text:
Theorems

Bers simultaneous uniformisation

quasifuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. A quasifuchsian group of the first kind is one whose limit set is the whole of the invariant Jordan curve. The choice of the name “quasifuchsian” is justified by the following

Proposition: A Klenian group $\Gamma$ is quasifuchsian iff it is quasiconformally conjugate (in $\mathrm{PSL}_2\mathbb{C}$) to a Fuchsian group.

Proof: The reverse direction is easy: boundary extensions of quasiconformal maps send limit sets of Fuchsian groups, which are round circles, to invariant Jordan curves.

The forward direction appears in a paper of Bers, and proceeds by noting that the domain of discontinuity $\Omega(\Gamma)$ has two disjoint components, conformally mapping any one of them (with the boundary) to a round disc, and proving that we can extend this to a quasiconformal mapping of the other component.

Given a closed genus-g surface $\Sigma_g$, let $QF(\Sigma_g)$ denote the space of quasifuchsian deformations of the fundamental group $\pi_1(\Sigma_g)$, i.e. all conjugates of $\pi_1(\Sigma_g)$ by quasiconformal maps.

Theorem (Bers). $QF(\Sigma_g) \cong \mathcal{T}(\Sigma_g) \times \mathcal{T}(\overline{\Sigma_g})$ (for $g \geq 2$.)

In other words, given any two Riemann surfaces of the same genus $g \geq 2$, there is some (hyperbolizable) 3-manifold which simultaneously uniformizes them, in the sense that its conformal boundary consists of two components, which are precisely the given Riemann surfaces.

(John Hubbard in his tome on Teichmüller theory compares the result to the works of Hieronymous Bosch: somehow unnatural and horrifying, but still a work of art. I don’t know if I agree, but it’s an interesting description anyhow … )

Proof: Let $\Gamma := \pi_1(\Sigma_g)$, and define a map $\Theta: QF(\Sigma_g) \to \mathcal{T}(\Sigma_g) \times \mathcal{T}(\overline{\Sigma_g})$ by $\phi\Gamma\phi^{-1} \mapsto \left( (\phi(S), \phi), (\bar\phi(S), \bar\phi) \right)$.

It is fairly straightforward to show that $\Theta$ is surjective: given $((X,g), (Y,h)) \in \mathcal{T}(\Sigma_g) \times \mathcal{T}(\overline{\Sigma_g})$, $g \coprod h$ lifts to a quasiconformal map $f: U \coprod L \to \mathbb{H}^2 \coprod \overline{\mathbb{H}^2}$, where U and L denote the upper and lower half-planes resp. We may then check that the quasifuchsian group corresponding to f is sent to the point $((X,g), (Y,h)) \in \mathcal{T}(\Sigma_g) \times \mathcal{T}(\overline{\Sigma_g})$ we started with.

To show $\Theta$ is injective: suppose $\Theta(\rho_1) = \Theta(\rho_2)$, so that there exists some conformal map j taking the conformal boundary of $N_{\rho_1}$ to that of $N_{\rho_2}$, which lifts to a conformal map $\tilde{j}: \Omega(\rho_1) \to \Omega(\rho_2)$ between the respective domains of discontinuity.

Now recall the limit sets $\Lambda(\rho_1)$ and $\Lambda(\rho_2)$ are both invariant Jordan curves, so we may canonically extend $\tilde{j}$ over the limit sets by sending a point on one limit set (thought of as a direction at infinity) to the the corresponding point (direction at infinity) on the other limit set. This gives us a map $\hat{j}: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$, which we may argue is quasiconformal.

Now an extension theorem of Douady-Earle allows us to naturally extend this quasiconformal map $\hat{j}$, or more specifically its quasisymmetric restriction to the limit sets, to a quasiconformal map of the interior disc (i.e. the quasiconformal copy of the hyperbolic plane) which it bounds. That such a map existed we already knew, but the naturality properties that come with the Douady-Earle extension, plus the fact that the quasiconformal $\hat{j}$ is conformal on a subset of $\hat{\mathbb{C}}$ of full measure and hence conformal, tells us that this induced extension is in fact conformal, i.e. in fact $\rho_1(\Gamma)$ is conformally conjugate to $\rho_2(\Gamma)$, as desired.

It is clear (or, at least, reasonable) that the map is continuous, as is its inverse, and we are done (a more rigorous proof of this last part would involve going through the construction of the map more carefully, and probably invoking the continuous dependence given by the measurable Riemann mapping theorem at some point.)


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# What happens to a photon as it experiences an increasing amount of force opposite to its momentum? Could it be frozen?

Image that a photon is emitted from a sphere, moving orthogonally away from the sphere's surface. As the weight of this sphere increased, how would the increasing gravitational force affect the motion of the photon? Imagine that the mass of the sphere was sufficiently high to form a black hole - but the photon is emitted outside the black hole's event horizon - what happens to the photon then?

• It gets redshifted, therefore it loses momentum. Other than that, it just moves forward in a straight line. – Wood Oct 18 '16 at 9:08
• So as the force approach infinity, would the only thing that occured be the photon losing energy? – Honwang Oct 18 '16 at 9:14
• Additionally, where does this energy go? – Honwang Oct 18 '16 at 9:14
• I think it just goes to gravitational potential energy, just like any massive particle. – Wood Oct 18 '16 at 9:33

• @Honwang: more or less; the gravitational frequency shift when increasing the distance from the center of mass from $r_0\to r$ is given by $1+z = \sqrt{(1-r_s/r)/(1-r_s/r_0)}$, which goes to infinity at the Schwarzschild radius – Christoph Oct 18 '16 at 16:38

format: plain


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# Rowcolor, Multicolumn, and Tabu

Ok, I want to create an environment that prepares my table in such a way that I have consistent results.

\documentclass[10pt]{article}

\usepackage[table]{xcolor}

\usepackage{graphics}
\usepackage{colortbl} %make alternating row colours

\usepackage{tabu} % \rowfont

\usepackage{multirow} % for multirow and multicol
\usepackage{booktabs}
\usepackage{hhline} %to make hlines over a certain number of columns

\usepackage{array} % adds more features on how tables are used
\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} %left-align with specific size
\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}%centre-align with specific size
\newcolumntype{R}[1]{>{\raggedleft\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}

\rowcolors{2}{blue!15}{white}
\arrayrulecolor{blue}

\newenvironment{myTable}[4][htbp]
{
\begin{table}[#1]
\begin{center}
\caption{#4}\label{#3}

\begin{tabu}{#2}
\hline
}{
\end{tabu}
\end{center}
\end{table}
}

\begin{document}

\begin{myTable}[htb]{*{4}{c |}|*{2}{C{1.5cm} |}| C{3cm}}%
{tbl:label}%
{Caption Text}
\theader$a$ & $b$ & $c$ & $d$ & $(a\land b)$ & $(c\land d)$ & $(a\land b)\lor(c\land d)$\\\hline
0 & 0 & 0 & 0 & 0 & 0 & 0\\
\multicolumn{4}{@{}c@{}}{\upbracefill} & \multicolumn{2}{@{}c@{}}{\upbracefill} & \upbracefill\\
\end{myTable}
\end{document}

I create an environment myTable that contains a table and tabu environment. However, whenever I use \multicolumn together with \rowcolor. I get the following compiler error:

! Emergency stop. \multispan ->\omit @multispan l.46 \end{myTable}

I had the same within a tabular environment and it worked fine. When I remove the multicolumn commands, it works fine. When I remove the rowcolor command, it works fine.

I can work around it by placing a \cellcolor{white} in each multicolumn command. But that's not a solution.

Does anybody has any ideas on how to fix it??? Any help is appreciated!

• Tabu is currently unmaintained and causes some undesired behavior especially when convinced with color. I'd suggest switching to an alternative such as tabularx. May 5, 2020 at 15:09
• Welcome to TeX.SE... It is not advisable to use the package tabu... Better switch over to normal tabular with colortbl support.... May 5, 2020 at 16:16
• OK, [colortbl] is fine but I need tabu for the [\rowfont] command. (I need to set the font color for the entire row). Is there an alternative? May 6, 2020 at 21:04

Another option would be the new tabularray package. The package allows you to completely separate the format from the content. It support cells spanning multiple columns and rows and coloring.

\documentclass[10pt]{article}

\usepackage{xcolor}
\usepackage{tabularray}

\NewColumnType{C}[1]{Q[c,wd=#1]}

\newenvironment{myTable}[4][htbp]
{
\begin{table}[#1]
\centering
\caption{#4}\label{#3}
\begin{tblr}{
colspec={#2},
hline{1-Y}={blue},
vline{1-Z}={1}{1-Y}{blue}, vline{5, 7}={2}{1-Y}{blue},
row{even}={white}, row{odd}={blue!15},
row{1}={bg=blue, fg=white, mode=math},
row{Z}={white, rowsep=0pt},
}
}{
\end{tblr}
\end{table}
}

\begin{document}

\begin{myTable}{*{4}{c}*{2}{C{1.5cm}}C{3cm}}%
{tbl:label}%
{Caption Text}
a & b & c & d & (a\land b) & (c\land d) & (a\land b)\lor(c\land d) \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\SetCell[c=4]{c} \raisebox{2mm}{$\underbrace{\hspace{2cm}}$} & & & & \SetCell[c=2]{c} \raisebox{2mm}{$\underbrace{\hspace{4cm}}$} & & \raisebox{2mm}{$\underbrace{\hspace{3cm}}$} \\
\end{myTable}

\end{document}

Which produces the output:

You can specify the format using row and column parameters, e.q. I used the setting mode=math so you don't have to write  every time in the first row.

• this is a really good package!.. instructions are a bit happy.. but I worked it out :) Apr 5 at 0:45


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# Thread: Separable Equations DFQ #2

1. ## Separable Equations DFQ #2

Ok so I really do think that this one is a separable equation but if not maybe someone could let me know that too haha

I need to separate this equation to have the dy and y's on one side and and the dx and x's on the other so I can integrate and solve the problem.

[(y*e^(xy))-(1/y)]dx + [(x*e^(y))+(x/(y^2))]dy=0

I have no idea how to separate the equation!

Any hints or first steps would be nice!!! THANK YOU!!!!

2. Originally Posted by cheertcc101
Ok so I really do think that this one is a separable equation but if not maybe someone could let me know that too haha

I need to separate this equation to have the dy and y's on one side and and the dx and x's on the other so I can integrate and solve the problem.

[(y*e^(xy))-(1/y)]dx + [(x*e^(y))+(x/(y^2))]dy=0

I have no idea how to separate the equation!

Any hints or first steps would be nice!!! THANK YOU!!!!
It's not separable, the term $e^{xy}$ stops this. It is however exact meaning the is an $F$ such that

$dF = F_x \,dx + F_y,dy = \left( y e^{xy} - \frac{1}{y}\right) dx +\left( x e^{ x y} + \frac{x}{y^2} \right) dy = 0$

(I'm guessing a typo - the $e^{y}$ should be $e^{x y}$ in the second term).

3. Alright so i agree that it would make way more sense for there to be a typo in the second part of the equation and if there is then I know how to do the problem BUT the teacher didnt say anything about a typo SOO i was wondering if for some reason there was NOT a typo then is there a way to still work the problem. I looked for special integration factors but found nothing .. maybe Im missing something or if there is no way to work the problem without a typo then I would like to know that too.

THANKS for the help thus far!


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# Notifying users in comments

I am a little bit confused by the @username syntax in comments. It says that both @peter and @PeterSmith will work, which in my opinion should be changed to something more clear.

Does this mean I can call user Peter Smith by @peter or do I have to address him with his full username including spaces?

• My understanding is that going up to the first space in the username works, e.g. @Peter will work on Peter Smith but not on PeterSmith. – Qiaochu Yuan Feb 23 '11 at 13:30
• @Qiaochu That would make sense. Nonetheless, I think you will agree with me that the explanation provided is confusing. – Milosz Wielondek Feb 23 '11 at 14:00
• Even @Mil is supposed to work. – Aryabhata Feb 23 '11 at 15:32
• I would say that ambiguity depends on context.If we had comments by a Peter Smith and by a Peter Jones, a reply starting @Peter would leave the adressee not well defined (!), but if only Peter Smith cared to post a comment, where is the problem? – Andrea Mori Feb 26 '11 at 9:12
• @Andrea Mori it's not about ambiguity as you understood it. It's about the automized notification system. – Milosz Wielondek Feb 26 '11 at 17:47

## 2 Answers

The authorative source with possibly more than you want to know is on meta.SO: How do comment @replies work?

My understanding is that at least the first three letters are needed, or enough to make a unique match, and that it's only matching against the limited set of users involved somehow in the post being commented on, such as the original poster, the person asking the question, and any previous commenters.


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Modular Square Root - Maple Help

NumberTheory

 ModularSquareRoot
 modular square root

 Calling Sequence ModularSquareRoot(x, n)

Parameters

 x - integer n - positive integer

Description

 • The ModularSquareRoot function computes a non-negative integer $y$ such that ${y}^{2}=x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}n$ if possible. If not possible, an error message is displayed.
 • When x has more than one square root, any one of them may be returned.

Examples

 > $\mathrm{with}\left(\mathrm{NumberTheory}\right):$

The follow numbers have square roots modulo $37$.

 > $\mathrm{residues}≔\left\{\mathrm{seq}\left({i}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}37,i=0..36\right)\right\}$
 ${\mathrm{residues}}{≔}\left\{{0}{,}{1}{,}{3}{,}{4}{,}{7}{,}{9}{,}{10}{,}{11}{,}{12}{,}{16}{,}{21}{,}{25}{,}{26}{,}{27}{,}{28}{,}{30}{,}{33}{,}{34}{,}{36}\right\}$ (1)

$30$ has a square root modulo $37$.

 > $\mathrm{evalb}\left(30\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{residues}\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{ModularSquareRoot}\left(30,37\right)$
 ${17}$ (3)
 > ${17}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}37$
 ${30}$ (4)

$24$ does not have a square root modulo $37$ and so an error message is displayed.

 > $\mathrm{evalb}\left(24\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{residues}\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{ModularSquareRoot}\left(24,37\right)$

Compatibility

 • The NumberTheory[ModularSquareRoot] command was introduced in Maple 2016.


text:
Thanks:  0

# Thread: Regarding mixing time of a Markov chain

1. Let $X=X_0, X_1, X_2, \ldots$ be a Markov chain on a finite state space $S$, and let $P$ denote the transition matrix.
Assume that there is an $\varepsilon>0$ such that whenever $\mu_0$ and $\nu_o$ are point distributions on $S$ (in other words, $\mu_0$ and $\nu_0$ are Direac masses) we have
$$\|\mu_0P-\nu_0P\|_{TV}\leq \varepsilon$$

Now let $Y=Y_0, Y_1, Y_2, \ldots$ be an independent copy of $X$.

Question. What can we say about the magnitude of $P[X_1\neq Y_1|X_0=x_0, Y_0=y_0]$, where $x_0$ and $y_0$ are two different states in $S$.

I intuitively think that we should be above to bound the above quantity by $\varepsilon$ up to multiplication by an absolute constant. But I am unable to prove it.

We have that
$$P[X_1\neq Y_1|X_0=x_0, Y_0=y_0] = \sum_{x\in S}\sum_{y\in S, y\neq x}P[X_1=x, Y_1=y|X_0=x_0, Y_0=y_0]$$
which is equal to
$$\sum_{x\in S} p(x_0, x)(1-p(y_0, x))$$
but I am unable to make progress.

2.

3. Really a standard exercise/result for countable state random variables (with emphasis on markov chains) is to show, where, for notational reasons it is understood in your particular case that $X_0 = 0$ and $Y_0 = 0$

$P[X_1\neq Y_1|X_0=x_0, Y_0=y_0] = P[X_1\neq Y_1] \geq d_{TV}(X_1,Y_1)= \|\mu_0P-\nu_0P\|_{TV}$
with $\mu := \mathbf e_0$ $\nu := \mathbf e_0$

and the inequality is reached with equality in the case of a maximal coupling. This is somewhat trivial here but useful for other couplings e.g. if $\nu_0$ was the steady state vector.

(nit: there are couple different formulations of total variation distance I am using the 1/2 L1 norm of difference between probability vectors interpretation for countable state r.v.v)

the inequality you're trying to prove 'points in the wrong way' so to speak. More context / background information is needed to make sense of what you're trying to do. E.g. if you don't know what coupling, that is a big deal and my post won't make any sense.

Originally Posted by steep
Really a standard exercise/result for countable state random variables (with emphasis on markov chains) is to show, where, for notational reasons it is understood in your particular case that $X_0 = 0$ and $Y_0 = 0$

$P[X_1\neq Y_1|X_0=x_0, Y_0=y_0] = P[X_1\neq Y_1] \geq d_{TV}(X_1,Y_1)= \|\mu_0P-\nu_0P\|_{TV}$
with $\mu := \mathbf e_0$ $\nu := \mathbf e_0$

and the inequality is reached with equality in the case of a maximal coupling. This is somewhat trivial here but useful for other couplings e.g. if $\nu_0$ was the steady state vector.

(nit: there are couple different formulations of total variation distance I am using the 1/2 L1 norm of difference between probability vectors interpretation for countable state r.v.v)

the inequality you're trying to prove 'points in the wrong way' so to speak. More context / background information is needed to make sense of what you're trying to do. E.g. if you don't know what coupling, that is a big deal and my post won't make any sense.
I realize now that the statement I want to prove does not hold for the random walk on a big complete graph.


text:
• ### Spin dynamics and exchange interactions in CuO measured by neutron scattering(1801.09473)

April 5, 2018 cond-mat.str-el
The magnetic properties of CuO encompass several contemporary themes in condensed matter physics, including quantum magnetism, magnetic frustration, magnetically-induced ferroelectricity and orbital currents. Here we report polarized and unpolarized neutron inelastic scattering measurements which provide a comprehensive map of the cooperative spin dynamics in the low temperature antiferromagnetic (AFM) phase of CuO throughout much of the Brillouin zone. At high energies ($E \gtrsim 100$\,meV) the spectrum displays continuum features consistent with the des Cloizeax--Pearson dispersion for an ideal $S=\frac{1}{2}$ Heisenberg AFM chain. At lower energies the spectrum becomes more three-dimensional, and we find that a linear spin-wave model for a Heisenberg AFM provides a very good description of the data, allowing for an accurate determination of the relevant exchange constants in an effective spin Hamiltonian for CuO. In the high temperature helicoidal phase, there are features in the measured low-energy spectrum that we could not reproduce with a spin-only model. We discuss how these might be associated with the magnetically-induced multiferroic behavior observed in this phase.
• ### Coupling of magnetic order and charge transport in the candidate Dirac semimetal EuCd$_2$As$_2$(1803.07061)

March 23, 2018 cond-mat.str-el
EuCd$_2$As$_2$ is a magnetic semimetal with unusual transport properties. It features similar networks of edge-sharing CdAs$_4$ tetrahedra to those found in the bulk Dirac semimetal Cd$_3$As$_2$, but in a layered, rather than three-dimensional, geometry and with interleaved planes of magnetic Eu$^{2+}$ ions. Here we use resonant elastic x-ray scattering to determine the evolution of magnetic order in EuCd$_2$As$_2$ below $T_\textrm{N}=9.5\,\textrm{K}$, as a function of temperature and applied magnetic field. We find an A-type antiferromagnetic structure with in-plane magnetic moments, and we observe dramatic magnetoresistive effects associated with field-induced changes in the magnetic structure and domain populations. Our ab initio electronic structure calculations indicate that the Dirac dispersion found in Cd$_3$As$_2$ is also present in EuCd$_2$As$_2$, but is gapped for $T < T_\textrm{N}$ due to the breaking of $C_3$ symmetry by the magnetic structure.
• ### The Final Chapter In The Saga Of YIG(1705.06594)

May 18, 2017 cond-mat.str-el
The magnetic insulator Yttrium Iron Garnet can be grown with exceptional quality, has a ferrimagnetic transition temperature of nearly 600 K, and is used in microwave and spintronic devices that can operate at room temperature. The most accurate prior measurements of the magnon spectrum date back nearly 40 years, but cover only 3 of the lowest energy modes out of 20 distinct magnon branches. Here we have used time-of-flight inelastic neutron scattering to measure the full magnon spectrum throughout the Brillouin zone. We find that the existing model of the excitation spectrum, well known from an earlier work titled "The Saga of YIG", fails to describe the optical magnon modes. Using a very general spin Hamiltonian, we show that the magnetic interactions are both longer-ranged and more complex than was previously understood. The results provide the basis for accurate microscopic models of the finite temperature magnetic properties of Yttrium Iron Garnet, necessary for next-generation electronic devices.
• ### Doping Dependence of Collective Spin and Orbital Excitations in Spin 1 Quantum Antiferromagnet La$_{2-x}$Sr$_x$NiO$_4$ Observed by X-rays(1612.06903)

We report the first empirical demonstration that resonant inelastic x-ray scattering (RIXS) is sensitive to \emph{collective} magnetic excitations in $S=1$ systems by probing the Ni $L_3$-edge of La$_{2-x}$Sr$_x$NiO$_4$ ($x = 0, 0.33, 0.45$). The magnetic excitation peak is asymmetric, indicating the presence of single and multi spin-flip excitations. As the hole doping level is increased, the zone boundary magnon energy is suppressed at a much larger rate than that in hole doped cuprates. Based on the analysis of the orbital and charge excitations observed by RIXS, we argue that this difference is related to the orbital character of the doped holes in these two families. This work establishes RIXS as a probe of fundamental magnetic interactions in nickelates opening the way towards studies of heterostructures and ultra-fast pump-probe experiments.
• ### Spin resonance in the superconducting state of Li$_{1-x}$Fe$_{x}$ODFe$_{1-y}$Se observed by neutron spectroscopy(1607.05588)

Oct. 25, 2016 cond-mat.supr-con
We have performed inelastic neutron scattering measurements on a powder sample of the superconductor lithium iron selenide hydroxide Li$_{1-x}$Fe$_{x}$ODFe$_{1-y}$Se ($x \simeq 0.16, y \simeq 0.02$, $T_{\rm c} = 41$\,K). The spectrum shows an enhanced intensity below $T_{\rm c}$ over an energy range $0.64\times2\Delta < E < 2\Delta$, where $\Delta$ is the superconducting gap, with maxima at the wave vectors $Q_1 \simeq 1.46$\,\AA$^{-1}$ and $Q_2 \simeq 1.97$\,\AA$^{-1}$. The behavior of this feature is consistent with the spin resonance mode found in other unconventional superconductors, and strongly resembles the spin resonance observed in the spectrum of the molecular-intercalated iron selenide, Li$_{0.6}$(ND$_{2}$)$_{0.2}$(ND$_{3}$)$_{0.8}$Fe$_{2}$Se$_{2}$. The signal can be described with a characteristic two-dimensional wave vector $(\pi, 0.67\pi)$ in the Brillouin zone of the iron square lattice, consistent with the nesting vector between electron Fermi sheets.
• ### Commensurate lattice distortion in the layered titanium oxypnictides Na$_{2}$Ti$_{2}Pn_{2}$O ($Pn =$ As, Sb) determined by X-ray diffraction(1604.07284)

Oct. 25, 2016 cond-mat.supr-con
We report single crystal X-ray diffraction measurements on Na$_2$Ti$_{2}Pn_{2}$O ($Pn$ = As, Sb) which reveal a charge superstructure that appears below the density wave transitions previously observed in bulk data. From symmetry-constrained structure refinements we establish that the associated distortion mode can be described by two propagation vectors, ${\bf q}_{1} = (1/2, 0, l)$ and ${\bf q}_{2} = (0, 1/2, l)$, with $l=0$ (Sb) or $l = 1/2$ (As), and primarily involves in-plane displacements of the Ti atoms perpendicular to the Ti--O bonds. The results provide direct evidence for phonon-assisted charge density wave order in Na$_2$Ti$_{2}Pn_{2}$O and identify a proximate ordered phase that could compete with superconductivity in doped BaTi$_{2}$Sb$_{2}$O.
• ### Direct evidence for charge stripes in a layered cobalt oxide(1605.07087)

Recent experiments indicate that static stripe-like charge order is generic to the hole-doped copper oxide superconductors and competes with superconductivity. Here we show that a similar type of charge order is present in La$_{5/3}$Sr$_{1/3}$CoO$_4$, an insulating analogue of the copper oxide superconductors containing cobalt in place of copper. The stripe phase we have detected is accompanied by short-range, quasi-one-dimensional, antiferromagnetic order, and provides a natural explanation for the distinctive hourglass shape of the magnetic spectrum previously observed in neutron scattering measurements of La$_{5/3}$Sr$_{1/3}$CoO$_4$ and many hole-doped copper oxide superconductors. The results establish a solid empirical basis for theories of the hourglass spectrum built on short-range, quasi-static, stripe correlations.
• ### All-in all-out magnetic order and propagating spin-waves in Sm2Ir2O7(1604.06401)

April 21, 2016 cond-mat.str-el
Using resonant magnetic x-ray scattering we address the unresolved nature of the magnetic groundstate and the low-energy effective Hamiltonian of Sm$_2$Ir$_2$O$_7$, a prototypical pyrochlore iridate with a finite temperature metal-insulator transition. Through a combination of elastic and inelastic measurements, we show that the magnetic ground state is an all-in all-out (AIAO) antiferromagnet. The magnon dispersion indicates significant electronic correlations and can be well-described by a minimal Hamiltonian that includes Heisenberg exchange ($J=27.3(6)$ meV) and Dzyaloshinskii-Moriya interaction ($D=4.9(3)$ meV), which provides a consistent description of the magnetic order and excitations. In establishing that Sm$_2$Ir$_2$O$_7$ has the requisite inversion symmetry preserving AIAO magnetic groundstate, our results support the notion that pyrochlore iridates may host correlated Weyl semimetals.
• ### Magnetic phase diagram of La$_{2-x}$Sr$_{x}$CoO$_{4}$ revised using muon-spin relaxation(1510.06244)

Oct. 21, 2015 cond-mat.str-el
We report the results of a muon-spin relaxation ($\mu$SR) investigation of La$_{2-x}$Sr$_{x}$CoO$_{4}$, an antiferromagnetic insulating series which has been shown to support charge ordered and magnetic stripe phases and an hourglass magnetic excitation spectrum. We present a revised magnetic phase diagram, which shows that the suppression of the magnetic ordering temperature is highly sensitive to small concentrations of holes. Distinct behavior within an intermediate $x$ range ($0.2 \leq x \lesssim 0.6$) suggests that the putative stripe ordered phase extends to lower $x$ than previously thought. Further charge doping ($0.67 \leq x \leq 0.9$) prevents magnetic ordering for $T \gtrsim 1.5~{\rm K}$
• ### The importance of XY anisotropy in Sr2IrO4 revealed by magnetic critical scattering experiments(1506.07720)

June 25, 2015 cond-mat.str-el
The magnetic critical scattering in Sr$_2$IrO$_4$ has been characterized using X-ray resonant magnetic scattering (XRMS) both below and above the 3D antiferromagnetic ordering temperature, T$_{\text{N}}$. The order parameter critical exponent below T$_{\text{N}}$ is found to be \beta=0.195(4), in the range of the 2D XYh$_4$ universality class. Over an extended temperature range above T$_{\text{N}}$, the amplitude and correlation length of the intrinsic critical fluctuations are well described by the 2D Heisenberg model with XY anisotropy. This contrasts with an earlier study of the critical scattering over a more limited range of temperature which found agreement with the theory of the isotropic 2D Heisenberg quantum antiferromagnet, developed to describe the critical fluctuations of the conventional Mott insulator La$_2$CuO$_4$ and related systems. Our study therefore establishes the importance of XY anisotropy in the low-energy effective Hamiltonian of Sr$_2$IrO$_4$, the prototypical spin-orbit Mott insulator.
• ### Evidence of quantum dimer excitations in Sr$_3$Ir$_2$O$_7$(1506.04877)

June 16, 2015 cond-mat.str-el
The magnetic excitation spectrum in the bilayer iridate Sr$_3$Ir$_2$O$_7$ has been investigated using high-resolution resonant inelastic x-ray scattering (RIXS) performed at the iridium L$_3$ edge and theoretical techniques. A study of the systematic dependence of the RIXS spectrum on the orientation of the wavevector transfer, $\mathbf{Q}$, with respect to the iridium-oxide bilayer has revealed that the magnon dispersion is comprised of two branches well separated in energy and gapped across the entire Brillouin zone. Our results contrast with those of an earlier study which reported the existence of a single dominant branch. While these earlier results were interpreted as two overlapping modes within a spin-wave model of weakly coupled iridium-oxide planes, our results are more reminiscent of those expected for a system of weakly coupled dimers. In this latter approach the lower and higher energy modes find a natural explanation as those corresponding to transverse and longitudinal fluctuations, respectively. We have therefore developed a bond-operator theory which describes the magnetic dispersion in Sr$_3$Ir$_2$O$_7$ in terms of quantum dimer excitations. In our model dimerisation is produced by the leading Heisenberg exchange, $J_c$, which couples iridium ions in adjacent planes of the bilayer. The Hamiltonian also includes in plane exchange, $J$, as well as further neighbour couplings and relevant anisotropies. The bond-operator theory provides an excellent account of the dispersion of both modes, while the measured $\mathbf{Q}$ dependence of the RIXS intensities is in reasonable qualitative accord with the spin-spin correlation function calculated from the theory. We discuss our results in the context of the quantum criticality of bilayer dimer systems in the presence of anisotropic interactions derived from strong spin-orbit coupling.
• ### Divacancy superstructures in thermoelectric calcium-doped sodium cobaltate(1505.02635)

May 11, 2015 cond-mat.str-el
We have grown single crystals of Na$_x$Ca$_y$CoO$_2$ and determined their superstructures as a function of composition using neutron and x-ray diffraction. Inclusion of Ca$^{2+}$ stabilises a single superstructure across a wide range of temperatures and concentrations. The superstructure in the Na$^+$ layers is based on arrays of divacancy clusters with Ca$^{2+}$ ions occupying the central site, and it has an ideal concentration Na$_{4/7}$Ca$_{1/7}$CoO$_2$. Previous measurements of the thermoelectric properties on this system are discussed in light of this superstructure. Na$_{4/7}$Ca$_{1/7}$CoO$_2$ corresponds to the maximum in thermoelectric performance of this system.
• ### Strong $(\pi,0)$ spin fluctuations in $\beta$-FeSe observed by neutron spectroscopy(1502.03838)

We have performed powder inelastic neutron scattering measurements on the unconventional superconductor $\beta$-FeSe ($T_{\rm c} \simeq 8\,\mathrm{K}$). The spectra reveal highly dispersive paramagnetic fluctuations emerging from the square-lattice wave vector $(\pi,0)$ extending beyond 80 meV in energy. Measurements as a function of temperature at an energy of $\sim 13\,\mathrm{meV}$ did not show any variation from $T_{\rm c}$ to $104\,\mathrm{K}$. The results show that FeSe is close to an instability towards $(\pi,0)$ antiferromagnetism characteristic of the parent phases of the high-$T_{\rm c}$ iron arsenide superconductors, and that the iron paramagnetic moment is neither affected by the orthorhombic-to-tetragonal structural transition at $T_{\rm s} \simeq 90\,\mathrm{K}$ nor does it undergo a change in spin state over the temperature range studied.
• ### Crystal field states of Tb3+ in the pyrochlore spin liquid Tb2Ti2O7 from neutron spectroscopy(1501.04927)

Jan. 20, 2015 cond-mat.str-el
We report time-of-flight neutron scattering measurements of the magnetic spectrum of Tb3+ in Tb2Ti2O7. The data, which extend up to 120 meV and have calibrated intensity, enable us to consolidate and extend previous studies of the single-ion crystal field spectrum. We successfully refine a model for the crystal field potential in Tb2Ti2O7 without relying on data from other rare earth titanate pyrochlores, and we confirm that the ground state is a non-Kramers doublet with predominantly |+/-4> components. We compare the model critically with earlier models.
• ### Mechanism of spin crossover in LaCoO3 resolved by shape magnetostriction in pulsed magnetic fields(1411.2757)

In the scientific description of unconventional transport properties of oxides (spin-dependent transport, superconductivity etc.), the spin-state degree of freedom plays a fundamental role. Because of this, temperature- or magnetic field-induced spin-state transitions are in the focus of solid-state physics. Cobaltites, e.g. LaCoO3, are prominent examples showing these spin transitions. However, the microscopic nature of the spontaneous spin crossover in LaCoO3 is still controversial. Here we report magnetostriction measurements on LaCoO3 in magnetic fields up to 70 T to study the sharp, field-induced transition at Hc ~ 60 T. Measurements of both longitudinal and transversal magnetostriction allow us to separate magnetovolume and magnetodistortive changes. We find a large increase in volume, but only a very small increase in tetragonal distortion at Hc. The results, supported by electronic energy calculations by the configuration interaction cluster method, provide compelling evidence that above Hc LaCoO3 adopts a correlated low spin/high spin state.
• ### An anisotropic local modification of crystal field levels in Pr-based pyrochlores: a muon-induced effect modelled using density functional theory(1408.5041)

Aug. 21, 2014 cond-mat.str-el
Muon spin relaxation measurements on some quantum spin ice candidate materials, the insulating pyrochlores Pr2B2O7 (B = Sn, Zr, Hf), have been performed for temperatures in the range 0.05-280 K. The results are indicative of a static distribution of magnetic moments which appears to grow on cooling and whose size at low temperatures is significantly larger than that expected for Pr nuclear moments. Using density functional theory we show how this effect can be explained via a hyperfine enhancement arising from a splitting of the non-Kramers doublet ground states on Pr ions close to the muon which itself causes a highly anisotropic distortion field. We provide a quantitative relationship between this effect and the measured temperature dependence of the muon relaxation and discuss the relevance of these observations to muon experiments in other frustrated magnetic materials.
• ### Coupling of magnetic order to planar Bi electrons in the anisotropic Dirac metals AMnBi2 (A = Sr, Ca)(1311.0308)

July 26, 2014 cond-mat.str-el
We report powder and single crystal neutron diffraction measurements of the magnetic order in AMnBi2 (A = Sr and Ca), two layered manganese pnictides with anisotropic Dirac fermions on a Bi square net. Both materials are found to order at TN approx 300 K in k = 0 antiferromagnetic structures, with ordered Mn moments at T = 10 K of approximately 3.8 muB aligned along the c axis. The magnetic structures are Neel-type within the Mn--Bi layers but the inter-layer ordering is different, being antiferromagnetic in SrMnBi2 and ferromagnetic in CaMnBi2. This allows a mean-field coupling of the magnetic order to Bi electrons in CaMnBi2 but not in SrMnBi2. We find clear evidence that magnetic order influences electrical transport. First principles calculations explain the experimental observations and suggest that the mechanism for different inter-layer ordering in the two compounds is the competition between the anteiferromagnetic superexchange and ferromagnetic double exchange carried by itinerant Bi electrons.
• ### Magnetic excitation spectrum of LuFe2O4 measured with inelastic neutron scattering(1406.7418)

June 28, 2014 cond-mat.str-el
We report neutron inelastic scattering measurements and analysis of the spectrum of magnons propagating within the Fe2O4 bilayers of LuFe2O4. The observed spectrum is consistent with six magnetic modes and a single prominent gap, which is compatible with a single bilayer magnetic unit cell containing six spins. We model the magnon dispersion by linear spin-wave theory and find very good agreement with the domain-averaged spectrum of a spin-charge bilayer superstructure comprising one Fe3+ -rich monolayer and one Fe2+ -rich monolayer. These findings indicate the existence of polar bilayers in LuFe2O4, contrary to recent studies that advocate a charge-segregated non-polar bilayer model. Weak scattering observed below the magnon gap suggests that a fraction of the bilayers contain other combinations of charged monolayers not included in the model. Refined values for the dominant exchange interactions are reported.
• ### Crystal field splitting in Sr$_{n+1}$Ir$_n$O$_{3n+1}$ ($n$ = 1, 2) iridates probed by x-ray Raman spectroscopy(1405.6522)

May 26, 2014 cond-mat.str-el
Non-resonant Raman spectroscopy in the hard X-ray regime has been used to explore the electronic structure of the first two members of the Ruddlesden-Popper series Sr$_{n+1}$Ir$_n$O$_{3n+1}$ of iridates. By tuning the photon energy transfer around 530 eV we have been able to explore the oxygen K near edge structure with bulk sensitivity. The angular dependence of the spectra has been exploited to assign features in the 528-535 eV energy range to specific transitions involving the Ir 5d orbitals. This has allowed us to extract reliable values for both the t2g-eg splitting arising from the cubic component of the crystal field (10Dq), in addition to the splitting of the eg orbitals due to tetragonal distortions. The values we obtain are (3.8, 1.6) eV and (3.55, 1.9) eV for Sr$_2$IrO$_4$ and Sr$_3$Ir$_2$O$_7$, respectively.
• ### Stripe disorder and dynamics in the hole-doped antiferromagnetic insulator La5/3Sr1/3CoO4(1301.5124)

Jan. 16, 2014 cond-mat.str-el
We report on an investigation into the dynamics of the stripe phase of La5/3Sr1/3CoO4, a material recently shown to have an hour-glass magnetic excitation spectrum. A combination of magnetic susceptibility, muon-spin relaxation and nuclear magnetic resonance measurements strongly suggest that the physics is determined by a disordered configuration of charge and spin stripes whose frustrated magnetic degrees of freedom are strongly dynamic at high temperature and which freeze out in a glassy manner as the temperature is lowered. Our results broadly confirm a recent theoretical prediction, but show that the charge quenching remains incomplete well below the charge ordering temperature and reveal, in detail, the manner in which the magnetic degrees of freedom are frozen.
• ### Circularly Polarised X-ray Scattering Investigation of Spin-Lattice Coupling in TbMnO$_3$ in Crossed Electric and Magnetic Fields(1311.6617)

Jan. 10, 2014 cond-mat.str-el
We present a study of the magnetic and crystallographic structure of TbMnO$_3$ in the presence of crossed electric and magnetic fields using circularly polarised X-ray non-resonant scattering. A comprehensive account is presented of the scattering theory and data analysis methods used in our earlier studies, and in addition we present new high magnetic field data and its analysis. We discuss in detail how polarisation analysis was used to reveal structural information, including the arrangement of Tb moments which we proposed for $H = 0$ T, and how the diffraction data for $H<H_C$ can be used to determine specific magnetostrictively induced atomic displacements with femto-metre accuracy. The connection between the electric polarisation and magnetostrictive mechanisms is discussed. Similar magnetostrictive displacements have been observed for $H > H_C$ as for $H < H_C$. Finally some observations regarding the kinetics and the conservation of domain population at the transition are described.
• ### Crystal field states in a Quantum Spin Ice]{Crystal field states of Pr$^{3+}$ in the candidate quantum spin ice Pr$_{2}$Sn$_{2}$O$_{7}$(1306.0759)

Oct. 25, 2013 cond-mat.str-el
Neutron time-of-flight spectroscopy has been employed to study the crystal-field splitting of Pr$^{3+}$ in the pyrochlore stannate Pr$_{2}$Sn$_{2}$O$_{7}$. The crystal field has been parameterized from a profile fit to the observed neutron spectrum. The single-ion ground state is a well isolated non-Kramers doublet of $\Gamma_3^+$ symmetry with a large Ising-like anisotropy, $\chi_{zz} /\chi_\perp \approx 60$ at 10 K, but with a significant admixture of terms $| M_J \neq \pm J >$ which can give rise to quantum zero-point fluctuations. This magnetic state satisfies the requirements for quantum spin ice behavior.
• ### Absence of strong magnetic fluctuations in the iron phosphide superconductors LaFePO and Sr2ScO3FeP(1302.5057)

Oct. 24, 2013 cond-mat.supr-con
We report neutron inelastic scattering measurements on polycrystalline LaFePO and Sr2ScO3FeP, two members of the iron phosphide families of superconductors. No evidence is found for any magnetic fluctuations in the spectrum of either material in the energy and wavevector ranges probed. Special attention is paid to the wavevector at which spin-density-wave-like fluctuations are seen in other iron-based superconductors. We estimate that the magnetic signal, if present, is at least a factor of four (Sr2ScO3FeP) or seven (LaFePO) smaller than in the related iron arsenide and chalcogenide superconductors. These results suggest that magnetic fluctuations are not as influential on the electronic properties of the iron phosphide systems as they are in other iron-based superconductors.
• ### The Hour-Glass Magnetic Spectrum Arising from a Striped, Cluster Spin Glass Ground State in La1.75Sr0.25CoO4(1308.2141)

Aug. 9, 2013 cond-mat.str-el
We report inelastic neutron scattering results that reveal an hour-glass magnetic excitation spectrum in La1.75Sr0.25CoO4. The magnetic spectrum is similar to that observed previously in La1.67Sr0.33CoO4, but the spectral features are broader. We show that the spectrum of La1.75Sr0.25CoO4 can be modeled by the spin dynamics of a system with a disordered cluster spin glass ground state. Bulk magnetization measurements are presented which support the proposed glassy ground state. The observations reiterate the importance of quasi-one-dimensional magnetic correlations and disorder for the hour-glass spectrum, and suggest that disordered spin and charge stripes exist at lower doping in La2-xSrxCoO4 than previously thought.
• ### Spin fluctuations away from (pi,0) in the superconducting phase of molecular-intercalated FeSe(1305.4898)

Magnetic fluctuations in the molecular-intercalated FeSe superconductor Li{x}(ND2){y}(ND3){1-y}Fe2Se2 (Tc = 43K) have been measured by inelastic neutron scattering from a powder sample. The strongest magnetic scattering is observed at a wave vector Q ~ 1.4 A^{-1}, which is not consistent with the (pi,0) nesting wave vector that characterizes magnetic fluctuations in several other iron-based superconductors, but is close to the (pi, pi/2) position found for A{x}Fe{2-y}Se2 systems. At the energies probed (~ 5kB Tc), the magnetic scattering increases in intensity with decreasing temperature below Tc, consistent with the superconductivity-induced magnetic resonance found in other iron-based superconductors.


text:
# MIKE

Popular questions and responses by MIKE
1. ## Chem

Consider the balanced chemical equation: 3Sr(OH)2(aq) + 2H3PO4(aq) → Sr3(PO4)2(s) + 6H2O(l). How many moles of Sr(OH)2 remain if 0.10 mol of H3PO4 react with 0.40 mol of Sr(OH)2?

2. ## Chemistry Help

The ΔHvap of a certain compound is 34.68 kJ·mol–1 and its ΔSvap is 93.60 J·mol–1·K–1. What it the boiling point of this compound?

3. ## Calculus - Need help please

Given the following function, y=tan(pi-3x) 1)determine the interval for the principal cycle. (^ Type your answer in interval notation) 2)Then for the principal cycle, determine the equations of the vertical asymptotes 3)find the coordinates of the

4. ## chemistry

For a particular isomer of C8H18, the following reaction produces 5113.3 kJ of heat per mole of C8H18(g) consumed, under standard conditions. C8H18(g) + 25/2(O2)(g) -> 8CO2(g) + 9H2O(g) DeltaH= -5113.3 kJ What is the standard enthalpy of formation of this

5. ## Physics

A bullet with a mass of 28 g is fired from a 2.8-kg gun that is stationary, but free to recoil. After the bullet is fired, the gun is observed to be moving at 1.4 m/s [left]. What is the velocity of the bullet?

6. ## maths

a car weighing 1600kg moving with a velocity of 30 meter per second retarded uniformly coming to rest in 20 seconds calculate 1, initial momentum 2, final momentum 3, rate of change of momentum 4,acceleration of the car 5, magnitude of the force apply

7. ## physics

a popgun uses a spring for which k = 20 N/cm. when cocked, the spring is compressed 3.0 cm. how high can the gun shoot a 5.0-g projectile

8. ## covalent or ionic bond

Is Mercury (ii) Chloride a covalent bond or ionic?

9. ## mathematics

Two questions here: 1. In a right-angled triangle ABC, catheter AC is 9.4 cm long and catheter BC is 6.2 cm long. D is a point on the catheter AC so that DA = DB. Determine the length of the CD. 2. In a right-angled triangle, the ratio of catheters is 2:

10. ## chemistry

calculate the change in pH when 9.00ml of 0.100M HCl(aq) is added to 100.0mL of a buffer solution that is 0.100M in NH3(aq) and 0.100M NH4Cl(aq). ...Calculate the change in pH when 9.00,L if 0.100M NaOH(aq) is added to the original buffer solution.

11. ## math

what is 22 2/9% as a fraction?

12. ## Calculus

The demand function for the Luminar desk lamp is given by the following function where x is the quantity demanded in thousands and p is the unit price in dollars. p = f(x) = -0.1x2 - 0.3x + 39 (a) Find f '(x). f '(x) = (b) What is the rate of change of the

13. ## Chemistry

In a mountainous location, the boiling point of pure water is found to be 95 degrees Celsius. How many grams of sodium chloride much be added to 1 kg of water to bring the boiling point back to 100 degrees Celsius? Assume that sodium chloride behaves

14. ## algebra

Determine K and M if x=y Does this mean m=0 k=1?

15. ## spanish

We sing a song entitled He drinks tequila,and she talks dirty in spanish ^ there are a couple of phrases that I don't know the meaning of could you help please. the first is "a brasa mi puerta" The second is "to dal ay nochay" I hope this is not offensive

16. ## mathematics

Jessica has received a scholarship. She invests the money in a fund that provides 3.5% in annual interest. By what percentage does the investment grow over five years?

17. ## Physics

A wooden block of mass m = 9 kg starts from rest on an inclined plane sloped at an angle θ from the horizontal. The block is originally located 5m from the bottom of the plane. If the block, undergoing constant acceleration down the ramp, slides to the

18. ## Math

The table below shows the scores of a group of students on a 10-point quiz. Test Score Frequency 3 1 4 1 5 2 6 3 7 1 8 1 9 1 10 4 The mean score on this test is: The median score on this test is:

19. ## Math

A colony of bees doubles in population every 5 weeks.hpw long does it take for the population to triple ?

20. ## math-

A new model of car is available in 4 exterior colors and 3 interior colors. Use a tree diagram and specific colors to show how many color schemes are possible for the car. Solve the problem using an alternative method as well.Note that you are asked to

21. ## calculus

A potter forms a piece of clay into a right circular cylinder. As she rolls it, the height h of the cylinder increases and the radius r decreases. Assume that no clay is lost in the process. Suppose the height of the cylinder is increasing by 0.5

22. ## physics

Seat belts and air bags save lives by reducing the forces exerted on the driver and passengers in an automobile collision. Cars are designed with a "crumple zone" in the front of the car. In the event of an impact, the passenger compartment decelerates

23. ## Algebra

For the function f (x) = C ∙ a^x the following relation applies: f(6) / f(4) = 0.25 f(2) - f(3) = 20000 Solve f(5) Solve f(x) = 625

24. ## Calculus check my answers

1. The general solution of the differential equation dy - 0.2x dx = 0 is a family of curves. These curves are all: a. lines b. hyperbolas c. parabolas (my answer) d. ellipses 2. The table below gives selected values for the function f(x). Use a trapezoidal

I have this question: Which is the smallest unit of measurement for temperature? One Celsius degree: one Kelvin:one Fahrenheit. I think it is Kelvin but I can't find anywhere in my book where it refers to that. When I google all three it says Celsius &

26. ## Physics- torque

A man ties one end of a strong rope 8.35 m long to the bumper of his truck, 0.506 m from the ground, and the other end to a vertical tree trunk at a height of 3.92 m. He uses the truck to create a tension of 8.32x10^2 N in the rope. Compute the magnitude

27. ## chemistry

In a particular redox reaction, Cr is oxidized to CrO4^2– and Cu2^+ is reduced to Cu^+ . Complete and balance the equation for this reaction in acidic solution. Cr + Cu^2+ ---> CrO4^2-+ Cu^+ i started off by doing this 2 x SO3^2- ---> SO4^2- + 2e 2 x

28. ## Financial Management

TLB EXHIBIT 1 SUMMARY OUTPUT Regression Statistics Multiple R 0.397871332 R Square 0.158301597 Adjusted R Square 0.146771482 Standard Error 0.099101922 Observations 75 ANOVA df SS MS F Significance F Regression 1 0.134839089 0.134839089 13.72940301

29. ## English

In “A Sound of Thunder,” Travis’s dialogue with Lesperance reveals how upset he is about Eckels’ having left the path. This is an example of direct characterization.******** indirect characterization.

30. ## statistics

Two dice are rolled and two fair coins are tossed. Let X be the sum of the number of spots that show on the top faces of the dice and the number of coins that land heads up. The expected value of X is _____ .

31. ## Art 17th Century Europe

Contrast the "Elevation of the Cross" by Rubens with Rembrandt's Return of the Prodigal Son. How does each work demonstrate the Baroque style? How does each work mirror the social and political climate of the time? **Hope someone can Help,Please! =)

32. ## math

3/5 of a revolution represents approximately how many radians? Question 12 options: 6.28 3.14 3.77 108.00

33. ## Algebra

Which is the degree measure of an angle whose tangent is 5.67? Round the answer to the nearest whole number.

34. ## Statistics

A 95% confidence interval for a population mean, ?, is given as (18.985, 21.015). This confidence interval is based on a simple random sample of 36 observations. Calculate the sample mean, x ?, and the sample standard deviation, s. Assume that all

35. ## math

There are 50 jellybeans which are red, orange, green, and yellow, how many ways can you place the beans in a row providing each color are identical? And, how many handfuls of 12 are possible?

36. ## Physics,Science

Two point charges Q1 = 10µc and Q2 = -2µc are arranged along the x-axis at x = 0 and x =4m respectively. Find the positions along the x-axis where v = 0

37. ## Math

The time t required to drive a certain distance varies inversely with the speed r. If it takes 4 hours to drive the distance at 35 miles per hour, how long will it take to drive the same distance at 45 miles per hour? about 3.11 hours********* 140 hours

38. ## Chemistry

Why are some solutes soluble in water, but others are soluble in cyclohexane?

62. ## calculus

A conical water tank with vertex down has a radius of 12 feet at the top and is 23 feet high. If water flows into the tank at a rate of 20 {\rm ft}^3{\rm /min}, how fast is the depth of the water increasing when the water is 17 feet deep?

63. ## physics 209

A mass m = 0.5 kg is hung from a pulley with moment of inertia I = 0.20kg⋅m2 and radius R = 10cm. There is friction in the pulley. The mass is dropped from rest, and after 2s it traveled 0.4m. What is the torque due to friction?

64. ## Math

Two fire towers are 30 kilometers apart, tower A being due west of tower B. A fire is spotted from the towers, and the bearings from A and B are E 14 degrees N and W 34 degrees N, respectively. Find the distance d of the fire from the line segment AB.

A firm has invested in corporate bonds it may engage in a financial futures contract in order to protect itself from what.

The question is: Find the following quantities the volume, in liters, of 835 g SO3 at STP My answer is: 1 mol of SO3=80g 835g/80g = 10.44 mols v = (10.44 x 8.314 x 273)/100 = 236 L Can someone tell me if I did it correct. Thanks very much.

67. ## GP

two consecutive terms of a GP are the 2nd ,4th and 7th terms of an A.P respectively.find the common ration of the GP

68. ## Physics

An interacting system of two objects has no external force acting on the system. The first has a mass of 2.00kg and is moving with a velocity (2.00m/s)i + (1.00m/s)j at time t=0s. The second object has a mass of 3.00kg and is moving with a velocity

71. ## science

After an athlete has completed a 100 meter sprint, his/her breathing rate will be greatly increased, and they need time to "catch their breath." Can you identify the process that leads to a person needing to catch their breath? Ans: Hypertrophy

72. ## geometry

A 4-ft vertical post casts a 20-in shadow at the same time a nearby cell phone tower casts a 115-ft shadow. How tall is the cell phone tower?

73. ## English/Language Arts

Which of the following parenthetical citations is placed and punctuated correctly? A)The only casualty on Lewis and Clark's trip died from appendicitis (Davis 104). B)(Davis, 104) Kenneth Davis explains that the only casualty on Lewis and Clark's trip died

74. ## economics

3. The following facts characterize the furniture industry in the United States:39 a. The industry has been very fragmented, so that few companies have the financial backing to make heavy investments in new technology and equipment. b. In 1998, only three

75. ## English II B

Deana created a computer-assisted slide show. To make the slide show legible, what is one thing she needs to avoid? A)typing bullets in a small font B)adding color and effects to the font C)printing handouts on paper D)adding tables and graphics

76. ## statistics

a quiz consists of 30 true or false questions. if the student guesses on each question, what is the standard deviation of the number of correct answers

77. ## Physics

During the Olympic ice competition, Boris (m = 75 kg) glides at 1.8 m/s to a stationary Juliette (52 kg) and hangs on. How far will the pair slide after the “collision” if the coefficient of kinetic friction between the ice and their skates is .042? My

78. ## Physics

Question: The amount of internal energy needed to raise the temperature of .25 kg of water by 0.2 degrees Celsius is 209.3 J. How fast must a 0.25 kg baseball travel in order for its kinetic energy to equal this internal energy? KE = (.5)(m)(v^2) 209.3 J =

79. ## Math

Tom Schriber, a director of personnel of Management Resources, Inc., is in the process of designing a program that its customers can use in the job-finding process. Some of the activities include preparing resumés, writing letters, making appointments to

80. ## Math-college

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. n=4; i and 3 i are zero; f(-2)=65 f(x)= An expression using x as the variable. Simplify your answer.

81. ## Math

Let r(t) be a differentiable function that is positive and increasing. The rate of increase of r^3 is equal to 12 times the rate of increase of r when r(t)=?

82. ## Statistics

The coding department of a large physician clinic is interested in purchasing a software program that will edit claims before they are sent out to billing. The license fee for the software costs \$34,000 per year. The software is expected to reduce the

83. ## Calculus

A car leaves an intersection traveling west. Its position 5 sec later is 22 ft from the intersection. At the same time, another car leaves the same intersection heading north so that its position 5 sec later is 27 ft from the intersection. If the speed of

84. ## Physics

A 5000 kg African elephant has a resting metabolic rate of 2500 W. On a hot day, the elephant's environment is likely to be nearly the same temperature as the animal itself, so cooling by radiation is not effective. The only plausible way to keep cool is

85. ## algebra

3 times the difference of 7 and d

86. ## General Physics

During heavy rain, a section of a mountainside measuring 2.5 km horizontally, 0.80 km up along the slope, and 2.0m deep slips into a valley in a mud slide. Assume that the mud ends up uniformly distributed over a surface area of the valley measuring 0.40km

87. ## Algebra 1

Write an equation for for each translation of y=|x| 1. left 2 units y = |x| consists of two 45 degree curves joined at the origin: y = x for x>0, and y = -x for x < 0. To move the curves to the left 2 units, you want the sign of the term inside the

88. ## math

2x + 3y = 78 How do i solve for x and y? 2x=78 78/2=39 so x=39 3y=78 78/3=26 so y=26 Can't be, because 2(39)+ 3(26) does not equl 78 You need TWO equations when you have TWO variables, such as x and y. It can't be done with one equation. BUT, you say, what

89. ## french

write a letter in french to your penpal inviting him or her to your birthday party

90. ## Math

The equation xy-1=x+y^2 defines a function y=f(x) in a neighborhood of (-1,0). Determine f(-1), fx(-1), and fxx(-1) and the second -taylor polynomial t2 (x) based around x=-1

91. ## Math

Help! I need the work and how to check it too! Victoria and Georgetown are 36.2 mi from each other. How far apart would the cities be on a map that has a scale of 0.9 in: 10.5 mi?

92. ## American Gov.

Which of the following does the free exercise clause of the first amendment protect? a. Teacher-led prayers in schools b. attending church c. the establishment of a state religion d. nurses reading the Bible to patients in a public hospital. My answer is

93. ## HVAC

A gas is compressed inside a compressors cylinder. When the piston is at its bottom dead center , the gas is initially at 10 psig , 65 degree and 10.5 in. After compression and when the piston is at top dead center the gas is 180 degree and occupies 1.5

94. ## ALGEBRA

Assume that a dart is randomly thrown at the dartboard shown here and strikes the board every time. The payoffs are listed on the board. How much should you be willing to pay for the opportunity to play this game? (Give your answer to the nearest cent.)

95. ## statistics

The U.S. Department of Agriculture claims that the mean consumption of coffee by the person in the United States is 24.2 gallons per year. A random sample of 120 people in the United States shows that the mean coffee consumption is 23.5 gallons per year

96. ## Physics

A spring is hung from the ceiling. When a coffee mug is attached to its end, it stretches 3.0 cm before reaching its new equilibrium length. The block is then pulled down slightly and released. What is the frequency of oscillation?

97. ## science

(a) Find the concentration of H+, HCO3 - and CO3 2-, in a 0.01M solution of carbonic acid if the pH of this is 4.18. Ka1(H2CO3)=4.45 10–7 andKa2 =4.69 10–11 (b).Calculate the pH at the equivalence point of the titration between 0.1M CH3- COOH ( 25 ml)

98. ## Algebra

If a and b are integers such that a^2− b^2 = 100, what is the greatest possible value of a?


text:
# 3 dimenional figure

1. ### Solution and explanation needed

I was working on problems and I came across one. I’m not quite sure why this is 48. Anyone have any input?


text:
# How do I decrypt message with RSA algorithm with modulus n = 302446877 and encryption key e 677?

Message: 196512625 25825210 250970710 140420423 22130212 288711674 26343056

Plaintext is grouped in blocks of 4 letters.

What I have so far =(

Fold[#1 n + #2 &, 0, %];
IntegerDigits[%, 4];
FromCharacterCode[%]

• What does Plaintext is grouped in blocks of 4 letters. mean precisely? I have decrypted your string but I have no idea how it's encoded because you haven't properly specified it (e.g ASCII? Numbers 1-26? Base-8, Base-256 etc.) – flinty Oct 27 '20 at 10:30

This is how to decrypt your message, however I need to know more how to decode the final numbers until I can produce a string, as mentioned in the comments:

n = 302446877;
publicExponent = 677;
{p, q} = FactorInteger[n][[All, 1]];

(* this is the same as EulerPhi[n] *)
phi = (p - 1) (q - 1);

privateExponent = PowerMod[publicExponent, -1, phi];

decryptInteger[c_, d_, n_] := PowerMod[c, d, n]

message = {196512625 , 25825210 , 250970710 , 140420423 , 22130212 ,
288711674 , 26343056};
decr = decryptInteger[#, privateExponent, n] & /@ message

(* result: {10040415, 2001112, 130303, 14132012, 1041719, 7041417, 236259039} *)

Update: based on @WReach's observation, there is a slight problem with your final number, but essentialy the message is "keep calm and do number theory!"

indices = FromDigits[#, 10] & /@
Partition[Flatten[IntegerDigits[#, 10, 8] & /@ decr], 2];
StringJoin @@ (indices /. Thread[Range[0, 99] -> CharacterRange[97, 97+99]])

(* result: "keepcalmanddonumbertheorz»" *)

• The digit pair 00 is a, 01 is b, etc. So the digit pairs 10 04 04 15 are read as keep. keepcalmanddonumbertheor.... The last group appears malformed as it contains four pairs plus a surplus digit. I believe your decryption logic is correct, so either the padding scheme eludes me or there is a transcription error in the message or a mistake was made during encryption. – WReach Oct 27 '20 at 15:20
• Thank you very much for the explanation. – suffering undergrad Nov 1 '20 at 4:27
• @sufferingundergrad please accept the answer if you feel it solved the problem, otherwise please let me know if you have any other requirements. – flinty Nov 1 '20 at 15:36


text:
# Urgent Help!

The idea for the first parts is that if $\epsilon > 0$ and $x$ is a fixed point then we can take $\delta > 0$ small enough such that $|x-y|^{\alpha} < \epsilon$ for all $|x-y| < \delta$. Therefore, $|f(x)-f(y)| \leq |x-y|^{\alpha} < \epsilon$. Thus, $f$ is continous.
In the second part if $\alpha > 1$ then $|f(x)-f(y)| \leq |x-y|^{\alpha}$ then $| [f(x)-f(y)]/[x-y] | \leq |x-y|^{\alpha - 1}$. Notice the LHS has form of a derivative. Thus, what can you conclude?


text:
# Intuition behind Conditional Expectation

I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know.

Let me get more specific. Let $$\left(\Omega,\mathcal{A},P\right)$$ be a probability space and $$X$$ an integrable real random variable defined on $$(\Omega,\mathcal{A},P)$$. Let $$\mathcal{F}$$ be a sub-$$\sigma$$-algebra of $$\mathcal{A}$$. Then $$E[X|\mathcal{F}]$$ is the a.s. unique random variable $$Y$$ such that $$Y$$ is $$\mathcal{F}$$-measurable and for any $$A\in\mathcal{F}$$, $$E\left[X1_A\right]=E\left[Y1_A\right]$$.

The common interpretation seems to be: "$$E[X|\mathcal{F}]$$ is the expectation of $$X$$ given the information of $$\mathcal{F}$$." I'm finding it hard to get any meaning from this sentence.

1. In elementary probability theory, expectation is a real number. So the sentence above makes me think of a real number instead of a random variable. This is reinforced by $$E[X|\mathcal{F}]$$ sometimes being called "conditional expected value". Is there some canonical way of getting real numbers out of $$E[X|\mathcal{F}]$$ that can be interpreted as elementary expected values of something?

2. In what way does $$\mathcal{F}$$ provide information? To know that some event occurred, is something I would call information, and I have a clear picture of conditional expectation in this case. To me $$\mathcal{F}$$ is not a piece of information, but rather a "complete" set of pieces of information one could possibly acquire in some way.

Maybe you say there is no real intuition behind this, $$E[X|\mathcal{F}]$$ is just what the definition says it is. But then, how does one see that a martingale is a model of a fair game? Surely, there must be some intuition behind that!

I hope you have got some impression of my misconceptions and can rectify them.

• This is not the definition of conditional expectation with which I'm familiar. Do you have a reference? – Qiaochu Yuan Feb 24 '11 at 20:51
• @Qiaochu: I'm using Klenke's Probability Theory, but it's the same on Wikipedia. – Stefan Feb 24 '11 at 21:19
• You may want to read the answer to this question, math.stackexchange.com/questions/23093/…, where user joriki explains what it means for event A to be conditionally dependent on event B. – Uticensis Feb 24 '11 at 22:48
• A wonderful explanation about conditional expectation can be found here ma.utexas.edu/users/gordanz/notes/conditional_expectation.pdf – Broken_Window Aug 12 '16 at 14:08
• For your first question consider the expected value of the years you have left to live given your current age. Or rather that same data for a random person you don't know. If it were a real (positive) number $N$ then you'd basically expect to live $N$ more years "forever". It must be a function of the given variable – John Cataldo Apr 16 '19 at 14:51

Maybe this simple example will help. I use it when I teach conditional expectation.

(1) The first step is to think of ${\mathbb E}(X)$ in a new way: as the best estimate for the value of a random variable $X$ in the absence of any information. To minimize the squared error $${\mathbb E}[(X-e)^2]={\mathbb E}[X^2-2eX+e^2]={\mathbb E}(X^2)-2e{\mathbb E}(X)+e^2,$$ we differentiate to obtain $2e-2{\mathbb E}(X)$, which is zero at $e={\mathbb E}(X)$.

For example, if I throw a fair die and you have to estimate its value $X$, according to the analysis above, your best bet is to guess ${\mathbb E}(X)=3.5$. On specific rolls of the die, this will be an over-estimate or an under-estimate, but in the long run it minimizes the mean square error.

(2) What happens if you do have additional information? Suppose that I tell you that $X$ is an even number. How should you modify your estimate to take this new information into account?

The mental process may go something like this: "Hmmm, the possible values were $\lbrace 1,2,3,4,5,6\rbrace$ but we have eliminated $1,3$ and $5$, so the remaining possibilities are $\lbrace 2,4,6\rbrace$. Since I have no other information, they should be considered equally likely and hence the revised expectation is $(2+4+6)/3=4$".

Similarly, if I were to tell you that $X$ is odd, your revised (conditional) expectation is 3.

(3) Now imagine that I will roll the die and I will tell you the parity of $X$; that is, I will tell you whether the die comes up odd or even. You should now see that a single numerical response cannot cover both cases. You would respond "3" if I tell you "$X$ is odd", while you would respond "4" if I tell you "$X$ is even". A single numerical response is not enough because the particular piece of information that I will give you is itself random. In fact, your response is necessarily a function of this particular piece of information. Mathematically, this is reflected in the requirement that ${\mathbb E}(X\ |\ {\cal F})$ must be $\cal F$ measurable.

I think this covers point 1 in your question, and tells you why a single real number is not sufficient. Also concerning point 2, you are correct in saying that the role of $\cal F$ in ${\mathbb E}(X\ |\ {\cal F})$ is not a single piece of information, but rather tells what possible specific pieces of (random) information may occur.

• Great answer, thank you. "In fact, your response is necessarily a function of this particular piece of information." But then, why is $E(X|\mathcal{F})$ defined on $\Omega$ and not on $\mathcal{F}$? This is central to my not understanding. If $\mathcal{F}$ is not generated by a disjoint family of sets, there is no such interpretation of $E(X|\mathcal{F})(\omega)$, is there? – Stefan Feb 24 '11 at 22:51
• The conditional expectation $E(X|{\cal F})$ is characterized by being $\cal F$ measurable and having certain integrals over $\cal F$ sets. This function is therefore only defined $P$ almost everywhere. Therefore, you are right; the pointwise value $E(X|{\cal F})(\omega)$ has no particular meaning. – user940 Feb 25 '11 at 15:02
• Wonderful explanation. – Jonas Teuwen Apr 20 '11 at 23:05
• I want to add that if a r.v. $Z$ (defined on $\Omega$) is measurable with respect to $\mathcal F$, then one can indeed interpret $Z$ as a function of information of $\mathcal F$ in the following sense: if $\mathcal F$ represents the information for knowing outcome of another r.v. $Y$ (also defined on $\Omega$), i.e., if $\mathcal F = \sigma(Y)$, then $Z$ is a measurable function of $Y$, i.e., $Z = \phi(Y)$ (a.e.) for some $\phi$ where $\phi$ is some measurable function from the codomain of $Y$ (which would be $\mathbb R$ if $Y$ were a real-valued r.v.) to the codomain of $Z$. and... – Jisang Yoo Apr 26 '14 at 16:04
• .. If $\mathcal F$ represents the information for knowing joint outcome of a (finite or countable) family of r.v.s $Y_i$ ($i \in I$), i.e., if $\mathcal F = \sigma(Y_i, i \in I)$, then $Z$ is a measurable function of $Y$, i.e., $Z = \phi(Y_i, i \in I)$ (a.e.) for some $\phi$ that is a measurable function from the product of codomains of $Y_i$ to the codomain of $Z$. – Jisang Yoo Apr 26 '14 at 16:04

I think a good way to answer question 2 is as follows.

I am performing an experiment, whose outcome can be described by an element $\omega$ of some set $\Omega$. I am not going to tell you the outcome, but I will allow you to ask certain questions yes/no questions about it. (This is like "20 questions", but infinite sequences of questions will be allowed, so it's really "$\aleph_0$ questions".) We can associate a yes/no question with the set $A \subset \Omega$ of outcomes for which the answer is "yes".

Now, one way to describe some collection of "information" is to consider all the questions which could be answered with that information. (For example, the 2010 Encyclopedia Britannica is a collection of information; it can answer the questions "Is the dodo extinct?" and "Is the elephant extinct?" but not the question "Did Justin Bieber win a 2011 Grammy?") This, then, would be a set $\mathcal{F} \subset 2^\Omega$.

If I know the answer to a question $A$, then I also know the answer to its negation, which corresponds to the set $A^c$ (e.g. "Is the dodo not-extinct?"). So any information that is enough to answer question $A$ is also enough to answer question $A^c$. Thus $\mathcal{F}$ should be closed under taking complements. Likewise, if I know the answer to questions $A,B$, I also know the answer to their disjunction $A \cup B$ ("Are either the dodo or the elephant extinct?"), so $\mathcal{F}$ must also be closed under (finite) unions. Countable unions require more of a stretch, but imagine asking an infinite sequence of questions "converging" on a final question. ("Can elephants live to be 90? Can they live to be 99? Can they live to be 99.9?" In the end, I know whether elephants can live to be 100.)

I think this gives some insight into why a $\sigma$-field can be thought of as a collection of information.

• Great job on this answer! – Don Shanil Jul 11 '16 at 12:16
• The Borel $\sigma$-field $\mathcal{B}$ contains every singleton $\{\omega\}$, so with the information in $\mathcal{B}$ we'd be able to answer every question $A\subseteq \mathbb{R}$ by asking the oracle whether $\omega\in\{\omega\}$. So we should add the caveat that we can only decide questions that we can understand. A Vitali set, for instance, is not intelligible. – Detached Laconian Sep 24 '18 at 21:59
• @LaconianThinker: Yeah, in the case of $\mathbb{R}$, you can start by saying that you can ask whether $\omega < b$ for any $b$ (i.e. the set $(-\infty, b)$). This is a "computable" question in the sense that if you start computing decimal digits of $\omega$, you will eventually be able to tell whether it is less than $b$. And if you're able to ask a countable number of such questions, then you can determine whether $\omega$ is in your favorite Borel set $B$. But a countable number of such questions can never tell you whether $\omega$ is in the Vitali set. – Nate Eldredge Sep 24 '18 at 22:29
• @NateEldredge I'm not sure you understand my point (maybe I don't, either). Naively, we'd say a set $A$ is measurable if, regardless of which $\omega$ was sampled, we can decide whether $\omega\in A$ by asking the oracle countably many questions of the form "is $\omega$ less than $b$? or "is $\omega$ equal to $b$?". (I know the second case is redundant, but it will simplify my explanation). (continued) – Detached Laconian Sep 24 '18 at 23:13
• The problem is that any set (even a Vitali set!) would be measurable: if say, $\omega = 5$, we can ask the question "is $\omega$ equal to $5$?", and the answer ("yes") will give us complete information about $\omega$. Similarly for any other value $\omega$ could possibly have. – Detached Laconian Sep 24 '18 at 23:14

An example. Suppose that $X \sim {\rm binomial}(m,p)$ and $Y \sim {\rm binomial}(n,p)$ are independent ($0 < p < 1$). For any integer $0 \leq s \leq m+n$, it holds $${\rm E}[X|X + Y = s] = \frac{{m }}{{m + n }}s.$$ This means that $${\rm E}[X|X + Y] = \frac{{m }}{{m + n }}(X+Y).$$ Note that ${\rm E}[X|X + Y]$ is a random variable which is a function of $X+Y$.

Note that, in general, the conditional expectation of $X$ given $Z$, denoted ${\rm E}[X|Z]$, is defined as ${\rm E}[X|\sigma(Z)]$, where $\sigma(Z)$ is the $\sigma$-algebra generated by $Z$.

EDIT. In response to the OP's request, I note that the binomial distribution (which is discrete) plays no special role in the above example. For completely analogous results for the normal and gamma distributions (both are continuous) see this and this, respectively; for a substantial generalization, see this.

• @Shai: +1, nice answer! I was wondering whether the conditional expectation of a r.v. $X$ given another r.v. $Y$ is defined on the common domain $\Omega$ of both $X$ and $Y$, or on the codomain of $Y$, or can be either? The Wikipedia article says it is defined on the codomain of $Y$. See en.wikipedia.org/wiki/… – Tim Feb 24 '11 at 22:10
• @Tim: The conditional expectation is defined on the common domain $\Omega$. – Shai Covo Feb 24 '11 at 22:23
• Thanks for your answer. In this example, there is a connection between conditional expectation and elementary expectation. But I fail to see how this generalizes a) to non-discrete random variables and b) to $\sigma$-algebras $\mathcal{F}$ which are not of the form $\sigma(Y)$ for any random variable $Y$. – Stefan Feb 24 '11 at 22:25
• @Shai: Define $f$ by $E[X|Z]=f(Z)$. In your example $E[X|Z=s]=f(s)$. This is absurd if $P[Z=s]=0$. If you have something similar in that case, it would be helpful, though it probably won't answer all my questions. – Stefan Feb 24 '11 at 23:26
• @Shai: I didn't see your edit before posting my comment. So is it correct that $E[X|Z=s]=f(s)$ defines $E[X|Z=s]$ with the terminology of my last comment? – Stefan Feb 24 '11 at 23:37

I happened to read an article on Wikipedia today on Conditional Expectation. That clarified a lot of my questions. Hope it helps!

1. For your first question, in the linked article, there is the definition for conditional expectation of a r.v. $X: \Omega \rightarrow \mathbb{R}$ given a sub sigma algebra $\mathcal{F}$ of the one $\mathcal{A}$ on domain $\Omega$. It is a $\mathcal{F}$-measurable function $: \Omega \rightarrow \mathbb{R}$, denoted as $E(X \vert \mathcal{F})$. If you evaluate this conditional expectation at a point $\omega \in \Omega$, you will get a value $E(X \vert \mathcal{F})(\omega)$, which is called the conditional expectation of $X$ given $\mathcal{F}$ at $\omega$.

When the r.v. $X$ is an indicator function on some measurable subset say $A \in \mathcal{A}$, its conditional expectation given the sub sigma algebra is called the conditional probability of the subset $A$ given the sub sigma algebra $\mathcal{F}$, denoted as $P( A \vert \mathcal{F})$. It is a mapping: $\Omega \rightarrow \mathbb{R}$.

If we let $A$ vary within $\mathcal{A}$, the conditional probability $P( \cdot \vert \mathcal{F})$ is a mapping: $\mathcal{A} \times \Omega \rightarrow \mathbb{R}$. In some cases, $\forall \omega \in \Omega$, $P( \cdot \vert \mathcal{F})(\omega)$ is a probability measure on $(\Omega, \mathcal{F})$, in which case $P(\cdot \vert \mathcal{F})$ is called a regular conditional probability.

When $\mathcal{F}$ is generated by another r.v. $Y$, then the conditional expectation and conditional probability will be called the ones given the r.v. $Y$.

2. For your second question, I am still wondierng what kind of information a sigma algebra (of a r.v.) can provide, and how it is provided?

You can think of the conditional expectation as the orthogonal projection onto the closed subspace of $\mathcal F$-measurable random variables in the Hilbert space of square integrable random variables.

This is a detailed and elementary discussion of this viewpoint.

I was only able to understand the notion of the conditional expectation with respect to a sub-$$σ$$-algebra $$\mathcal F$$, when I realized that this game is only interesting when $$\mathcal F$$ is "not Hausdorff", meaning that there might be points $$x$$ and $$y$$ which cannot be separated by an $$\mathcal F$$-measurable set. Any $$\mathcal F$$-measurable function must therefore coincide on $$x$$ and $$y$$, so $$E(X|\mathcal F)$$ tries to be the best photograph of the random variable $$X$$ which coincides on $$x$$ and $$y$$, as well as on any other similar pairs of points.

In the event that $$\mathcal F$$ is the smallest sub-$$σ$$-algebra possible, namely $$\mathcal F = \{\emptyset, \Omega\}$$, only constant functions are measurable. So $$E(X|\mathcal F)$$ must be a constant function, and that constant turns out to be the average of $$X$$, a.k.a. the expectation of $$X$$.

PS. This is a comment I made in a recent question (Refference for conditional expectation) which in turn brought me here to this 10 year old question when I clicked on a "Related" post. Reading the answers I did not find anyone referring to the above point of view, so I hope my little contribution will be useful to someone.


text:
## Algebra: A Combined Approach (4th Edition)

Using the absolute property rules, we can solve the problem by removing the bars and solving the equation. $-6|x|+44=-10$ $-6x+44=-10$ $-6x=-10-44$ $-6x=-54$ $x=-54/-6$ $x=9$ Since this is an absolute value equation, we must the positives and negatives of the number. So the solution is {9,-9}.


text:
# A question about INMO 2017, Question 3

Find all triples $$(x,a,b)$$ where $$x$$ is a real number, and $$a,b$$ are integers belonging to $$\{1,2,\dots,9\}$$ such that $$x^2-a\{x\}+b=0$$ Here $$\{x\}$$ denotes the fractional part of $$x$$.

My contention, which I know is wrong, is that the equation has no solutions.

Reason: If $$x$$ has $$0$$ fractional part, this clearly has no solutions. Let us suppose now that $$x$$ has a fractional part that runs upto $$n$$ decimals. Then $$\{x\}$$ also has a fractional part which runs up to $$n$$ decimals. Consequently, $$x^2$$ has a fractional part that runs up to $$2n$$ decimals, and $$a\{x\}$$ has a fractional part that runs only up to $$n$$ decimal places (as $$a$$ is an integer). Hence, $$x^2-a\{x\}$$ has a non-zero fractional part (at least in the $$n+1$$ to $$2n$$ decimal places), which implies $$x^2-a\{x\}+b$$ can never be $$0$$ as it has a fractional part.

Where in this argument am I going wrong?

• I think you are assuming that $x$ is rational. – rogerl Feb 21 at 1:29
• @rogerl You are likely correct as irrational numbers never have a fixed # of digits. However, note that even rational numbers are often repeating decimals, so they also don't necessarily have just $n$ digits for some natural number $n$. – John Omielan Feb 21 at 1:31
• @JohnOmielan True, of course. But there are no solutions in rational numbers (this is not hard to see - if $p/q$ is a rational solution, write $p = dq+r$, $0\le r<q$, and simplify). But you are correct, I probably should have said that the OP is assuming that $x$ is a terminating decimal. – rogerl Feb 21 at 1:35
• x²-3x+1 has a solution between 0 and 1. So there is at least 1 solution to the problem. Try a graphical approach ? – ama Feb 21 at 1:54

As for where in your argument you went wrong, as stated in the comments, you can't necessarily assume that $$x$$ is a terminating decimal.

As to determine what values will work, let

$$x = c + r \text{ where } c \in \mathbb{Z} \text{ and } 0 \le r \lt 1 \tag{1}\label{eq1}$$

This gives

$$\left(c + r\right)^2 - ar + b = 0 \tag{2}\label{eq2}$$

Expanding this and collecting into powers of $$r$$ terms gives

$$r^2 + \left(2c - a\right)r + \left(c^2 + b\right) = 0 \tag{3}\label{eq3}$$

First, note that as $$r \ge 0$$, $$c^2 \ge 0$$ and $$b \gt 0$$, this means that

$$2c - a \lt 0 \tag{4}\label{eq4}$$

Solving for $$r$$ using the quadratic formula gives

$$r = \frac{-\left(2c - a\right) \pm \sqrt{\left(2c - a\right)^2 - 4\left(c^2 + b\right)}}{2} \tag{5}\label{eq5}$$

The $$r$$ inequality in \eqref{eq1} gives that

$$0 \le -\left(2c - a\right) \pm \sqrt{\left(2c - a\right)^2 - 4\left(c^2 + b\right)} \lt 2 \tag{6}\label{eq6}$$

Due to \eqref{eq4}, plus that $$c^2 + b \gt 0$$, the left side of \eqref{eq6} will always be true. As such, consider the right side. From \eqref{eq4}, to be able to add the square root means that $$2c - a = -1$$. However, this makes the discriminant $$1 - 4\left(c^2 + b\right) \lt 0$$, so $$r$$ won't be real. Thus, only subtracting the square root can possibly work. Thus, moving the square root term to the right side and the $$2$$ to the left gives

$$-\left(2c - a\right) - 2 \lt \sqrt{\left(2c - a\right)^2 - 4\left(c^2 + b\right)} \tag{7}\label{eq7}$$

Next, since $$2c - a < -1$$, as discussed above, then both sides are non-negative so we can square each side to get

$$\left(2c - a\right)^2 + 8c - 4a + 4 \lt \left(2c - a\right)^2 - 4\left(c^2 + b\right) \tag{8}\label{eq8}$$

Simplifying by removing the common first term on both sides and dividing by $$4$$ gives

$$2c - a + 1 \lt -c^2 - b \tag{9}\label{eq9}$$

This becomes

$$c^2 + 2c + 1 \lt a - b \Rightarrow \left(c + 1\right)^2 \lt a - b \tag{10}\label{eq10}$$

As $$\left(c + 1\right)^2 \ge 0$$, this means $$a \ge b$$ and it limits the possible values of $$c$$ as $$-2 \le c + 1 \le 2$$ since $$a - b \lt 9$$, so $$c$$ can only be one of $$-3, -2, -1, 0, 1$$. You should be able to finish this now by checking these few values of $$c$$ to determine the appropriate values for $$r$$, thus $$x$$, plus $$a$$ and $$b$$, where the value $$r$$ obtained in \eqref{eq5} is real, with this requiring that $$a^2 - 4ac - 4b \ge 0$$.

• Just an added question. Can a number of the form $f(f+n)$, where $0<f<1$ is a fraction and $n$ is an integer, ever be an integer? – Anju George Feb 21 at 3:03
• @AnjuGeorge Yes, it can quite easily be any integer $1 \le m \le n$ if you allow $f$ to be a real value between $0$ and $1$. However, if you restrict $f$ to be a rational number, then this is not so simple to determine. Is this what you are actually asking about? – John Omielan Feb 21 at 3:06
• @AnjuGeorge Note that for $f$ being restricted to rational values, it'll never be true. A hint is to express $f$ as a fraction in lowest terms. See if you can determine the rest yourself. – John Omielan Feb 21 at 3:20


text:
#### Volume 19, issue 6 (2019)

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Categories and orbispaces

### Stefan Schwede

Algebraic & Geometric Topology 19 (2019) 3171–3215
##### Abstract

Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a “classifying space”, the geometric realization of the nerve. Up to weak homotopy equivalence, every space is the classifying space of a small category. More is true: the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors. We establish a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types of orbispaces whose underlying coarse moduli space is the traditional homotopy type hitherto considered.

A global equivalence is a functor $\Phi :\mathsc{C}\to \mathsc{D}$ between small categories with the following property: for every finite group $G\phantom{\rule{0.3em}{0ex}}$, the functor $G\Phi :G\mathsc{C}\to G\mathsc{D}$ induced on categories of $G$–objects is a weak equivalence. We show that the global equivalences are part of a model structure on the category of small categories, which is moreover Quillen equivalent to the homotopy theory of orbispaces in the sense of Gepner and Henriques. Every cofibrant category in this global model structure is opposite to a complex of groups in the sense of Haefliger.

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text:
# Borel $\sigma$-algebras and topologies

If $id:(M,\rho_1)\to(M,\rho_2)$ is continuous, then $\cal{B}_2$$(M)\subseteq \cal{B}_1$$(M)$

where $\cal{B}_1, \cal{B}_2$ are the Borel $\sigma$-algebras induced by the metrics $\rho_1 ,\rho_2$ respectively.

I've understood that by continuity of f, since the pre-image of an open set is open, then open sets $U$ that are open w.r.t $\rho_2$ are also open w.r.t $\rho_1$. This means that if $\tau_1,\tau_2$ are topologies w.r.t $\rho_1, \rho_2$ then $\tau_2 \subseteq \tau_1$

But how am I able to obtain/prove that $\cal{B}_2$$(M)\subseteq \cal{B}_1$$(M)$ must follow?

• If $U$ is open w.r.t. $\rho_2$, then $f^{-1}(U)$ is open w.r.t. $\rho_1$. How does this imply $U$ is open w.r.t. $\rho_1$? – parsiad May 5 '17 at 17:48
• Regardless, if you know $\tau_2 \subset \tau_1$, then trivially $\mathcal{B}_2(M) \subset \mathcal{B}_1(M)$ since $\mathcal{B}_i(M)=\sigma(\tau_i)$ where $\sigma(X)$ is the sigma algebra generated by $X$. – parsiad May 5 '17 at 17:49
• I forgot to add that f is the identity function in this case – Malcolm May 5 '17 at 17:55
• Is there someway to prove that if $\tau_2 \subseteq \tau_1 \Rightarrow \sigma(\tau_2) \subseteq \sigma(\tau_1)$? I'm not at the stage where I can visualize it – Malcolm May 5 '17 at 17:57
• Clearly $\tau_2 \subseteq \sigma(\tau_1)$. Hence $\sigma(\tau_1)$ is a $\sigma$-algebra which contains $\tau_2$. By definition, $\sigma(\tau_2)$ is the smallest $\sigma$-algebra containing $\tau_2$, so necessarily it is smaller than $\sigma(\tau_1)$, i.e. we have $\sigma(\tau_2) \subseteq \sigma(\tau_1)$. – Nate Eldredge May 5 '17 at 18:02

Theorem: If $X \subset Y$, then $\sigma(X) \subset \sigma(Y)$.
Proof: Since $$\sigma(X)\equiv\bigcap_{\substack{\mathcal{M}\text{ is a }\sigma\text{-algebra}\\ X\subset\mathcal{M} } }\mathcal{M} \text{ and } \sigma(Y)\equiv\bigcap_{\substack{\mathcal{M}\text{ is a }\sigma\text{-algebra}\\ Y\subset\mathcal{M} } }\mathcal{M},$$ the desired result follows.