text:
**Galois connection a.k.a. Adjunction Formula**
$f(p) \le q \iff p \le g(q) \\ \text{Where g is right-adjoint to f and f is left-adjoint to g}$

1. Choose a nontrivial partition \$$c : S \twoheadrightarrow P \$$ and let \$$g_!(c) \$$ be its push forward partition on T.
2. Choose any coarser partition \$$d : T \twoheadrightarrow P' \$$, i.e. where \$$g_!(c) \le d \$$ .
3. Choose any non-coarser partition \$$e : T \twoheadrightarrow Q \$$, i.e. where \$$g_!(c) \nleq e \$$. (If you can’t do this, revise your answer for #1.)
4. Find \$$g^\*(d) \text{ and } g^\*(e) \$$ .
5. The adjunction formula, in this case, says that since \$$g_!(c) \le d \text{ and } g_!(c) \nleq e \$$ , we should have \$$c \le g^\*(d) \text{ and } c \nleq g^\*(e) \$$ . Show that this is true.

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# graph

## English

Wikipedia has an article on:
Wikipedia
a demographic graph.

### Etymology

Shortening of graphic formula.

### Noun

graph (plural graphs)

1. in the common language
1. A diagram displaying data; in particular one showing the relationship between two or more quantities, measurements or indicative numbers that may or may not have a specific mathematical formula relating them to each other.
• 2012 March 1, Brian Hayes, “Pixels or Perish”, in American Scientist[1], volume 100, number 2, page 106:
Drawings and pictures are more than mere ornaments in scientific discourse. Blackboard sketches, geological maps, diagrams of molecular structure, astronomical photographs, MRI images, the many varieties of statistical charts and graphs: These pictorial devices are indispensable tools for presenting evidence, for explaining a theory, for telling a story.
2. in mathematics
1. (mathematics) A diagram displaying data, in particular one showing the relationship between two or more variables; specifically, for a function ${\displaystyle f(x_{1},x_{2},\ldots ,x_{n})}$, the set of all tuples ${\displaystyle (x_{1},x_{2},\ldots ,x_{n},f(x_{1},x_{2},\ldots ,x_{n}))}$.
2. (graph theory) An ordered pair ${\displaystyle (V,E)}$, where ${\displaystyle V}$ is a set of elements called vertices (or nodes) and ${\displaystyle E}$ is a set of pairs of elements of ${\displaystyle V}$, called edges; informally, a set of vertices together with a set edges that join these vertices.
3. (category theory) Said to be of a morphism f: a morphism ${\displaystyle \Gamma _{f}}$ from the domain of f to the product of the domain and codomain of f, such that the first projection applied to ${\displaystyle \Gamma _{f}}$ equals the identity of the domain, and the second projection applied to ${\displaystyle \Gamma _{f}}$ is equal to f.
3. in linguistics
1. (linguistics, typography) A graphical unit on the token-level, the abstracted fundamental shape of a character or letter as distinct from its ductus (realization in a particular typeface or handwriting on the instance-level) and as distinct by a grapheme on the type-level by not fundamentally distinguishing meaning.
• 2003, J. Richard Andrews, Introduction to Classical Nahuatl, Revised Edition, University of Oklahoma Press, page 10:
A graph is a token-level nondistinctive representation of a grapheme. It can differ from the other variants of its grapheme with regard to upper case, lower case, script, print, typeface style, typeface size, etc.

### Verb

graph (third-person singular simple present graphs, present participle graphing, simple past and past participle graphed)

1. (transitive) To draw a graph.
2. () To draw a graph of a function.

format: plain


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# About the infinite dimensional skew and obliquely reflected Ornstein-Uhlenbeck process

Röckner M, Trutnau G (2016)
Infinite Dimensional Analysis Quantum Probability and Related Topics 18(4): 1550031.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
Based on an integration by parts formula for closed and convex subsets Gamma of a separable real Hilbert space H with respect to a Gaussian measure, we first construct and identify the infinite dimensional analogue of the obliquely reflected Ornstein-Uhlenbeck process (perturbed by a bounded drift B) by means of a Skorokhod type decomposition. The variable oblique reflection at a reflection point of the boundary. G is uniquely described through a reflection angle and a direction in the tangent space (more precisely through an element of the orthogonal complement of the normal vector) at the reflection point. In case of normal reflection at the boundary of a regular convex set and under some monotonicity condition on B, we prove the existence and uniqueness of a strong solution to the corresponding SDE. Subsequently, we consider an increasing sequence (Gamma(alpha k))(k) (is an element of Z) of closed and convex subsets of H and the skew reflection problem at the boundaries of this sequence. We present concrete examples and obtain as a special case the infinite dimensional analogue of the p-skew reflected Ornstein-Uhlenbeck process.
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Infinite Dimensional Analysis Quantum Probability and Related Topics
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18
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4
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1550031
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### Zitieren

Röckner M, Trutnau G. About the infinite dimensional skew and obliquely reflected Ornstein-Uhlenbeck process. Infinite Dimensional Analysis Quantum Probability and Related Topics. 2016;18(4): 1550031.
Röckner, M., & Trutnau, G. (2016). About the infinite dimensional skew and obliquely reflected Ornstein-Uhlenbeck process. Infinite Dimensional Analysis Quantum Probability and Related Topics, 18(4), 1550031. doi:10.1142/S0219025715500319
Röckner, M., and Trutnau, G. (2016). About the infinite dimensional skew and obliquely reflected Ornstein-Uhlenbeck process. Infinite Dimensional Analysis Quantum Probability and Related Topics 18:1550031.
Röckner, M., & Trutnau, G., 2016. About the infinite dimensional skew and obliquely reflected Ornstein-Uhlenbeck process. Infinite Dimensional Analysis Quantum Probability and Related Topics, 18(4): 1550031.
M. Röckner and G. Trutnau, “About the infinite dimensional skew and obliquely reflected Ornstein-Uhlenbeck process”, Infinite Dimensional Analysis Quantum Probability and Related Topics, vol. 18, 2016, : 1550031.
Röckner, M., Trutnau, G.: About the infinite dimensional skew and obliquely reflected Ornstein-Uhlenbeck process. Infinite Dimensional Analysis Quantum Probability and Related Topics. 18, : 1550031 (2016).
Röckner, Michael, and Trutnau, Gerald. “About the infinite dimensional skew and obliquely reflected Ornstein-Uhlenbeck process”. Infinite Dimensional Analysis Quantum Probability and Related Topics 18.4 (2016): 1550031.

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Required field
Required field
Required field

# How to Predict Linear and Exponent Coefficients for a Power Law Porous Media?

## Objective

In this article, you will learn how to predict linear and exponent coefficients for a power law porous media using experimental data.

## Overview

In our simulations, often times we have to model a structure as a porous media. These structures can be as simple as perforated plates and as complex as radiators. The objective, however, is the same: save mesh cells while still accounting for local pressure losses.

In the Power Law formulation, pressure losses are given by:

$$\Delta P = C_0 \cdot \rho \cdot L \cdot U^{C_1}$$

• ΔP is the pressure drop across the porous media (Pa);
• ρ is the density of the fluid (kg/m³);
• L is the thickness of the porous zone (m);
• U is the fluid velocity (m/s);
• C0 is the linear coefficient and;
• C1 is the exponent coefficient.

Note: with the power law formulation, pressure drop will always be isotropic within the medium. If your media is anisotropic, make sure to check out the Darcy-Forchheimer approach. For compressible flows, check the Fixed Coefficient model.

## Power curve fitting approach

Using experimental data for pressure versus velocity, we can extract the coefficients.

For example, let’s take a 0.01 meters thick perforated plate and water as fluid. The following case was simulated on SimScale at 6 different velocities.

The table below shows the results from the simulations:

\begin{array} {|r|r|} \hline Velocity (m/s) & ΔP (Pa) \\ \hline 0.5 & 960 \\ \hline 0.7 & 1840 \\ \hline 1 & 3720 \\ \hline 1.2 & 5390 \\ \hline 1.5 & 8240 \\ \hline 2 & 14690 \\ \hline \end{array}

By fitting this data set with a power curve, we get the following:

The expression below is extracted from the plot:

$$\Delta P = 3736.2 \cdot U^{1.9685}$$

From that, we obtain C0 and C1:

$$C_0 = {3736.2 \over \rho \cdot L} = {3736.2 \over 997.33 \cdot 0.01} = 374.62$$

$$C_1 = 1.9685$$

In the platform, you can readily create a porous media and assign it to a volume.

Now to compare experimental data with simulation results:

The power law model proves to be accurate in accounting for pressure losses.

Note


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# Ilyashenko algebras based on transserial asymptotic expansions

Our preprint extending my earlier construction of Ilyashenko algebras is now on the arXiv.

The purpose of this paper is to extend Ilyashenko’s construction of the class of germs at $+\infty$ of almost regular functions to obtain a Hardy field containing them.  In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is an LE-series as defined by van den Dries et al.  and a transseries as defined by van der Hoeven.  As these series generally have support of order type larger than $\omega$, the notion of asymptotic expansion itself needs to be generalized.  This can be done naturally in the context of a quasianalytic algebra, leading to our definition of \textit{quasianalytic asymptotic algebra}, or \textit{qaa algebra} for short.  Any qaa algebra constructed by generalizing Ilyashenko’s construction will be called an \textbf{Ilyashenko algebra}.
The Hardy field $\H = \Hanexp$ of all unary germs at $+\infty$ of unary functions definable in the o-minimal structure $\Ranexp$ is an example of an Ilyashenko field; see van den Dries and Miller  and van den Dries et al.  My earlier paper contains a first attempt at constructing an Ilyashenko field $\F$ containing Ilyashenko’s almost regular germs.  The implied non-oscillatory properties of its germs were used in Belotto et al.’s recent solution of the strong Sard conjecture.  However, this field $\F$ does not contain $\H$; the Ilyashenko field constructed here is a Hardy field that contains both $\F$ and $\H$, implying non-oscillatory behaviour with respect to all $\log$-$\exp$-analytic germs.
Our main motivation for generalizing Ilyashenko’s construction in this way is the conjecture that the class of almost regular germs generates an o-minimal structure over the field of real numbers.  This conjecture, in turn, might lead to locally uniform bounds on the number of limit cycles in subanalytic families of real analytic planar vector fields all of whose singularities are hyperbolic.  Establishing such uniform bounds for planar polynomial vector fields follows Roussarie’s approach to Hilbert’s 16th problem (part 2); see Ilyashenko’s survey for an overview on the latter.  Our conjecture implies a generic instance of Roussarie’s finite cyclicity conjecture; see my preprint explaining this connection. In Kaiser et al.  we gave a positive answer to our conjecture in the special case where all singularities are, in addition, non-resonant.  (For a different approach to the general hyperbolic case, see Mourtada’s preprint.)
The almost regular germs also play a role in the description of Riemann maps and solutions of Dirichlet’s problem on semianalytic domains; see Kaiser’s papers here and here  for details.  Finally, in the spirit of the concluding remark of van den Dries et al., this paper provides a rigorous construction of a Hardy subfield of Ecalle’s field of “fonctions analysables” that properly extends $\H$, and we do so without the use of “acc\’el\’ero-sommation”; for more details on this, see the concluding remarks in our paper.
We plan to eventually settle our o-minimality conjecture by adapting the procedure used here, which requires three main steps:
1. extend the class of almost regular germs into an Ilyashenko field;
2. construct corresponding algebras of germs of functions in several variables, such that the resulting system of algebras is stable under various operations (such as blowings-up, say);
3. obtain o-minimality using a normalization procedure.
While my earlier paper contains a first successful attempt at Step (1), Step (2) poses some challenges.  For instance, it is not immediately obvious what the nature of LE-series in several variables should be; they should at least be stable under all the operations required for Step (3).  They should also contain the series used in Mourtada’s preprint to characterize parametric transition maps in the hyperbolic case, which use so-called Ecalle-Roussarie compensators as monomials.
Our approach to this problem is to enlarge the set of monomials used in asymptotic expansions.  A first candidate for such a set of monomials is the set of all (germs of) functions definable in the o-minimal structure $\Ranexp$ (in any number of variables).  This set of germs is obviously closed under the required operations, because the latter are all definable, and it contains the Ecalle-Roussarie compensators.  However, it is too large to be meaningful for use as monomials in asymptotic expansions, as it is clearly not $\RR$-linearly independent (neither in the additive nor the multiplicative sense) and contains many germs that have “similar asymptotic behavior” such as, in the case of unary germs, belonging to the same archimedean class.  More suitable would be to find a minimal subclass $\la_n$ of all definable $n$-variable germs such that every definable $n$-variable germ is piecewise given by a convergent Laurent series (or, if necessary, a convergent generalized Laurent series in a finite tuple of germs in $\la_n$.
Thus, the purpose of this paper is to determine such a minimal set of monomials $\la = \la_1$ contained in the set $\H$ of all unary germs at $+\infty$ definable in $\Ranexp$, and to further adapt the construction in my earlier paper to corresponding generalized series in one variable.  Recalling that $\H$ is a Hardy field, we can summarize the results of this paper as follows:

### Main Theorem

There is a multiplicative subgroup $\la$ of $\H$ such that the following hold:
1. no two germs in $\la$ belong to the same archimedean class;
2. every germ in $\H$ is given by composing a convergent Laurent series with a tuple of germs in $\la$;
3. the construction in my earlier paper generalizes, after replacing the finite iterates of $\,\log$ with germs in $\la$, to obtain a corresponding Ilyashenko field $\K$.
The resulting Ilyashenko field $\K$ is a Hardy field extending $\H$ as well as the Ilyashenko field $\F$ constructed in my earlier paper.
We obtain this set $\la$ of monomials by giving an explicit description of the Hardy field $\H$ as the set of all convergent LE-series, with $\la$ being the corresponding set of \textit{convergent LE-monomials}.  The proof that the construction in my earlier paper generalizes to this set $\la$ relies heavily on our recent paper on $\log$-$\exp$-analytic germs; indeed, our construction here was our main motivation for this paper.

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# 38=4w+-26

## Simple and best practice solution for 38=4w+-26 equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so dont hesitate to use it as a solution of your homework.

If it's not what You are looking for type in the equation solver your own equation and let us solve it.

## Solution for 38=4w+-26 equation:

Simplifying
38 = 4w + -26
Reorder the terms:
38 = -26 + 4w
Solving
38 = -26 + 4w
Solving for variable 'w'.
Move all terms containing w to the left, all other terms to the right.
Add '-4w' to each side of the equation.
38 + -4w = -26 + 4w + -4w
Combine like terms: 4w + -4w = 0
38 + -4w = -26 + 0
38 + -4w = -26
Add '-38' to each side of the equation.
38 + -38 + -4w = -26 + -38
Combine like terms: 38 + -38 = 0
0 + -4w = -26 + -38
-4w = -26 + -38
Combine like terms: -26 + -38 = -64
-4w = -64
Divide each side by '-4'.
w = 16
Simplifying
w = 16`

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text:
# Integral of cscx.cotx formula

## Formula

$\displaystyle \int{\csc{x}\cot{x} \,}dx \,=\, -\csc{x}+c$

### Introduction

Assume, $x$ as a variable, and represents an angle of a right triangle. The cosecant and cotangent functions are written in terms of $x$ as $\csc{x}$ or $\operatorname{cosec}{x}$ and $\cot{x}$ respectively. The indefinite integral of product of $\csc{x}$ and $\cot{x}$ functions with respect to $x$ is written in the following mathematical form in integral calculus.

$\displaystyle \int{\csc{x}\cot{x} \,} dx$

The indefinite integration of product of cosecant and cot functions with respect to $x$ is equal to the sum of negative cosecant function and an integral constant.

$\displaystyle \int{\csc{x}\cot{x} \,}dx \,=\, -\csc{x}+c$

#### Alternative forms

The integral of product of cosecant and cot functions formula can be written in terms of any variable in integral calculus.

$(1) \,\,\,$ $\displaystyle \int{\csc{(k)}\cot{(k)} \,}dk \,=\, -\csc{(k)}+c$

$(2) \,\,\,$ $\displaystyle \int{\csc{(m)}\cot{(m)} \,}dm \,=\, -\csc{(m)}+c$

$(3) \,\,\,$ $\displaystyle \int{\csc{(z)}\cot{(z)} \,}dz \,=\, -\csc{(z)}+c$

### Proof

Learn how to derive the integration rule for the product of cosecant and cotangent functions in integral calculus.

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Disks

MAPS targeted five well-known protoplanetary disks for an in-depth chemical exploration. The properties of these disks are listed below and are available in a convenient python dictionary via our data download page.

IM Lup

IM Lup is a young (~1 Myr), Solar-mass star that hosts a massive and radially-extended disk. IM Lup has the largest pebble disk (~260 au) in the DSHARP sample (Huang et al. 2018a) and a CO disk out to ~900 au (Cleeves et al. 2016). Continuum images reveal the presence of spiral arms in mm grains (Huang et al. 2018b), while scattered light observations show a highly flared, multi-ringed disk in micron-sized dust grains (Avenhaus et al. 2018). Despite its youth, the IM Lup disk shows evidence of grain growth (Lommen et al. 2007). The outer portion of the CO gas disk is diffuse and envelope-like, potentially indicating the presence of an external photoevaporative wind (Haworth et al. 2017). IM Lup has been been observed in several molecular lines (Öberg et al. 2011) and shows a striking double-ring structure in DCO+, due to a change in gas-phase C and O abundance at the pebble disk edge (Öberg et al. 2015b). Modeling indicates that CO is underabundant by a factor of ~20 compared to molecular clouds due to missing gas-phase carbon and oxygen (Cleeves et al. 2016). No depletions were found in nitrogen, resulting in an elevated gas-phase N/O ratio (Cleeves et al. 2018). Spitzer observations showed weak OH and H2O emission in the inner disk, suggesting intermediate water depletion (Banzatti et al. 2017).

 Distance 158 pc Gaia Collaboration et al. 2018 Inclination 47.5 deg Huang et al. 2018a P.A. 144.5 deg Huang et al. 2018a SpT K5 Alcalá et al. 2017 T$$_{\rm{eff}}$$ 4266 K Alcalá et al. 2017 L$$_*$$ 2.57 L$$_{\odot}$$ Alcalá et al. 2017 Age $${\sim}$$1 Myr Mawet et al. 2012 M$$_*$$ 1.1 M$$_{\odot}$$ MAPS XVIII, Teague et al. 2021 log$$_{10}$$ $$\small{\rm{\dot{M}}}$$ -7.9 M$$_{\odot}$$ yr$$^{-1}$$ Alcalá et al. 2017 V$$_{\rm{sys}}$$ 4.5 km s$$^{-1}$$ Pinte et al. 2018a

GM Aur

GM Aur is a ~3-10 Myr T Tauri star in the Taurus-Auriga star-forming region hosting a transition disk, i.e., a disk large with a large inner gap or cavity. The large dust cavity (~35 au) was first inferred from SED modeling (Marsh & Mahoney 1992) and later confirmed by direct imaging using mm observations (Hughes et al. 2009). High resolution continuum observations at cm and mm wavelengths showed that the outer disk consists of several nested rings surrounded by diffuse continuum emission to ~270 au (Macías et al. 2018) and a small inner disk inside this central gap (Huang et al. 2020). The GM Aur disk has been observed in several gas tracers, with indications of non-Keplerian CO emission (Dutrey et al. 2008) and bright lines associated with photochemistry and small organics (Öberg et al. 2010). The MAPS observations show the presence of spectacular, large-scale CO spiral arms (MAPS XIX, Huang et al. 2021). Herschel observations show modestly depleted oxygen (Keane et al. 2014), but due to its low IR flux, GM Aur was not included in Spitzer surveys (Pontoppidan et al. 2010) so its inner disk molecular content is unknown.

 Distance 159 pc Gaia Collaboration et al. 2018 Inclination 53.2 deg Huang et al. 2020 P.A. 57.2 deg Huang et al. 2020 SpT K6 Espaillat et al. 2010 T$$_{\rm{eff}}$$ 4350 K Espaillat et al. 2010 L$$_*$$ 1.2 L$$_{\odot}$$ Macías et al. 2018 Age $${\sim}$$3-10 Myr Kraus & Hillenbrand 2009 M$$_*$$ 1.1 M$$_{\odot}$$ MAPS XVIII, Teague et al. 2021 log$$_{10}$$ $$\small{\rm{\dot{M}}}$$ -8.1 M$$_{\odot}$$ yr$$^{-1}$$ Ingleby et al. 2015 V$$_{\rm{sys}}$$ 5.6 km s$$^{-1}$$ Huang et al. 2020

AS 209

AS 209 is a young (~1 Myr) T Tauri star in the Ophiuchus star-forming region. The disk is highly structured in mm continuum emission with at least 7 nested rings (e.g., Huang et al. 2018a, Guzmán et al. 2018), some of which are associated with ongoing planet formation (Favre et al. 2019). Scattered light observations show three rings in micron-sized dust grains (Avenhaus et al. 2018). The radial gas surface density and CO abundance profiles are also highly structured. Several CO line emission features are spatially related to peaks or gaps in dust emission (Guzmán et al. 2018), while continuum and NIR structures are associated with gas surface density perturbations (Teague et al. 2018). The AS 209 disk has a rich molecular line inventory at millimeter wavelengths (e.g., Öberg et al. 2011, Huang et al. 2017, Bergner et al. 2018, Bergner et al. 2019). Most of the millimeter molecular emission appears depleted towards the disk center but whether this due to dust opacity, excitation, C/N/O abundance, or gas surface density is not yet resolved (but see MAPS XV, Bosman et al. 2021). The inner disk chemistry of AS 209 has not been well characterized, with only upper limits on the water reservoir from Herschel (Du et al. 2017).

 Distance 121 pc Gaia Collaboration et al. 2018 Inclination 35.0 deg Huang et al. 2018a P.A. 85.8 deg Huang et al. 2018a SpT K5 Andrews et al. 2018 T$$_{\rm{eff}}$$ 4266 K Andrews et al. 2018 L$$_*$$ 1.41 L$$_{\odot}$$ Andrews et al. 2018 Age $${\sim}$$1-2 Myr Andrews et al. 2018 M$$_*$$ 1.2 M$$_{\odot}$$ MAPS XVIII, Teague et al. 2021 log$$_{10}$$ $$\small{\rm{\dot{M}}}$$ -7.3 M$$_{\odot}$$ yr$$^{-1}$$ Salyk et al. 2013 V$$_{\rm{sys}}$$ 4.6 km s$$^{-1}$$ Huang et al. 2017

HD 163296

HD 163296 is one of the most well studied Herbig Ae star-disk systems at millimeter wavelengths due to its relative proximity and massive disk. The HD 163296 disk presents multiple features that suggest the presence of planets and ongoing planet formation, including dust rings, azimuthal asymmetries, deviations from Keplerian velocities due to gas pressure variations, "kinks" in CO emission, and meridional flows (e.g., Isella et al. 2016, Pinte et al. 2018b, Teague et al. 2019). Observations in scattered light at optical and infrared wavelengths also reveal substantial radial substructure (Monnier et al. 2017, Muro-Arena et al. 2018, Rich et al. 2020). HD 163296 has been observed in CO and other molecular tracers at millimeter wavelengths in a number of single-dish, SMA, and ALMA projects (e.g., Thi et al. 2004, Qi et al. 2011, Öberg et al. 2011, Rosenfeld et al. 2013, Salinas et al. 2017, Pegues et al. 2020). Its popularity in previous surveys can be explained by its bright molecular lines, not only in CO isotopologues, but also in HCN, HCO+, deuterated species, and larger organics. The HD 163296 disk is relatively line poor at IR wavelengths (Pontoppidan et al. 2010, Salyk et al. 2011), but H2O and OH have been detected (Fedele et al. 2012, Banzatti et al. 2017), revealing a disk atmosphere that is not completely dry.

 Distance 101 pc Gaia Collaboration et al. 2018 Inclination 46.7 deg Huang et al. 2018a P.A. 133.3 deg Huang et al. 2018a SpT A1 Fairlamb et al. 2015 T$$_{\rm{eff}}$$ 9332 K Fairlamb et al. 2015 L$$_*$$ 17.0 L$$_{\odot}$$ Fairlamb et al. 2015 Age $$\small{\gtrsim}$$ 6 Myr Fairlamb et al. 2015 M$$_*$$ 2.0 M$$_{\odot}$$ MAPS XVIII, Teague et al. 2021 log$$_{10}$$ $$\small{\rm{\dot{M}}}$$ -7.4 M$$_{\odot}$$ yr$$^{-1}$$ Fairlamb et al. 2015 V$$_{\rm{sys}}$$ 5.8 km s$$^{-1}$$ Teague et al. 2019

MWC 480

MWC 480 is a ~7 Myr old Herbig Ae star in the Taurus-Auriga star-forming region. Its mm continuum disk has a centrally-peaked profile (Piétu et al. 2006) with two dust rings and low-level emission extending beyond 200 au (Long et al. 2018, Liu et al. 2019). MWC 480 was included in several early disk chemistry programs (i.e., CID, DISCS; Dutrey et al. 2007, Öberg et al. 2010, Henning et al. 2010, Dutrey et al. 2011), which showed that the disk is very bright in some lines, e.g., HCO+ and HCN, but weak in others, including N2H+ and H2CO (e.g., Pegues et al. 2020). The disk has also been probed in simple and complex nitriles, deuterated molecules, and sulfur-bearing species (Öberg et al. 2015a, Guzmán et al. 2015, Huang et al. 2017, Le Gal et al. 2019) and was the target of an unbiased spectral line survey (Loomis et al. 2020), which resulted in detections of several new disk molecules, including C2D and H2CS. The MWC 480 water content was probed by the Herschel WISH key program, but was not detected (van Dishoeck et al. 2011). Otherwise, the inner disk chemistry of MWC 480 is unknown.

 Distance 162 pc Gaia Collaboration et al. 2018 Inclination 37.0 deg Liu et al. 2019 P.A. 148.0 deg Liu et al. 2019 SpT A5 Montesinos et al. 2009 T$$_{\rm{eff}}$$ 8250 K Montesinos et al. 2009 L$$_*$$ 21.9 L$$_{\odot}$$ Montesinos et al. 2009 Age $${\sim}$$7 Myr Montesinos et al. 2009 M$$_*$$ 2.1 M$$_{\odot}$$ Simon et al. 2019 log$$_{10}$$ $$\small{\rm{\dot{M}}}$$ -6.9 M$$_{\odot}$$ yr$$^{-1}$$ Mendigutia et al. 2013 V$$_{\rm{sys}}$$ 5.1 km s$$^{-1}$$ Piétu et al. 2007

Note: Stellar luminosities and accretion rates extracted prior to Gaia have been updated with DR2 distances following Andrews et al. 2018 for IM Lup, AS 209, and HD 163296 and Macías et al. 2018 for GM Aur. Due to their large relative uncertainties, those for MWC 480 have not been rescaled. The accretion rate of GM Aur is variable (Ingleby et al. 2015), so we adopted an average value, following Macías et al. 2018.


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# Lesson 9

Equations of Lines

### Problem 1

Select all the equations that represent the graph shown.

A:

$$3x-2y=6$$

B:

$$y=\frac{3}{2}x+3$$

C:

$$y=\frac{3}{2}x-3$$

D:

$$y-3=\frac{3}{2}(x-4)$$

E:

$$y-6=\frac{3}{2}(x-2)$$

### Solution

For access, consult one of our IM Certified Partners.

### Problem 2

A line with slope $$\frac32$$ passes through the point $$(1,3)$$.

1. Explain why $$(3,6)$$ is on this line.
2. Explain why $$(0,0)$$ is not on this line.
3. Is the point $$(13,22)$$ on this line? Explain why or why not.

### Solution

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### Problem 3

Write an equation of the line that passes through $$(1,3)$$ and has a slope of $$\frac{5}{4}$$.

### Solution

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### Problem 4

A parabola has focus $$(3,\text{-}2)$$ and directrix $$y=2$$. The point $$(a,\text{-}8)$$ is on the parabola. How far is this point from the focus?

A:

6 units

B:

8 units

C:

10 units

D:

cannot be determined

### Solution

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(From Unit 6, Lesson 8.)

### Problem 5

Write an equation for a parabola with each given focus and directrix.

1. focus: $$(5, 2)$$; directrix: $$x$$-axis
2. focus: $$(\text{-}2, 3)$$; directrix: the line $$y=7$$
3. focus: $$(0,7)$$; directrix: $$x$$-axis
4. focus: $$(\text{-}3, \text- 4)$$; directrix: the line $$y=\text-1$$

### Solution

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(From Unit 6, Lesson 8.)

### Problem 6

A parabola has focus $$(\text{-}1,6)$$ and directrix $$y=4$$. Determine whether each point on the list is on this parabola. Explain your reasoning.

1. $$(\text{-}1,5)$$
2. $$(1 ,7)$$
3. $$(3, 9)$$

### Solution

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(From Unit 6, Lesson 7.)

### Problem 7

Select the center of the circle represented by the equation $$x^2 + y^2 - 8x + 11y - 2 = 0$$.

A:

$$(8, 11)$$

B:

$$(4, 5.5)$$

C:

$$(\text-4, \text-5.5)$$

D:

$$(4, \text-5.5)$$

### Solution

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(From Unit 6, Lesson 6.)

### Problem 8

Reflect triangle $$ABC$$ over the line $$x=\text-6$$.

Translate the image by the directed line segment from $$(0,0)$$ to $$(5,\text-1)$$.

What are the coordinates of the vertices in the final image?

### Solution

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(From Unit 6, Lesson 1.)


text:
July 1998

UCB-PTH-98/38

LBNL-42104

Super Yang-Mills on the Noncomutative Torus **To appear in the Arnowitt Festschrift Volume “Relativity, Particle Physics, and Cosmology”, Texas A&M University, April 1998, published by World Scientific This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-95-14797

Department of Physics

University of California

and

Theoretical Physics Group

Lawrence Berkeley National Laboratory

University of California

Berkeley, California 94720

After a brief review of matrix theory compactification leading to noncommutative supersymmetric Yang-Mills gauge theory, we present solutions for the fundamental and adjoint sections on a two-dimensional twisted quantum torus in two different gauges. We also give explicit transformations connecting different representations which have appeared in the literature. Finally we discuss the more mathematical concept of Morita equivalence of $C^{*}$ -algebras as it applies to our specific case.

Disclaimer

This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial products process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, or The Regents of the University of California.

Lawrence Berkeley National Laboratory is an equal opportunity employer.

## 1 Introduction

It was conjectured in [1]that the infinite momentum frame description of M-theory is given by the large $n$ limit of supersymmetric quantum mechanics (SQM) [2, 3, 4], obtained as the dimensional reduction of the $9+1$ dimensional $U(n)$ supersymmetric Yang-Mills (SYM) gauge field theory. Shortly afterwards Susskind took this a step further [5], conjecturing that the discrete light cone quantization (DLCQ) of M-theory is equivalent to the finite $n$ matrix theory.

Toroidal compactification of M-theory can then be obtained by first considering matrix theory on the covering space and then imposing a periodicity condition on the matrix variable [1, 6, 7], also known as the quotient condition. The result is a SYM field theory on a dual torus.

If we consider the DLCQ of M-theory and compactify on a torus $T^{d}$ for $d\geq 2$ there are additional moduli coming from the three-form of 11-dimensional supergravity. For example, if we compactify on $T^{2}$ along $X_{1}$ and $X_{2}$ then $C_{-12}$ cannot be gauged away, and is a modulus of the compactification. It was conjectured in [8]that turning on this modulus corresponds to deforming the SYM theory on the dual torus to a noncommutative SYM on a quantum torus [9]with deformation parameter $\theta$ given by

 $\theta=\int~{}C_{-12}\,dX_{-}dX_{1}dX_{2}.$

Evidence for this conjecture comes from comparison of the BPS mass spectra of the two theories and of their duality groups. Further evidence and discussions of this conjecture followed in [10, 11, 12, 13, 14, 16, 15, 17, 18, 19].

In Section 2we present a review of matrix theory compactification leading to noncommutative SYM gauge theory on trivial quantum bundles. We follow the elementary treatment of [17]with an emphasis on giving explicit formulae that closely resemble the commutative case. We further present an explicit realization of the algebra of the quantum torus ${\cal A}(T^{2}_{\theta})$ in terms of quantum plane coordinates.

In Section 3we introduce non-trivial quantum bundles as in [17]corresponding to compactified DLCQ of M-theory in the presence of transversely wrapped membranes. We also explain in some detail how to solve the boundary conditions for sections in the fundamental and adjoint quantum bundle. Finally using the special form of the transition functions in the given gauge we find an equivalent but simpler form of the general solution for fundamental sections.

In Section 4we discuss the more abstract language of projective modules, as presented in [8]and references therein, and we then give the explicit map between this formulation and the more elementary formulation in [17]. We also explain the notion of Morita equivalence [20, 8, 18, 19, 25]applied to our specific case. For an expanded coverage of noncommutative geometry see [24]and for a brief description see [26].

Finally in Section 5we discuss the general theory of gauge transformations on the noncommutative torus and find an explicit gauge transformation that trivializes one of the transition functions. With trivial transition functions T-duality transformations take the standard form, allowing us to interpret the gauge field as D-strings on the dual torus.

## 2 Review of Matrix Compactification

In this section we present a review of matrix theory compactification closely following the description given by Ho in [17]. The $P_{-}=n/R$ sector of the DLCQ of uncompactified M-theory is given by the $U(n)$ SQM [2, 3, 4]whose action in the temporal gauge is given by

 $S=\frac{1}{2R}\int dt\,{\rm Tr}\left(\dot{X}^{\mu}\dot{X}_{\mu}+\sum_{\mu>\nu}% [X_{\mu},X_{\nu}]^{2}+i\Theta^{T}\dot{\Theta}-\Theta^{T}\Gamma_{\mu}[X^{\mu},% \Theta]\right)$ (1)

where $\mu,\nu=0,\ldots,9$ . We will compactify matrix theory on a rectangular 2-torus of radii $R_{1}$ and $R_{2}$ . First let us consider matrix variables on the covering space and impose the quotient condition

 $\displaystyle U_{i}^{-1}X_{j}U_{i}$ $\displaystyle=$ $\displaystyle X_{j}+2\pi R_{j}\delta_{ij},$ (2) $\displaystyle U_{i}^{-1}X_{a}U_{i}$ $\displaystyle=$ $\displaystyle X_{a},$ $\displaystyle U_{i}^{-1}\Theta~{}U_{i}$ $\displaystyle=$ $\displaystyle\Theta,~{}~{}~{}i,j=1,2~{}~{}~{}a=3,\ldots,9.$

Here the $U_{i}$ are unitary operators. The consistency of these equation requires

 $U_{1}U_{2}=e^{2\pi i\theta}U_{2}U_{1}.$

Before solving the quotient condition ( 2), it is convenient to introduce two more unitary operators $\widetilde{U}_{i},~{}~{}i=1,2$ which commute with the $U_{i}$ ’s and satisfy the relation

 $\widetilde{U}_{1}\widetilde{U}_{2}=e^{-2\pi i\theta}\widetilde{U}_{2}% \widetilde{U}_{1}.$ (3)

One way to realize this algebra is by using canonical variables $\sigma_{i}$ satisfying

 $\left[\sigma_{1},\sigma_{2}\right]=2\pi i\theta.$ (4)

Then $\widetilde{U}_{i}\stackrel{\rm def}{=}e^{i\sigma_{i}}$ satisfy ( 3). The variables $\sigma_{i}$ are noncommutative coordinates on the quantum plane which is the covering space of the quantum torus. The algebra of functions on the quantum torus denoted ${\cal A}(T^{2}_{-\theta})$ is generated by $\widetilde{U}_{i}$ . Similarly the $U_{i}$ operators generate the algebra denoted ${\cal A}(T^{2}_{\theta})$ . To realize them we introduce partial derivative operators on the quantum plane, satisfying the following algebra

 $\left[\partial_{i},\sigma_{j}\right]=\delta_{ij},$ (5)
 $\left[\partial_{i},\partial_{j}\right]=0.$

Then, we realize $U_{i}$ as

 $U_{1}=e^{i\sigma_{1}}e^{2\pi\theta\,\partial_{2}},~{}~{}U_{2}=e^{i\sigma_{2}}e% ^{-2\pi\theta\,\partial_{1}}.$

For $\theta=0$ we have $U_{i}=\widetilde{U}_{i}=e^{i\sigma_{i}}$ , all generators commute allowing us to use either $U_{i}$ ’s or $\widetilde{U}_{i}$ ’s to generate the algebra of functions on the classical torus. It is then easy to check that

 $U_{i}^{-1}~{}\frac{1}{i}\partial_{j}~{}U_{i}=\frac{1}{i}\partial_{j}+\delta_{% ij}.$ (6)

This and many other formulae in this paper can be proven using the Campbell-Baker-Hausdorff formula which can be written in closed form since commutators like ( 4) and ( 5) are $c$ -numbers. Equation ( 6) is very similar to the quotient condition ( 2) so one can write a solution as a sum of the partial derivative and a fluctuating part that commutes with the $U_{i}$ ’s. However this is just the definition of the covariant derivative

 $\displaystyle X_{i}$ $\displaystyle=$ $\displaystyle-2\pi iR_{j}\delta_{ij}\partial_{j}+A_{i}(\widetilde{U}_{i}),$ (7) $\displaystyle X_{a}$ $\displaystyle=$ $\displaystyle X_{a}(\widetilde{U}_{i}),$ $\displaystyle\Theta$ $\displaystyle=$ $\displaystyle\Theta(\widetilde{U}_{i}),$

where $A_{i}$ , $X_{a}$ and each spinorial component of $\Theta$ are $n\times n$ hermitian matrices with operator valued entries. Note that since the partial derivatives already satisfy the cocycle condition, the gauge fields $A_{i}$ and the scalar fields $X_{a}$ must satisfy a homogeneous quotient condition like the second relation in ( 2). Hence $A_{i}$ and $X_{a}$ must depend only on $\widetilde{U}_{i}$ . Hidden in this dependence is the fact that we are working on a trivial bundle over the quantum torus.

If one inserts ( 7) into the SQM action ( 1) the result is a noncommutative SYM gauge field theory in $2+1$ dimensions, with the space part given by the above quantum torus and a commutative time. For the commutative case, matrix compactification on $T^{d}$ results in a SYM gauge theory in $d+1$ dimensions on the dual torus. In the limit when the size of the original torus vanishes the dual torus becomes ${\bf R}^{d}$ , therefore we obtain the opposite of dimensional reduction. If one starts from a Euclidean 10-dimensional SYM and dimensionally reduces in all directions including the Euclidean time one obtains the IKKT [27]functional. Matrix compactification of one direction in the IKKT functional results in the finite temperature action of the original theory ( 1).

## 3 Twisted Quantum Bundles

We can consider more general solutions of the quotient condition ( 2) which are connections on twisted bundles. They correspond to compactification of the DLCQ of M-theory in the presence of transversely wrapped membranes. Again the solution is a sum of two terms, a constant curvature connection $D_{i}$ and a fluctuating part

 $\displaystyle X_{i}$ $\displaystyle=$ $\displaystyle-2\pi iR_{j}\delta_{ij}D_{j}+A_{i}(Z_{j}),$ (8) $\displaystyle X_{a}$ $\displaystyle=$ $\displaystyle X_{a}(Z_{i}),$ $\displaystyle\Theta$ $\displaystyle=$ $\displaystyle~{}\Theta(Z_{i}).$

Here the $Z_{i}$ ’s are $n\times n$ matrices with operator entries and, just like the $\widetilde{U}_{i}$ ’s for the trivial bundle, commute with the $U_{i}$ ’s, but now are sections of the twisted bundle whose exact form will be discussed shortly. However, while for the trivial bundle $A_{i}$ , $X_{a}$ and the spinorial components of $\Theta$ are $n\times n$ matrix functions, in ( 8) $A_{i}$ , $X_{a}$ and the components of $\Theta$ are one-dimensional functions but with matrix arguments. Later, this will allow us to establish a relationship between a SYM on a twisted $U(n)$ bundle and one on a $U(1)$ bundle.

Following [17], up to a gauge transformation the constant curvature connection can be written as

 $D_{1}=\partial_{1},~{}D_{2}=\partial_{2}-if\sigma_{1},$ (9)

where $f$ is the constant field strength

 $[D_{1},D_{2}]=-if.$

Such a gauge field can only exist in a non-trivial bundle. One can introduce transition functions $\Omega_{i}$ such that the sections of the fundamental bundle satisfy the twisted boundary conditions

 $\displaystyle\Phi(\sigma_{1}+2\pi,\sigma_{2})$ $\displaystyle=$ $\displaystyle\Omega_{1}(\sigma_{1},\sigma_{2})~{}\Phi(\sigma_{1},\sigma_{2}),$ (10) $\displaystyle\Phi(\sigma_{1},\sigma_{2}+2\pi)$ $\displaystyle=$ $\displaystyle\Omega_{2}(\sigma_{1},\sigma_{2})~{}\Phi(\sigma_{1},\sigma_{2}).$

 $\displaystyle\Psi(\sigma_{1}+2\pi,\sigma_{2})=\Omega_{1}(\sigma_{1},\sigma_{2}% )~{}\Psi(\sigma_{1},\sigma_{2})~{}\Omega_{1}(\sigma_{1},\sigma_{2})^{-1},$ (11) $\displaystyle\Psi(\sigma_{1},\sigma_{2}+2\pi)=\Omega_{2}(\sigma_{1},\sigma_{2}% )~{}\Psi(\sigma_{1},\sigma_{2})~{}\Omega_{2}(\sigma_{1},\sigma_{2})^{-1}.$

Consistency of the transition functions of the bundle requires that

 $\Omega_{1}(\sigma_{1},\sigma_{2}+2\pi)~{}\Omega_{2}(\sigma_{1},\sigma_{2})=% \Omega_{2}(\sigma_{1}+2\pi,\sigma_{2})~{}\Omega_{1}(\sigma_{1},\sigma_{2}).$ (12)

This relation is known in the mathematical literature as the cocycle condition. The covariant derivatives transform just as the adjoint sections

 $D_{i}(\sigma_{1}+2\pi,\sigma_{2})=\Omega_{1}(\sigma_{1},\sigma_{2})~{}D_{i}(% \sigma_{1},\sigma_{2})~{}\Omega_{1}(\sigma_{1},\sigma_{2})^{-1},$
 $D_{i}(\sigma_{1},\sigma_{2}+2\pi)=\Omega_{2}(\sigma_{1},\sigma_{2})~{}D_{i}(% \sigma_{1},\sigma_{2})~{}\Omega_{2}(\sigma_{1},\sigma_{2})^{-1}.$

A particular solution for the transition functions compatible with the constant curvature connection ( 9) and satisfying the cocycle condition is given by

 $\Omega_{1}=e^{im\sigma_{2}/n}U,~{}\Omega_{2}=V,$ (13)

where $U,~{}V$ are $n\times n$ unitary matrices satisfying

 $UV=e^{-2\pi im/n}VU$

and $m$ is an integer. For simplicity, here we will only consider the case when $n$ and $m$ are relatively prime. For the general case see [8, 14, 31]. Using the representation given in [17]one has

 $U_{kl}=e^{2\pi ikm/n}\delta_{k,l},~{}~{}V_{kl}=\delta_{k+1,l},$

where the subscripts are identified with period $n$ .

We can express the above matrices in terms of the standard ’t Hooft matrices [28, 29]denoted here by $U^{\prime}$ and $V^{\prime}$ and satisfying

 $U^{\prime}V^{\prime}=e^{-2\pi i/n}V^{\prime}U^{\prime},~{}~{}U^{\prime n}=V^{% \prime n}=1.$

The relation is given by

 $U=e^{2\pi im/n}U^{\prime m},~{}~{}V=V^{\prime}.$ (14)

The phase in ( 14) is due to the nonstandard definition of $U$ used in [17]. This has certain advantages but similar phases will appear when comparing the results of [17]with similar results where the standard ’t Hooft matrices were used. We also introduce a unitary matrix $K$ which changes the representation so that $V^{\prime}$ is diagonal, and satisfies

 $KU^{\prime}K^{-1}=V^{\prime-1},~{}~{}KV^{\prime}K^{-1}=U^{\prime}.$ (15)

Note that $n$ is quantized since we are considering a $U(n)$ gauge theory and $m$ is quantized since the magnetic flux $f$ through $T_{2}$ is quantized

 $2\pi f=\frac{m}{n-m\theta}.$

In M-theory $m$ is the transversal membrane wrapping number.

One can solve the boundary conditions ( 10) for the fundamental sections as in [17]generalizing a previous result for $m=1$ in the commutative case presented in [7]. Using the ordered exponential explained below, the general solution has the form

 $\Phi_{k}(\sigma_{1},\sigma_{2})=\sum_{s\in{\bf Z}}\sum_{j=1}^{m}E\left(\frac{m% }{n}\left(\frac{\sigma_{2}}{2\pi}+k+ns\right)+j,i\sigma_{1}\right)\widehat{% \phi}_{j}\left(\frac{\sigma_{2}}{2\pi}+k+ns+\frac{nj}{m}\right).$

The ordered exponential [17]is defined for two variables whose commutator is a $c$ -number

 $E(A,B)=\frac{1}{1-[A,B]}~{}\sum_{l=0}^{\infty}\frac{1}{l!}A^{l}B^{l}.$

The normalization is such that

 $E(-B,A)E(A,B)=1$

and it has the following desirable properties similar to the usual exponential

 $\displaystyle E(A+c,B)=E(A,B)e^{cB},$ (16) $\displaystyle E(A,B+c)=e^{cA}E(A,B).$

The $\widehat{\phi}_{j}$ functions are defined on the whole real axis and are unrestricted except for the behavior at infinity. They should be considered as vectors in a Hilbert space on which all the elements of the algebra are represented.

Next we explain in some detail how to obtain this result. First we define

 $\phi(\sigma_{1},\sigma_{2})\stackrel{\rm def}{=}\Phi_{k}(\sigma_{1},\sigma_{2}% -2\pi(k-1)).$

The second boundary condition ( 10) implies that the definition of $\phi$ is consistent, i.e. $k$ -independent. Using $V^{n}=1$ we also find that $\phi$ is a periodic function in $\sigma_{2}$

 $\phi(\sigma_{1},\sigma_{2}+2\pi n)=\phi(\sigma_{1},\sigma_{2}).$

The other boundary condition gives

 $\phi(\sigma_{1}+2\pi,\sigma_{2})=e^{im(\sigma_{2}+2\pi)/n}\phi(\sigma_{1},% \sigma_{2}).$

It is convenient to separate out a factor to eliminate the above twist

 $\phi(\sigma_{1},\sigma_{2})=f(\sigma_{1},\sigma_{2})\check{\phi}(\sigma_{1},% \sigma_{2})$

and to require a simpler periodicity condition for $\check{\phi}$

 $\check{\phi}(\sigma_{1}+2\pi,\sigma_{2})=\check{\phi}(\sigma_{1},\sigma_{2}).$

Then the function $f$ must satisfy

 $f(\sigma_{1}+2\pi,\sigma_{2})=e^{im(\sigma_{2}+2\pi)/n}f(\sigma_{1},\sigma_{2}).$

This is satisfied exactly for

 $f(\sigma_{1},\sigma_{2})=E\left(\frac{m}{n}\left(\frac{\sigma_{2}}{2\pi}+1% \right),i\sigma_{1}\right),$

where in the right hand side we used the ordered exponential defined above. Now we can Fourier transform $\check{\phi}$ in $\sigma_{1}$

 $\check{\phi}(\sigma_{1},\sigma_{2})=\sum_{p\in{\bf Z}}~{}e^{ip\sigma_{1}}~{}% \phi_{p}(\sigma_{2})$

and using the property ( 16) of the ordered exponential we obtain

 $\phi(\sigma_{1},\sigma_{2})=\sum_{p\in{\bf Z}}E\left(\frac{m}{n}\left(\frac{% \sigma_{2}}{2\pi}+1\right)+p,i\sigma_{1}\right)\phi_{p}(\sigma_{2}).$

Let $p=ms+j$ with $j=1,\ldots,m$ and $s$ is an integer. Then the solution can be written as

 $\phi(\sigma_{1},\sigma_{2})=\sum_{s\in{\bf Z}}\sum_{j=1}^{m}E\left(\frac{m}{n}% \left(\frac{\sigma_{2}}{2\pi}+1\right)+ms+j,i\sigma_{1}\right)\phi_{s,j}(% \sigma_{2}),$

where $\phi_{s,j}\stackrel{\rm def}{=}\phi_{ms+j}$ . Periodicity in $\sigma_{2}$ then implies $\phi_{s-1,j}(\sigma_{2}+2\pi n)=\phi_{s,j}(\sigma_{2})$ so that using this recursively we have $\phi_{s,j}(\sigma_{2})=\phi_{0,j}(\sigma_{2}+2\pi ns)$ . Finally after defining $\widetilde{\phi}_{j}(x)\stackrel{\rm def}{=}\phi_{0,j}(2\pi(x-1))$ we obtain

 $\Phi_{k}(\sigma_{1},\sigma_{2})=\sum_{s\in{\bf Z}}\sum_{j=1}^{m}E\left(\frac{m% }{n}\left(\frac{\sigma_{2}}{2\pi}+k+ns\right)+j,i\sigma_{1}\right)\widetilde{% \phi}_{j}\left(\frac{\sigma_{2}}{2\pi}+k+ns\right).$

This is the result mentioned above up to another redefinition

 $\widetilde{\phi}_{j}(x)=\widehat{\phi}_{j}(x+\frac{n}{m}j).$

While the solutions for the sections of the fundamental bundle given in [17]are suitable for showing the equivalence to the projective modules of [8]as we will discuss in Section 4, the appearance of the ordered exponential is somewhat inconvenient. Using the special form of the transition functions we were able to rewrite the solution in an equivalent but simpler form. The transition functions in this gauge do not contain $\sigma_{1}$ and it is convenient to order all $\sigma_{1}$ to the right in the solution. Using $V^{n}=1$ in the second condition ( 10) one can express all $n$ components of $\Phi$ in terms of a single function with period $2\pi n$ in $\sigma_{2}$ . After Fourier transforming in $\sigma_{2}$ and imposing both boundary conditions ( 10) we obtain the general solution

 $\Phi_{k}(\sigma_{1},\sigma_{2})=\sum_{p\in{\bf Z}}~{}e^{2\pi i(\sigma_{2}/2\pi% +k)p/n}~{}e^{2\pi i(\sigma_{1}/2\pi-p/m)m/n}~{}\widehat{\psi}_{p}(\sigma_{1}/2% \pi-p/m),$

where only $m$ of the $\widehat{\psi}_{p}$ functions are independent, since

 $\widehat{\psi}_{p+m}(x)=\widehat{\psi}_{p}(x).$

Using the same technique one can show that an arbitrary adjoint section has the following expansion

 $\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}~{}c_{st}Z_{1}^{s}Z_{2}^{-t}.$ (17)

Here $c_{s,t}$ are $c$ -numbers and

 $Z_{1}=e^{i\sigma_{1}/(n-m\theta)}V^{b},~{}~{}Z_{2}=e^{i\sigma_{2}/n}U^{-b},$

where $b$ is an integer, such that we can find another integer $a$ satisfying $an-bm=1$ . For $n$ and $m$ relatively prime one can always find integer solutions to this equation. Again, we emphasize that the $Z_{i}$ ’s commute with the $U_{i}$ ’s. They are generators of the algebra of functions on a new quantum torus

 $Z_{1}Z_{2}=e^{2\pi i\theta^{\prime}}Z_{2}Z_{1},$

where $\theta^{\prime}$ is obtained by an $SL(2,{\bf Z})$ fractional transformation from $-\theta$

 $\theta^{\prime}=\frac{a(-\theta)+b}{m(-\theta)+n}.$

Now we outline how to obtain this result. Note first that

 $\Psi(\sigma_{1}+2\pi n,\sigma_{2})=\Omega_{1}^{n}~{}\Psi(\sigma_{1},\sigma_{2}% )~{}\Omega_{1}^{-n}=\Psi(\sigma_{1}+2\pi\theta m,\sigma_{2}).$

In the last equality we used the fact that $U^{n}=1$ , and we also used the exponential formula to shift $\sigma_{1}$ . Using both boundary conditions we have

 $\Psi(\sigma_{1}+2\pi(n-m\theta),\sigma_{2})=\Psi(\sigma_{1},\sigma_{2}),$
 $\Psi(\sigma_{1},\sigma_{2}+2\pi n)=\Psi(\sigma_{1},\sigma_{2}).$

We can expand the section as

 $\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}e^{is\sigma_{1}/(n-m\theta)}e^% {-it\sigma_{2}/n}\Psi_{s,t},$

where $\Psi_{s,t}$ is a $n\times n$ matrix and can be expanded as

 $\Psi_{s,t}=\sum_{i=i_{0}}^{n+i_{0}}\sum_{j=j_{0}}^{n+j_{0}}c_{s,t,i,j}V^{% \prime i}U^{\prime j}.$ (18)

Here $i_{0},j_{0}$ are two arbitrary integers, allowing us to freely shift the summation limits assuming that $c_{s,t,i+n,j}=c_{s,t,i,j+n}=c_{s,t,i,j}$ . Then one can obtain further restrictions on the $c_{s,t,i,j}$ coefficients using the boundary conditions ( 11). For example using the first equation ( 11) and comparing like coefficients in the Fourier expansion we have

 $c_{s,t,i,j}e^{2\pi is/(n-m\theta)}=c_{s,t,i,j}e^{-2\pi imi/n}e^{2\pi ism\theta% /[n(n-,\theta)]}.$

From this and the similar relation obtained by imposing the second equation ( 11) we have that $c_{s,t,i,j}$ vanish unless $(s+mi)/n=k$ and $(t+j)/n=s$ for $k$ and $s$ two integers. These equations have multiple solutions. However, if $(i,j)$ and $(i^{\prime},j^{\prime})$ are two solutions then $i-i^{\prime}\in n{\bf Z}$ and $j-j^{\prime}\in n{\bf Z}$ . This ensures that only one term survives in the sum ( 18) over $i$ and $j$ . Choosing for later convenience $i_{0}=sb$ and $j_{0}=mbt$ we have

 $\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}~{}e^{is\sigma_{1}/(n-m\theta)% }e^{-it\sigma_{2}/n}\sum_{i=sb}^{n+sb}~{}\sum_{j=mbt}^{n+mbt}c_{s,t,i,j}~{}V^{% \prime i}U^{\prime j}.$

Since $n$ and $m$ are relatively prime let $a,b\in{\bf Z}$ such that $an-bm=1$ . Then

 $k=as,~{}l=at,~{}i=bs,~{}j=mbt$

is an integer solution inside the $i$ and $j$ summation range. Dropping the $i,j$ indices since they are determined by $s$ and $t$ we have

 $\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}c_{s,t}\left(e^{i\sigma_{1}/(n% -m\theta)}V^{\prime b}\right)^{s}\left(e^{i\sigma_{2}/n}U^{\prime-mb}\right)^{% -t},$

which is just ( 17) after an additional phase redefinition of $c_{s,t}$ to accommodate the phase difference between $U$ and $U^{\prime m}$ .

## 4 Projective Modules and Morita Equivalence

A classic mathematical result of Gel’fand states that compact topological spaces are in one to one correspondence with commutative $C^{*}$ -algebras. In one direction, to a topological space $X$ we associate the algebra of continuous functions $C(X)$ . Conversely and rather nontrivially, the spectrum of a commutative $C^{*}$ -algebra is equivalent to a compact topological space. This important result allows for a dual description of topological spaces and brings powerful algebraic methods into the realm of topology. On the other hand, if we drop the commutativity requirement, a $C^{*}$ -algebra ${\cal A}$ describe what is called by correspondence a quantum space. To illustrate, consider the algebra of the quantum torus ${\cal A}(T^{2}_{\theta})$ generated by the $U_{i}$ ’s. An arbitrary element $a$ has the form

 $a=\sum_{k,l\in{\bf Z}}a_{k,l}U_{1}^{k}U_{2}^{l},$ (19)

where some restrictions (which we do not discuss here) are imposed on the $c$ -number coefficients $a_{k,l}$ . For $\theta=0$ , formula ( 19) reduces to the Fourier expansion of functions on $T^{2}$ . Thus we can read the compact space from the commutative algebra.

Using the same strategy one can describe other spaces of classical geometry in commutative algebraic terms and then remove the commutativity requirement. A quantum vector bundle is a projective ${\cal A}$ -module ${\cal E}$ . First consider the classical commutative picture. The set ${\cal E}$ of global sections of a vector bundle over a base space $X$ has the structure of a projective module over the algebra $C(X)$ . Having a module essentially means that we can add sections and can multiply them by functions. Not all modules over a commutative algebra are vector bundles. For example the set of sections on a space consisting of a collection of fibers of different dimensions over a base space also form a module. However, projective modules over the algebra of functions on a topological space are in one to one correspondence with vector bundles over that space. By definition a projective module is a direct summand in a free module. A free module ${\cal E}_{0}$ over an algebra ${\cal A}$ is a module isomorphic to a direct sum of a finite number of copies of the algebra

 ${\cal E}_{0}={\cal A}\oplus\ldots\oplus{\cal A}.$

Trivial bundles correspond to free modules since the description of their sections in terms of components is global, and each component is an element of $C(X)$ . For every vector bundle we can find another one such that their direct sum is a trivial bundle. In dual language this implies that the module of sections ${\cal E}$ is projective

 ${\cal E}_{0}={\cal E}\oplus{\cal E}^{\prime}.$

Again it is nontrivial to show the converse, that every projective module is isomorphic to the set of sections of some vector bundle. Finally projective modules over noncommutative algebras are the quantum version of vector bundles.

In the noncommutative case we distinguish between left and right projective modules. Multiplying fundamental sections from the right with elements of ${\cal A}(T^{2}_{-\theta})$ preserves the boundary conditions ( 10) while multiplication on the left gives something that no longer is a global section. Thus the set of sections of the fundamental bundle form a right projective module over the ${\cal A}(T^{2}_{-\theta})$ algebra which we denote ${\cal F}_{n,m}^{\theta}$ . This is no longer true for the adjoint sections since in ( 11) the transition functions multiply from both the left and right. However one can check that the fundamental and the adjoint are both left and right projective modules over the ${\cal A}(T^{2}_{\theta})$ algebra. This is because the exponents of the $U_{i}$ ’s satisfy

 $[i\sigma_{1}+2\pi\theta\partial_{2},\sigma_{i}]=0,~{}~{}[i\sigma_{2}-2\pi% \theta\partial_{1},\sigma_{i}]=0;$ (20)

thus the $U_{i}$ ’s can be commuted over the transition functions in ( 10) and ( 11). Additionally, the fact that ${\cal F}_{n,m}^{\theta}$ is both a left ${\cal A}(T^{2}_{\theta})$ -module and a right ${\cal A}(T^{2}_{-\theta})$ -module can be understood as follows. Since $[U_{i},\sigma_{j}]=0$ we have

 $U_{i}\Phi(\sigma_{1},\sigma_{2})=\Phi(\sigma_{1},\sigma_{2})\widetilde{U}_{i},$

where we dropped the derivatives when there was nothing to their right. Thus multiplying on the left with $a$ is equivalent to multiplying on the right with $\widetilde{a}$

 $a\Phi=\Phi\widetilde{a},$ (21)

where $\widetilde{a}=\sum_{k,l\in{\bf Z}}a_{k,l}\widetilde{U}_{2}^{l}\widetilde{U}_{1% }^{k}$ is the same function as $a$ but with $\widetilde{U}_{i}$ ’s as arguments and with all the factors written in reversed order.

As mentioned in [17]the construction in Section 2is equivalent to the projective modules discussed in [8]. By solving the boundary conditions we went from a local to a global description. Here we present explicit formulae for this equivalence. First one has to express the left actions on the fundamental sections as actions on the Hilbert space [17]. For example the action of the $Z_{i}$ generators is given by

 $(Z_{1}\widehat{\phi})_{j}(x)=\widehat{\phi}_{j-a}(x-\frac{1}{m}),~{}~{}(Z_{2}% \widehat{\phi})_{j}(x)=e^{-2\pi ij/m}e^{2\pi ix/(n-m\theta)}\widehat{\phi}_{j}% (x).$

This can be written as

 $Z_{1}=W_{1}^{a}V_{1},~{}~{}Z_{2}=W_{2}V_{2},$

where $V_{i}$ and $W_{i}$ are operators acting on the Hilbert space as

 $(V_{1}\widehat{\phi})_{j}(x)=\widehat{\phi}_{j}(x-\frac{1}{m}),~{}~{}(V_{2}% \widehat{\phi})_{j}(x)=e^{2\pi ix/(n-m\theta)}\widehat{\phi}_{j}(x),$
 $(W_{1}\widehat{\phi})_{j}(x)=\widehat{\phi}_{j-1}(x),~{}~{}(W_{2}\widehat{\phi% })_{j}(x)=e^{-2\pi ij/m}\widehat{\phi}_{j}(x).$

These operators satisfy the following relations

 $V_{1}V_{2}=e^{-2\pi i/[m(n-m\theta)]}V_{2}V_{1},~{}~{}W_{1}W_{2}=e^{2\pi i/m}W% _{2}W_{1},~{}~{}[V_{i},W_{j}]=0$

and can be used to express other operators acting in the Hilbert space. For example we have $U_{1}=W_{1}V_{1}^{n-m\theta}$ and $U_{2}=W_{2}^{n}V_{2}^{n-m\theta}$ .

We can now present the correspondence between [8]and [17]. The two integers $p$ and $q$ and the angular variable $\theta_{\rm CDS}$ labeling the projective module ${\cal H}_{p,q}^{\theta_{\rm CDS}}$ of [8], and $\theta^{\prime}_{\rm CDS}$ can be expressed in terms of the quantities used in this paper or in [17]

 $p=n,~{}~{}q=-m,~{}~{}\theta_{\rm CDS}=-\theta,~{}~{}\theta^{\prime}_{\rm CDS}=% \theta^{\prime}.$

Then ${\cal F}_{n,m}^{\theta}\cong{\cal H}_{n,-m}^{-\theta}$ . The Hilbert space representation of [8]written in terms of the function $f(s,k)$ with $s\in{\bf R}$ and $k\in{\bf Z}_{q}$ is linearly related to the $\widehat{\phi}_{k}(x)$ representation

 $\widehat{\phi}_{k}(x)=\sum_{l=1}^{m}{\cal K}_{kl}~{}({\cal S}_{(-\frac{m}{n-m% \theta})}f)(x,l).$

Here ${\cal K}$ is an $m\times m$ representation changing matrix defined as in ( 15) but for $m$ -dimensional ’t Hooft matrices, and ${\cal S}_{\lambda}$ is the rescaling operator $({\cal S}_{\lambda}f)(x,k)=f(\lambda x,k)$ which can be expressed using the ordered exponential

 ${\cal S}_{\lambda}=\lambda E((\lambda-1)x,\partial_{x}).$

Also, using lower case to distinguish them from our current notation which follows [17], the operators in [8]represented in the $\hat{\phi}_{k}(x)$ basis are given by

 $v_{0}=V_{2}^{n-m\theta},~{}~{}v_{1}=V_{1}^{n-m\theta},~{}~{}w_{0}=e^{2\pi in/m% }W_{2}^{n},~{}~{}w_{1}=e^{2\pi i/m}W_{1}$
 $z_{0}=e^{2\pi i/m}Z_{2},~{}~{}z_{1}=e^{-2\pi ia/m}Z_{1}^{-1},~{}~{}u_{0}=e^{2% \pi in/m}U_{2},~{}~{}u_{1}=e^{2\pi i/m}U_{1}.$

Next we introduce the Morita equivalence of two algebras [20, 21, 22, 23, 18, 19], which can be used to describe a subgroup of the T-duality group of the M-theory compactification in the language of noncommutative SYM gauge theory.

Two $C^{*}$ -algebras ${\cal A}$ and ${\cal A}^{\prime}$ are Morita equivalent if there exists a right ${\cal A}$ -module ${\cal E}$ such that the algebra $End_{\cal A}{\cal E}$ is isomorphic to ${\cal A}^{\prime}$ . Here $End_{\cal A}{\cal E}$ denotes the set of endomorphisms of the ${\cal A}$ -module ${\cal E}$ . It consists of linear maps $T$ on ${\cal E}$ where linearity is not only with respect to $c$ -numbers but also with respect to right multiplication by elements of ${\cal A}$

 $T(\Phi f)=T(\Phi)f,~{}~{}\Phi\in{\cal E},~{}~{}f\in{\cal A}.$

An example of Morita equivalent algebras is ${\cal A}(T^{2}_{-\theta})$ and ${\cal A}(T^{2}_{\theta^{\prime}})$


text:
## A community for students. Sign up today

Here's the question you clicked on:

## soccergal12 2 years ago solve: ( 1/(e^x) ) = 5

• This Question is Closed
1. Romero

This is tricky one you will have to use natural log.

2. Romero

$\frac{1}{e^x}=5$$1=5*e^x$$\frac{1}{5}=e^x$

3. zzr0ck3r

1/e^(x) = 5 e^(x) = 1/5 ln(e^(x)) = ln(1/5) x = ln(1/5) = ln(1) - ln(5) = 0 - ln(5) = -ln(5)

4. Romero

from there you simply natural log both sides.

5. Spacelimbus

$e^{-x}=5$ does this look better? or maybe $e^{x}= \frac{1}{5}$

6. Romero

The natural log will cancel the e and bring down the x

7. Romero

You understand?

8. soccergal12

so the answer would be ln(1/5) ?

9. lgbasallote

haha so many answers

10. Romero

Yes that is the answer. You can simplify it however you want. I would just leave it as that.

11. soccergal12

okay thank you. that makes sense!

12. zzr0ck3r

notice that 1/5 = 5^(-1) and a property of logs is a*log(b) = log(b^a) so log(1/5) = log(5^(-1)) = -1*log(5) this is the great thing about logs, you can really "play around" with them. notice how i got to the same thing two compelety different ways.

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# Naming Excel Sheets automatically

After performing some calculations with excel sheets I end with a tridimensional matrix a to be export to a .xlsx format and a vector q that contains the name of all the sheets (so $Length[q] = Length[a]$). In the documentation the usual way to export this named excel data is to do something like

Export["file name.xlsx", q[[1]] -> a[[1]], ..., q[[n]] -> a[[n]]] ]

But since the $Length[a]]$ is variable I do not know an automatic way to do that. I tried something like

q -> a

But Mathematica can't process in this way.

How can I do that?

Thanks

-
You say the usual way to export this named and then the command says Import? Is this a typo? –  Nasser Jan 27 '14 at 22:38
Yes, thanks, @Nasser. –  Zhozer Jan 28 '14 at 1:18

a = {{{1}}, {{2}}};
q = {"one", "two"};
Export["c:\\test.xlsx", Rule @@@ Transpose@{q, a}]

-
Fantastic, thanks, @belisarius. I was trying something more complicate. I need to use more the rule method. ^^ –  Zhozer Jan 28 '14 at 22:34


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Thread: Computing Integrals View Single Post

## Computing Integrals

I was having trouble starting up this question.

Let
------| 2, if x<6
f(x) ={ 7, if x=6
------|-3, if x>6.

Compute (the integral from 3 to 11) f and prove that your right.
The problem i was having was choosing a partition for this integral.
For example, [3,x,y,11], what type of values could i use that are close enough to 6. Is 5 and 7 close enough? or something like +/- epsilon/2(this part is what confuses me, the epsilon)?
 PhysOrg.com science news on PhysOrg.com >> Leading 3-D printer firms to merge in $403M deal (Update)>> LA to give every student an iPad;$30M order>> CIA faulted for choosing Amazon over IBM on cloud contract


text:
Free Version
Easy

# Probability from a Graph

SATSTM-QZCETZ

If a pet is selected at random from the pet show, then according to this graph, what is the probability that the pet is a fish?

A

$\cfrac{5}{41}$

B

$\cfrac{5}{6}$

C

$\cfrac{5}{46}$

D

$\cfrac{1}{9}$

E

$\cfrac{5}{56}$


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# Why photons have no mass

Tags:
1. Jan 13, 2016

### Evenus1

hi I was wondering if some one could give me a simpler explanation as to why photons have zero rest mass and of any subject areas I should read up on to better understand this. can we please bear in mind I'm just in 15 so do not have a university grade knowledge. but this is an are of interest that I wish to explore farther.
many thanks
Ewen

2. Jan 13, 2016

### BvU

Hello Ewen,

Difficult to answer without bringing in some Physics (with a capital P). In particular: special relativity. Lots of introductions to that, so look around.
One of the 'results' from special relativity is the Einstein $E = mc^2$ formula. The $m$ in there is not the rest mass $m_0$, but $$m = \gamma m_0$$ with $$\gamma = {1\over \sqrt{1 - v^2/c^2}}$$ and so this breaks down for something that moves at speed $v = c$, unless $m_0 = 0$.

'A particle with non-zero rest mass that moves with the speed of light would swallow up all the energy in the unverse'

PS
simpler than what ?

3. Jan 13, 2016

### mathman

Why anything is almost impossible to answer. As others have answered, the rest mass of a photon is zero because special relativity forces it to be.

4. Jan 13, 2016

### ellipsis

'why' - You can't explain small things with big things, just like you can't explain addition in terms of multiplication (but you can vice-versa). 'Why' is one of those big things.

As far as figuring out facts, though: We recognize patterns (equations) and extrapolate them to the situations we can't directly observe. A lot of this stuff in modern physics is way too big or too small to observe directly - instead we work with things we can play with, figure out models for them, and see what those models output in certain other circumstances.

It turns out the only way you can avoid a division by zero is if the rest mass of a photon is zero. It's fun to think, that's how the math talks to us.

5. Jan 14, 2016

### vanhees71

That the photon is massless is an empirical fact. To be precise, the upper limit of the empirically determined photon mass is very small:

http://pdglive.lbl.gov/Particle.action?node=S000

6. Jan 14, 2016

### A. Neumaier

How can the photon mass be nonzero? It would imply that light moves with less than the speed of light, which sounds contradictory. (At least on the level of ''basic'' questions, such as #1.)

Last edited: Jan 14, 2016
7. Jan 14, 2016

### jerromyjon

Doesn't sit well with me either... would this be a consistent mass irrelevant of the energy of the photon or would a photon of higher energy have a higher but still negligible mass?

8. Jan 14, 2016

### ellipsis

By "the empirically established upper limit is very small" only means we've proven it is less than a certain size. This says nothing about our established empirical lower limit, which is zero. He is saying:

"Everything else is on the scale of 1, 2, 3. We have proven a photon's mass is less than 0.00000000000000000000001. Based on how the equations work out, we think the mass is in fact zero."

9. Jan 14, 2016

### jerromyjon

Well energy does have mass which is why I ask if there any proven theories relating the energy of a photon as a real massive effect or if the light speed "outruns" its gravitational effect.

When I think about it sensibly photons travel along side each other for billions of years without "attracting" each other, which I think pretty much rules out "mass" in the gravitational attraction sense.

Last edited: Jan 14, 2016
10. Jan 14, 2016

### sheaf

Some work has been done on these issues: https://en.wikipedia.org/wiki/Bonnor_beam
If the photons are travelling parallel but in opposite directions, they allegedly attract! Of course these calculations are presumably restricted to classical light, so the use of the term "photons" in this context needs some caveating!

11. Jan 14, 2016

### vanhees71

It just means that light moves with a speed less than the limiting speed of Minkowski space. There is no fundamental principle in the Standard model that forbids a non-zero photon mass. You even have an Abelian renormalizable gauge theory with a massive gauge boson without Higgs mechanism (Stückelberg model). Only for non-Abelian gauge theories you need the Higgs mechanism to have massive gauge bosons without violating unitarity and causality.

12. Jan 14, 2016

### Staff: Mentor

Electromagnetic radiation is a form of energy, so it does contribute to the stress energy tensor and have gravitational effects. For example, if I had a perfectly mirrored box such that light could bounce around in it forever, that box would weigh more and produce a stronger gravitational field when it was "full of light" then when it was not.

However, photons are not little objects that can "travel along side each other", so you can't just assign them a mass and a position and expect that they'll gravitate as if that mass was at that position. They don't.

13. Jan 14, 2016

### A. Neumaier

I think you meant ''Abelian renormalizable quantum field theory with a massive vector boson''. A theory without massless vector bosons is never a gauge theory.

14. Jan 14, 2016

### Staff: Mentor

And I think neither of you noticed the "B" tag on this thread :)
It's always good to remind people that the easy math-free answers are just a sad and pale shadow of the real thing... but OP is asking for a "simpler" explanation.

15. Jan 14, 2016

### dipstik

16. Jan 14, 2016

### BvU

In the spirit of Nugatory: I have a smart nephew -- same age (15) and sharp enough -- with more or less the same question and I don't know how to bring this without talking nonsense while still helping out ...

17. Jan 14, 2016

### andresB

The mass of the photon is zero because that is what experiment (and the whole way we analyze them) seems to tell us with a high degree of accuracy. IHMO, It is really hard to go beyond that. What can be said is that the current theories made around the experimental results work very well.

18. Jan 14, 2016

### HomogenousCow

There's more to it than that, you can't write down a massive Lagrangian with the properties you want for the electromagnetic interaction. (I think, could be wrong)

19. Jan 14, 2016

### jerromyjon

This was indicating that in parallel light beams the electric and magnetic forces cancel out, but in antiparallel beams there is an attractive element since the fields don't cancel.
I guess that does make simple sense to me that a photon coming towards you could have some very slight amount of mass, but one moving away from you is just "gone", never to be seen by your eyes or sensed as massive, unless it gets reflected! And then that is another whole "can of worms"...

20. Jan 14, 2016

### andresB

Yes, but the properties we want from the lagrangian are the ones that are ok with the experiments.

For example, at classical level, you can write a lagrangian with a combination of any power of the relativistic scalar and even powers of the relativistic pseudoscalars and that lagrangian will be relativistic invariant and gauge invariant (for example, Euler-Heisenber or the Born-infled ones). But for classical electrodynamics we choose a lagrangian linear in the scalar. The reasons is because that is the one that gives Maxwell's equation, and we want Maxwell's equation because that's the one that works .

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# SQLite.jl Documentation

## High-level interface

DBInterface.executeFunction
DBInterface.execute(db::SQLite.DB, sql::String, [params])
DBInterface.execute(stmt::SQLite.Stmt, [params])

Bind any positional (params as Vector or Tuple) or named (params as NamedTuple or Dict) parameters to an SQL statement, given by db and sql or as an already prepared statement stmt, execute the query and return an iterator of result rows.

Note that the returned result row iterator only supports a single-pass, forward-only iteration of the result rows. Calling SQLite.reset!(result) will re-execute the query and reset the iterator back to the beginning.

The resultset iterator supports the Tables.jl interface, so results can be collected in any Tables.jl-compatible sink, like DataFrame(results), CSV.write("results.csv", results), etc.

Passing strict=true to DBInterface.execute will cause the resultset iterator to return values of the exact type specified by SQLite.

source
SQLite.load!Function
source |> SQLite.load!(db::SQLite.DB, tablename::String; temp::Bool=false, ifnotexists::Bool=false, replace::Bool=false, on_conflict::Union{String, Nothing} = nothing, analyze::Bool=false)
SQLite.load!(source, db, tablename; temp=false, ifnotexists=false, replace::Bool=false, on_conflict::Union{String, Nothing} = nothing, analyze::Bool=false)

Load a Tables.jl input source into an SQLite table that will be named tablename (will be auto-generated if not specified).

• temp=true will create a temporary SQLite table that will be destroyed automatically when the database is closed
• ifnotexists=false will throw an error if tablename already exists in db
• on_conflict=nothing allows to specify an alternative constraint conflict resolution algorithm: "ABORT", "FAIL", "IGNORE", "REPLACE", or "ROLLBACK".
• replace=false controls whether an INSERT INTO ... statement is generated or a REPLACE INTO .... This keyword argument exists for backward compatibility, and is overridden if an algorithm is selected using the on_conflict keyword.
• analyze=true will execute ANALYZE at the end of the insert
source

## Types/Functions

SQLite.DBType
SQLite.DB() => in-memory SQLite database
SQLite.DB(file) => file-based SQLite database

Constructors for a representation of an sqlite database, either backed by an on-disk file or in-memory.

SQLite.DB requires the file string argument in the 2nd definition as the name of either a pre-defined SQLite database to be opened, or if the file doesn't exist, a database will be created. Note that only sqlite 3.x version files are supported.

The SQLite.DB object represents a single connection to an SQLite database. All other SQLite.jl functions take an SQLite.DB as the first argument as context.

To create an in-memory temporary database, call SQLite.DB().

The SQLite.DB will be automatically closed/shutdown when it goes out of scope (i.e. the end of the Julia session, end of a function call wherein it was created, etc.)

source
SQLite.StmtType
SQLite.Stmt(db, sql; register = true) => SQL.Stmt

Prepares an optimized internal representation of SQL statement in the context of the provided SQLite3 db and constructs the SQLite.Stmt Julia object that holds a reference to the prepared statement.

Note: the sql statement is not actually executed, but only compiled (mainly for usage where the same statement is executed multiple times with different parameters bound as values).

The SQLite.Stmt will be automatically closed/shutdown when it goes out of scope (i.e. the end of the Julia session, end of a function call wherein it was created, etc.). One can also call DBInterface.close!(stmt) to immediately close it.

The keyword argument register controls whether the created Stmt is registered in the provided SQLite3 database db. All registered and unclosed statements of a given DB connection are automatically closed when the DB is garbage collected or closed explicitly after calling close(db) or DBInterface.close!(db).

source
SQLite.bind!Function
SQLite.bind!(stmt::SQLite.Stmt, values)

bind values to parameters in a prepared SQLite.Stmt. Values can be:

• Vector or Tuple: where each element will be bound to an SQL parameter by index order
• Dict or NamedTuple; where values will be bound to named SQL parameters by the Dict/NamedTuple key

Additional methods exist for working individual SQL parameters:

• SQLite.bind!(stmt, name, val): bind a single value to a named SQL parameter
• SQLite.bind!(stmt, index, val): bind a single value to a SQL parameter by index number

From the SQLite documentation:

Usually, though, it is not useful to evaluate exactly the same SQL statement more than once. More often, one wants to evaluate similar statements. For example, you might want to evaluate an INSERT statement multiple times though with different values to insert. To accommodate this kind of flexibility, SQLite allows SQL statements to contain parameters which are "bound" to values prior to being evaluated. These values can later be changed and the same prepared statement can be evaluated a second time using the new values.

In SQLite, wherever it is valid to include a string literal, one can use a parameter in one of the following forms:

• ?
• ?NNN
• :AAA
• \$AAA
• @AAA

In the examples above, NNN is an integer value and AAA is an identifier. A parameter initially has a value of NULL. Prior to calling sqlite3_step() for the first time or immediately after sqlite3_reset(), the application can invoke one of thesqlite3bind()interfaces to attach values to the parameters. Each call tosqlite3bind() overrides prior bindings on the same parameter.

source
SQLite.createtable!Function
SQLite.createtable!(db::SQLite.DB, table_name, schema::Tables.Schema; temp=false, ifnotexists=true)

Create a table in db with name table_name, according to schema, which is a set of column names and types, constructed like Tables.Schema(names, types) where names can be a vector or tuple of String/Symbol column names, and types is a vector or tuple of sqlite-compatible types (Int, Float64, String, or unions of Missing).

If temp=true, the table will be created temporarily, which means it will be deleted when the db is closed. If ifnotexists=true, no error will be thrown if the table already exists.

source
SQLite.drop!Function
SQLite.drop!(db, table; ifexists::Bool=true)

drop the SQLite table table from the database db; ifexists=true will prevent an error being thrown if table doesn't exist

source
SQLite.dropindex!Function
SQLite.dropindex!(db, index; ifexists::Bool=true)

drop the SQLite index index from the database db; ifexists=true will not return an error if index doesn't exist

source
SQLite.createindex!Function
SQLite.createindex!(db, table, index, cols; unique=true, ifnotexists=false)

create the SQLite index index on the table table using cols, which may be a single column or vector of columns. unique specifies whether the index will be unique or not. ifnotexists=true will not throw an error if the index already exists

source
SQLite.removeduplicates!Function
SQLite.removeduplicates!(db, table, cols)

Removes duplicate rows from table based on the values in cols, which is an array of column names.

A convenience method for the common task of removing duplicate rows in a dataset according to some subset of columns that make up a "primary key".

source
SQLite.enable_load_extensionFunction
SQLite.enable_load_extension(db, enable::Bool=true)

Enables extension loading (off by default) on the sqlite database db. Pass false as the second argument to disable.

source
SQLite.registerFunction
SQLite.register(db, func)
SQLite.register(db, init, step_func, final_func; nargs=-1, name=string(step), isdeterm=true)

Register a scalar (first method) or aggregate (second method) function with a SQLite.DB.

source
SQLite.@registerMacro
SQLite.@register db function

User-facing macro for convenience in registering a simple function with no configurations needed

source
SQLite.@sr_strMacro
sr"..."

This string literal is used to escape all special characters in the string, useful for using regex in a query.

source
SQLite.transactionFunction
SQLite.transaction(db, mode="DEFERRED")
SQLite.transaction(func, db)

Begin a transaction in the specified mode, default = "DEFERRED".

If mode is one of "", "DEFERRED", "IMMEDIATE" or "EXCLUSIVE" then a transaction of that (or the default) mutable struct is started. Otherwise a savepoint is created whose name is mode converted to AbstractString.

In the second method, func is executed within a transaction (the transaction being committed upon successful execution)

source

## User Defined Functions

### SQLite Regular Expressions

SQLite provides syntax for calling the regexp function from inside WHERE clauses. Unfortunately, however, sqlite does not provide a default implementation of the regexp function. It can be easily added, however, by calling SQLite.@register db SQLite.regexp

The function can be called in the following ways (examples using the Chinook Database)

julia> using SQLite

julia> db = SQLite.DB("Chinook_Sqlite.sqlite")

julia> # using SQLite's in-built syntax

julia> DBInterface.execute(db, "SELECT FirstName, LastName FROM Employee WHERE LastName REGEXP 'e(?=a)'") |> DataFrame
1x2 ResultSet
| Row | "FirstName" | "LastName" |
|-----|-------------|------------|
| 1   | "Jane"      | "Peacock"  |

julia> # explicitly calling the regexp() function

julia> DBInterface.execute(db, "SELECT * FROM Genre WHERE regexp('e[trs]', Name)") |> DataFrame
6x2 ResultSet
| Row | "GenreId" | "Name"               |
|-----|-----------|----------------------|
| 1   | 3         | "Metal"              |
| 2   | 4         | "Alternative & Punk" |
| 3   | 6         | "Blues"              |
| 4   | 13        | "Heavy Metal"        |
| 5   | 23        | "Alternative"        |
| 6   | 25        | "Opera"              |

julia> # you can even do strange things like this if you really want

julia> DBInterface.execute(db, "SELECT * FROM Genre ORDER BY GenreId LIMIT 2") |> DataFrame
2x2 ResultSet
| Row | "GenreId" | "Name" |
|-----|-----------|--------|
| 1   | 1         | "Rock" |
| 2   | 2         | "Jazz" |

julia> DBInterface.execute(db, "INSERT INTO Genre VALUES (regexp('^word', 'this is a string'), 'My Genre')") |> DataFrame
1x1 ResultSet
| Row | "Rows Affected" |
|-----|-----------------|
| 1   | 0               |

julia> DBInterface.execute(db, "SELECT * FROM Genre ORDER BY GenreId LIMIT 2") |> DataFrame
2x2 ResultSet
| Row | "GenreId" | "Name"     |
|-----|-----------|------------|
| 1   | 0         | "My Genre" |
| 2   | 1         | "Rock"     |

Due to the heavy use of escape characters, you may run into problems where julia parses out some backslashes in your query, for example "\y" simply becomes "y". For example, the following two queries are identical:

julia> DBInterface.execute(db, "SELECT * FROM MediaType WHERE Name REGEXP '-\d'") |> DataFrame
1x1 ResultSet
| Row | "Rows Affected" |
|-----|-----------------|
| 1   | 0               |

julia> DBInterface.execute(db, "SELECT * FROM MediaType WHERE Name REGEXP '-d'") |> DataFrame
1x1 ResultSet
| Row | "Rows Affected" |
|-----|-----------------|
| 1   | 0               |

This can be avoided in two ways. You can either escape each backslash yourself or you can use the sr"..." string literal that SQLite.jl exports. The previous query can then successfully be run like so:

julia> # manually escaping backslashes

julia> DBInterface.execute(db, "SELECT * FROM MediaType WHERE Name REGEXP '-\\d'") |> DataFrame
1x2 ResultSet
| Row | "MediaTypeId" | "Name"                        |
|-----|---------------|-------------------------------|
| 1   | 3             | "Protected MPEG-4 video file" |

julia> # using sr"..."

julia> DBInterface.execute(db, sr"SELECT * FROM MediaType WHERE Name REGEXP '-\d'") |> DataFrame
1x2 ResultSet
| Row | "MediaTypeId" | "Name"                        |
|-----|---------------|-------------------------------|
| 1   | 3             | "Protected MPEG-4 video file" |

The sr"..." currently escapes all special characters in a string but it may be changed in the future to escape only characters which are part of a regex.

### Custom Scalar Functions

SQLite.jl also provides a way that you can implement your own Scalar Functions. This is done using the SQLite.register function and macro.

SQLite.@register takes a SQLite.DB and a function. The function can be in block syntax:

julia> SQLite.@register db function add3(x)
x + 3
end

inline function syntax:

julia> SQLite.@register db mult3(x) = 3 * x

and previously defined functions:

julia> SQLite.@register db sin

The SQLite.register function takes optional arguments; nargs which defaults to -1, name which defaults to the name of the function, isdeterm which defaults to true. In practice these rarely need to be used.

The SQLite.register function uses the SQLite.sqlreturn function to return your function's return value to SQLite. By default, sqlreturn maps the returned value to a native SQLite type or, failing that, serializes the julia value and stores it as a BLOB. To change this behaviour simply define a new method for sqlreturn which then calls a previously defined method for sqlreturn. Methods which map to native SQLite types are

sqlreturn(context, ::NullType)
sqlreturn(context, val::Int32)
sqlreturn(context, val::Int64)
sqlreturn(context, val::Float64)
sqlreturn(context, val::UTF16String)
sqlreturn(context, val::String)
sqlreturn(context, val::Any)

As an example, say you would like BigInts to be stored as TEXT rather than a BLOB. You would simply need to define the following method:

sqlreturn(context, val::BigInt) = sqlreturn(context, string(val))

Another example is the SQLite.sqlreturn used by the regexp function. For regexp to work correctly, it must return it must return an Int (more specifically a 0 or 1) but occursin (used by regexp) returns a Bool. For this reason the following method was defined:

sqlreturn(context, val::Bool) = sqlreturn(context, int(val))

Any new method defined for sqlreturn must take two arguments and must pass the first argument straight through as the first argument.

### Custom Aggregate Functions

Using the SQLite.register function, you can also define your own aggregate functions with largely the same semantics.

The SQLite.register function for aggregates must take a SQLite.DB, an initial value, a step function and a final function. The first argument to the step function will be the return value of the previous function (or the initial value if it is the first iteration). The final function must take a single argument which will be the return value of the last step function.

julia> dsum(prev, cur) = prev + cur

julia> dsum(prev) = 2 * prev

julia> SQLite.register(db, 0, dsum, dsum)

If no name is given, the name of the first (step) function is used (in this case "dsum"). You can also use lambdas; the following does the same as the previous code snippet

julia> SQLite.register(db, 0, (p,c) -> p+c, p -> 2p, name="dsum")`


text:
# Integrate AFTER differentiate exponent to power exponent

#### sjaffry

Hi all,

I need to differentiate the following

f(t) = \frac{\partial }{\partial t}e^{-ne^{-\lambda t}}

$$\int_{0}^{+Inf} t dt = ?$$

#### hollywood

To take the derivative in the first equation, use the chain rule:

$$\displaystyle \frac{\partial}{\partial t}e^{-ne^{-\lambda t}}=\frac{\partial}{\partial t}e^{-n(e^{-\lambda t})}=-ne^{-ne^{-\lambda t}}\frac{\partial}{\partial t}e^{-\lambda t}=n\lambda e^{-ne^{-\lambda{t}}}e^{-\lambda t}$$

The integral diverges to infinity:

$$\displaystyle \int_0^\infty t\,dt=\lim_{s\to\infty}\int_0^s t\,dt=\lim_{s\to\infty}\frac{1}{2}s^2=\infty$$

- Hollywood


text:
## Cryptology ePrint Archive: Report 2020/811

Groth16 SNARKs are Randomizable and (Weakly) Simulation Extractable

Markulf Kohlweiss and Mikhail Volkhov

Abstract: Due to its simplicity, succinctness, and performance, Groth16 is currently the most widely deployed succinct (zero-knowledge) argument of knowledge (SNARK) system. Groth16 is known to be perfectly zero-knowledge and knowledge sound in the generic (and algebraic) group model. However, the existing security arguments for Groth16 are silent about the soundness of the proof system in the presence of simulated proofs --- a common requirement for both the composable security and game-hopping style security analysis of protocols built using such argument systems. This important gap let to a line of work on simulation-extractable, also called simulation knowledge sound, succinct proof systems. Groth16 itself cannot satisfy the strongest notion of simulation-extractability that implies proof non-malleability --- in fact, proofs are perfectly randomizable. Surprisingly, in this short note we show that Groth16 does satisfy a weaker notion of simulation-extractability implying statement non-malleability. This property is often sufficient for typical applications that motivate the use of strong simulation-extractability. Notably, one can obtain UC security using efficient compilers.

Category / Keywords: cryptographic protocols / zero knowledge, NIZK, zk-SNARK, simulation extractability, QAP, algebraic group model

Date: received 29 Jun 2020, last revised 1 Jul 2020

Contact author: mikhail volkhov at ed ac uk

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2020/811

[ Cryptology ePrint archive ]


text:
# Follow up: GKL similarity and social choice

The previous post discusses a “distance minimizing” way of picking a compromise between agents with a diverse set of utilities. If you measure distance (better: divergence) between utility functions by square Euclidean distance, then a utilitarian compromise pops out.

I wanted now to discuss briefly a related set of results (I’m grateful for pointers and discussion with Richard Pettigrew here, though he’s not to blame for any goofs I make along the way). The basic idea here is to use a different distance/divergence measure between utilities, and look at what happens. One way to regard what follows is as a serious contender (or serious contenders) for measuring similarity of utilities. But another way of looking at this is as an illustration that the choice of similarity I made really has significant effects.

I borrowed the square Euclidean distance analysis of similarity from philosophical discussions of similarity of belief states. And the rival I now discuss is also prominent in that literature (and is all over the place in information theory). It is (generalized) Kullback-Leibler relative entropy (GKL), and it gets defined, on a pair of real valued vectors U,V in this way:

$D_{KL}(U,V):=\sum_{p\in P} U(p)\log \frac{U(p)}{V(p)} - U(p)+V(p)$

Note that when the vectors are each normalized to the same quantity, the sum of U(p) over all p is equal to the sum of V(p) over all p, and so two latter summands cancel. In the more general case, they won’t. Kullback-Leibler relative entropy is usually applied with U and V being probability functions, which are normalized, so you normally find it in the form where it is a weighted sum of logs. Notoriously, GKL is not symmetric: the distance from U to V can be different from the distance to U from V. This matters; more anon.

(One reason I’m a little hesitant with using this as a measure of similarity between utilities in this context is the following. When we’re using it to measure similarity between beliefs or probability functions, there’s a natural interpretation of it as the expectation from U’s perspective of difference between the log of U and the log of V. But when comparing utilities rather than probabilities means we can’t read the formula this way. It feels to me a bit more of a formalistic enterprise for that reason. Another thing to note is that taking logs is well defined only when the relevant utilities are positive, which again deserves some scrutiny. Nevertheless….)

What happens when we take GKL as a distance (divergence) measure, and then have a compromise between a set of utilities by minimizing total sum distance from the compromise point to the input utilities? This article by Pettigrew gives us the formal results that speak to the question. The key result is that the compromise utility $U_C$ that emerges from a set of m utility functions $U_i$ is the geometrical mean:

$U_C(p)= (\prod_{i\in A} U_i(p))^{\frac{1}{m}}$.

Where the utilitarian compromise utilities arising from squared euclidean distance similarity look to the sum of individual utilities, this compromise looks at the product of individual utilities. It’s what’s called in the social choice literature a symmetrical “Nash social welfare function” (that’s because it can be viewed as a special case of a solution to a bargaining game that Nash characterized: the case where the “threat” or “status quo” point is zero utility for all). It has some interesting and prima facie attractive features—it prioritizes the worse off, in that a fixed increment of utility will maximize the product of everyone’s utilities if awarded to someone who has ex ante lowest utility. It’s also got an egalitarian flavour, in that you maximize the product of a population’s utilities by dividing up total utility evenly among the population (contrast utilitarianism, where you can distribute utility in any old way among a population and get the same overall sum, and so any egalitarian features of the distribution of goods have to rely on claims about diminishing marginal utility of those goods; which by the same token leaves us open to “utility monsters” in cases where goods have increasing utility for one member of the population). Indeed, as far as I can tell, it’s a form of prioritarianism, in that it ranks outcomes by way of a sum of utilities which are discounted by the application of a concave function (you preserve the ranking of outcomes if you transform the compromise utility function by a monotone increasing function, and in this case we can first raise it to the mth power, and then take logs, and the result will be the sum of log utilities. And since log is itself a concave function this meets the criteria for prioritarianism). Anyway, the point here is not to evaluate Nash social welfare, but to derive it.

The formal result is proved in the Pettigrew paper, as a corollary to a very general theorem. Under the current interpretation that theorem also has the link between squared Euclidean distance and utilitarianism of the previous post as another special case. However, it might be helpful to see how the result falls out of elementary minimization (it was helpful for me to work through it, anyway, so I’m going to inflict it on you). So we start with the following characterization, where A is the set of agents whose utilities we are given:

$U_C=\textsc{argmin}_X \sum_{i\in A} D_{KL}(X,U_i)$

To find this we need to find X which makes this sum minimal (P being the set of n propositions over which utilities are defined, and A being the set of m agents):

$\sum_{i\in A} \sum_{p\in P} X(p)\log \frac{X(p)}{U_i(p)} - X(p)+U_i(p)$

Rearrange as a sum over p:

$\sum_{p\in P} \sum_{i\in A} X(p)\log \frac{X(p)}{U_i(p)} - X(p)+U_i(p)$

Since we can assign each X(p) independently of the others, we minimize this sum by minimizing each summand. Fixing p, and writing $x:=X(p)$ and $u_i:=U_i(p)$, our task now is to find the value of u which minimizes the following:

$\sum_{i\in A} x\log \frac{x}{u_i} - x+u_i$

We do this by differentiating and setting the result to zero. The result of differentiating (once you remember the product rule and that differentiating logs gives you a reciprocal) is:

$\sum_{i\in A} \log \frac{x}{u_i}$

But a sum of logs is the log of the product, and so the condition for minimization is:

$0=\log \frac{x^m}{\prod_{i\in A}u_i}$

Taking exponentials we get:

$1=\frac{x^m}{\prod_{i\in A}u_i}$

That is:

$x^m=\prod_{i\in A}u_i$

Unwinding the definitions of the constants and variables gets us the geometrical mean/Nash social welfare function as promised.

So that’s really neat! But there’s another question to ask here (also answered in the Pettigrew paper). What happens if we minimize sum total distance, not from the compromise utility to each of the components, but from the components to the compromise? Since GKL distance/divergence is not symmetric, this could give us something different. So let’s try it. We swap the positions of the constant and variables in the sums above, and the task becomes to minimize the following:

$\sum_{i\in A} u_i\log \frac{x}{u_i} - u_i+x$

When we come to minimize this by differentiating, we no longer have a product of functions in x to differentiate with respect to x. That makes the job easier, and ends up with us with the constraint:

$\sum_{i\in A} 1-\frac{u_i}{x}$

Rearranging we get:

$x= \frac{1}{n} \sum_{i\in A} u_i$

and we’re back to the utilitarian compromise proposal again! (That is, this distance-minimizing compromise delivers the arithmetical mean rather than the geometrical mean of the components).

Stepping back: what we’ve seen is that if you want to do distance-minimization (similarity-maximization, minimal-mutilation) compromise on cardinal utilities then the precise way distance you choose really matters. Go for squared euclidean distance and you get utilitarianism dropping out. Go for the log distance of the GKL, and you get either utilitarianism or the Nash social welfare rule dropping out, depending on the “direction” in which you calculate the distances. These results are the direct analogues of results that Pettigrew gives for belief-pooling. If we assume that the way of measuring similarity/distance for beliefs and utilities should be the same (as I did at the start of this series of posts) then we may get traction on social welfare functions through studying what is reasonable in the belief pooling setting (or indeed, vice versa).