot more nuance to mathematics than I had ever realized, and learning those nuances opened up an entire new level of understanding, unlocking all sorts of links between concepts that had previously seemed worlds apart. So it's no wonder that "Infinity Is Not A Number" (which I will occasionally abbreviate as "IINAN") was one of the first claims to which I took a fine-tooth comb. What I learned changed my stance on infinity and firmly cemented it as my favorite number - not just concept, but honest-to-god
number.*
The most common argument made IINAN proponents involves the curious property that \(\infty +1=\infty\). This, they say, leads to all sorts of contradictions, because all one has to do is simply subtract \(\infty\) from both sides:
\[
\infty+1=\infty\\
\underline{-\infty\ \ \ \ \ \ \ \ -\infty}\ \ \\
\ \ \ \ \ \ 1=0\]Oh no! We know that the statement \(1=0\) is obviously false, so there must be a false assumption somewhere. Many IINAN defenders claim that the false assumption was that we tried to treat \(\infty\) as a number. But that's not actually where the problem with infinity lies.
The problem is that we tried to do algebra with it.
For mathematicians, the most convenient place to do algebra is in a structure called a
field. If you're already familiar with what a field is, great, but if not, you can think of a field as a number system in which the age-old operations of addition, subtraction, multiplication, and division — the four operations that my father often notes are the only ones he ever needs when I talk about the kinds of math I teach — work exactly as we'd like them to. The fields with which we are most familiar are the rational numbers (\(\mathbb{Q}\)), the badly-named so-called "real" numbers (\(\mathbb{R}\)), and often the complex numbers (\(\mathbb{C}\)). One basic thing about a field is that the subtraction property of equalityholds: For any numbers \(a\), \(b\), and \(c\) in our field, if \(a=b\), then \(a-c=b-c\).
What about \(\infty\) though? When we attempted to use \(\infty\) in an algebra problem, we got back complete garbage. And we know that the subtraction property of equality should hold for any numbers in a field. What this means, then, is that \(\infty\)
is not part of that field(or any field as far as I'm aware). So, when someone says "Infinity is not a number", what they reallymean is "Infinity is not a real number." (It's not a complex number, either, for that matter.) It doesn't follow the same rules that the real numbers do.
But that doesn't mean it's not a number at all.
We've seen this sort of thing happen before. The Greek mathematician Diophantus, when faced the equation \(4x+20=0\), called its solution of \(-5\) "absurd" — yet now students learn about negative numbers as early as elementary school, and we barely blink an eye at their use in everyday life. Square roots of negative numbers seemed equally preposterous to the Italian mathematician Gerolamo Cardano, and the French mathematician René Descartes called them "imaginary", a term that we're unfortunately stuck with today. But imaginary numbers — and the complex numbers we build from them — are a vital part of physics, from alternating currents to quantum mechanics.
So what makes infinity any different from \(i\)?
Sure, it seems bizarre that a number plus one could equal itself. But it's equally bizarre that the square of a number could be negative. And sure, we can get a contradiction if we do certain things to infinity. But that happens with \(i\) as well! If we attempt to use the identity \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\), we can arrive at a similar contradiction:
\[\sqrt{-1}\cdot\sqrt{-1}=\sqrt{-1\cdot-1}\\
\ \ \ i\cdot i=\sqrt{1}\\
-1=1\ \]When this equation fails, you don't see mathematicians clamoring that "\(i\) isn't a number"! Instead, the response is that the original equation doesn't work like we thought it did when we extend our real number system to include the complex numbers — instead, the square root function takes on a new life as a multi-valued function. There's that nuance again! For the same reason, infinity makes us look closer at something as simple as subtraction, at which point we find that \(\infty-\infty\) is an
indeterminate form, something that we need the tools of calculus to properly deal with.
The truth is, mathematicians have been treating \(\infty\) as a number** for quite some time now.
In
real analysis, which was developed to give the techniques of calculus a rigorous footing, points labelled \(+\infty\) and \(-\infty\) can be added to either end of the real number line to give what we call the extended real number line, often denoted \(\overline{\mathbb{R}}\) or \(\left[-\infty,+\infty\right]\). The extended real number line is useful in describing concepts in measure theory and integration, and it has algebraic rules of its own, though analysts are still careful to mention that these two extra points are not realnumbers. What's more, the extended real line is not a field, because it doesn't satisfy all the nice properties that a field does. (But that just makes us appreciate working in a field that much more!) Projective geometrygives us a different sort of infinity, what I like to call an "unsigned infinity", one that is obtained by letting \(-\infty\) and \(+\infty\) overlap and creating what is known as the real projective line. And complex analysis, which extends calculus to the complex plane, takes it even further, letting allthe different infinities in all directions overlap to create a sort of "complex infinity", sometimes written \(\tilde{\infty}\), sitting atop the Riemann sphere. What I particularly like about these projective infinities is that, using them, you can actually divide by zero!***
So, since there are actually a number of different kinds of infinity that can be referred to, I would say that, more specifically,
complex infinity is my favorite number.
The tough thing about this situation is that the concept of "number" is a very difficult one to precisely and universally define — similar to how linguists still struggle to come up with a universal definition of "word". By trying to come up with such a description, you end up either including things that you don't want to be numbers (such as matrices) or excluding things that you do want to be numbers (such as complex numbers). The best we can really do is keep an open mind about what a "number" is.
After all, there's infinitely many of them already — so there's bound to be new ones we haven't seen yet sooner or later.
∎
*
I'm not saying that infinity isn't a concept. When it really comes down to it, every number is a concept. That's the beauty of having abstracted the number "two" as an adjective, as in "two sheep", to "two" as a noun. ** There's an argument to be made that treatingsomething like a number doesn't mean it isa number. But at some point, the semantic distinction between these two becomes somewhat blurred. *** Don't worry, I'll make a post about how to legitimately divide by zero in the near future!
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We are trying to build a discrete model for each SLE (Schramm-Loewner evolution) and one key step is solving the following question:
Q: Finding a two-dimensional $\mathbb{H}$-conformally invariant (details below) process $X=(X_{1,\beta},X_{2,\beta})$ in the upper half-plane such that $$c\int^{arg(z)}_{0}\sin(\theta)^{\beta}d\theta= P_{z}[(X_{1,\beta}(T_{\mathbb{H}}),X_{2,\beta}(T_{\mathbb{H}}) )\in \mathbb{R}^{-}],$$ for constant $c=1/(\int^{\pi}_{0}\sin(\theta)^{\beta}d\theta)$ and arbitrary $\beta\in [0,2]$ and $T_{\mathbb{H}}$ the exit time from the upper half-plane.
By $\mathbb{H}$-conformally invariant we mean that if for $D\subset \mathbb{H}$ the $f:D\to \mathbb{H}$ is a conformal map and $X\in D$ then $f(X)$ has the law of X but with possibly different variance/time change.
To get a flavor for it set $\beta=0$, then we simply set X=2d Brownian motion to obtain $P_{z}[X_{T_{\mathbb{H}}}\in \mathbb{R}^{-}]=\frac{1}{\pi}arg(z)$. Then this probability uniquely characterizes SLE(4).
Attempts
1)Finding a generator:
The $f(s):=a\int^{s}_{0}\sin(\theta)^{\beta}d\theta$ satisfies the ode
$$-\beta f'cot(s)+f''=0,$$
with boundary $f(0)=0,f(\pi)=1$, where $a:=1/(\int^{\pi}_{0}\sin(\theta)^{\beta}d\theta)$. So one idea is to turn the ode into a pde: $\Delta f(x,y)=\frac{\beta}{y}f_{y}(x,y)$ with boundary $1_{\mathbb{R}^{-}}$.
Q2: Next we want to apply Feynman Kac to this pde but the boundary data is not continuous, so the rest is just speculation.
The diffusion we get is $$dX_{1}=dB_{1}, dX_{2}=\frac{\beta}{X_{2}}dt+dB_{2},$$
which interestingly has shown up in the literature under the keyword Bessel-Brownian diffusions. As we can see from the pde it is not conformally invariant. However, it would still be interesting if:
Q3: $$c\int^{arg(z)}_{0}\sin(\theta)^{\beta}d\theta= P_{z}[(X_{1,T_{\mathbb{H}}},X_{2,T_{\mathbb{H}} })\in \mathbb{R}^{-}],$$
2)Finding other conformally invariant processes
Q4: Is conformal invariance unique to Brownian motion?
In "Conformal mapping of some non-harmonic functions in transport theory" Bazant identified other pdes that are also conformally invariant:
$$a(u) \nabla^{2}u+b(u) |\nabla u|^{2} u=0,$$ for possibly nonlinear functions a,b.
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Title A cyclic integral on k-Minkowski noncommutative space-time Publication Type Journal Article Year of Publication 2006 Authors Agostini, A, Amelino-Camelia, G, Arzano, M, D'Andrea, F Journal Int. J. Mod. Phys. A 21 (2006) 3133-3150 Abstract
We examine some alternative possibilities for an action functional for $\\\\kappa$-Minkowski noncommutative spacetime, with an approach which should be applicable to other spacetimes with coordinate-dependent commutators of the spacetime coordinates ($[x_\\\\mu,x_\\\\nu]=f_{\\\\mu,\\\\nu}(x)$). Early works on $\\\\kappa$-Minkowski focused on $\\\\kappa$-Poincar\\\\\\\'e covariance and the dependence of the action functional on the choice of Weyl map, renouncing to invariance under cyclic permutations of the factors composing the argument of the action functional. A recent paper (hep-th/0307149), by Dimitrijevic, Jonke, Moller, Tsouchnika, Wess and Wohlgenannt, focused on a specific choice of Weyl map and, setting aside the issue of $\\\\kappa$-Poincar\\\\\\\'e covariance of the action functional, introduced in implicit form a cyclicity-inducing measure. We provide an explicit formula for (and derivation of) a choice of measure which indeed ensures cyclicity of the action functional, and we show that the same choice of measure is applicable to all the most used choices of Weyl map. We find that this ``cyclicity-inducing measure\\\'\\\' is not covariant under $\\\\kappa$-Poincar\\\\\\\'e transformations. We also notice that the cyclicity-inducing measure can be straightforwardly derived using a map which connects the $\\\\kappa$-Minkowski spacetime coordinates and the spacetime coordinates of a ``canonical\\\'\\\' noncommutative spacetime, with coordinate-independent commutators.
URL http://hdl.handle.net/1963/2158 DOI 10.1142/S0217751X06031077 Alternate Journal Action functional for kappa-Minkowski Noncommutative Spacetime A cyclic integral on k-Minkowski noncommutative space-time
Research Group:
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Finite-Sample Expressivity of Neural Networks
In the present blog post I would like to present a slighlty altered version of the proof of a theorem in [1]. The statement is about the
finite-sample expressivity of neural networks. You can basically take the approach and present it in a arguably more direct and compact fashion, and explicitly writing down the desired network. But let us first take a look at said theorem. (see [1, Theorem 1], Section 4 on page 8). There exists a two-layer neural network with ReLU activations and $2n+d$ weights that can represent any function on a sample of size $n$ in $d$ dimensions. Theorem
That means if we choose $n$ mutually distinct samples $z_1, \ldots, z_n \in \mathbb{R}^d$ and real valued labels $y_1, \ldots, y_n \in \mathbb{R}$ there is a $2$-layer neural network $C$ depending only on $2n+d$ weight parameters such that for $i=1, \ldots , n$ we have $$ C(z_i) = y_i. $$ Moreover the network is of the form $\mathbb{R}^d \stackrel{F}{\to} \mathbb{R}^n \stackrel{w}{\to} \mathbb{R} $ with activation function given by $ \alpha(x) = \max \{ x ,0 \}$.
Remark and Question. What is somewhat interesting is that $d$ parameters can be chosen generically, i.e. a random choice will almost certainly give the desired result. This is not immediately clear from the presentation in [1]. If someone has a comment on this I’d be happy to hear about it! Alternative version of the proof (cf. [1]).
First we are going to reduce the problem to the case where the samples are chosen from $\mathbb{R}$. Choose $a \in \mathbb{R}^d$ such that $x_i := a \cdot z_i$ are mutually distinct, i.e. we have
\[
x_i \neq x_j \text{, for $i \neq j$.}
\]
Relabel the data points and add a value $x_0$ such that we have
$$
x_0 < x_1 < \ldots < x_n.
$$
(Note that a generic $a$ will do the job.)
We are now in the one-dimensional case, and will proceed by defining a family $\{f_i\}$ of affine-linear functions, that will only depend on $n$ parameters, which combined with the previous projection, and the rectifier will give the first layer of the desired network. We define $f_i \colon\thinspace \mathbb{R} \to \mathbb{R}$ by
$$
f_i(x) := \tfrac{1}{x_i – x_{i-1}} (x – x_{i-1}),
$$
and note that we have $f_i(x_i) = 1$, and $f_i(x) \leq 0$ for $x \leq x_{i-1}$. We are now ready to define the final layer of the network. Set $w_1 := y_1$ and define iteratively
$$
w_j := y_j – \sum_{i=1}^{j-1} f_i(x_j)\cdot w_i.
$$
One easily computes that
$$
y_j = \sum_{i=1}^{n} \max\big\{ f_i(a \cdot z_j) , 0 \big\} \cdot w_i.
$$
With $F(z) = ( f_1(a \cdot z), \ldots, f_n(a \cdot z) )$ this can be expressed as $\max\{ F(z_j),0 \}^T \cdot w = y_j$. Note that $F$ is a affine linear function $\mathbb{R}^d \to \mathbb{R}^n$ that depends on $d + n$ weights, namely those coming from $a$ and $x_1,\ldots,x_n$. With the additional $n$ weights defining $w$ we conclude the proof.
QED References [1]Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht and Oriol Vinyals, “Understanding deep learning requires rethinking generalization”, under review as a conference paper at ICLR 2017
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№ 9
All Issues Shevchuk I. A.
Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 147-150
↓ Abstract
Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 230-245
We give an estimate of the general divided differences $[x_0, ..., x_m; f]$, where some points xi are allowed to coalesce (in this case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated Whitney and Marchaud inequalities and their generalization to the Hermite interpolation. For example, one of the numerous corollaries of this estimate is the fact that, given a function $f \in C(r)(I)$ and a set $Z = \{ z_j\}^{\mu}_{j=0}$ such that $z_{j+1} - z_j \geq \lambda | I|$ for all $0 \leq j \leq \mu 1$, where $I := [z_0, z_{\mu} ], | I|$ is the length of $I$, and $\lambda$ is a positive number, the Hermite polynomial $\scrL (\cdot ; f;Z)$ of degree $\leq r\mu + \mu + r$ satisfying the equality $\scrL (j)(z\nu ; f;Z) = f(j)(z\nu )$ for all $0 \leq \nu \leq \mu$ and $0 \leq j \leq r$ approximates $f$ so that, for all $x \in I$, $$| f(x) \scr L (x; f;Z)| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z))^{r+1} \int^{2| I|}_{dist (x,Z)}\frac{\omega_{m-r}(f^{(r)}, t, I)}{t^2}dt,$$ where $m := (r + 1)(\mu + 1), C = C(m, \lambda )$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z) := \mathrm{m}\mathrm{i}\mathrm{n}0\leq j\leq \mu | x zj | $.
↓ Abstract
Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 379-403
We introduce the moduli of smoothness with Jacobi weights $(1 x)\alpha (1+x)\beta$ for functions in the Jacobi weighted spaces $L_p[ 1, 1],\; 0 < p \leq \infty $. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces $L_p$. If $1 \leq p \leq \infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian – Totik moduli of smoothness for these $p$), while for $0 < p < 1$ they are equivalent to certain “Realization functionals”.
Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 624-630
For natural $k$ and $n \geq 2k$, we determine the exact constant $c(n, k)$ in the Dzyadyk inequality $$|| P^{\prime}_n\varphi^{1-k}_n ||_{C[ 1,1]} \leq c(n, k)n\| P_n\varphi^{-k}_n \|_{C[ 1,1]}$$ for the derivative $P^{\prime}_n$ of an algebraic polynomial $P_n$ of degree $\leq n$, where $$\varphi_n(x) := \sqrt{n^{-2} + 1 - x_2,} .$$ Namely, $$c(n, k) = \biggl( 1 + k \frac{\sqrt{ 1 + n^2} - 1}{n} \biggr)^2 - k.$$
Fedorenko V. V., Ivanov А. F., Khusainov D. Ya., Kolyada S. F., Maistrenko Yu. L., Parasyuk I. O., Pelyukh G. P., Romanenko O. Yu., Samoilenko V. G., Shevchuk I. A., Sivak A. G., Tkachenko V. I., Trofimchuk S. I.
Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 257-260
Boichuk О. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.
Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144
Babenko V. F., Davydov O. V., Kofanov V. A., Parfinovych N. V., Pas'ko A. N., Romanyuk A. S., Ruban V. I., Samoilenko A. M., Shevchuk I. A., Shumeiko A. A., Timan M. P., Trigub R. M., Vakarchuk S. B., Velikin V. L.
Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 995-999
Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 674-684
Let $P: X \rightarrow V$ be a projection from a real Banach space $X$ onto a subspace $V$ and let $S \subset X$. In this setting, one can ask if $S$ is left invariant under $P$, i.e., if $PS \subset S$. If $V$ is finite-dimensional and $S$ is a cone with particular structure, then the occurrence of the imbedding $PS \subset S$ can be characterized through a geometric description. This characterization relies heavily on the structure of $S$, or, more specifically, on the structure of the cone $S^{*}$ dual to $S$. In this paper, шє remove the structural assumptions on $S^{*}$ and characterize the cases where $PS \subset S$. We note that the (so-called) $q$-monotone shape forms a cone which (lacks structure and thus) serves as an application for our characterization.
Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 369–386
In Part I of the paper, we have proved that, for every $α > 0$ and a continuous function $f$, which is either convex $(s = 0)$ or changes convexity at a finite collection $Y_s = \{y_i\}^s_i = 1$ of points $y_i ∈ (-1, 1)$, $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha,s) \sup \left\{n^{\alpha}E_n(f):\; n \geq 1 \right\},$$ where $E_n (f)$ and $E^{(2)}_n (f, Y_s)$ denote, respectively, the degrees of the best unconstrained and (co)convex approximations and $c(α, s)$ is a constant depending only on $α$ and $s$. Moreover, it has been shown that $N^{∗}$ may be chosen to be 1 for $s = 0$ or $s = 1, α ≠ 4$, and that it must depend on $Y_s$ and $α$ for $s = 1, α = 4$ or $s ≥ 2$. In Part II of the paper, we show that a more general inequality $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha, N, s) \sup \left\{n^{\alpha}E_n(f):\; n \geq N \right\},$$ is valid, where, depending on the triple $(α,N,s)$ the number $N^{∗}$ may depend on $α,N,Y_s$, and $f$ or be independent of these parameters.
Berezansky Yu. M., Bojarski B., Gorbachuk M. L., Kopilov A. P., Korolyuk V. S., Lukovsky I. O., Mitropolskiy Yu. A., Portenko N. I., Reshetnyak Yu. G., Samoilenko A. M., Sharko V. V., Shevchuk I. A., Skorokhod A. V., Tamrazov P. M., Zelinskii Yu. B.
Ukr. Mat. Zh. - 2008. - 60, № 5. - pp. 701 – 703
Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1722-1724
Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1722
Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1200-1212
Assume that a function
f ∈ C[−1, 1] changes its convexity at a finite collection Y := { y 1, ... y s} of s points y i ∈ (−1, 1). For each n > N( Y), we construct an algebraic polynomial P n of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points y i as f and $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$ where c is an absolute constant, ω 2( f, t) is the second modulus of smoothness of f, and if s = 1, then N( Y) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness.
Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 579-580
Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 463-473
We present a survey of the scientific results obtained by E. A. Storozhenko and related results of her disciples and give brief information about the seminar on the theory of functions held under her guidance.
Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 1002–1004
For every
n, we compute the Lebesgue constant of Rogosinski kernel with any preassigned accuracy.
Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 130 – 132
On a constructive characterization of functions from the classes D( r H ω t) on closed sets with a piecewise smooth boundary rH ω
Ukr. Mat. Zh. - 1973. - 25, № 1. - pp. 81—90
Ukr. Mat. Zh. - 1972. - 24, № 5. - pp. 601–617
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Owing to the overwhelming excess of \(H_2O\) molecules in aqueous solutions, a bare hydrogen ion has no chance of surviving in water.
Free Hydrogen Ions do not Exist in Water
The hydrogen ion in aqueous solution is no more than a proton, a bare nucleus. Although it carries only a single unit of positive charge, this charge is concentrated into a volume of space that is only about a hundred-millionth as large as the volume occupied by the smallest atom. (Think of a pebble sitting in the middle of a sports stadium!) The resulting extraordinarily high
charge density of the proton strongly attracts it to any part of a nearby atom or molecule in which there is an excess of negative charge. In the case of water, this will be the lone pair (unshared) electrons of the oxygen atom; the tiny proton will be buried within the lone pair and will form a shared-electron (coordinate) bond with it, creating a hydronium ion, \(H_3O^+\). In a sense, \(H_2O\) is acting as a base here, and the product \(H_3O^+\) is the conjugate acid of water:
Although other kinds of dissolved ions have water molecules bound to them more or less tightly, the interaction between H
+ and \(H_2O\) is so strong that writing “H + ” hardly does it justice, although it is formally correct. The formula \(H_3O^+\) more adequately conveys the sense that it is both a molecule in its own right, and is also the conjugate acid of water. (aq)
The equation "HA → H
+ + A –" is so much easier to write that chemists still use it to represent acid-base reactions in contexts in which the proton donor-acceptor mechanism does not need to be emphasized. Thus, it is permissible to talk about “hydrogen ions” and use the formula H + in writing chemical equations as long as you remember that they are not to be taken literally in the context of aqueous solutions.
Interestingly, experiments indicate that the proton does not stick to a single \(H_2O\) molecule, but changes partners many times per second. This molecular promiscuity, a consequence of the uniquely small size and mass the proton, allows it to move through the solution by rapidly hopping from one \(H_2O\) molecule to the next, creating a new \(H_3O^+\) ion as it goes. The overall effect is the same as if the \(H_3O^+\) ion itself were moving. Similarly, a hydroxide ion, which can be considered to be a “proton hole” in the water, serves as a landing point for a proton from another \(H_2O\) molecule, so that the OH
– ion hops about in the same way.
Because hydronium and hydroxide ions can “move without actually moving” and thus without having to plow their way through the solution by shoving aside water molecules, as do other ions, solutions which are acidic or alkaline have extraordinarily high
electrical conductivities. Reaction
The
hydronium ion is an important factor when dealing with chemical reactions that occur in aqueous solutions. Its concentration relative to hydroxide is a direct measure of the pH of a solution. It can be formed when an acid is present in water or simply in pure water. It's chemical formula is \(H_3O^+\). It can also be formed by the combination of a H + ion with an \(H_2O\) molecule. The hydronium ion has a trigonal pyramidal geometry and is composed of three hydrogen atoms and one oxygen atom. There is a lone pair of electrons on the oxygen giving it this shape. The bond angle between the atoms is 113 degrees.
\[H_2O_{(l)} \rightleftharpoons OH^-_{(aq)} + H^+_{(aq)}\]
As H
+ ions are formed, they bond with \(H_2O\) molecules in the solution to form \(H_3O^+\) (the hydronium ion). This is because hydrogen ions do not exist in aqueous solutions, but take the form of the hydronium ion, \(H_3O^+\). A reversible reaction is one in which the reaction goes both ways. In other words, the water molecules dissociate while the OH - ions combine with the H + ions to form water. Water has the ability to attract H + ions because it is a polar molecule. This means that it has a partial charge, in this case the charge is negative. The partial charge is caused by the fact that oxygen is more electronegative than hydrogen. This means that in the bond between hydrogen and oxygen, oxygen "pulls" harder on the shared electrons thus causing a partial negative charge on the molecule and causing it to be attracted to the positive charge of H + to form hydronium. Another way to describe why the water molecule is considered polar is through the concept of dipole moment. The electron geometry of water is tetrahedral and the molecular geometry is bent. This bent geometry is asymmetrical, which causes the molecule to be polar and have a dipole moment, resulting in a partial charge. Figure \(\PageIndex{1}\): The picture above illustrates the electron density of hydronium. The red area represents oxygen; this is the area where the electrostatic potential is the highest and the electrons are most dense.
An overall reaction for the dissociation of water to form
hydronium can be seen here:
\[2 H_2O_{(l)} \rightleftharpoons OH^-_{(aq)} + H_3O^+_{(aq)}\]
With Acids
Hydronium not only forms as a result of the dissociation of water, but also forms when water is in the presence of an acid. As the acid dissociates, the H
+ ions bond with water molecules to form hydronium, as seen here when hydrochloric acid is in the presence of water:
\[HCl (aq) + H_2O \rightarrow H_3O^+ (aq) + Cl^-(aq)\]
pH The pH of a solution depends on its hydronium concentration. In a sample of pure water, the hydronium concentration is \(1 \times 10^{-7}\) moles per liter (0.0000001 M) at room temperature. The equation to find the pH of a solution using its hydronium concentration is:
\[pH = -\log {(H_3O^+)}\]
Using this equation, we find the pH of pure water to be 7. This is considered to be neutral on the pH scale. The pH can either go up or down depending on the change in
hydronium concentration. If the hydronium concentration increases, the pH decreases, causing the solution to become more acidic. This happens when an acid is introduced. As H + ions dissociate from the acid and bond with water, they form hydronium ions, thus increasing the hydronium concentration of the solution. If the hydronium concentration decreases, the pH increases, resulting in a solution that is less acidic and more basic. This is caused by the \(OH^-\) ions that dissociate from bases. These ions bond with H + ions from the dissociation of water to form \(H_2O\) rather than hydronium ions.
A variation of the equation can be used to calculate the hydronium concentration when a pH is given to us:
\[H_3O^+ = 10^{-pH} \]
When the pH of 7 is plugged into this equation, we get a concentration of 0.0000001 M as we should.
Learning to use mathematical formulas to calculate the acidity and basicity of solutions can be difficult. Here is a video tutorial on the subject of calculating hydronium ion concentrations:
Configuration of Hydronium in Water
It is believed that, on average, every hydronium ion is attracted to six water molecules that are not attracted to any other hydronium ions. This topic is still currently under debate and no real answer has been found.
Problems Determine the pH of a solution that has a hydronium concentration of 2.6x10 -4M. Determine the hydronium concentration of a solution that has a pH of 1.7. If a solution has a hydronium concentration of 3.6x10 -8M would this solution be basic or acidic? What is the pH of a solution that has 12.2 grams of hydrochloric acid in 500 ml of water? Why do acids cause burns? Answers
1. Remembering the equation: pH = -log[H
3O]
Plug in what is given: \(pH = -\log[2.6 \times 10^{-4}\;M]\)
When entered into a calculator: pH = 3.6
2. Remembering the equation: [H
3O] = 10 -pH
Plug in what is given: [H
3O] = 10 -1.7
When entered into a calculator: 1.995x10
-2M
3. Determine pH the same way we did in question one: pH = -log[3.6x10
-8]
pH = 7.4
Because this pH is above 7 it is considered to be basic.
4. First write out the balanced equation of the reaction: \[HCl_{(aq)} + H_2O_{(l)} \rightleftharpoons H_3O^+_{(aq)} + Cl^-_{(aq)}\]
Notice that the amount of HCl is equal to the amount of \(H_3O^+\) produced due to the fact that all of the stoichiometric coefficents are one.
So if we can figure out concentration of HCl we can figure out concentration of hydronium.
Notice that the amount of HCl given to us is provided in grams. This needs to be changed to moles in order to find concentration:
\[12.2\;g\; HCl \times \dfrac{1 \;mol\; HCl}{36.457\; g} = 0.335\; mol\; HCl\]
Concentration is defined as moles per liter so we convert the 500mL of water to liters and get .5 liters.
\[\dfrac{0.335\; mol\; HCl}{0.5\; L} = 0.67\;M\]
Using this concentration we can obtain pH: pH = -log[.67M]
\[pH = 0.17\]
5. Acids cause burns because they dehydrate the cells they are exposed to. This is caused by the dissociation that occurs in acids where H
+ ions are formed. These H + ions bond with water in the cell and thus dehydrate them to cause cell damage and burns. References Petrucci, Harwood, Herring, Madura. General Chemistry Principles & Modern Applications. Prentice Hall. New Jersey, 2007. Marx, Tuckerman, Hutter, J. & Parrinello, M. (1999) The nature of the hydrated excess proton in water Petrucci, Ralph H. General Chemistry: Principles and Modern Applications.Upper Saddle River, NJ: Pearson/Prentice Hall, 2007. Print. http://www.youtube.com/watch?v=Y9E_ZlOqk4o
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Hi, I'm trying to solve a nonlinear PDE that looks similar to a time-dependent eikonal equation. Can anyone provide me some advise for the best Julia-ecosystem approach to solving it, please?
I'm trying to find $$v:[0,1]\times \mathbb R_{\geq 0}^2$$, such that $$ v_t - \frac{\sigma^2}{2} g^2 v_{gg} - gq_2{\left(\frac{\frac{q_1}{q_2}-v_s}{2}\right)}^2 = 0 $$
With boundary conditions
$$ v(t,s,g) = -Cs $$ on $${t=0}$$ or $${s=0}$$ or $${g=0}$$. $$v_g(t,s,\infty) = 0 $$ $$v_s(t,\infty,g) = -C $$
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In philosophy,
Ramsey sentences refer to an attempt by logical positivist philosopher Rudolf Carnap to reconstruct theoretical propositions such that they gained empirical content.
For Carnap, questions such as: “Are electrons real?” and: “Can you prove electrons are real?” were not legitimate questions implying great philosophical/metaphysical import. They were meaningless "pseudo-questions without cognitive content,” asked from outside a language framework of science. Inside this framework, entities such as electrons or sound waves, and relations such as mass and force not only exist and have meaning, but are "useful" to the scientists who work with them. To accommodate such internal questions in a way that would justify their theoretical content empirically – and to do so while maintaining a distinction between analytic and synthetic propositions – Carnap set out to develop a systematized way to consolidate theory and empirical observation in a meaningful language formula.
Carnap began by differentiating observable things from non-observable things. Immediately, a problem arises: neither the German nor the English language naturally distinguish predicate terms on the basis of an observational categorization. As Carnap admitted, "The line separating observable from non-observable is highly arbitrary." For example, the predicate "hot" can be perceived by touching a hand to a lighted coal. But "hot" might take place at such a microlevel (e.g., the theoretical "heat" generated by the production of proteins in a eukaryotic cell) that it is virtually non-observable (at present). Physicist-philosopher Moritz Schlick characterized the difference linguistically, as the difference between the German verbs "kennen" (knowing as being acquainted with a thing – perception) and "erkennen" (knowing as understanding a thing – even if non-observable). This linguistic distinction may explain Carnap’s decision to divide the vocabulary into two artificial categories: a vocabulary of non-observable ("theoretical") terms (hereafter "V
T"): i.e., terms we know of but are not acquainted with (erkennen), and a vocabulary of observable terms ("V O"), those terms we are acquainted with (kennen) and will accept arbitrarily. Accordingly, the terms thus distinguished were incorporated into comparable sentence structures: T-terms into Theoretical sentences ( T-sentences); O-terms into Observational sentences (O-sentences).
The next step for Carnap was to connect these separate concepts by what he calls "Correspondence Rules" (C-rules), which are "mixed" sentences containing both T- and O-terms. Such a theory can be formulated as: T + C = df: the conjunction of T-postulates + the conjunction of C-rules – i.e., [(T_1 \land T_2 \land \dots \land T_n ) + ( C_1 \land C_2 \land \dots \land C_m )]. This can be further expanded to include class terms such as for the class of all molecules, relations such as "betweenness," and predicates: e.g., TC ( t
1, t 2, . . ., t n, o 1, o 2, . . ., o m). Though this enabled Carnap to establish what it means for a theory to be "empirical," this sentence neither defines the T-terms explicitly nor draws any distinction between its analytic and its synthetic content, therefore it was not yet sufficient for Carnap’s purposes.
In the theories of Frank P. Ramsey, Carnap found the method he needed to take the next step, which was to substitute variables for each T-term, then to quantify existentially all T-terms in both T-sentences and C-rules. The resulting "Ramsey sentence" effectively eliminated the T-terms as such, while still providing an account of the theory’s empirical content. The evolution of the formula proceeds thus:
Step 1 (empirical theory, assumed true): TC ( t 1 . . . t n, o 1 . . . o m) Step 2 (substitution of variables for T-terms): TC (x 1 . . . x n, o 1 . . . o m) Step 3 (\exists-quantification of the variables): \exists x_1 . . . \exists x_n TC ( x_1 . . . x_n, o_1 . . . o_m).
Step 3 is the complete Ramsey sentence, expressed "
RTC," and to be read: "There are some (unspecified) relations such that TC (x 1 . . . x n, o 1 . . . o m) is satisfied when the variables are assigned these relations. (This is equivalent to an interpretation as an appropriate model: there are relations r 1 . . . r n such that TC (x 1 . . . x n, o 1 . . . o m) is satisfied when x i is assigned the value r i, and 1 \leq i \leq m.)
In this form, the Ramsey sentence captures the factual content of the theory. Though Ramsey believed this formulation was adequate to the needs of science, Carnap disagreed, with regard to a comprehensive reconstruction. In order to delineate a distinction between analytic and synthetic content, Carnap thought the reconstructed sentence would have to satisfy three desired requirements:
The factual (F T) component must be observationally equivalent to the original theory (TC). The analytic (A T) component must be observationally uninformative. The combination of F T and A T must be logically equivalent to the original theory – that is, F_T + A_T \Leftrightarrow TC.
Requirement 1 is satisfied by
RTC in that the existential quantification of the T-terms does not change the logical truth (L-truth) of either statement, and the reconstruction FT has the same O-sentences as the theory itself, hence RTC is observationally equivalent to TC : (i.e., for every O-sentence: O, [TC \models O \Leftrightarrow ^{R}TC \models O] ). As stated, however, requirements 2 and 3 remain unsatisfied. That is, taken individually, A T does contain observational information (such-and-such a theoretical term is observed to do such-and-such, or hold such-and-such a relation); and A T does not necessarily follow from F T. Carnap’s solution is to make the two statements conditional. If there are some relations such that [TC (x1 . . . xn, o1 . . . om)] is satisfied when the variables are assigned some relations, then the relations assigned to those variables by the original theory will satisfy [TC (t1 . . . tn, o1 . . . om)] – or: RTC → TC. This important move satisfies both remaining requirements and effectively creates a distinction between the total formula’s analytic and synthetic components. Specifically, for requirement 2: The conditional sentence does not make any information claim about the O-sentences in TC, it states only that "if" the variables in are satisfied by the relations, "then" the O-sentences will be true. This means that every O-sentence in TC that is logically implied by the sentence RTC → TC is L-true (i.e., every O-sentence in AT is true or not-true: the metal expands or it does not; the chemical turns blue or it does not, etc.). Thus TC can be taken as the non-informative (i.e., non-factual) component of the statement, or A T. Requirement 3 is satisfied by inference: given A T, infer F T → A T. This makes A T + F T nothing more than a reformulation of the original theory, hence A T Ù F T ó TC.
Finally, the all-important requirement for an analytic–synthetic distinction is clearly met by using two distinct processes in the formulation: drawing an empirical connection between the statement’s factual content and the original theory (observational equivalence), and by requiring the analytic content to be observationally non-informative. Of course, Carnap’s reconstruction as it is given here is not intended to be a literal method for formulating scientific propositions. To capture what Pierre Duhem would call the entire "holistic" universe relating to any specified theory would require long and complicated renderings of
RTC → TC. Instead, it is to be taken as demonstrating logically that there is a way that science could formulate empirical, observational explications of theoretical concepts – and in that context the Ramsey and Carnap construct can be said to provide a formal justificatory distinction between scientific observation and metaphysical inquiry. The Ramsey and Carnap formulation is, of course, not inviolate. Among its critics are John Winnie, who extended the requirements to include an "observationally non-creative" restriction on Carnap’s A T – and both Quine and Hempel attacked Carnap’s initial assumptions by emphasizing the ambiguity that persists between observable and non-observable terms. Nonetheless, the Ramsey and Carnap construct was an interesting attempt to draw a substantive line between science and metaphysics. Works cited Carnap, R. "Theoretical Concepts in Science," with introduction by Psillos, S. Studies in History and Philosophy of Science Part A Carnap, R. (1966) An Introduction to the Philosophy of Science (esp. Parts III, and V), ed. Martin Gardner. Dover Publications, New York. 1995. Carnap, R. (1950) "Empiricism, Semantics, and Ontology," in Moser & Nat, Human Knowledge Oxford Univ. Press. (2003). Demopoulos, W. "Carnap on the Reconstruction of Scientific Theories," The Cambridge Companion to Carnap, eds. R. Creath and M. Friedman. Moser, P.K. and vander Nat, A. (2003) Human Knowledge Oxford Univ. Press. Schlick, Moritz (1918) General Theory of Knowledge (Allegemeine Erkenntnislehre). Trans. Albert Blumberg. Open Court Publishing, Chicago/La Salle, IL. (2002). Hallvard Lillehammer, D. H. Mellor (2005), Ramsey's legacy, Oxford University Press, pg 109. Stathis Psillos, "CARNAP, THE RAMSEY-SENTENCE AND REALISTIC EMPIRICISM", 2000 External links
Epistemic Structural Realism and Ramsey Sentences
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The Perron–Frobenius Theorem states the following.
Let $A = (a_{ij})$ be an $n \times n$ irreducible, non-negative matrix ($a_{ij} \geq 0, \forall i,j: 1\leq i,j \leq n$). Then the following statements are true.
$A$ has a real eigenvalue $c \geq 0$ such that $c > |c'|$ for all other eigenvalues $c'$. There is an eigenvector $v$ with non-negative real components corresponding to the largest eigenvalue $c: Av = cv, v_i \ge 0, 1 \leq i \leq n$, and $v$ is unique up to multiplication by a constant. If the largest eigenvalue $c$ is equal to $1$, then for any starting vector $x^{\langle 0\rangle} \neq 0$ with non-negative components, the sequence of vectors $A^k x^{\langle 0\rangle}$ converge to a vector in the direction of $v$ as $k \rightarrow \infty$.
But the theorem does not say
the sequence of vectors $A^k x^{\langle 0\rangle}$ will converge. Are there any known results on the rate of convergence? What are some good, polynomial-time algorithms to compute this limiting vector? how fast
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Let's begin with a little review of
unweighted median filtering.
Suppose I have a list of $N$ real-valued numbers, $x=x_1,...,x_N$. Let $m_i$ be the median of $K$ consecutive values: $m_i=$ median$(x_i,...,x_{i+K})$. Let $m=(m_1,...,m_{N-K+1})$. The act of transforming $x$ to $m$ is called (unweighted)
median filtering. We usually imagine $N\gg K$, and frequently we assume that $K$ is odd (so the median is unambiguously defined).
Median filtering is a useful smoothing technique in signal processing because it is robust against outliers; apparently the 2 dimensional analogue is useful in image processing because it smooths images while keeping edges relatively intact.
There is a naive $O(NK\log(K))$ algorithm to compute $m$ from $x$ (take each of the $O(N)$ windows $(x_i,...,x_{i+K})$, sort in $O(K\log(K))$ time, and report the medians). However, this performance can be substantially improved (both in theory and practice). The true complexity is $\Omega(N\log(K))$; see [Suomela, 2014] for an overview of algorithms that can achieve this (plus a clever, new algorithm).
I am interested in a closely (?) related problem,
weighted median filtering. Given a set of $K$ weights, $w_1,...,w_K$, such that $\sum_i w_i=1$, and $w_i\geq 0$, the weighted median of a set $z_1,...,z_K$ is the $z_i$ such that$$\sum_{j: z_j<z_i}w_j<1/2$$ and$$\sum_{j: z_j\geq z_i}w_j\leq 1/2$$
Note that if all the $w_i=1/K$, then the weighted median reduces to the regular median.
The naive algorithm (where we treat each window of length $K$ separately) can compute the weighted median in $O(NK\log(K))$. As Emil points out, it is possible to compute the weighted median of a single set in $O(K)$ time (as outlined on the wikipedia page on weighted median), so one can improve this to $O(NK)$. My question is:
Is there a more efficient algorithm?
A few notes:
If you assume that the $x_i$ take values over some small set, there are various special algorithms one can use. Unfortunately for me, my values will probably be largely distinct. There is a paper by Zhang et al in 2014 with the promising title "100+ Times Faster Weighted Median Filter". If I'm reading their paper correctly, the results only apply to the 2 dimensional problem, and (in the language of this post) reduces the complexity from $O(NK)$ to $O(N\sqrt{K})$ (although it's not clear to me if this result holds in one dimension). In typical usage, the weights form a window function of some sort, which typically has a largest weight in the middle of the window, and then shrinks towards either end. I would be very interest in a solution even if it only applied to this special case.
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Here’s batch 2 of my old google+ posts on ‘Inter Universal Teichmuller theory’, or rather on the number theoretic examples of Frobenioids.
June 5th, 2013
Mochizuki’s categorical prime number sieve
And now for the interesting part of Frobenioids1: after replacing a bunch of arithmetic schemes and maps between them by a huge category, we will reconstruct this classical picture by purely categorical means.
Let’s start with the simplest case, that of the ‘baby arithmetic Frobenioid’ dismantling $\mathbf{Spec}(\mathbb{Z}) (that is, the collection of all prime numbers) and replacing it by the category having as its objects
all $(a)$ where $a$ is a strictly positive rational number
and morphisms labeled by triples $(n,r,z)$ where $n$ and $z$ are strictly positive integers and $r$ is a strictly positive rational number and connecting two objects
\[
(n,r,z) : (a) \rightarrow (b) \quad \text{ if and only if } \quad a^n.z=b.r \]
Composition of morphisms is well-defined and looks like $(m,s,v) \circ (n,r,u) = (m.n,r^m.s,u^m.v)$ as one quickly checks.
The challenge is to recover all prime numbers back from this ‘Frobenioid’. We would like to take an object $(a)$ and consider the maps $(1,1,p)$ from it for all prime numbers $p$, but cannot do this as categorically we have to drop all labels of objects and arrows. That is, we have to recognize the map $(1,1,p)$ among all maps starting from a given object.
We can identify all isomorphisms in the category and check that they are precisely the morphisms labeled $(1,r,1)$. In particular, this implies that all objects are isomorphic and that there is a natural correspondence between arrows leaving $(a)$ and arrows leaving $(b)$ by composing them with the iso $(1,b/a,1) : (b) \rightarrow (a)$.
Another class of arrows we can spot categorically are the ‘irreducibles’, which are maps $f$ which are not isos but have the property that in any factorization $f=g \circ h$ either $g$ or $h$ must be an iso. One easily verifies from the composition rule that these come in two flavours:
– those of Frobenius type : $(p,r,1)$ for any prime number $p$
– those of Order type : $(1,r,p)$ for any prime number $p$
We would like to color the froBs Blue and the oRders Red, but there seems to be no way to differentiate between the two classes by purely categorical means, until you spot Mochizuki’s clever little trick.
start with a Red say $(1,r,q)$ for a prime number $q$ and compose it with the Blue $(p,1,1)$, then you get the morphism $(p,r^p,q^p)$ which you can factor as a composition of $p+1$ irreducibles
\[
(p,r^p,q^p) = (1,r,q) \circ (1,r,q) \circ …. \circ (1,r,q)o(p,1,1) \]
and if $p$ grows, so will the number of factors in this composition.
On the other hand, if you start with a Blue and compose it with either a Red or a Blue irreducible, the obtained map cannot be factored in more irreducibles.
Thus, we can identify the Order-type morphisms as those irreducibles $f$ for which there exists an irreducible $g$ such that the composition $g \circ f$ can be factored as the composition in at least $n$ irreducibles, where we can take n arbitrarily large.
Finally we say that two Reds out of $(a)$ are equivalent iff one is obtained from the other by composing with an isomorphism and it is clear that the equivalence classes are exactly the arrows labeled $(1,r,p)$ for fixed prime number $p$.
So we do indeed recover all prime numbers from the category.
Similarly, we can see that equivalence classes of Frobs from $(a)$ are of the form $(p,r,1)$ for fixed prime $p$. An amusing fact is that we can recover the prime $p$ for a Frob by purely categorical ways using the above long factorization of a composition with a Red.
There seems to be no categorical way to determine the prime number associated to an equivalence class of Order-morphisms though… Or, am i missing something trivial?
June 7th, 2013
Mochizuki’s Frobenioids for the Working Category Theorist
Many of you, including +David Roberts +Charles Wells +John Baez (and possibly others, i didn’t look at all comments left on all reshares of the past 4 posts in this
MinuteMochizuki project) hoped that there might be a more elegant category theoretic description of Frobenioids, the buzz-word apparently being ‘Grothendieck fibration’ …
Hence this attempt to deconstruct Frobenioids. Two caveats though:
– i am not a category theorist (the few who know me IRL are by now ROFL)
– these categories are meant to include all arithmetic information of number fields, which is a messy business, so one should only expect clear cut fibrations in easy situation such as principal ideal domains (think of the integers $\mathbb{Z}$).
Okay, we will try to construct the Frobenioid associated to a number field $K$ (that is, a finite dimensional extension of the rationals $\mathbb{Q}$) with ring of integers $R$ (the integral closure of $\mathbb{Z}$ in $K$). For a concrete situation, look at the quadratic case.
The objects will be
fractional ideals of K which are just the R-submodules $I$ of $K$ such that there in an $r$ in $R$ such that $I.r$ is a proper ideal of $R$. Dedekind showed that any such thing can be written uniquely as a product
\[
I = P_1^{a_1} … P_k^{a_k} \]
where the $P_i$ are prime ideals of $R$ and the $a_i$ are integers (if they are all natural numbers, I will be a proper ideal of $R$). Clearly, if one multiplies two fractional ideals $I$ and $J$, the result $I.J$ is again a fractional ideal, so they form a group and by Dedekind’s trick this group is the free Abelian group on all prime ideals of $R$.
Next, we define an
equivalence relation on this set, calling two fractional ideals $I$ and $J$ equivalent if there is a $k$ in $K$ such that $I=J.k$ (or if you prefer, if they are isomorphic as $R$-modules).
We have a set with an equivalence relation and hence a
groupoid where these is a unique isomorphism between any two equivalent objects. This groupoid is precisely the groupoid of isomorphisms of the Frobenioid we’re after.
The number of equivalence classes is
finite and these classes correspond to the element of a finite group $Cl(R)$ called the class group of $R$ which is the quotient group of ideals modulo principal ideals (so if your $R$ is a principal ideal domain there is just one component). The ‘groups’ corresponding to each connected component of the groupoid are all isomorphic to the quotient group of the units in $K$ by the units in $R$.
Next, we will add the other morphisms. By definition they are all compositions of
irreducibles which come in 2 flavours:
– the
order-morphisms $P$ for any prime ideal $P$ of $R$ sending $I$ to $I.P$. Typically, these maps switch between different equivalence classes (unless $P$ itself is principal). We can even explicitly compute small norm prime ideals which will generate all elements in the class group $Cl(R)$. – the power-maps $[p]$ for any prime number $p$ which sends $I$ to $I^p$. The nature of these maps really depend on the order of the component in the finite group $Cl(R)$.
Well, that’s it basically for the layer of the Frobenioid corresponding to the number field $K$. (You have to repeat all this for any subfield between $\mathbb{Q}$ and $K$). A cute fact is that all endomorphism-monoids of objects in the layer of K are all isomorphic as abstract monoid to the skew-monoid
$\mathbb{N}^x_{>0} x Prin(R)$ of the multiplicative group of all strictly positive integers $n$ with the monoid of all principal ideals in R with multiplication defined by
\[
(n,Ra).(m,Rb)=(nm,Rab^n) \]
The only extra-type morphisms we still have to include are those between the different layers of the Frobenioid, the green ones which M calls the pull-back morphisms.
They are of the following form: if $R_1$ and $R_2$ are rings of integers in the fields $K_1$ contained in $K_2$, then for any ringmorphism $\sigma : R_1 \rightarrow R_2$ one can extend a fractional ideal $I$ of $R_1$ to $K_2$ by considering $R_2.\sigma(I)$. These then give the morphism $r_2\sigma(I) \rightarrow I$ and as we will see in a next instalment, they encode the splitting behaviour of prime ideals.
a question for category people
Take the simplest situation, that of the integers $\mathbb{Z}$. So, we have just a groupoid with extra morphisms generated by the order-maps $o_p$ and the power maps $f_p$. The endo-ring of any object is then isomorphic top the abstract group generated by the $f_p$ and $o_p$ and satisfying following relations
\[
o_p.o_q=o_q.o_p \] \[ f_p.f_q=f_q.f_p \] \[ f_p.o_q=o_q^p.f_p \]
My question now is: if for two different primes $p$ and $q$ i switch their role in the endo-ring of 1 object and propagate this via all isos to all morphisms, do i get a category equivalence? (or am i missing something?). (tbc)
June 11th, 2013
my problem with Mochizuki’s Frobenioid1
Let us see how much arithmetic information can be reconstructed from an arithmetic Frobenioids. Recall that for a fixed finite Galois extension $L$ of $\mathbb{Q}$ this is a category with objects all fractional ideals in subfields of $L$, and maps generated by multiplication-maps with ideals in rings of integers, power-maps and Galois-extension maps.
When all objects and morphisms are labeled it is quite easy to reconstruct the Galois field $L$ from it as well as all maps between prime spectra of rings of integers in intermediate fields, which after all was the intended use of Frobenioids, to ‘dismantle’ these arithmetic schemes and endow them with extra structure given by the power-maps.
However, in this reconstruction process we are only allowed t use the category structure, so all objects and morphisms are unlabelled (the situation top left) and we want to reconstruct from it the different layers of the Frobenioid (corresponding to the different subfields) and divide all arrows according to their type (situation bottom left).
First we can look at all isomorphisms. They will divide the category in the dashed regions, some of them will be an entire layer (for example for $\mathbb{Q}$) but in general a finite number of these regions will make up the full layer of a subfield (the regions labeled by the elements of the ideal class group).
Another categorical notion we can use are ‘irreducible morphisms’, that is a morphism $f$ which is not an iso but having the property that in each factorisation $f = g \circ h$ either $g$ or $h$ must be an iso. If we remember the different types of morphisms in our Frobenioid we see that the irreducibles come in 3 flavours:
– oRder-maps (Red) : multiplication by a prime ideal $P$ of the ring of integers of the subfield
– froBenius or power-maps (Blue) sendingg a fractional ideal $I$ to $I^p$ for a prime number $p$ – Galois-maps (Green) extending ideals for a subfield $K$ to $K’$ having no intermediate field.
We would like to determine the colour of these irreducibles purely categorical. The idea is that reds have the property that they can be composed with another irreducible (in fact, of power type) such that the composition can again be decomposed in irreducibles and that there is no a priori bound on the number of these terms (this uses the fact that $[p] \circ Q = Q \circ Q \circ … Q \circ [p]$ and that there are infinitely many prime numbers $p$). One checks that compositions of order or Galois maps with irreducibles have factorisation with a bounded number of irreducibles.
The most interesting case is the composition of a Galois map with an order map $P$, this can be decomposed alternatively as order maps in the bigger field followed by a Galois map, the required order maps are the bigger primes $Q_i$ occurring in the decomposition of the extended ideal
\[
S.P = Q_1.Q_2….Q_k \]
but the number $k$ of this decomposition is bounded by the dimension of the bigger field over the smaller one. Summarizing:
– we can determine all the red maps, which will then give us also the different layers
– we can determine the green maps as they move between different layers – to the remaining blue ones we can even associate their label $[p]$ by the observed property of composition with order maps.
Taking an object in a layer, we get the set of prime ideals of the ring on integers in that field as the set of all red arrows leaving that object unto equivalence (by composing with an isomorphism), so we get the prime spectra $\mathbf{Spec}(S)$.
For a ring-extension $R \rightarrow S$ we also can recover the cover map $\mathbf{Spec}(S) \rightarrow \mathbf{Spec}(R)$
Indeed, composing the composition of the Galois map with the order-map $P$ and decomposing it alternatively will give us the finite number of prime ideals $Q_i$ of $S$ lying over $P$. That is, we get all splitting behaviour of prime ideals in intermediate field-extensions.
Let $K$ be a Galois subfield of $L$ then we have a way to see how a prime ideals in $\mathbf{Spec}(\mathbb{Z})$ splits, ramifies or remains inert in $K$ and so by Chebotarev density this gives us the dimension of $K$ as well as the Galois group. And, if we could label the prime ideal by a prime number $p$, we could even reconstruct $K$ itself as $K$ is determined once we know all prime numbers which completely split.
The problem i have is that i do not see a categorical way to label the red arrows in $Frob(\mathbb{Z})$ by prime numbers. Mochizuki says we can do this in the proof of Thm 6.4(iii) by using the fact that the $log(p)$ are linearly independent over $\mathbb{Q}$.
This suggests that one might use the ‘Arakelov information’ (that is the archimidean valuations) to do this (the bit i left out so far), but i do not see this in the case of $\mathbb{Q}$ as there is just 1 extra (real) valuation determined by the values of the nonarchimidean valuations.
Probably i am missing something so all sorts of enlightenment re welcome!
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
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Consider the equation $$ u'(t) = (Fu)(t) $$ where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type) nonlinear operator. It means that the value of $(Fu)(t_0)$ depends on values $u(t)$ for $t \in (0,t_0)$.
I need results about solvability of this problem. The book by Gajewski et al. contains some results when the operator $F$ fulfills Lipschitz condition: $$ (*)\;\; \|Fu - Fv\|_{L^2(0,T;\mathbb R^n)} \leq L\|u - v\|_{L^2(0,T;\mathbb R^n)}. $$
But if $Fu$ contains, for instance, square $u^2$ then it fulfills only local Lipshitz condition, i.e. $(*)$ is fulfilled only for $u, v \in B(u_0, r)$ where $B$ is a ball.
Where can I find results for solvability of this equation with local Lipschitz condition?
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In evaluating the vacuum structure of quantum field theories you need to find the minima of the effective potential including perturbative and nonperturbative corrections where possible.
In supersymmetric theories, you often see the claim that the Kähler potential is the suitable quantity of interest (as the superpotential does not receive quantum corrections). For simplicity, let's consider just the case of a single chiral superfield:$\Phi(x,\theta)=\phi(x)+\theta^\alpha\psi_\alpha(x) + \theta^2 f(x)$and its complex conjugate. The low-energy action functional that includes the Kähler and superpotential is$$
S[\bar\Phi,\Phi] = \int\!\!\!\mathrm{d}^8z\;K(\bar\Phi,\Phi)
+ \int\!\!\!\mathrm{d}^6z\;W(\Phi) + \int\!\!\!\mathrm{d}^6\bar{z}\;\bar{W}(\bar\Phi)
$$Keeping only the scalar fields and no spacetime derivatives, the components are$$\begin{align}
S[\bar\Phi,\Phi]\big|_{\text{eff.pot.}} = &\int\!\!\!\mathrm{d}^4x\Big(\bar{f}f\,\frac{\partial^2K(\bar\phi,\phi)}{\partial\phi\partial{\bar\phi}} + f\,W'(\phi) + \bar{f}\, W(\phi)\Big) \\
\xrightarrow{f\to f(\phi)}
-\!&\int\!\!\!\mathrm{d}^4x\Big(\frac{\partial^2K(\bar\phi,\phi)}{\partial\phi\partial{\bar\phi}}\Big)^{-1}|W'(\phi)|^2
=: -\!\int\!\!\!\mathrm{d}^4x \ V(\bar\phi,\phi)
\end{align}$$where in the second line we solve the (simple) equations of motion for the auxiliary field.The vacua are then the minuma of the effective potential $V(\bar\phi,\phi)$.
However, if you read the old (up to mid 80s) literature on supersymmetry they calculate the effective potential using all of the scalars in the theory, i.e. the Coleman-Weinberg type effective potential using the background/external fields $\Phi(x,\theta)=\phi(x) + \theta^2 f(x)$. This leads to an effective potential $U(\bar\phi,\phi,\bar{f},f)$ which is more than quadratic in the auxiliary fields, so clearly not equivalent to calculating just the Kähler potential. The equivalent superfield object is the Kähler potential + auxiliary fields' potential, as defined in "Supersymmetric effective potential: Superfield approach" (or here). It can be written as$$
S[\bar\Phi,\Phi] = \int\!\!\!\mathrm{d}^8z\;\big(K(\bar\Phi,\Phi) + F(\bar\Phi,\Phi,D^2\Phi,\bar{D}^2\bar{\Phi})\big)
+ \int\!\!\!\mathrm{d}^6z\;W(\Phi) + \int\!\!\!\mathrm{d}^6\bar{z}\;\bar{W}(\bar\Phi)
$$where $F(\bar\Phi,\Phi,D^2\Phi,\bar{D}^2\bar{\Phi})$ is at least cubic in $D^2\Phi,\bar{D}^2\bar{\Phi}$.The projection to low-energy scalar components of the above gives the effective potential $U(\bar\phi,\phi,\bar{f},f)$ that is in general non-polynomial in the auxiliary fields and so clearly harder to calculate and work with than the quadratic result given above.
This post has been migrated from (A51.SE)
So my question is: when did this shift to calculating only the Kähler potential happen and is there a good reason you can ignore the corrections of higher order in the auxiliary fields?
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Multidimensional Nonparametric Density Estimates: Minimax Risk with Random Normalizing Factor
Multidimensional Nonparametric Density Estimates: Minimax Risk with Random Normalizing Factor
Abstract
We consider nonparametric minimax problem of multidimensional density estimation. Using the concept of random normalizing factor, by considering the plausible hypothesis of independence, we improve the accuracy of minimax estimation $n^{-\frac{\beta}{2\beta+d}}$: with prescribed confidence level $\alpha_{n}$, we show that the best possible attainable (random) rate is $\displaystyle{\big\{\sqrt{\log(2/\alpha_{n})}/n\big\}^{\frac{2\beta}{4\beta+d}}}$. We construct an optimal estimator and an optimal random normalizing factor in the sense of Lepski.
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Existence of Non-Measurable Subset of Real Numbers Theorem Proof
We construct such a set.
For $x, y \in \left[{0 \,.\,.\, 1}\right)$, define the sum modulo 1:
$x +_1 y = \begin{cases} x + y & : x + y < 1 \\ x + y - 1 & : x + y \ge 1 \end{cases}$
Let $E \subset \left[{0 \,.\,.\, 1}\right)$ be a measurable set.
Let $E_1 = E \cap \left[{0 \,.\,.\, 1 - x}\right)$ and $E_2 = E \cap \left[{1 - x \,.\,.\, 1}\right)$.
So:
$m \left({E_1}\right) + m \left({E_2}\right) = m \left({E}\right)$ We have: $E_1 +_1 x = E_1 + x$ $m \left({E_1 +_1 x}\right) = m \left({E_1}\right)$
Also:
$E_2 +_1 x = E_2 + x - 1$, and so $m \left({E_2 +_1 x}\right) = m \left({E_2}\right)$
Then we have:
$m \left({E +_1 x}\right) = m \left({E_1 +_1 x}\right) + m \left({E_2 +_1 x}\right) = m \left({E_1}\right) + m \left({E_2}\right) = m \left({E}\right)$
So, for each $x \in \left[{0 \,.\,.\, 1}\right)$, the set $E +_1 x$ is measurable and:
$m \left({E + x}\right) = m \left({E}\right)$ Taking, as before, $x, y \in \left[{0 \,.\,.\, 1}\right)$, define the relation: $x \sim y \iff x - y \in \Q$
where $\Q$ is the set of rational numbers.
Let $P_i := P +_1 r_i$.
Then $P_0 = P$.
Let $x \in P_i \cap P_j$.
Then:
$x = p_i + r_i = p_j + r_j$
where $p_i, p_j$ are elements of $P$.
But then $p_i - p_j$ is a rational number.
$i = j$
The $P_i$ are pairwise disjoint.
$\displaystyle \bigcup P_i = \left[{0 \,.\,.\, 1}\right)$
But if this were the case, then:
$\displaystyle m \left[{0 \,.\,.\, 1}\right) = \sum_{i \mathop = 1}^\infty m \left({P_i}\right) = \sum_{i \mathop = 1}^\infty m \left({P}\right)$
Therefore:
$m \left({P}\right) = 0$ implies $m \left[{0 \,.\,.\, 1}\right) = 0$
and:
$m \left({P}\right) \ne 0$ implies $m \left[{0 \,.\,.\, 1}\right) = \infty$
This contradicts Measure of Interval is Length.
$\blacksquare$
Axiom of Choice
This proof depends on the Axiom of Choice.
Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.
However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.
Boolean Prime Ideal Theorem
This theorem depends on the Boolean Prime Ideal Theorem (BPI).
As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.
Sources 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications: $\S 1.1$
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A comprehensive review Buy this book eBook 118,99 € price for Spain (gross) ISBN 978-3-540-92792-1 Digitally watermarked, DRM-free Included format: PDF ebooks can be used on all reading devices Immediate eBook download after purchase Softcover 145,59 € price for Spain (gross) ISBN 978-3-642-10087-1 Free shipping for individuals worldwide Usually dispatched within 3 to 5 business days. The final prices may differ from the prices shown due to specifics of VAT rules About this book
This monograph treats the effectiveness of useing flavor physics in offering probes of the TeV scale, while providing a timely interface during the emerging LHC era. By concentrating only with the TeV-scale connection, a large part of the B factory output can be bypassed, and emphasis is placed on loop-induced processes, i.e. virtual, quantum processes that probe TeV-scale physics. The experimental perspective is taken, resulting in selecting processes, rather than the theories or models, as the basis to exploration. Two-thirds of the book is therefore concerned with b -> s or bs <-> sb transitions. The guiding principle is: unless it can be identified as the smoking gun, it is better to stick to the simplest, rather than elaborate, explanation of an effect that may call for New Physics. By focusing on heavy flavor as a probe of TeV-scale physics, technicalities can be employed to unveil their beauty, without getting ensnared in them, while aiming for the deeper, higher-scale physics that such probes provide.
This tract originated from a plenary talk at the SUSY 2007 conference in Karlsruhe, Germany.
Table of contents (10 chapters)
Introduction
Pages 1-9
CP Violation in Charmless $b \to s\bar qq$ Transitions
Pages 11-31
B s Mixing and $\sin2\Phi_{B_s}$
Pages 33-55
$H^+$ Probes: $b\to s\gamma$ and $B\to\tau\nu$
Pages 57-72
Electroweak Penguin: bsZ Vertex, Z′, Dark Matter
Pages 73-86
Table of contents (10 chapters) Buy this book eBook 118,99 € price for Spain (gross) ISBN 978-3-540-92792-1 Digitally watermarked, DRM-free Included format: PDF ebooks can be used on all reading devices Immediate eBook download after purchase Softcover 145,59 € price for Spain (gross) ISBN 978-3-642-10087-1 Free shipping for individuals worldwide Usually dispatched within 3 to 5 business days. The final prices may differ from the prices shown due to specifics of VAT rules Recommended for you Bibliographic Information Bibliographic Information Book Title Flavor Physics and the TeV Scale Authors George W. S. Hou Series Title Springer Tracts in Modern Physics Series Volume 233 Copyright 2009 Publisher Springer-Verlag Berlin Heidelberg Copyright Holder Springer-Verlag Berlin Heidelberg eBook ISBN 978-3-540-92792-1 DOI 10.1007/978-3-540-92792-1 Softcover ISBN 978-3-642-10087-1 Series ISSN 0081-3869 Edition Number 1 Number of Pages XIII, 143 Number of Illustrations 58 b/w illustrations Topics
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Density-based clustering in spatial data (1) This is the first of a series of posts on cluster-algorithms and ideas in data analysis (and related fields). Density-based spatial clustering of applications with noise (DBSCAN) is a data clustering algorithm proposed by Martin Ester, Hans-Peter Kriegel, Jörg Sander and Xiaowei Xu in 1996 [1]. It uses notions of connectivity and density of points in the data set to compute the connected components of dense enough regions of the data set. But let’s cut the intro and dive right into it. Definitions
Let $(X,d)$ be a finite discrete metric space, i.e. let $X$ be a data set on which we have a notion of similarity expressed in terms of a suitable distance function $d$. Assume we fixed a pair of parameters $\varepsilon > 0$ and $m \in \mathbb{N}$. We say that two points $x,x’ \in X$ are
$\varepsilon$-connected if there is a sequence of points $x=x_0,\ldots,x_n=x’$ such that \[ d(x_i,x_{i+1}) < \varepsilon, \text{ for $i=0,\ldots,n-1$}. \] Note that $\varepsilon$-connectivity defines an equivalence relation on $X$ which we denote by $\sim_\varepsilon$.
For a subset $A \subset X$ we define its
$\varepsilon$-boundary $\Delta_\varepsilon A$ to be the set of points whose distance to $A$ is at most $\varepsilon$, i.e. we set \[ \Delta A := \Delta_\varepsilon A := \{ x \in X \setminus A \ | \ d(A,x) \leq \varepsilon \}. \] Here $d(A,x)$ denotes the minimum $\text{min}_{a \in A} \{ d(a,x) \}$ over all points in $A$. This notion of boundary is borrowed from graph theory and slightly adapted to our situation. Don’t worry if you don’t like that notion it could also be omitted — I will elaborate on that later on. The choice of the parameters above allows us to define a discrete notion of density $\rho(x)$ of a point $x$ by \[ \rho(x) = \rho_{\varepsilon}(x) := |N_\varepsilon(x)|, \] where $|.|$ counts the number of elements of a set and $N_\varepsilon(x)$ denotes the $\varepsilon$-neighborhood of $p$ (i.e.~the set of points whose distance to $x$ is at most $\varepsilon$). A point is called $m$-dense if its ($\varepsilon$-)density is greater or equal to $m$, i.e.~its $\varepsilon$-neighborhood $N_\varepsilon(p)$ contains at least $m$ points. In the literature these points usually are referred to as core points. Sketch of the DBSCAN-algorithm
The idea behind DBSCAN can be formulated as follows. Choose pair of parameters $(\varepsilon, m) \in \mathbb{R}_{\geq 0} \times \mathbb{N}$. The first parameter, $\varepsilon > 0$, gives rise to the notions of connectivity, density, and the boundary of subset as described above. Let \[ X_{m \leq \rho} := \rho^{-1}\big( [m,\infty) \big) \subset X \] denote the points of density at least $m$ and let $C_1,\ldots,C_n$ denote its connected components with respect to the notion of $\varepsilon$-connectivity defined above, i.e. \[ \pi_0^\varepsilon(X_{m \leq \rho}) := \big( X_{m \leq \rho} \big)/_{x \sim_\varepsilon x’} = \big\{ C_1,\ldots,C_n \big\}. \] The components $C_1,\ldots,C_n$ can be understood as
dense cores of the final clusters. Note that these sets are mutually disjoint. The final clusters $\overline{C}_1,\ldots,\overline{C}_n$ are obtained from $C_1,\ldots,C_n$ by adding their respetive boundaries, i.e. for $i=1,\ldots,n$ we define \[ \overline{C}_i := C_i \cup \Delta C_i. \] Actually $\overline{C}_i$ is just the $\varepsilon$-neighborhood $N_\varepsilon(C_i)$ of $C_i$, but I like the formulation in terms of $\Delta_\varepsilon$. Note that that $\overline{C}_1,\ldots,\overline{C}_n$ are not necessarily pairwise disjoint. To achieve that we need to make a choice (usually following the order in which the points are visited) and in that sense an implementation of DBSCAN is usually not deterministic. References [1] M. Ester, H.-P. Kriegel, J. Sander, and X. Xu, A density-based algorithm for discovering clusters in large spatial databases with noise, Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (1996), AAAI Press, pp. 226–231
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Array Phasing
When we randomly dither the position of the emitters in a 3 dimensional phased array, it smears out the grating lobes. I am looking for a better function.
A position dither function to try:
D = L/2
k = 2 \pi / N * L
\Delta x = D * ( \sin( k z ) + \cos( k y ) )
\Delta y = D * ( \sin( k x ) + \cos( k z ) )
\Delta z = D * ( \sin( k y ) + \cos( k x ) )
... or some variation of that ( I originally tried D = \lambda/2 , with little effect). This assumes the spacing L >> \lambda , a sparse array, so that the antennas do not couple (much). Try scaling D and k, and also modifying amplitudes across the array like a Hamming window, and see how that changes the sidelobes.
This happens on top of the array of perhaps hundreds of emitters on the thinsat itself, which beamforms to a few degrees of angle, reducing power splattered far from the target.
The signal is broadband, so there is not a well defined \lambda . We may end up making k a function of x, y, and z as well.
5x5x5 skewed, dithered array
Annotated version with explanation coming soon.
16x16x16 skewed, dithered array
Annotated version with explanation coming soon.
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The way I know of to bound generalization error by Rademacher complexity is Theorem 2.4 in this lecture notes, http://ttic.uchicago.edu/~tewari/lectures/lecture9.pdf. Here the quantity on the LHS that Rademacher complexity is trying to upperbound is given as, $L_{\phi}(\hat{f}_{\phi}^*)-\min_{f \in F} L_{\phi}(f)$ where $F$ is some "hypothesis class" of functions mapping $ f :X \rightarrow D$, $\phi : D \times Y \rightarrow [0,1]$ is the "loss function", "$\phi-$"loss of any function $g : X \rightarrow D$ is defined as, $L_\phi(g) = \mathbb{E}[\phi(f(x),y)]$ - where the expectation is taken over some distribution over the points $(x,y) \in X \times Y$ and $\hat{f}_{\phi}^*$ is what the ERM returns over some $m$ samples i.e $\hat{f}_{\phi}^* = argmin_{f \in F} \frac{1}{m} \sum_{i=1}^m \phi(f(x_i),y_i)$.
The above setting is called ``agnostic" because at no point was it assumed that, $\exists$ any ground-truth labelling function $L \in F$ such that $y = L(x)$ but rather the class $F$ is to be seen to be trying to learn via empirical risk minimization a distribution, say ${\cal D}$, over $X \times Y$.
My question is 3 fold , is there any analogue of this Theorem $2.4$ when,
(a) an existence of a $L$ is assumed with $L$ may or maynot be in $F$. (the later is I guess often called the ``realizable setting") (...I have seen some papers trying to bound generalization error of a specific algorithm in the realizable setting but I somehow dont see Rademacher complexity defined in those settings!..)
(b) the loss function $\phi$ is not assumed to be bounded above but only assumed to be bounded below.
(c)
AND most importantly, say I have a class of labelling functions ${\cal L}$ mapping $X \rightarrow Y$ and I want to say the following, "Given a loss function $\phi$, irrespective of which member of ${\cal L}$ labels the data (maybe also irrespective of the distribution over $X$ used to measure $L_{\phi}$) the member of class $F$ obtained via ERM on the data, can never generalize well". Is there a version of Rademacher complexity which captures this?
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I just wanted to add that there is a fairly easy proof for your final question: Is every continuous homomorphism between Lie groups actually smooth?
The theorem we need is the closed subgroup theorem (also called the Cartan Theorem): If H is a topologically closed subgroup of a Lie group G, then H is actually an embedded Lie subgroup.
Granting this, one proves all continuous homomorphisms are smooth as follows:
Given Lie groups H and G with $f:H\rightarrow G$ a continuous homomorphism, consider the subgroup $K$ of $H\times G$ given by the graph of $f$. The graph is a closed subset of $H\times G$ precisely because $f$ is continuous, and hence, by the closed subgroup theorem, the graph is an embedded smooth submanifold of $H\times G$. Thus, the restriction of the two canonical projection maps $\pi_1:H\times G\rightarrow H$ and $\pi_2:H\times G\rightarrow G$ are smooth when restricted to K.
Now, $\pi_1$ restricted to $K$ is clearly* a diffeomorphism onto $H$, and hence has a smooth inverse and so is smooth. But then we find that $f = \pi_2\circ \pi_1^{-1}$ is a composition of smooth maps, and hence is smooth. (To be clearer, the $\pi_1^{-1}$ means the inverse of $\pi_1:K\rightarrow H$.)
*- (Edited in due to comments). One knows by Sard's theorem that there is a point $p\in K$ such that $d_p \pi_1$ is invertible (of full rank). I claim that this implies that for all $q\in K$, $d_q \pi_1$ is invertible. The point is that $\pi_1$ is group homomorphism, which is the same as saying $\pi_1\circ L_{qp^{-1}} = L_{\pi_1(qp^{-1})}\circ \pi_1$, where $L_g$ denotes left multiplication by $g: L_g(h) = gh$. Taking the differentials at p on each side of this equation and using the chain rule, one finds
$$d_q \pi_1 \circ d_p L_{qp^{-1}} = d_{\pi_1(p)}L_{\pi_1(qp^{-1})}\circ d_p \pi_1.$$
The fact that $L_g$ is a diffeomorphism (with inverse $L_{g^{-1}}$) implies that $dL$ is invertible at any point, and hence we see that
$$d_q\pi_1 = d_{\pi_1(p)}L_{\pi_1{qp^{-1}}}\circ d_p \pi_1\circ d_pL_{pq^{-1}};$$ i.e., that $d_q \pi_1$ is a composition of invertible maps, and hence is itself invertible.
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1. Background:
Lense-Thirring precession is the rotation undergone by the spin of a particle located in the gravitational field of a massive spinning body. In terms of asymptotically inertial coordinates $(t,\vec x)$ in a four-dimensional space-time, and if we denote by $\vec J$ the angular momentum of the source, the angular velocity of precession of a particle at position $\vec x$ is
$$\vec\Omega = \frac{1}{r^3}\left(-\vec J+3\frac{(\vec J\cdot\vec x)\vec x}{r^2}\right)$$
where $r\equiv\sqrt{\vec x\cdot\vec x}$ and the dot denotes the scalar product of spatial vectors. We use units such that $G=c=1$. (For the derivation of this formula, see e.g. Misner-Thorne-Wheeler, section 40.7.)
2. My question:
The dependence of the angular velocity $\vec\Omega$ on the source's angular momentum $\vec J$ and on the spatial position $\vec x$ is exactly the same as that of the electric field generated by an electric dipole. Under such an identification, $\vec\Omega$ is identified with the electric field while $\vec J$ is identified with the dipole moment. My question is the following: is there an
intuitive explanation for why the precessional angular velocity has to be of the same form as the field sourced by a dipole?
Just to make things clear: I'm not looking for a mathematical proof that the above formula for $\vec\Omega$ is correct. Instead, I'd like to find an intuitive (but nevertheless rigorous) argument that makes the above result obvious. Indeed, the standard derivation of the formula for $\vec\Omega$ relies on some relatively advanced mathematical tools, but the result is so simple and pretty that I suspect there's a deeper reason for the apparent coincidence with the formula from electrostatics. (Of course, my expectation may be wrong.)
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№ 9
All Issues Evtukhov V. M. On the asymptotic of solutions of second-order differential equations with rapidly varying nonlinearities
↓ Abstract
Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 73-91
We establish the conditions of existence for one class of monotone solutions of two-term nonautonomous differential equations of the second order with rapidly varying nonlinearities and the asymptotic representations of these solutions and their first-order derivatives as $t \uparrow \omega (\omega \leq +\infty )$.
Asymptotic behavior of the solutions of second-order differential equations with rapidly varying nonlinearities
↓ Abstract
Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1345-1363
We establish conditions for the existence of one class of solutions of two-term nonautonomous differential equations of the second-order with rapidly varying nonlinearities and the asymptotic representations for these solutions and their first-order derivatives as и $t \uparrow \omega (\omega \leq +\infty )$.
Asymptotic representation of solutions of differential equations with rightly varying nonlinearities
Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1198-1216
The conditions of existence of some types of power-mode solutions of a binomial nonautonomous ordinary differential equation with regularly varying nonlinearities are established.
Asymptotic Representations for Some Classes of Solutions of Ordinary Differential Equations of Order $n$ with Regularly Varying Nonlinearities
Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 354-380
Existence conditions and asymptotic (as $t \uparrow \omega (\omega \leq +\infty)$) representations are obtained for one class of monotone solutions of an $n$th-order differential equation whose right-hand side contains a sum of terms with regularly varying nonlinearities.
Asymptotics of Solutions of Nonautonomous Second-Order Ordinary Differential Equations Asymptotically Close to Linear Equations
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1346-1364
Asymptotic representations are obtained for a broad class of monotone solutions of nonautonomous binary differential equations of the second order that are close in a certain sense to linear equations.
Asymptotic representations of solutions of essentially nonlinear systems of ordinary differential equations with regularly and rapidly varying nonlinearities
Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1165-1185
We obtain asymptotic representations for one class of solutions of systems of ordinary differential equations more general than systems of the Emden – Fowler type.
Existence criteria and asymptotics for some classes of solutions of essentially nonlinear second-order differential equations
Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 924-938
We establish existence theorems and asymptotic representations for some classes of solutions of second-order differential equations whose right-hand sides contain nonlinearities of a more general form than nonlinearities of the Emden - Fowler type.
Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point
Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 52 - 80
We establish conditions for the existence of solutions vanishing at a singular point for various classes of systems of quasilinear differential equations appearing in the investigation of the asymptotic behavior of solutions of essentially nonlinear nonautonomous differential equations of higher orders.
Asymptotic representations of solutions of essentially nonlinear two-dimensional systems of ordinary differential equations
Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1597-1611
We establish asymptotic representations for one class of solutions of two-dimensional systems of ordinary differential equations that are more general than systems of the Emden–Fowler type.
Asymptotic representations of the solutions of essentially nonlinear nonautonomous second-order differential equations
Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 310–331
We establish asymptotic representations for the solutions of a class of nonlinear nonautonomous second-order differential equations.
Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order
Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1363–1375
We establish asymptotic representations for unbounded solutions of nonlinear nonautonomous differential equations of the third order that are close, in a certain sense, to equations of the Emden-Fowler type.
Conditions of oscillatory or nonoscillatory nature of solutions for a class of second-order semilinear differential equations
Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 458–466
For a class of second-order semilinear differential equations, we prove the theorems on oscillatory or nonoscillatory nature of all proper solutions. These theorems are analogs of the well-known Kneser theorems for linear differential equations.
Asymptotic behavior of unbounded solutions of essentially nonlinear second-order differential equations. II
Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 901–921
We establish asymptotic representations for one class of unbounded solutions of second-order differential equations whose right-hand sides contain a sum of terms with nonlinearities of a more general form than nonlinearities of the Emden-Fowler type.
Asymptotic Behavior of Unbounded Solutions of Essentially Nonlinear Second-Order Differential Equations. I
Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 338–355
We establish asymptotic representations for one class of unbounded solutions of second-order differential equations whose right-hand sides contain a sum of terms with nonlinearities of a more general form than nonlinearities of the Emden-Fowler type.
Asymptotic Representations for Solutions of One Class of Systems of Quasilinear Differential Equations
Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1658-1668
We establish asymptotic representations for solutions of one class of systems of differential equations appearing in the investigation of the asymptotic behavior of
nth-order quasilinear differential equations.
Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 20-42
We investigate smoothness properties of the roots of algebraic equations with almost constant coefficients and construct a transformation, which may be efficiently used for the investigation of the asymptotic behavior of a fundamental family of solutions of a broad class of nonautonomous linear differential equations of the
nth order. On conditions for oscillation and nonoscillation of the solutions of a semilinear second-order differential equation
Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 833–841
We establish sufficient conditions for oscillation and nonoscillation of regular solutions of the secondorder differential equation Sign y=0, where ?<1 and p: [a,?[? ?, ??< a < ??+? is a locally summable function.
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On Commutativity of Prime Γ-Rings with θ-Derivations
On Commutativity of Prime Γ-Rings with θ-Derivations
Abstract
Let $M$ be a prime $\Gamma-$ring, $I$ a nonzero ideal, $\theta$ an automorphism and $d$ a $\theta-$derivation of $M$. In this article we have proved the following result: (1) If $d([x,y]_{\alpha})=\pm([x,y]_{\alpha})$ or $d((x\circ y)_{\alpha})=\pm((x\circ y)_{\alpha})$ for $x, y\in I; \alpha\in \Gamma$, then $M$ is commutative. (2) Under the hypothesis $d\theta=\theta d$ and $Char M\neq2$, if $(d(x)\circ d(y))_{\alpha}=0$ or $[d(x),d(y)]_{\alpha}=0$ for all $x, y\in I;\alpha\in \Gamma$, then $M$ is commutative. (3) If $d$ acts as a homomorphism or an anti-homomorphism on $I$, then $d=0$ or $M$ is commutative. Moreover, an example is given to demonstrate that the primeness imposed on the hypothesis of the various results is essential.
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In mathematics, there are countless sequences such as arithmetic sequences, geometric sequences, and many more. The Fibonacci sequence is one of them, but it is different from other sequences in that it can be easily found in everyday life. Let’s take a look at patterns that can be discovered in Fibonacci numbers and how we can find them around us.
In a Fibonacci sequence, every number after the first two numbers is the sum of the two preceding ones.
0, 1, 1, 2, 3, 5, 8, 13,…
This is a Fibonacci sequence because 2 is found by adding up the two former numbers, 1 and 1. Likewise, 3 is found by adding 1 and 2, which are the preceding numbers. Based on this rule, we can figure out that the next number in this sequence would be 8 + 13 = 21.
Numbers that appear in the Fibonacci sequence such as 1, 2, 3, 5, 8, and so on are referred to as Fibonacci numbers. Here is a list of Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, … (A000045)
We can write this sequence by using symbols instead of numbers:
\begin{align*}
u_n &= \text{term number} (n)\\ u_{n-1} &= \text{former term} (n-1)\\ u_{n-2} &= \text{term before} (n-2) \end{align*}
$u_n= u_{n-1} + u_{n-2}$
$n$ 0 1 2 3 4 5 6 7 8 $u_n$ 0 1 1 2 3 5 8 13 21
The above table shows Fibonacci numbers for the $n$th term in the Fibonacci sequence. Let’s find out the next term, $u_9$, by using this equation.
\begin{align*}
u_9 &= u_8 + u_7\\ {} &= 21 + 13\\ {} &= 34 \end{align*} Fibonacci sequence formula
There is another way to figure out Fibonacci numbers, other than adding up two former terms. In fact, there is a formula that easily lets us find Fibonacci numbers, just by substituting numbers. Following Nevil Hopley‘s derivation of the Fibonacci sequence’s general term, we proceed as follows:
Let’s define a certain Fibonacci number as $u_n$. We will assume that $u_n = k x^n$ because Fibonacci numbers grow exponentially, and an exponential function can be described as $k x^n$. Since $u_n$ is a Fibonacci number, it is the sum of two former Fibonacci terms.
$u_{n+1} = u_n + u_{n-1}$
This can also be written as:
$k x^{n+1} = k x^n + k x^{n-1}$
We can eliminate $k$ from the equation above by dividing all terms by $k$.
$x^{n+1} = x^n + x^{n-1}.$
This can be simplified by dividing all the terms by $x^{n-1}$.
$x^2 = x + 1$
The value of $x$ can be measured with the quadratic formula.
$\displaystyle x = \frac{1 \pm \sqrt{5}}{2}$
$\displaystyle x_1 = \frac{1 + \sqrt{5}}{2}, x_2 = \frac{1 – \sqrt{5}}{2}$
Therefore, $\displaystyle u_n = k x^n = k_1 \left(\frac{1 + \sqrt{5}}{2}\right)^n$ or $\displaystyle u_n = k x^n = k_2 \left(\frac{1 – \sqrt{5}}{2}\right)^n$.
The sum of two possible values of $u_n$ would also work.
$\displaystyle u_n = k_1 \left(\frac{1 + \sqrt{5}}{2}\right)^n + k_2 \left(\frac{1 – \sqrt{5}}{2}\right)^n$
Since $u_n$ is a formula for Fibonacci numbers, $u_0$ should be 0 and $u_1$ should be 1.
\begin{align*} n&=0&&\implies& u_0 = k_1 \left(\frac{1 + \sqrt{5}}{2}\right)^0 + k_2 \left(\frac{1 – \sqrt{5}}{2}\right)^0 &= 0\\ &&&&k_1+k_2 &= 0\\ &&&&k_2 &= -k_1 \end{align*} \begin{align*} n &= 1 &&\implies&u_1 = k_1 \left(\frac{1 + \sqrt{5}}{2}\right)^1 + k_2 \left(\frac{1 – \sqrt{5}}{2}\right)^1 &= 1\\ &&&&k_1 \left(\frac{1 + \sqrt{5}}{2}\right) + k_2 \left(\frac{1 – \sqrt{5}}{2}\right) &= 1\\ \end{align*} Multipling all terms by $2$ in order to eliminate fractions, we get:
$$k_1 \left(1 + \sqrt{5}\right) + k_2 \left(1 – \sqrt{5}\right) = 2$$
From the previous calculation when we substituted $0$ for $n$, we derived the equation $k_2 = -k_1$. Thus we will rewrite $k_2$ as $-k_1$ in the equation above.
\begin{align*} k_1 \left(1 + \sqrt{5}\right) – k_1 \left(1 – \sqrt{5}\right) &= 2\\ k_1 + \sqrt{5} k_1 – k_1 + \sqrt{5} k_1 &= 2\\ 2\sqrt{5} k_1 &= 2 \end{align*} So we have found that: \begin{align*} k_1 &= \frac{1}{\sqrt{5}}\\ k_2 &= -\frac{1}{\sqrt{5}} \end{align*}
Therefore:
$$_n = k_1 \left(\frac{1 + \sqrt{5}}{2}\right)^n + k_2 \left(\frac{1 – \sqrt{5}}{2}\right)^n = \frac{1}{\sqrt{5}} \left(\frac{1 + \sqrt{5}}{2}\right)^n + \frac{1}{\sqrt{5}} \left(\frac{1 – \sqrt{5}}{2}\right)^n $$ What happens when the numbers are squared?
In fact, something much more interesting happens when the numbers are squared.
$u_n$ 1 1 2 3 5 8 13 21 34 55 $(u_n )^2$ 1 1 4 9 25 64 169 441 1156 3025
When we look at this new sequence closely, we can find a pattern.
$\displaystyle (u_n)^2 + (u_{n+1})^2 = u_{2n+1}$
This sequence can be derived not only by observation, but also by using algebra. Previously, we derived the $n$th term of the Fibonacci sequence:
$$u_n = \frac{1}{\sqrt{5}} \left(\frac{1 + \sqrt{5}}{2}\right)^n + \frac{1}{\sqrt{5}} \left(\frac{1 – \sqrt{5}}{2}\right)^n$$
In order to justify $\displaystyle (u_n)^2 + (u_{n+1})^2 = u_{2n+1}$ in a simpler way, I will write $\displaystyle \frac{1 + \sqrt{5}}{2}$ as $A$ and $\displaystyle \frac{1 – \sqrt{5}}{2}$ as $B$.
In other words, I will write $\displaystyle u_n = \frac{\left(\frac{1 + \sqrt{5}}{2}\right)^n – \left(\frac{1 – \sqrt{5}}{2}\right)^n}{\sqrt{5}} = \frac{A^n – B^n}{\sqrt{5}}$
$\displaystyle u_n^2$ and $\displaystyle u_{n+1}^2$ can be written by using $A$ and $B$:
\begin{align*} u_n^2 &= \left( \frac{A^n – B^n}{\sqrt{5}} \right)^2 = \frac{A^{2n} – 2 A^n B^n + B^{2n}}{5}\\ u_{n+1}^2 &= \left( \frac{A^{n+1} – B^{n+1}}{\sqrt{5}} \right)^2 = \frac{A^{2n+2} – 2 A^ {n+1}B^{n+1} + B^{2n+2}}{5} \\&= \frac{A^{2n} \times A^2 – 2 A^ {n}B^{n} \times A \times B + B^{2n} \times B^2}{5} \end{align*}
Here $A \times B$ equals $-1$, so:
\begin{align*} u_{n+1}^2 &= \frac{A^{2n} \times A^2 + 2 A^ {n}B^{n} + B^{2n} \times B^2}{5}\\ u_n^2 + u_{n+1}^2 &= \frac{A^{2n} + B^{2n} + A^{2n} \times A^2 + 2 A^ {n}B^{n} + B^{2n} \times B^2}{5} \\&= \frac{A^{2n}\left( 1 + A^2\right) + B^{2n}\left( 1 + B^2\right)}{5} \end{align*}
Here,
$$1 + A^2 = 1 + \left(\frac{1 + \sqrt{5}}{2}\right)^2 = \frac{5 + \sqrt{5}}{2}$$ and $$1 + B^2 = 1 + \left(\frac{1 – \sqrt{5}}{2}\right)^2 = \frac{5 – \sqrt{5}}{2}.$$ This means that: \begin{align*} u_n^2 + u_{n+1}^2 &= \frac{A^{2n}\left( \frac{5 + \sqrt{5}}{2} \right) + B^{2n}\left(\frac{5 – \sqrt{5}}{2} \right)}{5}\\ &= \frac{A^{2n} \times \sqrt{5} \left( \frac{\sqrt{5} + 1}{2} \right) – \sqrt{5} B^{2n}\left(\frac{1-\sqrt{5}}{2} \right)}{5}\\ &= \frac{A^{2n} \times \left( \frac{\sqrt{5} + 1}{2} \right) – B^{2n}\left(\frac{1-\sqrt{5}}{2} \right)}{ \sqrt{5}} \end{align*}
Plugging $A$ and $B$ back in gives:
\begin{align*} u_n^2 + u_{n+1}^2 &= \frac{\left( \frac{\sqrt{5} + 1}{2} \right)^{2n} \times \left( \frac{\sqrt{5} + 1}{2} \right) – \left(\frac{1-\sqrt{5}}{2} \right)^{2n}\left(\frac{1-\sqrt{5}}{2} \right)}{ \sqrt{5}}\\&= \frac{\left( \frac{\sqrt{5} + 1}{2} \right)^{2n+1} – \left(\frac{1-\sqrt{5}}{2} \right)^{2n+1}}{ \sqrt{5}}\\ u_{2n+1} &= \frac{\left( \frac{\sqrt{5} + 1}{2} \right)^{2n+1} – \left(\frac{1-\sqrt{5}}{2} \right)^{2n+1}}{ \sqrt{5}} \end{align*}
Therefore, $\displaystyle u_n^2 + u_{n+1}^2 = u_{2n+1}$.
Let’s verify if this equation is right by substituting some numbers.
$n=1 \Rightarrow \left(u_1\right)^2 + \left(u_2\right)^2 = u_3 \Rightarrow (1)^2 + (1)^2 = 2$
$n=2 \Rightarrow \left(u_2\right)^2 + \left(u_3\right)^2 = u_5 \Rightarrow (1)^2 + (2)^2 = 5$
$n=3 \Rightarrow \left(u_3\right)^2 + \left(u_4\right)^2 = u_7 \Rightarrow (2)^2 + (3)^2 = 2$
Another pattern can be found when adding up the first few Fibonacci numbers.
$ \left(u_1\right)^2 = 1$
$ \left(u_1\right)^2 + \left(u_2\right)^2 = 1 + 1 = 2$
$ \left(u_1\right)^2 + \left(u_2\right)^2 + \left(u_3\right)^2 = 1 + 1 +4 = 6$
$ \left(u_1\right)^2 + \left(u_2\right)^2 + \left(u_3\right)^2 + \left(u_4\right)^2 = 1 + 1 +4 + 9 = 15$
$ \left(u_1\right)^2 + \left(u_2\right)^2 + \left(u_3\right)^2 + \left(u_4\right)^2 + \left(u_5\right)^2= 1 + 1 +4 + 9 + 25 = 40$
There seems to be no pattern in the numbers we got from adding up the squares of Fibonacci numbers. However, take a look at the factors of these numbers:
$1 = 1 \times 1$
$2 = 1 \times 2$
$6 = 2 \times 3$
$15 = 3 \times 5$
$40 = 5 \times 8$
The factors are two consecutive Fibonacci numbers. An equation can be derived from this pattern.
$ \left(u_1\right)^2 + \left(u_2\right)^2 + \left(u_3\right)^2 +\ldots + \left(u_{n-2}\right)^2 + \left(u_{n-1}\right)^2 + \left(u_n\right)^2 = u_n \times u_{n+1}$
Applications of the Fibonacci sequence
The Fibonacci sequence can be found in mathematical situations as well as non-mathematical situations. As for mathematical situations, the Fibonacci sequence can be detected in Pascal’s triangle. Pascal’s triangle is usually used in probability, combinatorics or algebra. However, it is possible to find Fibonacci numbers in this triangle, although it is quite challenging to see them.
The Fibonacci numbers can be seen when adding up the numbers in the “shallow” diagonals in Pascal’s triangle.
Sum of the numbers in the first shallow diagonal: $1$
Sum of the numbers in the second shallow diagonal: $1$ Sum of the numbers in the third shallow diagonal: $1+1=2$ Sum of the numbers in the fourth shallow diagonal: $1+2=3$ Sum of the numbers in the fifth shallow diagonal:$1+3+1=5$ Sum of the numbers in the sixth shallow diagonal: $1+4+3=8$
1, 1, 2, 3, 5, and 8 are all consecutive Fibonacci numbers.
In fact, the Fibonacci sequence can be commonly found in nature.
For example, this pineapple has 13 spirals clockwise and 8 spirals anticlockwise. 13 and 8 are adjacent Fibonacci numbers.
The number of flower petals are often Fibonacci numbers.
Number of petals Type of flowers 3 Lilies, amaryllis, iris, tulips 5 Wild rose, columbine, larkspur, buttercup 8 Clematis, delphiniums 13 Cineraria, ragwort, corn marigold 21 Aster, black-eyed susan, chicory
The above table shows the type of flowers depending on the number of petals.
Other than pineapples and flower petals, there are many more situations in which you can encounter a Fibonacci sequence. They might be in your town, in your house, or even inside your room! Look carefully around you. You might see Fibonacci numbers hiding somewhere. Somewhere, maybe right next to you.
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Here we want to give an easy mathematical bootstrap argument why solutions to the time independent 1D Schrödinger equation (TISE) tend to be rather nice. First formally rewrite the differential form$$-\frac{\hbar^2}{2m} \psi^{\prime\prime}(x) + V(x) \psi(x) ~=~ E \psi(x) \tag{1}$$into the int...
[Some time travel comments] Since in the previous paragraph, we have explained how travelling to the future will not necessary result in you to arrive in the future that is resulted as if you have never time travelled (via twin paradox), what is the reason that the past you travelled back, has to be the past you learnt from historical records :?
@0ßelö7 Well, I'd omit the explanation of the notation on the slide itself, and since there seems to be two pairs of formulae, I'd just put one of the two and then say that there's another one with suitable substitutions.
I mean, "Hey, I bet you've always wondered how to prove X - here it is" is interesting. "Hey, you know that statement everyone knows how to prove but doesn't bother to write down? Here is the proof written down" significantly less so
Sorry I have a quick question: For questions like this physics.stackexchange.com/questions/356260/… where the accepted answer clearly does not answer the original question what is the best thing to do; downvote, flag or just leave it?
So this question says express $u^0$ in terms of $u^j$ where $u$ is the four-velocity and I get what $u^0$ and $u^j$ are but I'm a bit confused how to go about this one? I thought maybe using the space-time interval and evaluating for $\frac{dt}{d\tau}$ but it's not workin out for me... :/ Anyone give me a quickie starter please? :p
Although a physics question, this is still important to chemistry. The delocalized electric field is related to the force (and therefore the repulsive potential) between two electrons. This in turn is what we need to solve the Schrödinger Equation to describe molecules. Short answer: You can calculate the expectation value of the corresponding operator, which comes close to the mentioned superposition. — Feodoran13 hours ago
If we take an electron that's delocalised w.r.t position, how can one evaluate the electric field over some space? Is it some superposition or a sort of field with all the charge at the expectation value of the position?
@0ßelö7 I just looked back at chat and noticed Phase's question, I wasn't purposefully ignoring you - do you want me to look over it? Because I don't think I'll gain much personally from reading the slides.
Maybe it's just me having not really done much with Eigenbases but I don't recognise where I "put it in terms of M's eigenbasis". I just wrote it down for some vector v, rather than a space that contains all of the vectors v
If we take an electron that's delocalised w.r.t position, how can one evaluate the electric field over some space? Is it some superposition or a sort of field with all the charge at the expectation value of the position?
Honey, I Shrunk the Kids is a 1989 American comic science fiction film. The directorial debut of Joe Johnston and produced by Walt Disney Pictures, it tells the story of an inventor who accidentally shrinks his and his neighbor's kids to a quarter of an inch with his electromagnetic shrinking machine and throws them out into the backyard with the trash, where they must venture into their backyard to return home while fending off insects and other obstacles.Rick Moranis stars as Wayne Szalinski, the inventor who accidentally shrinks his children, Amy (Amy O'Neill) and Nick (Robert Oliveri). Marcia...
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Expectation Maximization (from here on called EM) is a method for finding parameter estimates in a probabilistic model where some of the random variables (data or latent variables) are unobserved (often called “missing” when referring to data, but “unobserved” or “hidden” when referring to latent variables). It is a frequentist method that computes a maximum likelihood estimate, and maybe that's part of the reason that it wasn't included in this course. It's not that far from the idea of just sampling from unobserved variables, in that you get repeated estimates for the values of the missing data/hidden nodes. I think that EM is easiest understood in the context of k-means clustering, so I will start with that analogy, then generalize the approach.
K-means is an algorithm for clustering some set of data into $K$ clusters. It is a very simple algorithm that implicitly makes the assumption of normally distributed data (so I suppose you could instead make the implicit assumption explicit and use Bayesian methods instead, but let's not get ahead of ourselves…). The idea is that you start by randomly picking initial centers (or means) for your $K$ clusters, then iteratively recompute where the centers should be. The algorithm is as follows:
Input: K, the number of clusters to find, and D, a set of data points Initialize the K clusters, either completely randomly, or by picking K points from D Let k_points be the set of points assigned to cluster k in K, initially empty Repeat until converged: For each point d in D: For each cluster mean k in K: Calculate the distance between d and k Assign d to the k_points set that had the minimum distance between d and k For each cluster k_points: Assign the cluster mean k to be the mean of the data points in k_points
Determining what “converged” means can be slightly sticky depending on the data, but I won't cover that issue here, as this is just meant for an analogy. The take home point is that K-means is a two step process. The first step is to assign all of the data points to clusters based on some closeness metric (Euclidean distance, in this case). The second step is to recompute the parameter k for each cluster (the mean of the cluster) based on the points that were assigned to it. The jump from here to EM is a very small one.
Instead of assigning each data point completely to one cluster, EM computes
expected assignments of the data to clusters (hence Expectation Maximization). To do EM instead of K-means clustering in our example, we really only have to change one part of the algorithm above, the part where we assign data to clusters. Where K-means gave a hard assignment of each data point to each cluster, we calculate the probability of each data point belonging to each cluster, and add each point to each cluster with a weighting for how likely that point is to end up in that cluster. Thus we again have a two step process, in EM commonly called the Expectation step (E-step) and the Maximization step (M-step).
In this step we compute expected assignments of data to clusters based on our current estimates of the model parameters:
For each point d in D: For each cluster mean k in K: Calculate the probability p_k that d is assigned to k Add d to each k_points set with a weighting equal p_k
Computing the probability that d is assigned to k can be done in a few ways. In our simple problem from above, we can just compute the distance from the point to each cluster and then normalize all of the distances by their sum, so that they all sum to one. A perhaps more principled approach would be to have a model for $p(d|k)$, then we can use Bayes law to compute $p(k|d)$ for each k, then normalize the probabilities. If we were assuming normally distributed data, $p(d|k)$ would be a likelihood from a Normal distribution with some mean and some variance whose values were from the current model parameter estimates.
Once we have completed this step, we have the
expected assignments of all of the data points to clusters. With these expected assignments, we are ready for the M-step.
In the M-step we take the expected assignments of the data and re-estimate the model parameters so as to maximize the likelihood of the expected assignments to each cluster:
For each cluster k_points: Assign the cluster mean k to be the mean of the expected data points in k_points (If your model also has a variance, re-estimate that too)
And then we repeat the process until we have converged. EM is guaranteed to increase the likelihood of the data at every step - it is a simple hill climbing approach that is subject to converge upon a local optimum. Thus EM is
not guaranteed to find the maximum likelihood estimate of the parameters given the data, though in practice it often does very well. Wikipedia has a nice example of using EM to estimate the mean and variance of a set of clusters in normally distributed data, similar to our K-means example. This is often called a “Gaussian mixture model.” However, the Wikipedia example will probably make more sense after I generalize the description of the algorithm and introduce the notation they use.
So far we have looked at a specific case of the expectation maximization algorithm. From the way I described it the algorithm may seem like it only is applicable in the context of clustering. That is not the case; clustering just provided an easy introduction to the basic idea of the algorithm. Here I will formalize the general EM algorithm.
What the EM algorithm does is iteratively maximize the expected value of the log likelihood function. This is commonly written down as follows.
The expected log likelihood function is first written down:
$Q(\theta|\theta^t) = E_{Z|\vec{x},\theta^t}[\log L(\theta; \vec{x}, Z)]$
Here $\vec{x}$ is the data that you have, $\theta$ is some set of parameters (the superscript $t$ denotes a particular estimation of them), $Z$ is your set of unobserved variables, and $Q$ is just a dummy name for the function. Let's break this equation down a little bit.
First, we have $\log L(\theta; \vec{x}, Z)$. This is just the log likelihood function that we've seen plenty of times throughout the class. It is a function of $\theta$, $\vec{x}$, and $Z$, though $\vec{x}$ and $Z$ are considered fixed, and $\theta$ is the only argument of interest (that's what the semicolon is supposed to represent). We know $\vec{x}$, because it's a set of observed data, but we don't know $Z$. But then how is $Z$ fixed? That's where the expectation comes in. Remember that an expectation is really just an integral, and we can see from the subscript on the expectation that $Z$ is the variable of integration. So, for any particular point in the integration, $Z$ is fixed, and our log likelihood function makes sense.
So why are we taking an expectation with respect to $Z|\vec{x},\theta^t$? Because we don't know $Z$, so we have to average it out somehow. Bayesian methods do some kind of sampling; here we just take an expected value (conditioned on the things that we know, which is the data and our current estimates of the parameters). It turns out that EM can actually give you a probability distribution over $Z$, just not over $\theta$, but we don't really need to worry about that here.
Ok, so, computing this equation gives us the log likelihood function of the
parameters given the data and the previous estimation of the parameters. Doing the calculations necessary for this equation constitutes the E-step mentioned above (if you were to look at the equations on Wikipedia for the Gaussian mixture model, you could convince yourself that this is true - in order to find the log likelihood that's desired, you have to sum over the data). Once we have found $Q(\theta|\theta^t)$, we can maximize it with respect to $\theta$:
$\theta^{t+1} = \mathrm{argmax}_\theta Q(\theta|\theta^t)$
This is the M-step. We are just re-estimating the parameters given the expected log likelihood function that we calculated previously. It hopefully is pretty clear how this relates to the specific M-step described above.
Now that we know the equations, we could go through the example again with the exact math, but Wikipedia does it better than I could, so I just refer you to that page again.
You have a Gaussian Mixture Model with two Gaussians. The variables are distributed as follows:
$X_i|Z_i=1\ \sim\ Normal(\mu_1, \Sigma_1)$
$X_i|Z_i=2\ \sim\ Normal(\mu_2, \Sigma_2)$
$P(Z_i = 1) = \tau$
$P(Z_i = 2) = 1 - \tau$
All $X_i$ are observed, all $Z_i$ are unobserved. Derive the formula $Q(\theta|\theta^t)$. Then derive the maximum likelihood update to that formula, thus giving you the new parameters $\theta^{t+1}$.
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Mapper – A discrete generalization of the Reeb graph This is the third of a series of posts on cluster-algorithms and ideas in data analysis. Mapper is a construction that uses a given cluster-algorithm to associate a simplicial complex to a reference map on a given data set. It was introduced by Carlsson–Mémoli–Singh in [1] and lies at the core of the of the topological data analysis startup Ayasdi. A good reference, I personally enjoyed reading, is [2]. Mapper is closely related to the Reeb graph of a real-valued function on a topological space. Just as the Reeb graph it is able to capture certain topological and geometrical features of the underlying space. The nerve of a cover
Let $X$ be a topological space. We associate to each open cover $\mathcal{U}=\{U_i\}_I$ a simplicial complex $\check N(\mathcal{U})$ called the
nerve of $\mathcal{U}$ defined as follows: there is a vertex $i$ for each $U_i$, and a $k$-simplex spanned by $i_0,…,i_k$ whenever the intersection $U_{i_0} \cap \ldots \cap U_{i_k}$ of the corresponding sets is nonempty. Reference maps, filters or lenses
Let $f:X \to Y$ be a continuous map between topological spaces $X$ and $Y$. Any open cover $\mathcal{V}$ of $Y$ induces an open cover $f^*\mathcal{V}$ of $X$ obtained by the pullback under $f$, i.e. \[ f^*\mathcal{V} := \big\{ f^{-1}(V) \ | \ {V \in \mathcal{V}} \big\}. \] For example set $Y = \mathbb{R}$ and think of $f$ as a height function on $X$. In the context of the Mapper construction these maps are called
filters or lenses. Cluster functions
For a given space $X$ let $\mathcal{P}(X)$ denote its power set. Then we call an assignment \[ X \leadsto \pi(X)\subset \mathcal{P}(X), \] that associates to a space $X$ a family $\pi(X)$ of subsets of $X$, a
cluster function or cluster algorithm — note that this is not really a function in the mathematical sense, but rather in the sense of computer science and programming. We refer to an element in $\pi(X)$ as a cluster. We don’t require $\pi(X)$ to satisfy any properties at that point. From a programming perspective think of it as the signature of a function implementing a certain cluster algorithm.
Given a family of open sets $\mathcal{U}=\{ U_i \}_I$ we define another family $\pi_*(\mathcal{U})$ of open sets by applying $\pi$ to each set $U_i$ and collecting the resulting clusters, i.e. we define \[ \pi_*(\mathcal{U}) := \bigcup_{i \in I} \pi(U_i). \] Finally we have everything in place to define the Mapper-construction.
Put it all together
Let $X$ be a topological space (the data set), $\pi$ be a cluster algorithm, $f:X \to Y$ a reference map to a topological space $Y$, and $\mathcal{V}$ an open cover of $Y$. Then the result of Mapper applied to this triple is the simplicial complex $\mathcal{M}(\pi, f, \mathcal{V})$ defined by \[ \mathcal{M}(\pi, f, \mathcal{V}) := \check{N} \big( \pi_*(f^*\mathcal{V}) \big). \]
References [1] G. Carlsson, F. Mémoli and G. Singh, Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition, Eurographics Symposium on Point Based Graphics (2007), pp. 91–100. [2] G. Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009), pp. 255–308.
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For any real $2\times 2$ matrix A, $$ \begin{pmatrix} a & b\\ c & d \end{pmatrix}\in \text{Mat}_{\mathbb{R}}(2) $$
Let $S$ be the set $$ S=\{ A\in \text{Mat}_{\mathbb{R}}(2) \mid a^2+b^2+c^2+d^2=1,\det(A)=0\}. $$ Using regular value theorem, I have proved that this is a two dimensional submanifold of $\text{Mat}_{\mathbb{R}}(2)$. And now I want to show that the map $$ \pi:S\to \mathbb{R}P(1) $$ Taking $A\in S$ to its range $ran(A)\subseteq \mathbb{R}^2$, is smooth. Since we know $A$ has rank $1$, so at least one of its one of its row or column is not a zero vector. I think it is possible to use the two charts where $U_0$ is the chart for which it has no zero rows and $U_1$ be the chart for which it has no zero columns. However, I have trouble think of the map associated with the chart. Is there any way such that we can avoid talking about the charts and prove the smoothness?
Intuitively speaking, the map $\pi$ just takes each vector to its one dimensional span, thus we can identify it with points in $\mathbb{R}P(1)$, but how can we be more a little bit more rigorous?
Also, is there a quick way of saying $S$ is a 2-torus?
Any hint would be appreciated, thanks!
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May 27th, 2017, 08:51 AM
# 31
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From: USA
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How can infinity exist inside of presized location.
0 - infinity - 1
0 - infinity - 0.1
0 - infinity - 0.001
As I am still able to pull frames of infinite from each interval.
0 - ( 0.09... 0.099... 0.0999... ) - 0.1
May 28th, 2017, 05:22 AM
# 32
Senior Member
Joined: Jun 2014
From: USA
Posts: 573
Thanks: 44
Quote:
We can have a finite set of real numbers: e.g. {0.5, 0.75}.
We can have a set of real numbers with cardinality $\aleph_0$: e.g. {0.1, 0.11, 0.111, 0.1111, ... }.
We can also have a set of real numbers with cardinality $2^{\aleph_0}$: e.g. (0, 1).
Now here is a million dollar question for you. Can we have a set of real numbers (or, any set really) that has a cardinality X such that $\aleph_0 < X < 2^{\aleph_0}$? A solution to this question would prove whether or not the continuum hypothesis is true.
https://en.wikipedia.org/wiki/Continuum_hypothesis
May 28th, 2017, 10:39 AM
# 34
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From: USA
Posts: 573
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Quote:
The cardinality of $\mathbb{N}$ is $\aleph_0$.
The cardinality of both $P( \,\mathbb{N}) \,$ and $\mathbb{R}$ is $2^{\aleph_0}$.
See my previous posts for what $\mathbb{N}$ and $P( \,\mathbb{N}) \,$ are if you are unsure (the natural numbers and the powerset of the natural numbers, respectively). The set of real numbers is denoted $\mathbb{R}$.
When referring to the cardinality of a set, you may enclose the set in "| |". For example, $|P( \,\mathbb{N}) \,| = |\mathbb{R}| = 2^{\aleph_0}$.
Don't go sinking into a black hole, as there is no need for that.
Infinite sets can have properties that are counter intuitive. Trying to solve the countinuum hypothesis (working "on X" as we put it) is probably too much, as the hypothesis has gone unsolved for a couple hundred years. Just learning the basics is a good start.
Maybe I should have shown you something simpler to start. For example, we can show that the cardinality of the set of positive even integers is the same as the cardinality of the set of natural numbers by showing that a bijective function exists between the two sets. To go from the positive even numbers to the natural numbers, our function could be defined as follows:
$$\text{Let } f : \{ x \in \mathbb{N} : x \text{ is even} \} \rightarrow \mathbb{N}$$
$$f( \,x) \, = \frac{x}{2}$$
This happens to be a bijective function because:
$$\{1 = \frac{2}{2}, 2 = \frac{4}{2}, 3 = \frac{6}{2}, ... \}$$
Knowing that there exists a bijective function $f$ from the set of positive even integers onto the natural numbers shows that the two sets' cardinalities are equal, so we write:
$$|\{ x \in \mathbb{N} : x \text{ is even}\}| = |\mathbb{N}|$$
May 28th, 2017, 11:19 AM
# 35
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Joined: May 2017
From: USA
Posts: 31
Thanks: 0
Thank you and I will study more farther, I really will.
I would like so any reader of this topic can understand it easy.
Little help here:
You asked to name number or set of numbers that is more than one set of numbers and less then another set. Am I right? Like:
N0<X<2N0
Where each set of numbers represents "infinity"?
Last edited by Microlab; May 28th, 2017 at 11:27 AM.
May 28th, 2017, 11:42 AM
# 36
Senior Member
Joined: Jun 2014
From: USA
Posts: 573
Thanks: 44
Quote:
More specifically, does there exist a set $S$ such that:
$$|\mathbb{N}| < |S| < |P( \,\mathbb{N}) \,| = |\mathbb{R}|$$
If we denote the cardinality of $S$ as simply $X$ (i.e., let $|S| = X$), an equivalent way of asking this question is to ask if there exists a cardinal number $X$ such that:
$$\aleph_0 < X < 2^{\aleph_0}$$
where (to help you with notation)
$$\aleph_0 = |\mathbb{N}| < X = |S| < 2^{\aleph_0} = |P( \,\mathbb{N}) \,| = |\mathbb{R}|$$
May 28th, 2017, 12:33 PM
# 39
Senior Member
Joined: Jun 2014
From: USA
Posts: 573
Thanks: 44
https://oeis.org/wiki/List_of_LaTeX_...atical_symbols
Is there some symbol in particular that you would like clarification on? Tip: If I've used the symbol, you can quote my post to see how I wrote it, then ask me what it means.
Last edited by AplanisTophet; May 28th, 2017 at 12:35 PM.
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Einstein probably did not say, "Everything should be made as simple aspossible, but _no simpler_." However, somebody did, and somebody wasright. One of the biggest problems in teaching programming is theconstant pretense that we are not doing complicated mathematics, andthe resulting attempt to hide the math.
There is a lovely little book called Mathematics Made Difficult, whosepremise is that refusing to tackle topics of modest complexity makesunderstanding far more difficult. With examples, of course.
I have a similar complaint about freshman college physics courses thatattempt to get by with no calculus.
One of my favorite matth examples is how the use of elementarydifferential equations and Taylor series simplifies the definition oftrigonometry. Define the exponential function by the equation
y' = y
which says that the growth rate of the function is proportional to itscurrent value. Bring in examples from compound interest, biologicalgrowth, inflationary cosmology,... The function we are looking for is
exp(x) = sum_0^\infty (x^n)/n!
or any multiple of it. If you know that d(x^n)/dx is nx^(n-1), thenyou can see that the derivative of the power series for exp is itself.
Now solve
y' = -y (negative of exp)y'' = y (hyperbolic sinh and cosh functions)y'' = -y (sin and cos functions)
in the same way, and look at the relations among their power series.
Now derive e^(i\pi)+1=0 from e^(i\theta) = cos \theta + i sin \theta,which follows directly from the power series above, and then switch tolinear algebra to get the sum, difference, and other formulae, andgeometry to get the solutions of triangles.
If anybody has difficulty with any of this I can point you totextbooks using these methods.
If you would like a bit more of a challenge, we can do this all overagain in elliptic and hyperbolic geometry, where we don't have similartriangles. ^_^
>> What is a Python module? #==========================> Common answer is "a file containing Python source code?",> but I'm questioning whether that's sufficient definition.> How about an importable .pyc or .pyd, with no .py in the> picture. That's a module too, no?> ** Import Star #====================================> When is> import *> a good idea?> There's all this righteous moralistic hoopla that gets built up> against specific idioms, to where eval( ) appears to be fighting> for its very existence... Mary wants to keep her little lambda.> So I'd rather phrase these in the positive, as in when IS it> a good idea... e.g. to use semi-colons between statements.> ** Another student question: #=========================> Why does all([]) return True by default? Is this a case> of half full versus half empty?> ** Correcting a misconception #========================> No, docstrings do NOT have to be triple quoted.> Kirby>> _______________________________________________> Edu-sig mailing list> [hidden email] > http://mail.python.org/mailman/listinfo/edu-sig >>
On Wed, Mar 30, 2011 at 11:57 AM, Edward Cherlin <[hidden email]> wrote:
Einstein probably did not say, "Everything should be made as simple aspossible, but _no simpler_." However, somebody did, and somebody wasright. One of the biggest problems in teaching programming is theconstant pretense that we are not doing complicated mathematics, andthe resulting attempt to hide the math.
There is a lovely little book called Mathematics Made Difficult, whosepremise is that refusing to tackle topics of modest complexity makesunderstanding far more difficult. With examples, of course.
I have a similar complaint about freshman college physics courses thatattempt to get by with no calculus.
One of my favorite matth examples is how the use of elementarydifferential equations and Taylor series simplifies the definition oftrigonometry. Define the exponential function by the equation
y' = y
which says that the growth rate of the function is proportional to itscurrent value. Bring in examples from compound interest, biologicalgrowth, inflationary cosmology,... The function we are looking for is
exp(x) = sum_0^\infty (x^n)/n!
or any multiple of it. If you know that d(x^n)/dx is nx^(n-1), thenyou can see that the derivative of the power series for exp is itself.
Now solve
y' = -y (negative of exp)y'' = y (hyperbolic sinh and cosh functions)y'' = -y (sin and cos functions)
in the same way, and look at the relations among their power series.
Now derive e^(i\pi)+1=0 from e^(i\theta) = cos \theta + i sin \theta,which follows directly from the power series above, and then switch tolinear algebra to get the sum, difference, and other formulae, andgeometry to get the solutions of triangles.
If anybody has difficulty with any of this I can point you totextbooks using these methods.
If you would like a bit more of a challenge, we can do this all overagain in elliptic and hyperbolic geometry, where we don't have similartriangles. ^_^
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Abstract for the talk on 09.02.2018 (11:00 h) Arbeitsgemeinschaft ANGEWANDTE ANALYSIS Tobias Ried(Karlsruher Institut für Technologie) Gevrey smoothing of weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules
We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with finite mass, energy and entropy, that is, $f_0 \in L^1_2(\R^d) \cap L \log L(\R^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity.
This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers.
(Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)
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Now showing items 31-40 of 182
Jet-hadron correlations relative to the event plane at the LHC with ALICE
(Elsevier, 2017-11)
In ultra relativistic heavy-ion collisions at the Large Hadron Collider (LHC), conditions are met to produce a hot, dense and strongly interacting medium known as the Quark Gluon Plasma (QGP). Quarks and gluons from incoming ...
Measurements of the dielectron continuum in pp, p-Pb and Pb-Pb collisions with ALICE at the LHC
(Elsevier, 2017-11)
Dielectrons produced in ultra-relativistic heavy-ion collisions provide a unique probe of the whole system evolution as they are unperturbed by final-state interactions. The dielectron continuum is extremely rich in physics ...
Exploring jet substructures with jet shapes in ALICE
(Elsevier, 2017-11)
The characterization of the jet substructure can give insight into the microscopic nature of the modification induced on high-momentum partons by the Quark-Gluon Plasma that is formed in ultra-relativistic heavy-ion ...
Measurements of the nuclear modification factor and elliptic flow of leptons from heavy-flavour hadron decays in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 and 5.02 TeV with ALICE
(Elsevier, 2017-11)
We present the ALICE results on the nuclear modification factor and elliptic flow of electrons and muons from open heavy-flavour hadron decays at mid-rapidity and forward rapidity in Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ ...
Multiplicity dependence of the average transverse momentum in pp, p-Pb, and Pb-Pb collisions at the LHC
(Elsevier, 2013-12)
The average transverse momentum <$p_T$> versus the charged-particle multiplicity $N_{ch}$ was measured in p-Pb collisions at a collision energy per nucleon-nucleon pair $\sqrt{s_{NN}}$ = 5.02 TeV and in pp collisions at ...
Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider
(American Physical Society, 2016-02)
The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has been studied using the ALICE detector at the LHC. The spectra exhibit a significant hardening with ...
Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV
(Elsevier, 2016-02)
Two-particle angular correlations between trigger particles in the forward pseudorapidity range ($2.5 < |\eta| < 4.0$) and associated particles in the central range ($|\eta| < 1.0$) are measured with the ALICE detector in ...
K$^{*}(892)^{0}$ and $\phi(1020)$ meson production at high transverse momentum in pp and Pb-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 2.76 TeV
(American Physical Society, 2017-06)
The production of K$^{*}(892)^{0}$ and $\phi(1020)$ mesons in proton-proton (pp) and lead-lead (Pb-Pb) collisions at $\sqrt{s_\mathrm{NN}} =$ 2.76 TeV has been analyzed using a high luminosity data sample accumulated in ...
Production of $K*(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV
(Springer, 2012-10)
The production of K*(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 N/dydp_T$ at midrapidity |y|<0.5 in ...
Directed flow of charged particles at mid-rapidity relative to the spectator plane in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV
(American Physical Society, 2013-12)
The directed flow of charged particles at midrapidity is measured in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV relative to the collision plane defined by the spectator nucleons. Both, the rapidity odd ($v_1^{odd}$) and ...
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Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
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№ 9
All Issues Gorbachuk V. I.
Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 579-587
Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 148–159
For uniformly stable bounded analytic $C_0$-semigroups $\{T(t)\} t ≥ 0$ of linear operators in a Banach space $B$, we study the behavior of their orbits $T (t)x, x ∈ B$, at infinity. We also analyze the relationship between the order of approaching the orbit $T (t)x$ to zero as $t → ∞$ and the degree of smoothness of the vector $x$ with respect to the operator $A^{-1}$ inverse to the generator A of the semigroup $\{T(t)\}_{t \geq 0}$. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than $e^{-at^{\alpha}}$, where $a > 0,\; 0 < \alpha < \pi/(2 (\pi - 0 )),\; \theta$ is the angle of analyticity of $\{T(t)\}_{t \geq 0}$, and the collection of these orbits is dense in the set of all orbits.
Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1120-1122
Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 3-11
Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 582–600
We give a brief survey of results on functional analysis obtained at the Institute of Mathematics of the Ukrainian National Academy of Sciences from the day of its foundation.
Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 608-615
We describe all weak solutions of a first-order differential equation in a Banach space on (0, ∞) and investigate their behavior in the neighborhood of zero. We use the results obtained to establish necessary and sufficient conditions for the essential maximal dissipativity of a dissipative operator in a Hilbert space.
Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 510–516
We study the dependence of the rate of growth of the extended spectral measure of a self-adjoint operator at infinity on the order of singularity of the vector on which this measure is considered.
Ukr. Mat. Zh. - 1995. - 47, № 10. - pp. 1416–1417
Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems
Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 55–62
Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1299–1313
Ukr. Mat. Zh. - 1986. - 38, № 5. - pp. 569–575
Ukr. Mat. Zh. - 1986. - 38, № 3. - pp. 309–314
Ukr. Mat. Zh. - 1985. - 37, № 3. - pp. 342–343
Ukr. Mat. Zh. - 1984. - 36, № 5. - pp. 559 – 567
A study of certain boundary properties of solutions of the equation Δ u - c ² u = 0 in the half plane
Ukr. Mat. Zh. - 1983. - 35, № 6. - pp. 681-688
Ukr. Mat. Zh. - 1983. - 35, № 5. - pp. 557—562
Ukr. Mat. Zh. - 1983. - 35, № 5. - pp. 617—621
Ukr. Mat. Zh. - 1982. - 34, № 2. - pp. 144-150
Estimates of the accuracy of an approximate solution of the cauchy problem for Laplace's equation in an infinite strip
Ukr. Mat. Zh. - 1980. - 32, № 6. - pp. 731–736
Conditions for the oscillation of solutions of a class of elliptic equations of high orders with constant coefficients
Ukr. Mat. Zh. - 1980. - 32, № 5. - pp. 593–600
Ukr. Mat. Zh. - 1979. - 31, № 2. - pp. 123–131
Ukr. Mat. Zh. - 1978. - 30, № 4. - pp. 452–461
Some questions of the spectral theory of differential equations of elliptic type in the space of vector-functions
Ukr. Mat. Zh. - 1976. - 28, № 3. - pp. 313–324
Some questions of the spectral theory of differential equations of elliptic type in the space of vector functions on a finite interval
Ukr. Mat. Zh. - 1976. - 28, № 1. - pp. 12–26
Ukr. Mat. Zh. - 1976. - 28, № 1. - pp. 79–82
Ukr. Mat. Zh. - 1975. - 27, № 1. - pp. 74–81
The spectrum of self-adjoint extensions of the minimal operator generated by a Sturm-Liouville equation with operator potential
Ukr. Mat. Zh. - 1972. - 24, № 6. - pp. 726—734
Ukr. Mat. Zh. - 1972. - 24, № 3. - pp. 291—304
Ukr. Mat. Zh. - 1972. - 24, № 2. - pp. 147—161
On the pithiness of bounds in converse theorems on the approximation of functions on regular compacta of the complex plane
Ukr. Mat. Zh. - 1972. - 24, № 1. - pp. 83–92
Problems of the spectral theory of the second order linear differential equation with unbounded operator coefficients
Ukr. Mat. Zh. - 1971. - 23, № 1. - pp. 3–14
Ukr. Mat. Zh. - 1967. - 19, № 4. - pp. 119–125
Ukr. Mat. Zh. - 1965. - 17, № 3. - pp. 43-58
Ukr. Mat. Zh. - 1964. - 16, № 2. - pp. 232-237
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Abstract: In this talk, we explore explicit cross-sections to the horocycle and geodesic flows on $\operatorname{SL}(2, \mathbb{R})/G_q$, with $q \geq 3$. Our approach relies on extending properties of the primitive integers $\mathbb{Z}_\text{prim}^2 := \{(a, b) \in \mathbb{Z}^2 \mid \gcd(a, b) = 1\}$ to the discrete orbits $\Lambda_q := G_q (1, 0)^T$ of the linear action of $G_q$ on the plane $\mathbb{R}^2$. We present an algorithm for generating the elements of $\Lambda_q$ that extends the classical Stern-Brocot process, and from that derive another algorithm for generating the elements of $\Lambda_q$ in planar strips in increasing order of slope. We parametrize those two algorithms using what we refer to as the \emph{symmetric $G_q$-Farey map}, and \emph{$G_q$-BCZ map}, and demonstrate that they are the first return maps of the geodesic and horocycle flows resp. on $\operatorname{SL}(2, \mathbb{R})/G_q$ to particular cross-sections. Using homogeneous dynamics, we then show how to extend several classical results on the statistics of the Farey fractions, and the symbolic dynamics of the geodesic flow on the modular surface to our setting using the $G_q$-BCZ and symmetric $G_q$-Farey maps. This talk is self-contained and does not assume any prior knowledge of Hecke triangle groups or homogeneous dynamics.
Abstract: We develop a correspondence between Borel equivalence relations induced by closed subgroups of $S_\infty$ and symmetric models of set theory without choice, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998).For example, we show that the equivalence relation $\cong^\ast_{\omega+1,0}$ is strictly below $\cong^\ast_{\omega+1,
Abstract: We present a novel algorithm to handle both equality and inequality constraints ininfinite dimensional optimization problems. The inequality constraints are tackledvia a nonstandard penalty. On the other hand, the equality constraints are handledusing trust region methods. The latter permits inexact PDE solves. As applications,we consider PDE constrained optimization (PDECO) problems with contact typeconstraints and topology optimization problems.
We will also introduce novel optimal control concepts within the realm of PDECOproblems with fractional/nonlocal PDEs as constraints and discuss their applicationsin geophysics and imaging sciences. We will further illustrate the role of fractionaloperators as a regularizer in machine learning.
We conclude this talk by introducing a general framework based on Gibbs posteriorto update the belief distributions for inverse problems governed by PDEs. Contrary totraditional Bayesian analysis, noise model is not assumed to be known.
Abstract: We consider transport equations on graphs, where mass is distributed over vertices and is transported along the edges. The first part of the talk will deal with the graph analogue of the Wasserstein distance, in the particular case where the notion of density along edges is inspired by the upwind numerical schemes. This natural notion of interpolation however leads to the fact that Wasserstein distance is only a quasi-metric. In the second part of the talk we will interpret the nonlocal-interaction equation equations on graphs as gradient flows with respect to the graph-Wasserstain quasi-metric of the nonlocal-interaction energy. We show that for graphs representing data sampled from a manifold, the solutions of the nonlocal-interaction equations on graphs converge to solutions of an integral equation on the manifold. We also show that the limiting equation is a gradient flow of the nonlocal-interaction energy with respect to a nonlocal analogue of the Wasserstein metric.
Abstract: Arithmetic Transfer conjectures are the analogues of WeiZhang's Arithmetic Fundamental Lemma conjecture in the presence oframification. Some such conjectures were given in two papers bySmithling, Zhang and myself. I will present more such conjectures. Thisis joint work in progress with S. Kudla, B. Smithling and W. Zhang.
Abstract: Network science is a rapidly expanding field, with a large and growing body of work on network-based dynamical processes. Most theoretical results in this area rely on the so-called "locally tree-like approximation" (which assumes that one can ignore small loops in a network). This is, however, usually an `uncontrolled' approximation, in the sense that the magnitudes of the error are typically unknown, although numerical results show that this error is often surprisingly small. In our work, we place this approximation on more rigorous footing by calculating the magnitude of deviations away from tree-based theories in the context of network cascades (i.e., a network dynamical process describing the spread of activity through a network). For this widely applicable problem, we discuss the conditions under which tree-like approximations give good results, and also explain the reasons for deviation from this approximation. More specifically, we show that these deviations are negligible for networks with a large number of network links, justifying why tree-based theories appear to work well for most real-world networks.
Abstract: We present recent results on time-scales separation in fluidmechanics. The fundamental mechanism to detect in a precisequantitative manner is commonly referred to as fluid mixing. Itsinteraction with advection, diffusion and nonlocal effects produces avariety of time-scales which explain many experimental and numericalresults related to hydrodynamic stability and turbulence theory.
Abstract: Graph embeddings, a class of dimensionality reduction techniques designed for relational data, have proven useful in exploring and modeling network structure. Most dimensionality reduction methods allow out-of-sample extensions, by which an embedding can be applied to observations not present in the training set. Applied to graphs, the out-of-sample extension problem concerns how to compute the embedding of a vertex that is added to the graph after an embedding has already been computed. In this talk, we will consider the out-of-sample extension problem for two graph embedding procedures: the adjacency spectral embedding and the Laplacian spectral embedding. In both cases, we prove that when the underlying graph is generated according to a latent space model called the random dot product graph, which includes the popular stochastic block model as a special case, an out-of-sample extension based on a least-squares objective obeys a central limit theorem. Our results also yield a convenient framework in which to analyze trade-offs between estimation accuracy and computational expenses, which we will explore briefly.
4176 Campus Drive - William E. Kirwan HallCollege Park, MD 20742-4015P: 301.405.5047 | F: 301.314.0827
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I’m currently taking a (meta)logic class. There are assigned problem sets. A lot of people either don’t know how to type logical symbols or else cannot be bothered to fight with Word. I’m a fan of LaTeX. I like it for several reasons, one of them being easy use of logical symbols.
There are a lot of guides to using LaTeX. To my knowledge, none start from nothing and end with just what’s needed for a logic class. So here I fill in that void. My goal is to be comprehensive enough to cover what’s needed to type up assignments for a logic class while not including anything else so someone can be up and running with just this guide in a few minutes.
Setting Up
First, you need something to edit your text and something to compile it to a PDF or whatever other format you like. I personally use Overleaf. It’s a free, online application that lets you type in one column with live updates to what it looks like on the page in the other column. It also has templates, allows collaboration, and has some other nice features that are not important to our purposes here. (Full disclosure: The link is a referral link. If you refer people, you get extra storage space and pro features for free. The default free features and space are fine, though.)
There are other popular options. If you need to compile offline, I suggest TeXmaker. If you go this route, you need to download MiKTeX. If you want to write something very long, you may want to type into a text editor and then copy and paste into Overleaf or TeXmaker. (By “long” I mean over fifty pages, give or take based on things like included pictures.)
Onto the actual typing process. If you’re using Overleaf, go to the “My Projects” page and then create a new project. Choose “blank paper”. Then you’ll have this code:
\documentclass{article}
\usepackage[utf8]{inputenc}
\begin{document}
(Type your content here.)
\end{document}
If you’re not using Overleaf, go ahead and put that code into your document.
There is a bit of tweaking to the basic template to make this better. Before the \begin{document} line, add a line containing just \usepackage{amsmath}. Then add lines with add \title{TITLE} and \author{NAME}. Then after the \begin{document} line, add a line saying \maketitle. If you want it to not be huge, type \small\maketitle\normalsize. (The \small makes it small. The \normalsize makes the stuff after it normal size.) At this point my document looks like this.
\documentclass{article}
\usepackage[utf8]{inputenc}
\title{Phil 125 Homework Set 2}
\author{Nichole Smith}
\begin{document}
\small\maketitle\normalsize
(Type your content here.)
\end{document} Typing the Document
Everything after this replaces “(Type your content here.)”.
Typing letters and numbers works as you would expect. Certain symbols are used by the code so typing them is not straightforward. (The & and squiggle brackets are the most notable here.) Single line breaks are ignored. So if you type some stuff, hit return/enter, and then type some more, it will show up as one paragraph. (This can be useful. I like to type every step of a proof in a new line. Then it compiles into a paragraph.) Double line breaks give you a new paragraph. If you want extra space, use \vspace{1cm} as its own paragraph. You can choose lengths other than 1cm if you want.
Onto the logic specific stuff. Of critical importance is math mode. Whenever you surround text with dollar signs ($) LaTeX treats it as mathematical symbols. So, if you type $x$ it will be italicized like a variable should be. Math mode does not have spaces. So $two words$ will not have a space between them. (If you need a space while in math mode for some reason, “\ ” gives you a space. That is a backslash with a space after it.)
Note all logical symbols have to be typed in math mode. The logical symbols:
\land gives you the and symbol \lor gives you the or symbol \lnot gives you the not symbol \rightarrow gives you the material conditional arrow \Rightarrow gives you the logical implication arrow \leftrightarrow gives you the biconditional arrow \Leftrightarrow gives you the logical equivalence arrow (So, capitalizing the arrow tags makes them the bigger arrows) = is the equal sign Parentheses are parentheses \subset gives you the strict subset symbol \subseteq gives you the subset symbol In general, typing \not immediately before another symbol puts a slash through it. E.g. \not\subseteq gives you the not a subset symbol \in gives you the element symbol \times gives you the times sign \neq gives you the not equal sign > and < can be typed directly. To get the or equal to versions, type \geq or \leq \emptyset gives you the empty set symbol \{ and \} give you squiggle brackets \& gives you the & symbol \top and \bot give you the tautology and contradiction symbols. \Alpha and \alpha give you upper and lower case alpha. The other Greek letters are similar. | gives you the Sheffer stroke and \downarrow gives you the Peirce dagger. An underscore gives you subscript. A caret gives you superscript. E.g. p sub 1 is typed $p_1$. \hdots gives you a nice ellipsis. Use \cdots if you want them elevated to the middle of the line. Anything on a line after % will not be compiled. So if you want to make a note to self, you can.
I think this covers it. Most of them are pretty straightforward. If you do need more, this webpage has a nifty list. Or, detexify lets you just draw what you want, and it gives you the code. At this point you’re ready to type stuff.
I will provide an example now. Say problem 2 asks you to symbolize “neither both
p and q, nor q only if p” with the and, material conditional, and nor operators. Then you type:
2. The sentence “neither both $p$ and $q$, nor $q$ only if $p$” symbolized with the and, material conditional, and nor operators is $(p\land q)\downarrow(q\rightarrow p)$.
Truth Tables
LaTeX can also handle tables very nicely. If you’re lazy, there are online tools to make tables. They have quite a few options. You’re probably fine using that.
I prefer more control for my truth tables. Again, you’re fine without. But in case anyone is interested, I’ll explain. Maybe you’ll want to be able to edit the code the generator spits out. (I often use a generator to start and then tweak as needed.) First, here’s the code for the truth table for
p_1 or not p_1:
\begin{tabular}{c|cccc}
$p_1$ & $p_1$ & $\lor$ & $\lnot$ & $p_1$ \\
\hline
T & & \textcolor{red}{T} & F & \\
F & & \textcolor{red}{T} & T & \\
\end{tabular}
How do you construct this thing? First set up the tabular environment:
\begin{tabular}{}
\end{tabular}
The second set of squiggle brackets after \begin let you set up the columns. Each c gives a center aligned column. If you want left or right aligned columned, use l or r instead of c. Yes, you can mix the three. The | gives a vertical line going down the entire table. Note for truth tables you want a column for every single symbol. That way nothing is under the variables and you can have a straight line of Ts and Fs under the connectives. So, for
p_1 or not p_1 we want a column for p_1, a bar, then columns for each of p_1, or, not, and p_1. That’s four more. So, we have:
\begin{tabular}{c|cccc}
\end{tabular}
We have the table set up. Now to fill it in. The first line of the table has the atomic sentences on the left and then the sentence in question on the right. Type the content of each column, separated by &. Then end the line with \\. So, to have the first line of the truth table:
\begin{tabular}{c|cccc}
$p_1$ & $p_1$ & $\lor$ & $\lnot$ & $p_1$ \\
\end{tabular}
To have the horizontal line, type \hline on its own line. Then more on to the next row, doing the same thing you did for the first row. Note that if you want nothing in a certain spot, just leave the space between the two &s empty. So, for the second row, you want a T under the first
p_1 (The one on the left side of the table), then nothing under the first one on the right, then a T under the and sign, an F under the not sign, and then nothing under the last p_1. The third line is similar. Now we have:
\begin{tabular}{c|cccc}
$p_1$ & $p_1$ & $\lor$ & $\lnot$ & $p_1$ \\
\hline
T & & T & F & \\
F & & T & T & \\
\end{tabular}
This is a fine truth table. But, maybe you want to bold the truth values for the main connective. To make T bold, type \textbf{T}. You can replace “T” with other text, of course. If you’re using Overleaf, highlighting the text and pressing Ctrl+B will put the tag in automatically.
This brings us to the complete table as quoted in the beginning of this section.
The comment section is open. Questions and suggestions are welcome.
(Edit notes: As Soren pointed out, I originally put the wrong symbol for commenting. I also realized the amsmath package is not needed, so I removed that. Since these are usually printed in black and white anyway, I got rid of color in favor of boldface type. This has the added benefit of avoiding the need for packages entirely. In the third edit I added the \leq and \geq tags as well as \hdots because I realized they’re needed for indexing variables. \hdots requires the amsmath package, so I added that line back in. Using bold instead of color still seems to be better.)
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Sinusoids (sines and cosines) are the eigenfunctions of the wave equation. That is if you look for a set of functions $f_{i,\omega}$ for which it is true that$$ \frac{\partial^2 f_{i,\omega}}{\partial x} \pm \frac{1}{c^2} \frac{\partial^2 f_{i,\omega}}{\partial t} = \lambda f_{i,\omega} \;, $$for some real number $\lambda$, then all the answers would be some combinations of $\sin(kx \mp \omega t)$ and $\cos (kx \mp \omega t)$.
1
This is a pretty special property. For any other periodic functions (such as triangle waves or square waves) there are no qualifying solutions at all. However, you
can write all those non-qualifying solutions as a sum of the ones that do qualify. Letting $F$ be a stand in for any non-qualifying periodic solution, then $$F = \sum_i \int \mathrm{d}\omega\, c_{i,\omega} \, f_{i,\omega} \;, $$for some selected set of coefficient $c_{i,\omega}$s. 2
So, long and mathematical story short: sinusoidal waves have the special properties of having unique frequency and a being usable to compose all periodic functions.
1 I included the $i$ subscript on $f_{i,\omega}$ so that we could distinguish $\sin$ and $\cos$. It turns out that $\lambda = k^2 - \frac{\omega^2}{c^2}$, and that by choosing $k$ (for a given $\omega$) so that $\lambda = 0$ you force the wave speed to be $c$ and have a description of a physical wave that matches the conditions of your apparatus.
2 There is even a way to find the correct $c_{i,\omega}$s from $F$, but writing it down here wouldn't actually explain any more about why the sinusoids are special.
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
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Six Numbers, Three Inequalities Problem
Solution
The answers to the questions are $\displaystyle \frac{1}{24}$ and $\displaystyle \frac{5}{24},$ respectively.
For the first problem, we choose four digits to place next to the inequality signs. These can be permuted in $4!=24$ ways of which only one satisfies all three inequalities. We conclude that the probability of this happening is $\displaystyle \frac{1}{24}.$
We can do a more complete counting: there are $\displaystyle {6\choose 4}=\frac{6!}{2!4!}$ to choose four numbers out of six, the other two can be permuted in $2!=2$ ways. Thus the sought probability is $\displaystyle \frac{6!2!}{2!4!6!}=\frac{1}{4!},$ as before.
For the second problem, observe that if four numbers $u,v,w,x$ fit the inequalities in the first question, i.e., $u\lt v\lt w\lt x$ then
$u\lt w\gt v\lt x\\ u\lt x\gt v\lt w\\ v\lt w\gt u\lt x\\ v\lt x\gt u\lt w\\ w\lt x\gt u\lt v$
Thus to every solution of the first question there correspond five solutions to the second. It follows that the probability of a random sequence satisfying the inequalities in the second question is $\displaystyle \frac{5}{24}.$
Generalization
We can establish several rules. Let's call a sequence of consecutive square cells whose inner (but not the two outer) sides carry a symbol of inequality a
. run Rule 1: The event of inequalities satisfied in one run are independent of the event of the inequalties being satisfied in any other run.
We call a run
, if its inequalities are all the same, either $\lt$ or $\gt.$ consistent Rule 2: For a grid of $n$ cells, the probability of satisfying a consistent run of length $k$ is $\displaystyle \frac{1}{n\choose k}.$
... to be continued ...
Dr. Pasquale Cirillo made a suggesion to make inequality symbols sparser which would allow to compare multidigit numbers.
An Extra
Practically it becomes a counting problem. Suppose there are $n-1$ inequality changes of $(x_1,x_2,\ldots,x_n)$.
Examples: suppose we have $a \lt b \gt c\lt d$, so $1$ rise followed by $1$ drop, then $1$ rise. We represent this by tuple $(1,1,1)$. If we have $a \gt b\lt c\lt d\lt e \gt f$, then this has representation $(1,3,1)$, i.e. one drop, $3$ raises, $1$ drop.It does not matter if we start the sequence with a drop or a raise. Also, the counts for symmetric sequences is the same.
Denote the number of sequences for the tuple $(m_1,\ldots,m_q)$ as $a_{m_1,\ldots,m_q}.$ By the definition, $a_{m_1,\ldots,m_q}=a_{m_q,\ldots,m_1}$.
Some numbers: $a_{1}=1,a_{1,1}=2,a_{2}=1.$
$a_{1,1,1}=5,a_{1,2}=3,a_{3}=1,$ $a_{1,1,1,1}=16,$ $a_{1,1,2}=9,$ $a_{1,2,1}=11,$ $a_{1,3}=4,$ $a_{2,2}=6,$ $a_{4}=1.$
Note: $a_{1,1,1}=5$ and $a_{3}=1$ appeared in connection with the Twitter problem.
Recurrence relation: An example will show it, the general case is similar. Suppose we need to calculate $a_{2,3,2,1,2,1}$. We may assume we start with a rise. We have two valleys (convexities) between $3$ falls and $2$ raises and between $1$ fall and $2$ raises. What changes by the addition of last number when we drop once? This last number can be the lowest one, in which case all the others can freely move in $a_{2,3,2,1,2}$ ways.
Or the lowest number can be the first and the others move freely in $a_{1,3,2,1,2,1}$ ways.
Or the lowest number can be in the first valley between $3$ and $2.$ In this case, we might have that the left neighbour is higher than right neighbour, in which case the counts are $a_{2,3,1,1,2,1}$ or the right neighbour is higher than the left neighbour, in which case the count is $a_{2,2,2,1,2,1}$.
Similarly, in the second valley the counts are (depending which of the neighbours is higher) $a_{2,3,4,1}$ and $a_{2,3,2,1,1,1}$.
In conclusion
$\begin{align}a_{2,3,2,1,2,1}&=a_{1,3,2,1,2,1}+a_{2,3,2,1,2}+a_{2,3,1,1,2,1}\\ &\qquad\qquad+a_{2,2,2,1,2,1}+a_{2,3,4,1}+a_{2,3,2,1,1,1}\end{align}$
Examples:$a_{1,1,1}=a_{1,1}+a_2+a_{1,1}\\ a_{2,2}=a_{2,1}+a_{1,2}\\ a_{2,1,1}=a_{1,1,1}+a_3+a_{2,1}\\ a_{1,1,1,1}=a_{1,1,1}+a_{1,1,1}+a_{2,1}+a_{2,1}\\ a_{1,1,1,2}=a_{1,1,1,1}+a_{1,1,2}+a_{2,2}+a_{1,3}\\ a_{1,1,1,1,1}=a_{1,1,1,1}+a_{2,1,1}+a_{1,2,1}+a_{1,2,1}+a_{1,1,1,1}$
Acknowledgment
These problems have been inspired by Donald Knuth's puzzles from the multi-author collection Puzzle Box, v 2.
Bogdan Lataianu is responsible for the Extra section.
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IX - Neural Networks: Learning 9.1 - Cost Function We use Lto denote the total number of layers. We use \(s_l\) = number of units in layer l (not counting the bias unit). We also use Kas the number of output units. With binary classification we would have only one output unit. (K = 1). With multiclass classification, we would have \(K \ge 3\). The cost function we use here is a generalization of the Logistic regression cost function (with \(h_\Theta(x) \in \mathbb{R}^K\) and \((h_\Theta(x))_i = i^{\text{th}}~\text{output}\)):
\[J(\Theta) = - \frac{1}{m} \left[ \sum\limits_{i=1}^m \sum\limits_{k=1}^K y_k^{(i)}~log((h_\Theta(x^{(i)}))_k) + (1 -y_k^{(i)})~log(1 - (h_\Theta(x^{(i)}))_k) \right] + \frac{\lambda}{2m} \sum\limits_{l=1}^{L-1} \sum\limits_{i=1}^{s_l} \sum\limits_{j=1}^{s_{l+1}} (\Theta_{ji}^{(l)})^2 \]
9.2 - Backpropagation Algorithm For each node, we are going to compute \(\delta_j^{(l)}\) representing the “error” of node j in layer l. So, from a simple 4 layers network, we could compute the error on the output layer as: \(\delta_j^{(4)} = a_j^{(4)} - y_j = (h_\Theta(x))_j - y_j\). Then we can continue with:
\[\delta^{(3)} = (\Theta^{(3)})^T\delta^{(4)}.*g'(z^{(3)}) \text{ where } g'(z^{(3)}) = a^{(3)} .* (1-a^{(3)})\] \[\delta^{(2)} = (\Theta^{(2)})^T\delta^{(3)}.*g'(z^{(2)}) \text{ where } g'(z^{(2)}) = a^{(2)} .* (1-a^{(2)})\]
Finally, it can be proved that, if we ignore the regularization term then we get:
\[\frac{\partial}{\partial\Theta_{ij}^{(l)}} J(\Theta) = a_j^{(l)}\delta_i^{(l+1)}\]
Backpropagation algorithm: We set \(\Delta_{ij}^{(l)} = 0\) for all i,j,l. For i=1 to m: Set \(a^{(1)} = x^{(i)}\) Perform forward propagation to compute \(a^{(l)}\) for l=2,3,…,L Using \(y^{(i)}\), compute \(\delta^{(L)} = a^{(L)} - y^{(i)}\) Then compute \(\delta^{(L-1)},\delta^{(L-2)},\dots,\delta^{(2)}\) Set \(\Delta_{ij}^{(l)} := \Delta_{ij}^{(l)} + a_j^{(l)}\delta_i^{(l+1)}\). Note that a vectorized implementation of this would be: \(\Delta^{(l)} := \Delta^{(l)} + \delta^{(l+1)}~(a^{(l)})^T\). Then we compute the following term \(D_{ij}^{(l)} := \begin{cases} \frac{1}{m}\Delta_{ij}^{(l)} + \frac{\lambda}{m} \Theta_{ij}^{(l)} & \text{ if } j \neq 0 \\ \frac{1}{m}\Delta_{ij}^{(l)} & \text{ if } j = 0 \end{cases}\)
⇒ Finally, it can be prooved that \(\frac{\partial}{\partial\Theta_{ij}^{(l)}} J(\Theta) = D_{ij}^{(l)}\).
9.3 - Backpropagation Intuition If we select a single example i, then, formally: \(\delta_j^{(j)} = \frac{\partial}{\partial z_j^{(l)}} cost(i)\). 9.4 - Implementation Note: Unrolling parameters for advanced optimization, we would need to write a function such as: function [jVal, gradient] = costFunction(theta) ... optTheta = fminunc(@costFunction,initialTheta, options); But note that in the previous code, gradient, theta and initialTheta would be assumed to be vectors from \(\mathbb{R}^{n+1}\) by the optimization routine fminunc. Here we have matrices instead, so we need to unrollthose matrices into vectors. To do so in Octave, we can write: thetaVec = [ Theta1(:); Theta2(:); Theta3(:) ]; DVec = [ D1(:); D2(:); D3(:)]; If we want to do the opposite, we use: Theta1 = reshape(thetaVec(1:110),10,11); Theta2 = reshape(thetaVec(111:220),10,11); Theta3 = reshape(thetaVec(221:231),1,11); 9.5 - Gradient checking We can compute an approximation of the derivative of \(J(\Theta)\) with (taking an example here with \(\Theta \in \mathbb{R}\): \(\frac{d}{d\Theta} \approx \frac{J(\Theta+\epsilon) - J(\Theta-\epsilon)}{2\epsilon}\) (this is called the 2 sided difference, which gives a slightly more accurate approximation than the 1 sided difference). Usually we can use \(\epsilon \approx 10^{-4}\). So in Octave, we would write: gradApprox = (J(theta + EPSILON) - J(theta - EPSILON))/(2*EPSILON) In the case \(\theta\) is a vector : \(\theta = [\theta_1,\theta_2,\theta_3,\dots,\theta_n]\). Then we can compute the gradient approximation as:
\[\frac{\partial}{\partial \theta_i} J(\theta) \approx \frac{J(\theta_1,\dots,\theta_i+\epsilon,\dots,\theta_n) - J(\theta_1,\cdots,\theta_i-\epsilon,\cdots,\theta_n)}{2\epsilon}\]
So in Octave, we would write: for i=1:n, thetaPlus = theta; thetaPlus(i) = thetaPlus(i) + EPSILON; thetaMinus = theta; thetaMinus(i) = thetaMinus(i) - EPSILON; gradApprox(i) = (J(thetaPlus) - J(thetaMinus))/(2*EPSILON); end; Then we need to check that \(gradApprox \approx DVec\). Note that we should ensurethat gradient checking is disabledwhen we want to run the backprop code for learning (otherwise, its going to be extremely slow). 9.6 - Random Initialization We need to initialize the initialThetaparameters. We can simply use \(initialTheta = zeros(n,1);\) for logistic regression. But this doesn't work for neural networks (otherwise the activation values will be the same and so will be the delta values). Random initialization for the neural network is used to ensure symmetry breaking. Concretely, we initialize each value \(\Theta_{ij}^{(l)}\) to a random value in the range \([-\epsilon, \epsilon]\); For instance in Octave: Theta1 = rand(10,11)*(2*INIT_EPSILON) - INIT_EPSILON; Theta2 = rand(1,11)*(2*INIT_EPSILON) - INIT_EPSILON; Note that in Octave rand(n,m)will create an nxm matrix will all elements set to random values between 0 and 1. 9.7 - Putting It Together When training a neural network, we need to choose: Number of input units : = number of features available. Number of output units: = number of classes. A reasonnable default is then to choose only 1 hidden layer. Or if we use multiple hidden layers, then a reasonnable choice is to have the same number of units in each hidden layer. (usually, the mode units, the better). 9.8 - Autonomous Driving This is just an example on autonomous driving with a neural network.
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:5579-5588, 2019.
Abstract
We present a provably optimal differentially private algorithm for the stochastic multi-arm bandit problem, as opposed to the private analogue of the UCB-algorithm (Mishra and Thakurta, 2015; Tossou and Dimitrakakis, 2016) which doesn’t meet the recently discovered lower-bound of $\Omega \left(\frac{K\log(T)}{\epsilon} \right)$ (Shariff and Sheffet, 2018). Our construction is based on a different algorithm, Successive Elimination (Even-Dar et al., 2002), that repeatedly pulls all remaining arms until an arm is found to be suboptimal and is then eliminated. In order to devise a private analogue of Successive Elimination we visit the problem of private stopping rule, that takes as input a stream of i.i.d samples from an unknown distribution and returns a multiplicative $(1 \pm \alpha)$-approximation of the distribution’s mean, and prove the optimality of our private stopping rule. We then present the private Successive Elimination algorithm which meets both the non-private lower bound (Lai and Robbins, 1985) and the above-mentioned private lower bound. We also compare empirically the performance of our algorithm with the private UCB algorithm.
@InProceedings{pmlr-v97-sajed19a,title = {An Optimal Private Stochastic-{MAB} Algorithm based on Optimal Private Stopping Rule},author = {Sajed, Touqir and Sheffet, Or},booktitle = {Proceedings of the 36th International Conference on Machine Learning},pages = {5579--5588},year = {2019},editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan},volume = {97},series = {Proceedings of Machine Learning Research},address = {Long Beach, California, USA},month = {09--15 Jun},publisher = {PMLR},pdf = {http://proceedings.mlr.press/v97/sajed19a/sajed19a.pdf},url = {http://proceedings.mlr.press/v97/sajed19a.html},abstract = {We present a provably optimal differentially private algorithm for the stochastic multi-arm bandit problem, as opposed to the private analogue of the UCB-algorithm (Mishra and Thakurta, 2015; Tossou and Dimitrakakis, 2016) which doesn’t meet the recently discovered lower-bound of $\Omega \left(\frac{K\log(T)}{\epsilon} \right)$ (Shariff and Sheffet, 2018). Our construction is based on a different algorithm, Successive Elimination (Even-Dar et al., 2002), that repeatedly pulls all remaining arms until an arm is found to be suboptimal and is then eliminated. In order to devise a private analogue of Successive Elimination we visit the problem of private \emph{stopping rule}, that takes as input a stream of i.i.d samples from an unknown distribution and returns a \emph{multiplicative} $(1 \pm \alpha)$-approximation of the distribution’s mean, and prove the optimality of our private stopping rule. We then present the private Successive Elimination algorithm which meets both the non-private lower bound (Lai and Robbins, 1985) and the above-mentioned private lower bound. We also compare empirically the performance of our algorithm with the private UCB algorithm.}}
%0 Conference Paper%T An Optimal Private Stochastic-MAB Algorithm based on Optimal Private Stopping Rule%A Touqir Sajed%A Or Sheffet%B Proceedings of the 36th International Conference on Machine Learning%C Proceedings of Machine Learning Research%D 2019%E Kamalika Chaudhuri%E Ruslan Salakhutdinov%F pmlr-v97-sajed19a%I PMLR%J Proceedings of Machine Learning Research%P 5579--5588%U http://proceedings.mlr.press%V 97%W PMLR%X We present a provably optimal differentially private algorithm for the stochastic multi-arm bandit problem, as opposed to the private analogue of the UCB-algorithm (Mishra and Thakurta, 2015; Tossou and Dimitrakakis, 2016) which doesn’t meet the recently discovered lower-bound of $\Omega \left(\frac{K\log(T)}{\epsilon} \right)$ (Shariff and Sheffet, 2018). Our construction is based on a different algorithm, Successive Elimination (Even-Dar et al., 2002), that repeatedly pulls all remaining arms until an arm is found to be suboptimal and is then eliminated. In order to devise a private analogue of Successive Elimination we visit the problem of private stopping rule, that takes as input a stream of i.i.d samples from an unknown distribution and returns a multiplicative $(1 \pm \alpha)$-approximation of the distribution’s mean, and prove the optimality of our private stopping rule. We then present the private Successive Elimination algorithm which meets both the non-private lower bound (Lai and Robbins, 1985) and the above-mentioned private lower bound. We also compare empirically the performance of our algorithm with the private UCB algorithm.
Sajed, T. & Sheffet, O.. (2019). An Optimal Private Stochastic-MAB Algorithm based on Optimal Private Stopping Rule. Proceedings of the 36th International Conference on Machine Learning, in PMLR 97:5579-5588
This site last compiled Mon, 16 Sep 2019 16:05:04 +0000
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About the Degenerate Spectrum of the Tension Field for Mappings into a Symmetric Riemannian Manifold
About the Degenerate Spectrum of the Tension Field for Mappings into a Symmetric Riemannian Manifold
Abstract
Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds, where $(N,h)$ is symmetric, $v\in W^{1,2}((M,g),(N,h))$, and $\tau $ is the tension field for mappings from $(M,g)$ into $(N,h)$. We consider the nonlinear eigenvalue problem $\tau (u)-\lambda \exp _{u}^{-1}v=0$, for $u$ $\in W^{1,2}(M,N)$ such that $u_{\left\vert \partial M\right. }=v_{\left\vert \partial M\right.}$, and $\lambda \in \mathbb{R}$. We prove, under some assumptions, that the set of all $\lambda $, such that there exists a solution $(u,\lambda )$ of this problem and a non trivial Jacobi field $V$ along $u$, is contained in $\mathbb{R}_{+}$, is countable, and has no accumulation point in $\mathbb{R}$. This result generalizes a well known one about the spectrum of the Laplace--Beltrami operator $\Delta $ for functions from $(M,g)$ into $\mathbb{R}$.
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The usual argument to show that the group of all orientation-preserving symmetries of the Klein quartic is the simple group $L_2(7)$ of order $168$ goes like this:
There are two families of $7$ truncated cubes on the Klein quartic. The triangles of one of the seven truncated cubes in the first family have as center the dots, all having the same colour. The triangles of one of the truncated cubes in the second family correspond to the squares all having the same colour.
If you compare the two colour schemes, you’ll see that every truncated cube in the first family is disjoint from precisely $3$ truncated cubes in the second family.
That is, we can identify the truncated cubes of the first family with the points in the Fano plane $\mathbb{P}^2(\mathbb{F}_2)$, and those of the second family with the lines in that plane.
The Klein quartic consists of $24$ regular heptagons, so its rotation symmetry group consists of $24 \times 7 = 168$ rotations,each preserving the two families of truncated cubes. This is exactly the same number as there are isomorphisms of the Fano plane, $PGL_3(\mathbb{F}_2) = L_2(7)$. Done!
For more details, check John Baez’ excellent page on the Klein quartic, or the Buckyball curve post.
Here’s another ‘look-and-see’ proof, starting from Klein’s own description of his quartic.
Look at the rotation $g$, counter-clockwise with angle $2\pi / 7$ fixing the center of the central blue heptagon, and a similar rotation $h$ fixing the center of one of the neighbouring red heptagons.
The two vertices of the edge shared by the blue and red heptagon are fixed by $g.h$ and $h.g$, respectively, so these rotations must have order three (there are $3$ heptagons meeting in the vertex).
That is, the rotation symmetry group $G$ of the Klein quartic has order $168$, and contains two elements $g$ and $h$ of order $7$, such that the subgroup generated by them contains elements of order $3$.
This is enough to prove that the $G$ must be simple and therefore isomorphic to $L_2(7)$!
The following elegant proof is often attributed to Igor Dolgachev.
If $G$ isn’t simple there is a maximal normal subgroup $N$ with $G/N$ simple .
The only non-cyclic simple group having less elements that $168$ is $A_5$ but this cannot be $G/N$ as $60$ does not divide $168$.
So, $G/N$ must be cyclic of order $2,3$ or $7$ (the only prime divisors of $168=2^3.3.7$).
Order $2$ is not possible as any group $N$ of order $84=2^2.3.7$ can just have one Sylow $7$-subgroup. Remember that the number of $7$-Sylows of $N$ must divide $2^2.3=12$ and must be equal to $1$ modulo $7$. And $G$ (and therefore $N$) has at least two different cyclic subgroups of order $7$.
Order $3$ is impossible as this would imply that the normal subgroup $N$ of order $2^3.7=56$ must contain all $7$-Sylows of $G$, and thus also an element of order $3$. But, $3$ does not divide $56$.
Order $7$ is a bit more difficult to exclude. This would mean that there is a normal subgroup $N$ of order $2^3.3=24$.
$N$ (being normal) must contain all Sylow $2$-subgroups of $G$ of which there are either $1$ or $3$ (the order of $N$ is $2^3.3=24$).
If there is just one $S$ it should be a normal subgroup with $G/S$ (of order $21$) having a (normal) unique Sylow $7$-subgroup, but then $G$ would have a normal subgroup of index $3$, which we have excluded.
The three $2$-Sylows are conjugated and so the conjugation morphism
\[ G \rightarrow S_3 \] is non-trivial and must have image strictly larger than $C_3$ (otherwise, $G$ would have a normal subgroup of index $3$), so must be surjective.
But, $S_3$ has a normal subgroup of index $2$ and pulling this back, $G$ must also have a normal subgroup of index two, which we have excluded. Done!
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Answer
$651.92 ft$ The angle increases the final depth gradually.
Work Step by Step
Here, we have $\tan 15^{\circ}=\dfrac{d}{1500}$ This gives: $ d \approx 1500 tan 15^{\circ}$ and $d \approx 401.92 ft$ Thus, $401.92 ft+ 250 ft =651.92 ft$ Hence, the angle increases the final depth gradually.
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Imagine doing a hypothetical experiment that would lead to the discovery of electron spin. Your laboratory has just purchased a microwave spectrometer with variable magnetic field capacity. We try the new instrument with hydrogen atoms using a magnetic field of 10
4 Gauss and look for the absorption of microwave radiation as we scan the frequency of our microwave generator.
Figure \(\PageIndex{1}\): Schematic diagram of a microwave spectrometer with the sample in a variable magnetic field. The strength of the magnetic field is set, and the sample’s absorption of microwave photons is measured for a range of microwave photon energies (or frequencies).
Finally we see absorption at a microwave photon frequency of \(28 \times 10^9\, Hz\) (28 gigahertz). This result is really surprising from several perspectives. Each hydrogen atom is in its ground state, with the electron in a 1s orbital. The lowest energy electronic transition that we predict based on existing theory (the electronic transition from the ground state (\(\Psi _{100}\) to \(\Psi _{21m}\)) requires an energy that lies in the vacuum ultraviolet, not the microwave, region of the spectrum. Furthermore, when we vary the magnetic field we note that the frequency at which the absorption occurs varies in proportion to the magnetic field. This effect looks like a Zeeman effect, but if you think about the situation, even if the 1s orbital were doubly degenerate, a \(1s\) orbital still has zero orbital angular momentum, no magnetic moment, and therefore no predicted Zeeman effect!
To discover new things, experimentalists sometimes must explore new areas in spite of contrary theoretical predictions. Our theory of the hydrogen atom at this point gives no reason to look for absorption in the microwave region of the spectrum. By doing this crazy experiment, we discovered that when an electron is in the \(1s\) orbital of the hydrogen atom, there are two different states that have the same energy. When a magnetic field is applied, this degeneracy is removed, and microwave radiation can cause transitions between the two states. In the rest of this section, we see what can be deduced from this experimental observation. This experiment actually could be done with electron spin resonance spectrometers available today.
In order to explain our observations, we need a new idea, a new model for the hydrogen atom. Our original model for the hydrogen atom accounted for the motion of the electron and proton in our three-dimensional world; the new model needs something else that can give rise to an additional Zeeman-like effect. We need a charged particle with angular momentum to produce a magnetic moment, just like that obtained by the orbital motion of the electron. We can postulate that our observation results from a motion of the electron that wasn’t considered in the last section - electron spin. We have a charged particle spinning on its axis. We then have charge moving in a circle, angular momentum, and a magnetic moment, which interacts with the magnetic field and gives us the Zeeman-like effect that we observed.
To describe electron spin from a quantum mechanical perspective, we must have spin wavefunctions and spin operators. The properties of the spin states are deduced from experimental observations and by analogy with our treatment of the states arising from the orbital angular momentum of the electron.
The important feature of the spinning electron is the spin angular momentum vector, which we label \(S\) by analogy with the orbital angular momentum \(L\). We define spin angular momentum operators with the same properties that we found for the rotational and orbital angular momentum operators. After all, angular momentum is angular momentum.
We found that (in Bra-ket notation)
\[ \hat {L}^2 | Y^{m_l} _l \rangle= l(l + 1) \hbar^2 | Y^{m_l}_l \rangle \label {8-41}\]
so by analogy for the spin states, we must have
\[ \hat {S}^2 | \sigma ^{m_s} _s \rangle = s( s + 1) \hbar ^2 | \sigma ^{m_s}_s \rangle \label {8-42}\]
where \(\sigma\) is a spin wavefunction with quantum numbers \(s\) and \(m_s\) that obey the same rules as the quantum numbers \(l\) and \(m_l\) associated with the spherical harmonic wavefunction \(Y\). We also found
\[ \hat {L}_z | Y^{m_l}_l \rangle = m_l \hbar | Y^{m_l}_l \rangle \label {8-43}\]
so by analogy, we must have
\[ \hat {S}_z | \sigma ^{m_s}_s \rangle = m_s \hbar | \sigma ^{m_s}_s \rangle\label {8-44}\]
Since \(m_l\) ranges in integer steps from \(-l\) to \(+l\), also by analogy \(m_s\) ranges in integer steps from \(-s\) to \(+s\). In our hypothetical experiment, we observed one absorption transition, which means there are two spin states. Consequently, the two values of \(m_s\) must be \(+s\) and \(-s\), and the difference in \(m_s\) for the two states, labeled f and i below, must be the smallest integer step, i.e. 1. The result of this logic is that
\[ \begin{align} m_{s,f} - m_{s,i} &= 1 \nonumber \\[4pt] (+s) - (-s) &= 1 \nonumber \\[4pt] 2s &= 1 \nonumber \\[4pt] s &= \dfrac {1}{2} \end{align} \label {8-45} \]
Therefore our conclusion is that the magnitude of the spin quantum number is 1/2 and the values for ms are +1/2 and -1/2. The two spin states correspond to spinning clockwise and counter-clockwise with positive and negative projections of the spin angular momentum onto the z-axis. The state with a positive projection, \(m_s\) = +1/2, is called \(\alpha\); the other is called \(\beta\). These spin states are arbitrarily labeled \(\alpha\) and \(\beta\), and the associated spin wavefunctions also are designated by \(\alpha\) and \(\beta\).
From Equation \ref{8-44} the magnitude of the z-component of spin angular momentum, \(S_z\), is given by
\[S_z = m_s \hbar \label {8-46}\]
so the value of \(S_z\) is +ħ/2 for spin state \(\alpha\) and -ħ/2 for spin state \(\beta\). Using the same line of reasoning we used for the splitting of the \(m_l\) states in Section 8.4 as represented by Equation (8-39), we conclude that the \(\alpha\) spin state, where the magnetic moment is aligned against the external field direction, has a higher energy than the \(\beta\) spin state.
Even though we don’t know their functional forms, the spin wavefunctions are taken to be normalized and orthogonal to each other.
\[ \int \alpha ^* \alpha d \tau _s = \int \beta ^* \beta d \tau _s = 1 \label {8-47}\]
and
\[ \int \alpha ^* \beta d \tau _s = \int \beta ^* \alpha d \tau _s = 0 \label {8-48}\]
where the integral is over the spin variable \(\tau _s\).
Now let's apply these deductions to the experimental observations in our hypothetical microwave experiment. Using Equation \ref{8-27}, we can account for the frequency of the transition (\(\nu\)= 28 gigahertz) that was observed in this hypothetical experiment in terms of the magnetic moment of the spinning electron and the strength of the magnetic field. The photon energy, \(h \nu\), is given by the difference between the energies of the two states, \(E_{\alpha}\) and \(E_{\beta}\)
\[ \Delta E = h \nu = E_{\alpha} - E_{\beta} \label {8-49}\]
Figure \(\PageIndex{2}\): Absorption of a photon to cause a transition from the \(\beta\) to the \(\alpha\) state.
The energies of these two states consist of the sum of the energy of an electron in a 1s orbital, \(E_{1s}\), and the energy due to the interaction of the spin magnetic dipole moment of the electron, \(\mu _s\), with the magnetic field, B (as in Section 8.4). The two states with distinct values for spin magnetic moment \(\mu _s\) are denoted by the subscripts \(\alpha\) and \(\beta\).
\[ E_{\alpha} = E_{1s} - \mu _{s,\alpha} \cdot B \label {8-50}\]
\[ E_{\beta} = E_{1s} - \mu _{s,\beta} \cdot B \label {8-51}\]
Substituting the two equations above into the expression for the photon energy gives
\[ h \nu = E_{\alpha} - E_{\beta} = (E_{1s} - \mu _{s, \alpha} \cdot B) - (E_{1s} - \mu_{s,\beta} \cdot B) \label {8-52} \]
\[ = ( \mu _{s, \beta} - \mu _{s, \alpha}) \cdot B \label {8-52B}\]
\[h \nu = (\mu _{s, \beta} - \mu _{s, \alpha}) \cdot B \label {8-53}\]
Again by analogy with the orbital angular momentum and magnetic moment discussed in Section 8.4, we take the spin magnetic dipole of each spin state, \(\mu _{s, \alpha}\) and \(\mu _{s, \beta}\), to be related to the total spin angular momentum of each state, \(S_{\alpha}\) and \(S_{\beta}\), by a constant spin gyromagnetic ratio, \(\gamma _s\), as shown below.
\[ \mu _s = \gamma _s S\]
\[\mu _{s, \alpha} = \gamma _s S_\alpha \]
\[\mu _{s, \beta} = \gamma _s S_\beta \label {8-54}\]
With the magnetic field direction defined as z, the scalar product in Equation \ref{8-53} becomes a product of the z-components of the spin angular momenta, \(S_{z, \alpha}\) and \(S_{z, \beta}\), with the external magnetic field.
Inserting the values for \(S_{z,\alpha} = +\dfrac {1}{2} \hbar \) and \( S_{z, \alpha} = -\dfrac {1}{2} \hbar\) from Equation \ref{8-46} and rearranging Equation \ref{8-55} yields
\[ \dfrac {h \nu}{B} = - \gamma _s \hbar \label {8-56}\]
Calculating the ratio \(\dfrac {h \nu}{B}\) from our experimental results, \(\nu = 28 \times 10^9\, Hz\) when \(B = 10^4\, gauss\), gives us a value for
\[- \gamma_s \hbar = 18.5464 \times 10^{-21}\, erg/gauss.\]
This value is about twice the Bohr magneton,\(-\gamma _e \hbar \), found in Equation \ref{8-40} i.e. \(\gamma _s \hbar = 2.0023, \gamma _e \hbar\), or
\[\gamma _s = 2.0023 \gamma _e \label {8-57}\]
The factor of 2.0023 is called the g-factor and accounts for the deviation of the spin gyromagnetic ratio from the value expected for orbital motion of the electron. In other words, it accounts for the spin transition being observed where it is instead of where it would be if the same ratio between magnetic moment and angular momentum held for both orbital and spin motions. The value 2.0023 applies to a freely spinning electron; the coupling of the spin and orbital motion of electrons can produce other values for \(g\).
Exercise \(\PageIndex{1}\)
Carry out the calculations that show that the g-factor for electron spin is 2.0023.
Interestingly, the concept of electron spin and the value g = 2.0023 follow logically from
Dirac's relativistic quantum theory, which is beyond the scope of this discussion. Electron spin was introduced here as a postulate to explain experimental observations. Scientists often introduce such postulates parallel to developing the theory from which the property is naturally deduced.
Now that we have discovered electron spin, we need to determine how the electron spin changes when radiation is absorbed or emitted, i.e. what are the selection rules for electron spin of a single electron? Unlike orbital angular momentum, which can have several values, the spin angular momentum can have only the value
\[ |S| = \sqrt {s (s + 1) \hbar } = \dfrac {\sqrt {3}}{2} \hbar \label {8-58}\]
Since s = ½, one spin selection rule is
\[\Delta s = 0 \label {8-59}\]
When a magnetic field is applied along the z-axis to remove the \(m_s\) degeneracy, another magnetic field applied in the x or y direction exerts a force or torque on the magnetic dipole to turn it. This transverse field can “flip the spin,” and change the projection on the z-axis from \(\dfrac {1}{+2} \hbar \) to \(\dfrac {1}{-2} \hbar \) or from \(\dfrac {1}{-2} \hbar \) to \(\dfrac {1}{+2} \hbar \). So the other spin selection rule for a single electron is
\[ \Delta m_s = \pm 1 \label {8-60}\]
Contributors Adapted from "Quantum States of Atoms and Molecules" by David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski
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Article
Keywords: least concave majorant; level function; spline approximation
Summary: The least concave majorant, $\hat F$, of a continuous function $F$ on a closed interval, $I$, is defined by \[ \hat F (x) = \inf \{ G(x)\colon G \geq F,\ G \text { concave}\},\quad x \in I. \] We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$. Given any function $F \in \mathcal {C}^4(I)$, it can be well-approximated on $I$ by a clamped cubic spline $S$. We show that $\hat S$ is then a good approximation to $\hat F$. \endgraf We give two examples, one to illustrate, the other to apply our algorithm.
References:
[1] Brudnyĭ, Y. A., Krugljak, N. Y.:
Interpolation Functors and Interpolation Spaces. Vol. 1
. North-Holland Mathematical Library 47, Amsterdam (1991). MR 1107298
| Zbl 0743.46082
[6] Härdle, W., Kerkyacharian, G., Picard, D., Tsybakov, A.:
Wavelets, Approximation, and Statistical Applications
. Lecture Notes in Statistics 129, Springer, Berlin (1998). DOI 10.1007/978-1-4612-2222-4
| MR 1618204
| Zbl 0899.62002
[9] Lorentz, G. G.:
Bernstein Polynomials
. Mathematical Expositions, no. 8. University of Toronto Press X, Toronto (1953). MR 0057370
| Zbl 0051.05001
|
A
definite description is a denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun. The definite description is proper if X applies to a unique individual or object. For example: "the first person in space" and "the 42nd President of the United States of America", are proper. The definite descriptions "the person in space" and "the Senator from Ohio" are improper because the noun phrase X applies to more than one thing, and the definite descriptions "the first man on Mars" and "the Senator from Washington D.C." are improper because X applies to nothing. Improper descriptions raise some difficult questions about the law of excluded middle, denotation, modality, and mental content.
Contents Russell's analysis 1 Generalized quantifier analysis 2 Fregean analysis 3 Mathematical logic 4 See also 5 References 6 External links 7 Russell's analysis
France is currently a republic, and has no king. Bertrand Russell pointed out that this raises a puzzle about the truth value of the sentence "The present King of France is bald."
The sentence does not seem to be true: if we consider all the bald things, the present King of France isn't among them, since there is no present King of France. But if it is false, then one would expect that the negation of this statement, that is, "It is not the case that the present King of France is bald," or its logical equivalent, "The present King of France is not bald," is true. But this sentence doesn't seem to be true either: the present King of France is no more among the things that fail to be bald than among the things that are bald. We therefore seem to have a violation of the Law of Excluded Middle.
Is it meaningless, then? One might suppose so (and some philosophers have; see below) since "the present King of France" certainly does fail to refer. But on the other hand, the sentence "The present King of France is bald" (as well as its negation) seem perfectly intelligible, suggesting that "the Present King of France" can't be meaningless.
Russell proposed to resolve this puzzle via his theory of descriptions. A definite description like "the present King of France", he suggested, isn't a referring expression, as we might naively suppose, but rather an "incomplete symbol" that introduces quantificational structure into sentences in which it occurs. The sentence "the present King of France is bald", for example, is analyzed as a conjunction of the following three quantified statements:
there is an x such that x is currently King of France: ∃x[PKoF(x)] (using 'PKoF' for 'currently King of France') for any x and y, if x is currently King of France and y is currently King of France, then x=y (i.e. there is at most one thing which is currently King of France): ∀x∀y[[PKoF(x) & PKoF(y)] → y=x] for every x that is currently King of France, x is bald: ∀x[PKoF(x) → B(x)] (using 'B' for 'bald')
More briefly put, the claim is that "The present King of France is bald" says that some x is such that x is currently King of France, and that any y is currently King of France only if y = x, and that x is bald:
∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]
This is
false, since it is not the case that some x is currently King of France.
The negation of this sentence, i.e. "The present King of France is not bald", is ambiguous. It could mean one of two things, depending on where we place the negation 'not'. On one reading, it could mean that there is no one who is currently King of France and bald:
~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]
On this disambiguation, the sentence is
true (since there is indeed no x that is currently King of France).
On a second reading, the negation could be construed as attaching directly to 'bald', so that the sentence means that there is currently a King of France, but that this King fails to be bald:
∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & ~B(x)]
On this disambiguation, the sentence is
false (since there is no x that is currently King of France).
Thus, whether "the present King of France is not bald" is true or false depends on how it is interpreted at the level of logical form: if the negation is construed as taking wide scope (as in ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]), it is true, whereas if the negation is construed as taking narrow scope (with the existential quantifier taking wide scope, as in ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & ~B(x)]), it is false. In neither case does it lack a truth value.
So we do
not have a failure of the Law of Excluded Middle: "the present King of France is bald" (i.e. ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]) is false, because there is no present King of France. The negation of this statement is the one in which 'not' takes wide scope: ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]. This statement is true because there does not exist anything which is currently King of France. Generalized quantifier analysis
Stephen Neale, among others, has defended Russell's theory, and incorporated it into the theory of generalized quantifiers. On this view, 'the' is a quantificational determiner like 'some', 'every', 'most' etc. The definite description 'the' has the following denotation (using lambda notation):
λf.λg.[∃x(f(x)=1 & ∀y(f(y)=1 → y=x) & g(x)=1)].
(That is, the definite article 'the' denotes a function which takes a pair of properties f and g to truth if, and only if there exists something that has the property f, only one thing has the property f, and that thing also has the property g.) Given the denotation of the predicates 'present King of France' (again PKoF for short) and 'bald (B for short)'
λx.[PKoF(x)] λx.[B(x)]
we then get the Russellian truth conditions via two steps of function application: 'The present King of France is bald' is true if, and only if ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]. On this view, definite descriptions like 'the present King of France' do have a denotation (specifically, definite descriptions denote a function from properties to truth values—they are in that sense not syncategorematic, or "incomplete symbols"); but the view retains the essentials of the Russellian analysis, yielding exactly the truth conditions Russell argued for.
Fregean analysis
The Fregean analysis of definite descriptions, implicit in the work of Frege and later defended by Strawson (1950) among others, represents the primary alternative to the Russellian theory. On the Fregean analysis, definite descriptions are construed as referring expressions rather than quantificational expressions. Existence and uniqueness are understood as a presupposition of a sentence containing a definite description, rather than part of the content asserted by such a sentence. The sentence 'The present King of France is bald', for example, isn't used to claim that there exists a unique present King of France who is bald; instead, that there is a unique present King of France is part of what this sentence
presupposes, and what it says is that this individual is bald. If the presupposition fails, the definite description fails to refer, and the sentence as a whole fails to express a proposition.
The Fregean view is thus committed to the kind of truth value gaps (and failures of the Law of Excluded Middle) that the Russellian analysis is designed to avoid. Since there is currently no King of France, the sentence 'The present King of France is bald' fails to express a proposition, and therefore fails to have a truth value, as does its negation, 'The present King of France is not bald'. The Fregean will account for the fact that these sentences are nevertheless
meaningful by relying on speakers' knowledge of the conditions under which either of these sentences could be used to express a true proposition. The Fregean can also hold on to a restricted version of the Law of Excluded Middle: for any sentence whose presuppositions are met (and thus expresses a proposition), either that sentence or its negation is true.
On the Fregean view, the definite article 'the' has the following denotation (using lambda notation):
λf: ∃x(f(x)=1 & ∀y(f(y)=1 → y=x)).[the unique y such that f(y)=1]
(That is, 'the' denotes a function which takes a property f and yields the unique object y that has property f, if there is such a y, and is undefined otherwise.) The presuppositional character of the existence and uniqueness conditions is here reflected in the fact that the definite article denotes a partial function on the set of properties: it is only defined for those properties f which are true of exactly one object. It is thus undefined on the denotation of the predicate 'currently King of France', since the property of currently being King of France is true of no object; it is similarly undefined on the denotation of the predicate 'Senator of the US', since the property of being a US Senator is true of more than one object.
Mathematical logic
In much formal work, authors use a definite description operator symbolized using \scriptstyle\iota x. The operator is usually defined so as to reflect a Russellian analysis of descriptions (though other authors, especially in linguistics, use the \scriptstyle \iota operator with a Fregean semantics). Thus
\iota x(\phi x),
means "the unique \scriptstyle x such that \scriptstyle\phi x", and
\psi(\iota x(\phi x))
is stipulated to be equivalent to "There is exactly one \scriptstyle\phi and it has the property \scriptstyle\psi":
\exists x\forall y (\phi(y) \iff y=x \and \psi(y)) See also References Donnellan, Keith, "Reference and Definite Descriptions," in Philosophical Review 75 (1966): 281-304. Neale, Stephen, Descriptions, MIT Press, 1990. Ostertag, Gary (ed.). (1998) Definite Descriptions: A Reader Bradford, MIT Press. (Includes Donnellan (1966), Chapter 3 of Neale (1990), Russell (1905), and Strawson (1950).) Reimer, Marga and Bezuidenhout, Anne (eds.) (2004), Descriptions and Beyond, Clarendon Press, Oxford Russell, Bertrand, "On Denoting," in Mind 14 (1905): 479-493. Online text Strawson, P. F., "On Referring," in Mind 59 (1950): 320-344. External links
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Honest answer for your question is
''No''! As in the standard model (SM), the number of fermion generations appears as an arbitrary parameter, meaning that a mathematically consistent theory can be built up using any number of fermion generations. Therefore, In order to answer the question perhaps we may need to beyond the standard model.
In the quest for such a
model(s), first lets rephrase your question is the following way:
Is there any extension of the standard model be possible where the
number of fermion generations can in any way be explained through the
internal consistency of the model? (since we don't have the answer within the framework of SM.)
Its good to start from SM itself. Lets go back in sixties. If we make a table of fermions, those were discovered up to 1965, it would looks like:
\begin{eqnarray}\text{Lepton}& : & \begin{pmatrix} \nu_{e}\\ e \end{pmatrix},\quad \begin{pmatrix} \nu_{\mu}\\ \mu\end{pmatrix} \\\text{Quark} & : & \begin{pmatrix} u \\ d \end{pmatrix},\qquad s\end{eqnarray}Anyone with naked eye can say how ''ugly'' this table is looks like! In fact it was James Bjorkenand Sheldon Glashowproposed the existance of chram ($c$) quark in order to restore the
''quark-lepton symmetry''. The table now looks more symmetric and beautiful:
\begin{eqnarray}\text{Lepton}& : & \begin{pmatrix} \nu_{e}\\ e \end{pmatrix},\quad \begin{pmatrix} \nu_{\mu}\\ \mu\end{pmatrix} \\\text{Quark} & : & \begin{pmatrix} u \\ d \end{pmatrix},\qquad \begin{pmatrix} c \\ s \end{pmatrix}\end{eqnarray}Which was later discovered during November Revolution 1974 . Lesson is, these two physicists were dictated by the
notion of symmetry in order to restore the order in the realm of fermions. Later GIM mechanism was given an explanation of the non-existent of FCNC in SM taking into account the charm quark.
The very existence of three generations of quarks is necessary for CP violation. And also for anomaly cancellations to make the SM mathematically consistant.But the undeylying symmetry (if it really exists) which may ensure the equal numbers of quarks and leptons, yet to be discovered.
Story that goes beyond SM:Back in nineties an extension of SM was proposed by F. Pisano here and V. Pleitez based on gauge group $SU(3)_{L}\times U(1)_{Y}$. Their model to accommodate the standard fermions into multiplets of this gauge group which must include some new fermions.
This model has remarkable feature. As we already know that, a consistent quantum field theory must be free from gauge anomaly. Without that, theory become ill. In case of SM the anomalies get cancelled in miraculous (or should we say in an ugly way?) way. But for the model with gauge group $SU(3)_{c}\times SU(3)_{L}\times U(1)_{Y}$, has the interesting featurethat each generation of fermions is anomalous, but that with three generations the anomalies cancelled.
In other words,
Electroweak interactions based on a gauge group $SU(3)_{L}\times U(1)_{Y}$, coupled to to the QCD gauge group $SU(3)_{c}$ can predict
the number of generations to be multiples of three.
(For technical detail one can read this paper).
But with the cost that we have to incorporate a right handed neutrino in the game. In fact, one may find other different models with the same features.
GUT considerations: In a recent paper Pritibhajan Byakti et al proposed a grand unified theory based on the gauge group $SU(9)$. The model uses fermions in antisymmetric representations only and the consistency of the model demands that the number of fermion generations is three. Nevertheless like all GUT, it also come up with some superheavy gauge bosons. Which can trigger baryon number non-conserving processes.
The upshot is, perhaps we would be able to explain the lepton-quark symmetry with the price of some new physics (may be new particles) which lives beyond the SM.This post imported from StackExchange Physics at 2016-11-10 22:33 (UTC), posted by SE-user AMS
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Rocky Mountain Journal of Mathematics Rocky Mountain J. Math. Volume 48, Number 7 (2018), 2311-2335. Dominating sets in intersection graphs of finite groups Abstract
Let $G$ be a group. The intersection graph $\Gamma (G)$ of $G$ is an undirected graph without loops and multiple edges, defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$, where $1$ denotes the trivial subgroup of $G$. In this paper, we study the dominating sets in intersection graphs of finite groups. We classify abelian groups by their domination number and find upper bounds for some specific classes of groups. Subgroup intersection is related to Burnside rings. We introduce the notion of an intersection graph of a $G$-set (somewhat generalizing the ordinary definition of an intersection graph of a group) and establish a general upper bound for the domination number of $\Gamma (G)$ in terms of subgroups satisfying a certain property in the Burnside ring. The intersection graph of $G$ is the $1$-skeleton of the simplicial complex. We name this simplicial complex intersection complex of $G$ and show that it shares the same homotopy type with the order complex of proper non-trivial subgroups of $G$. We also prove that, if the domination number of $\Gamma (G)$ is 1, then the intersection complex of $G$ is contractible.
Article information Source Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2311-2335. Dates First available in Project Euclid: 14 December 2018 Permanent link to this document https://projecteuclid.org/euclid.rmjm/1544756811 Digital Object Identifier doi:10.1216/RMJ-2018-48-7-2311 Mathematical Reviews number (MathSciNet) MR3892134 Zentralblatt MATH identifier 06999264 Subjects Primary: 20D99: None of the above, but in this section Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 05C69: Dominating sets, independent sets, cliques 20C05: Group rings of finite groups and their modules [See also 16S34] 55U10: Simplicial sets and complexes Citation
Kayacan, Selcuk. Dominating sets in intersection graphs of finite groups. Rocky Mountain J. Math. 48 (2018), no. 7, 2311--2335. doi:10.1216/RMJ-2018-48-7-2311. https://projecteuclid.org/euclid.rmjm/1544756811
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Linear/Eigenvalue Buckling analysis are fairly easy to set up and post process. Typically an Eigenvalue Buckling Analysis is preceded by a static structural analysis with perturbation load leading to compressive stress field being generated in the model. The pre-stressed model from the static structural analysis is then analyzed for buckling failure modes. Results of buckling analysis are eigenvalues or load multipliers for various buckling failure modes. The product of the eigenvalue and the perturbation load gives the critical buckling load for the analyzed mode. Typically the 1st buckling mode is the most critical one with the lowest eigenvalue, however its generally recommended to find atleast 3 buckling modes to be sure that any critical buckling mode is not missed.
Problem Statement:
Evaluate the following tower for compliance with respect to the Type-1 buckling criteria provided in paragraph 5.4.1.2.
Material – Shell and Heads = SA-516, Grade 70, Normalized Design Conditions = -14.7 psig at 300 oF Corrosion Allowance = 0.125 inches
ANALYSIS SUMMARY
A mid surface shell model was constructed using corroded dimensions from the above figure. The model was meshed with higher order shell elements.
Fixed Support was applied to the base of the skirt. External Pressure = 14.7 psi was applied as perturbation load for the static structural analysis.
The pre-stressed model from static structural analysis was used for eigenvalue buckling analysis. 3 buckling modes were requested using analysis setup and the numerical model was solved. Following Eigenvalues were found from the buckling analysis
Mode Eigenvalues 1 9.0009 2 9.0016 3 15.288
ASME PTB-3 Eigenvalues
Mode Eigenvalues 1 7.939 2 7.940 3 14.351
Note the ASME PTB-3 analysis is carried out using abaqus which calculates Buckling Load as Eigenvalue + Eigenvalue*Perturbation Load. But in ANSYS Buckling Load is calculated as Eigenvalue*Perturbation Load. Hence there is a difference of about 1 between the two eigenvalues.
First Mode Shape Plot
ASME PTB-3 First Mode Shape
The plots are quite similar. Note that values of deformations have no physical meaning for buckling analysis
CRITICAL BUCKLING LOAD EVALUATION
For Type 1 buckling analysis (Elastic Stress Analysis with small deformation theory) performed here ASME SEC VIII Div 2 gives a minimum design factor of \(\Phi_B=\frac{2}{\beta_{cr}}\).
For unstiffened cylinders under external pressure \(\beta_{cr} = 0.80 \) from eq 5.14.
Therefore \(\Phi_B=\frac{2}{\beta_{cr}}=\frac{2}{0.80}=2.5\)
Critical Buckling Load (Mode 1)
= Perturbation Load * Eigenvalue/Design Factor
= 14.7*9.0009/2.5
= 52.9 psi
Critical Buckling Load (Mode 1 ASME PTB-3)
= (Perturbation Load +Perturbation Load* Eigenvalue)/Design Factor
= (14.7 + 14.7*7.939)/2.5
= 52.6 psi
The values compare fairly well.
Since external pressure = 14.7 psi < buckling pressure = 52.9 psi the structure is safe w.r.t buckling under the design conditions.
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I’m trying to get into the latest Manin-Marcolli paper Quantum Statistical Mechanics of the Absolute Galois Group on how to create from Grothendieck’s dessins d’enfant a quantum system, generalising the Bost-Connes system to the non-Abelian part of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$.
In doing so they want to extend the action of the multiplicative monoid $\mathbb{N}_{\times}$ by power maps on the roots of unity to the action of a larger monoid on all dessins d’enfants.
Here they use an idea, originally due to Jordan Ellenberg, worked out by Melanie Wood in her paper Belyi-extending maps and the Galois action on dessins d’enfants.
To grasp this, it’s best to remember what dessins have to do with Belyi maps, which are maps defined over $\overline{\mathbb{Q}}$
\[ \pi : \Sigma \rightarrow \mathbb{P}^1 \] from a Riemann surface $\Sigma$ to the complex projective line (aka the 2-sphere), ramified only in $0,1$ and $\infty$. The dessin determining $\pi$ is the 2-coloured graph on the surface $\Sigma$ with as black vertices the pre-images of $0$, white vertices the pre-images of $1$ and these vertices are joined by the lifts of the closed interval $[0,1]$, so the number of edges is equal to the degree $d$ of the map.
Wood considers a very special subclass of these maps, which she calls Belyi-extender maps, of the form
\[ \gamma : \mathbb{P}^1 \rightarrow \mathbb{P}^1 \] defined over $\mathbb{Q}$ with the additional property that $\gamma$ maps $\{ 0,1,\infty \}$ into $\{ 0,1,\infty \}$.
The upshot being that post-compositions of Belyi’s with Belyi-extenders $\gamma \circ \pi$ are again Belyi maps, and if two Belyi’s $\pi$ and $\pi’$ lie in the same Galois orbit, then so must all $\gamma \circ \pi$ and $\gamma \circ \pi’$.
The crucial Ellenberg-Wood idea is then to construct “new Galois invariants” of dessins by checking existing and easily computable Galois invariants on the dessins of the Belyi’s $\gamma \circ \pi$.
For this we need to know how to draw the dessin of $\gamma \circ \pi$ on $\Sigma$ if we know the dessins of $\pi$ and of the Belyi-extender $\gamma$. Here’s the procedure
Here, the middle dessin is that of the Belyi-extender $\gamma$ (which in this case is the power map $t \rightarrow t^4$) and the upper graph is the unmarked dessin of $\pi$.
One has to replace each of the black-white edges in the dessin of $\pi$ by the dessin of the expander $\gamma$, but one must be very careful in respecting the orientations on the two dessins. In the upper picture just one edge is replaced and one has to do this for all edges in a compatible manner.
Thus, a Belyi-expander $\gamma$ inflates the dessin $\pi$ with factor the degree of $\gamma$. For this reason i prefer to call them
dessinflateurs, a contraction of dessin+inflator.
In her paper, Melanie Wood says she can separate dessins for which all known Galois invariants were the same, such as these two dessins,
by inflating them with a suitable Belyi-extender and computing the monodromy group of the inflated dessin.
This monodromy group is the permutation group generated by two elements, the first one gives the permutation on the edges given by walking counter-clockwise around all black vertices, the second by walking around all white vertices.
For example, by labelling the edges of $\Delta$, its monodromy is generated by the permutations $(2,3,5,4)(1,6)(8,10,9)$ and $(1,3,2)(4,7,5,8)(9,10)$ and GAP tells us that the order of this group is $1814400$. For $\Omega$ the generating permutations are $(1,2)(3,6,4,7)(8,9,10)$ and $(1,2,4,3)(5,6)(7,9,8)$, giving an isomorphic group.
Let’s inflate these dessins using the Belyi-extender $\gamma(t) = -\frac{27}{4}(t^3-t^2)$ with corresponding dessin
It took me a couple of attempts before I got the inflated dessins correct (as i knew from Wood that this simple extender would not separate the dessins). Inflated $\Omega$ on top:
Both dessins give a monodromy group of order $35838544379904000000$.
Now we’re ready to do serious work.
Melanie Wood uses in her paper the extender $\zeta(t)=\frac{27 t^2(t-1)^2}{4(t^2-t+1)^3}$ with associated dessin
and says she can now separate the inflated dessins by the order of their monodromy groups. She gets for the inflated $\Delta$ the order $19752284160000$ and for inflated $\Omega$ the order $214066877211724763979841536000000000000$.
It’s very easy to make mistakes in these computations, so probably I did something horribly wrong but I get for both $\Delta$ and $\Omega$ that the order of the monodromy group of the inflated dessin is $214066877211724763979841536000000000000$.
I’d be very happy when someone would be able to spot the error!
Leave a Comment
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The Helmholtz free energy \(A (N, V, T ) \) is a natural function of \(N, V \) and \(T\). The isothermal-isobaric ensemble is generated by transforming the volume \(V\) in favor of the pressure \(P\) so that the natural variables are \(N\), \(P\), and \(T\) (which are conditions under which many experiments are performed, e.g., `standard temperature and pressure'.
Performing a Legendre transformation of the Helmholtz free energy
\[ \tilde{A}(N,P,T) = A(N,V(P),T) - V(P) \frac {\partial A}{\partial V}\]
But
\[ \frac {\partial A}{\partial V} = -P\]
Thus,
\[\tilde{A}(N,P,T) = A(N,V(P),T) + PV \equiv G(N,P,T)\]
where \(G (N, P, T ) \) is the Gibbs free energy.
The differential of \(G\) is
\[ dG = \left(\frac {\partial G}{\partial P}\right)_{N,T} dP+ \left(\frac {\partial G}{\partial T}\right)_{N,P} dT+ \left(\frac {\partial G}{\partial N}\right)_{P,T} dN\]
But from \(G = A + PV \), we have
\[ dG = dA + PdV + VdP \]
but \(dA = - SdT - PdV + \mu dN \), thus
\[ dG = - SdT + VdP + \mu dN \]
Equating the two expressions for \(dG\), we see that
\[V=\left(\frac {\partial G}{\partial P}\right)_{N,T}\]
\[S=-\left(\frac {\partial G}{\partial T}\right)_{N,P}\]
\[\mu=\left(\frac {\partial G}{\partial N}\right)_{P,T}\]
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If polarization is interpreted as a pattern/direction of the electric-field in an electromagnetic wave and the frequency as the frequency of oscillation, how can we interpret polarization and frequency when we are dealing with one single photon?
Maxwell's equations
exactly define the propagation of a lone photon in free space. The state of a photon can be defined by a vector valued state in Hilbert space and this vector valued state is a precise mathematical analogy of the $\vec{E}$ and $\vec{H}$ fields of a macroscopic, classical field. That's not to say that, for a single photon, the $\vec{E}$ and $\vec{H}$ are to be interpreted as electric and magnetic field: the vector valued $\vec{E}$ and $\vec{H}$ state is the unitarily evolving quantum state before any measurement is made. But:
There is a one-to-one, onto correspondence between every classical electromagnetic field for a given system and a one photon quantum state for a photon propagating in that system.
This is the
first quantized description of the photon. To understand what measurements a photon state implies, one has to shift to a second quantized description where we have electic and magnetic field observables, whose measurements behave more and more like classical measurements as the number of photons gets bigger. A classical state is a coherent state of the second quantised field. But, given a photon can be described by a vector valued quantum state, it should be clear that polarization and all like "classical" attributes are meaningful for a lone photon.
In particular, a photon can be a
quantum superposition of eigenstates, so:
One photon can be spread over a range of frequencies and wavelengths (i.e. it can be in a superposition of energy eigenstates), with possibly different polarisation for all components of the superposition.
One can even broaden this concept to propagation through dielectric mediums: the light becomes a quantum superposition of free photons and excited matter states, and the lone, first quantized quasiparticle that results from this superposition (strictly speaking a "polariton" rather than a true, fundamental, photon) has a quantum state which evolves following Maxwell's equations solved for the medium. Thus, for example, we talk about lone photons propagating in the bound modes of optical fibres.
Another take on the one photon state is given in the first chapter of Scully and Zubairy "Quantum Optics". The one photon state $\psi$ can be defined by the
ensemble statistics derived from the second quantized electric and magnetic field observables:
$$\vec{E} = \left(\begin{array}{c}\left<0 | \hat{E}_x | \psi\right>\\\left<0 | \hat{E}_y | \psi\right>\\\left<0 | \hat{E}_z | \psi\right>\end{array}\right);\quad\quad \vec{B} = \left(\begin{array}{c}\left<0 | \hat{B}_x | \psi\right>\\\left<0 | \hat{B}_y | \psi\right>\\\left<0 | \hat{B}_z | \psi\right>\end{array}\right)$$
where $\hat{E}_j$ is the $j^{th}$ component of the vector valued electric field observable and $\hat{B}_j$ that of the magnetic induction observables. ($[\hat{E}_j, \hat{B}_j]=0$ for $j\neq k$ and, in the right units, $[\hat{E}_j, \hat{B}_j]=i\,\hbar\,I$). For a one-photon state $\psi$, these statistics:
Propagate exactlyfollowing Maxwell's equations; Unquely define the light field's quantum state for a one-photon state, even though they are not the state. This is in the same way that the mean of the classical Poisson probability distribution uniquely defines the distribution (even though it is a lone number, not a distribution).
Things are much more complicated for general, $N$ photon states so we need much more information than simple means to fully define the quantum state particularly with entangled states. Going back to our classical probability distribution analogy, the normal distribution needs
two independent parameters, mean and variance, to wholly specify it, so it's a more complicated thing than the Poisson distribution, which is defined by only its mean (which equals the variance).So quantum fields are hugely more complicated things than classical ones. But a coherent state of any photon is again uniquely defined by the mean values of the field observables, which means again propagate following the same Maxwell equations as the one-photon means: hence the one-to-one, onto correspondence between classical and one-photon states I spoke of - I like to call this the one photon correspondence principle ("OpCoP"). Why our macroscopic EM fields seem to behave like coherent quantum states rather than hugely more general, entangled ones (unless one goes to considerable experimental effort to observe entanglement) is still an open question. It is interesting to note, though, that the class of coherent states is the unique class of quantum harmonic oscillator states that achieve the lower bound of the Heisenberg uncertainty inequality.
Also see my answers to:
If photons carry 1 spin unit, why does visible light seem to have no angular momentum? and Electromagnetic radiation and quanta.
Incidentally, even though general, entangled light states are hugely more complicated than one-photon (and, equivalently through the OpCoP, classical) light states, in principle we can still decompose them into a quantum superposition of tensor products of coherent states and so represent a general state by a set of field observable means. This was one of the contributions of 2005 Nobel Laureate Roy Glauber, who showed the above in 1963 in:
The coherent state tensor products are, however,
overcomplete so the decomposition of a general quantum state into coherent states is highly not unique. Nonetheless, such a decomposition allows classical-like techniques to be brought to bear on entangled quantum states (in principle - in practice it is still complicated!).
If you google Iwo Bialynicki-Birula and his work on the photon wave function, he has heaps more to say about the one-photon wave function. He defines the photon wave function as the positive frequency part of left and right circularly polarized eigenfunctions $\vec{F}_\pm = \sqrt{\epsilon} \vec{E} \pm i \sqrt{\mu} \vec{H}$. Iwo Bialynicki-Birula's personal website is http://cft.edu.pl/~birula and all his publications are downloadable therefrom. $|\vec{F}_+|^2 + |\vec{F}_-|^2$ is the electromagnetic energy density. He defines the pair $(\vec{F}_+, \vec{F}_-)$, normalised so that $|\vec{F}_+|^2 + |\vec{F}_-|^2$ becomes a probability density to absorb the photon at a particular point, to be a first quantized photon wave function (without a position observable). There is special, nonlocal inner product to define the Hilbert space and in such a formalism the general Hamiltonian observable is $\hbar\, c\, \mathrm{diag}\left(\nabla\wedge, -\nabla\wedge\right)$. Please also see Arnold Neumaier's pithy summary (here) of a key result in section 7 of Bialynicki-Birula's "Photon wave function" in
Progress in Optics 36 V (1996), pp. 245-294 also downloadable from arXiv:quant-ph/0508202. The Hilbert space of Riemann Silberstein vector pairs that Bialynicki-Birula defines is acted on by an irreducible unitary representation, defined by Bialynicki-Birula's observables $\hat{H}$, $\hat{\mathbf{P}}$, $\hat{\mathbf{K}}$ and $\hat{\mathbf{J}}$, of the full Poincaré group presented in the paper.
If polarization is interpreted as a pattern/direction of the electric-field in an electromagnetic wave and the frequency as the frequency of oscillation, how can we interpret polarization and frequency when we are dealing with one single photon?
The classical wave is composed by a large ensemble of photons. Both the photon/particle equations and Maxwell's equations contain the state of the electric field in their solutions. Thus, it is not a matter of interpretation but a matter of showing how from single individual photons mathematically described by the equation of second quantization as such, one can derive for an ensemble of photons the electromagnetic wave.
This is not simple but it has been done. A demonstration is given in the article in this blog.
Hand waving an answer: the functions describing the photons have to be coherent (in phase), then the constants in their mathematical description which pertain to the electric and magnetic field "miraculously" build up a classical electromagnetic field which carries the frequency which is contained in the particle description in E=h*nu.
Well the energy of the photon is simply h.f , so if you can determine the energy of a single photon, you can determine its frequency. One way to determine the energy of a photon; assuming that you can generate one photon at a time, all of the same energy, would be to use the photo-electric effect, with adjustable band gap photo-cathode materials in a PMT, which can detect single photons. Variable bandgap photo-cathodes, can be made over limited ranges from III-V Ternary or quaternary compounds, such as GaAsP or InGaAsP. Smaller bandgap cathodes will emit a photo-electron; higher bandgap ones will not.
Now you didn't say you wanted to know a practical way of doing that, but if you have a need for single photons of known frequency and polarization, making the PMTs should be no problem for you.
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Density-based clustering in spatial data (2) This is the second of a series of posts on cluster-algorithms and ideas in data analysis (and related fields). Ordering points to identify the clustering structure (OPTICS) is a data clustering algorithm presented by Mihael Ankerst, Markus M. Breunig, Hans-Peter Kriegel and Jörg Sander in 1999 [1]. It builds on the ideas of DBSCAN, which I described in a previous post. But let’s cut the intro and dive right into it.
Let $(X,d)$ be a finite discrete metric space, i.e.~let $X$ be a data set on which we have a notion of similarity expressed in terms of a suitable distance function $d$. Given a positive integer $m \in \mathbb{N}$ we can define a notion of
density on points of our data set, which in turn can be used to define a perturbed metric. Given a starting point $x_0 \in X$ the OPTICS-algorithm iteratively defines a linear ordering on $X$ in terms of a filtration $X_0 \subset X_1 \subset \ldots \subset X_n$ of $X$, where $X_{n+1}$ is obtained from $X_n$ by appending the closest element (with respect to the perturbed metric) in its complement.
The original OPTICS-algorithm also takes an additional parameter $\varepsilon > 0$. However this is only needed to reduce the time complexity and is ignored in our discussion for now — to be more precise, we implicitly set $\varepsilon = \infty$.
Definitions
Let $(X,d)$ be a finite discrete metric space and $m \in \mathbb{N}$ a positive integer. We define the
(co-)density $\delta_m(x)$ of a point $x$ by \[ \delta_m(x) := d(x, \mathfrak{nn}_m(x) ), \] where $\mathfrak{nn}_m(x) $ is an $m$-nearest-neighbor of $x$. Loosely speaking: the lower the value $\delta_m(x)$ the closer the neighbors of $x$ are distributed around $x$. Note that with $\varepsilon = \delta_m(x)$ the point $x$ is a core point in the sense of [2] — in a previous post about DBSCAN we called such a point $\varepsilon$-dense. That is why in the literature $\delta_m(x)$ is referred to as core-distance. I however like to think of it, and hence refer to it, as a notion of co-density.
Given the co-density $\delta_m(.)$ we can define the
reachability distance $r_x(y)$ of $y$ from $x$ by \[ r_x(y) := \text{max} \big\{ \delta_m(x), d(x,y) \big\}. \] Note that $r_x(y)$ is not symmetric, since the density of $x$ and $y$ may differ. Sketch of the algorithm
Choose a starting point $x_0 \in X$. Then we can iteratively define a filtration $X_0 \subset \ldots \subset X_n$ of the data set $X$ by \[ X_0 := \{ x_0 \} \ \ \text{and} \ \ X_{k+1} := X_k \cup \{ x_{k+1} \}, \] where $x_{k+1}$ minimizes $r_{X_k}(.)$ over $X \setminus X_k$. In parallel we define a sequence $(r_n)$ of distances by \[ r_0 = 0 \ \ \text{and} \ \ r_{k+1} := r_{X_k}(x_{k+1}). \] Note that a small distance $r_k$ may be understood as $x_k$ being close to a rather dense region. Therefore the filtration tries to stay in dense regions for as long as possible, before it passes a less dense region. The cluster structure can now be extracted by analyzing the
reachability-plot, that is a $2$-dimensional plot, with the ordered $x_k$ on the $x$-axis and the associated distances $r_k$ on the $y$-axis. By the above considerations it should be clear that clusters show up as valleys in the reachability plot. The deeper the valley, the denser the cluster. Example
Consider the data points $X\subset \mathbb{R}^2$ in the euclidian plane given in the following figure:
Below you see the reachability plot corresponding the OPTICS-algorithm applied to the above data set with $m=4$. The starting point is chosen among the group of points on the left.
References [1] M. Ankerst, M.M. Breunig, H.-P. Kriegel, OPTICS: Ordering Points To Identify the Clustering Structure, ACM SIGMOD international conference on Management of data (1999), pp 49–60. [2] M. Ester, H.-P. Kriegel, J. Sander, and X. Xu, A density-based algorithm for discovering clusters in large spatial databases with noise, Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (1996), pp. 226–231.
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We know that the density matrix for a 2-qubit system can be written in the Pauli representation as :
$$\rho = \frac{1}{4}\sum_{ij}t_{ij}\sigma_i\otimes\sigma_j$$ where $\sigma_i$ are the Pauli operators with $\sigma_0 = I$ and $t_{ij} = \langle\sigma_i\otimes\sigma_j\rangle = Tr(\rho\sigma_i\otimes\sigma_j)$. Recently, I read in a book that if the qubits are entangled then $|t_{11}| + |t_{22}| + |t_{33}| > 1$, so the condition for the state to be written as a product of two 1-qubit states is that the sum must be less than or equal to 1. Is there any elementary proof for it ? The easiest and the most straightforward test for a 2-qubit state given by the column vector \begin{pmatrix} a \\ b \\ c \\ d \\ \end{pmatrix} w.r.t. the basis $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$, to be separable is that $ad = bc$. How is this related to the one given above ? Thanks.
We know that the density matrix for a 2-qubit system can be written in the Pauli representation as :
Just to be clear, you want to prove that $|t_{11}|+|t_{22}|+|t_{33}|>1$ $\Rightarrow$ ($\rho$ is entangled), or equivalently ($\rho$ is separable) $\Rightarrow$ $|t_{11}|+|t_{22}|+|t_{33}|\leq 1$.
The other direction is not true.
Let's prove this for a product state $\rho=\rho_A\otimes \rho_B$ first. In this case $$t_{ij}=(\mathrm{tr} \rho_A \sigma_i) (\mathrm{tr} \rho_B \sigma_j)=u_i v_j.$$ So $$|t_{11}|+|t_{22}|+|t_{33}|=|u_1| |v_1|+|u_2| |v_2|+|u_3||v_3| \leq |\vec{u}| |\vec{v}| \leq 1,$$ where we used the Cauchy-Schwarz inequality and that the Bloch vectors of $\rho_A$ and $\rho_B$ have norm less or equal one (otherwise the reduced states wouldn't be positive).
Now for a general separable state, a mixture of product states $\rho=\sum_k p_k \rho_A^{(k)}\otimes \rho_B^{(k)}$ with probabilities $p_k$, we find that $$\sum_i |t_{ii}|=\sum_i |\sum_k p_k t_{ii}^{(k)}|\leq\sum_k p_k \sum_i |t_{ii}^{(k)}|\leq 1.$$ In the last step we used that we proved the relation for product states already and that the probabilities sum up to one.
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1. Background:
Lense-Thirring precession is the rotation undergone by the spin of a particle located in the gravitational field of a massive spinning body. In terms of asymptotically inertial coordinates $(t,\vec x)$ in a four-dimensional space-time, and if we denote by $\vec J$ the angular momentum of the source, the angular velocity of precession of a particle at position $\vec x$ is
$$\vec\Omega = \frac{1}{r^3}\left(-\vec J+3\frac{(\vec J\cdot\vec x)\vec x}{r^2}\right)$$
where $r\equiv\sqrt{\vec x\cdot\vec x}$ and the dot denotes the scalar product of spatial vectors. We use units such that $G=c=1$. (For the derivation of this formula, see e.g. Misner-Thorne-Wheeler, section 40.7.)
2. My question:
The dependence of the angular velocity $\vec\Omega$ on the source's angular momentum $\vec J$ and on the spatial position $\vec x$ is exactly the same as that of the electric field generated by an electric dipole. Under such an identification, $\vec\Omega$ is identified with the electric field while $\vec J$ is identified with the dipole moment. My question is the following: is there an
intuitive explanation for why the precessional angular velocity has to be of the same form as the field sourced by a dipole?
Just to make things clear: I'm not looking for a mathematical proof that the above formula for $\vec\Omega$ is correct. Instead, I'd like to find an intuitive (but nevertheless rigorous) argument that makes the above result obvious. Indeed, the standard derivation of the formula for $\vec\Omega$ relies on some relatively advanced mathematical tools, but the result is so simple and pretty that I suspect there's a deeper reason for the apparent coincidence with the formula from electrostatics. (Of course, my expectation may be wrong.)
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On this page I am going to collect wordpress plugins and other resources that I use and like.
Contents
Plugins currently in use Akismet
This is probably the most important plugin, it protects your blog from spam. It comes with your wordpress installation (also when self hosted) and you only need to request a key. It is really good at detecting spam comments (since February no false positives or false negatives).
Email Subscribers
After some time I thought it would be good to provide email subscriptions. I tried some plugins (unfortunately I don’t recall the names) until I found this. I am still not really satisfied since it is not possible to have more than one subscription form (would be useful since I blog in German and English and also about completely different topics). Also the handling with creating groups is a little bit weird (you can not create new groups directly, you need to create a user and add him to a new group). I would be happy about suggestions for a better tool.
MathJax-LaTeX
This plugin is a must have if you want to write \(\LaTeX\) formulas. The syntax is exactly the same as in normal \(\LaTeX\), the only difference is that you need to use $$ instead of $, because a single $ is reserved for something else.
$$p(x) = \frac{1}{\sigma \times 2 \pi} \times e^{-\frac{(x-\mu)^2}{2 \sigma^2}}$$
Table of Contents Plus
Some of my posts tend to get really long, which looks even more so because of the very narrow design I use. “Table of Contents Plus” has really many settings (how many headings there should be at least, whether to insert in posts and/or pages, how many levels of headers it should contain, how it should look, …) and I think it is really cool that it gets included automatically into pages/posts that fulfil the criteria defined in the settings. I use mainly default settings and now it looks very similar to the Wikipedia table of content. But see for yourself on top of the post 😉
WordPress Related Posts
In other blogs I like it when they link to related posts or even better to related content somewhere on the web. For that purpose I installed “WordPress Related Posts” which detects (in an intelligent way) posts from your site that are relevant to the new post (and you can change them). It also finds posts over the web that seem to have similar content than your new post or seem to be somehow related. Currently I am not so happy with the second feature, because most of the time they are only very slightly related to a my post (share only one keyword).
It looks like this for related posts (the other feature I don’t use):
WP-Mail-SMTP
This plugin is only required if you have a self hosted wordpress blog. When I was looking for the “Email subscribers” plugin, I first thought none of the plugins was working, but then I found out that I needed to install an SMTP plugin. It’s very simple, you only need to add information about your mail server (gmail in my case) and then it works.
WP Code Highlight
This plugin is pretty ok for languages it knows like XML (but only since I changed the ugly pink/red line color to gray).
Unfortunately most of the time I write R code and then it does not know how to deal with it. I assume it should be possible to modify the css yourself but I am not sure if I will ever find time or just hope that a better plugin (with R support) comes along. Sidenote: I recently downloaded WP Codebox and I think it will work better with R code, but I currently don’t have enough time (The old posts also require updating then).
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One can compute the amount of twin primes below a positive integer $n$ by using the Mathematica command (taken from OEIS A001097):
Length[Select[Prime[Range[n]], PrimeQ[# + 2] &]]
The twin prime conjecture states that this value should approach $1.320323632\ldots\times\int_2^n \frac{dt}{\log^2 t}$. So I tried using
N[Integrate[Log[t]^(-2),{t,2,n}]]*1.320323632
However, I got vastly different results than I expected. For instance, for $n=1000$ and $n=10000$ I get, respectively, $45.8\ldots$ and $214.21\ldots$, while the real values are $174$ and $1270$. Obviously, there is something wrong with the Mathematica command above. But what is it?
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