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# Character table does not determine group Vs Tannaka duality

From the example $D_4$, $Q$, we see that the character table of a group doesn't determine the group up to isomorphism. On the other hand, Tannaka duality says that a group $G$ is determined by its representation ring $R(G)$.

What is the additional information contained in $R(G)$ as opposed to the character table?

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You might be interested in this related question: mathoverflow.net/questions/500/… – Steven Sam Jan 10 '10 at 10:31
The representation ring is determined by the character table, since direct sum of representations is pointwise sum of characters and tensor product of representations is pointwise multiplication of characters. – Qiaochu Yuan Jan 10 '10 at 11:21

If I'm not mistaken, the extra information is not contained in the representation ring, you have to look at the category of representations. In particular, you want to look at the representation category equipped with its forgetful functor to vector spaces. Then the group can be recovered as the automorphisms of this functor. Here's a blogpost I wrote which may be helpful: http://concretenonsense.wordpress.com/2009/05/16/tannaka%E2%80%93krein-duality/

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Let me try and give as low-level an explanation as I can, in case you're scared of all this "monoidal category" stuff.

A complex representation of a finite group $G$ is just a module for the group ring $R=\mathbf{C}[G]$. The two groups you mention with the same character table have isomorphic group rings as well (by some abstract structure theorem for simple $R$-modules), so, if you fix an isomorphism between the group rings you have a way of moving from a representation of one to a representation of the other! Let's call this way of moving between representations of the groups "the trick". The representation theory of both groups is very close because we can use the trick.

However the way to spot the difference in the representation theory of the two groups is to use the tensor product. If $M$ and $N$ are representations of $G$, then one can give $M\otimes_\mathbf{C} N$ the structure of a representation of $G$, but this construction must use "more than the abstract $R$-module structure of $M$ and $N$". What I mean by that is this: if $R$ is a random non-commutative $\mathbf{C}$-algebra and $M$ and $N$ are two $R$-modules, there's no natural way (that I know of) to give $M\otimes_\mathbf{C}N$ an $R$-module structure. The way this works with $R=\mathbf{C}[G]$ is that you let $\mathbf{C}$ act in the obvious way (i.e. on one of the factors) and let $G$ act diagonally (i.e. on both of the factors at once). This is a big asymmetry and we're using the fact that $R=\mathbf{C}[G]$ (i.e. that $G$ gives a basis of $R$) rather than just its abstract ring structure.

Because the groups you mention have the same character tables, it's true that if $M$ and $N$ are modules for one, then using the trick they're modules for the other. But the shocker is this: the trick commutes with tensor products! (at least up to isomorphism). Because the character of the tensor product is the product of the characters! So applying the trick to $M\otimes N$ gives a module that happens to be isomorphic to the tensor product of the tricks applied to $M$ and $N$. So now the representation theory of the groups is looking really close.

But the trick is not canonical, it involves a random choice of isomorphism between the two representation rings, and so when you actually start drawing bunches of natural isomorphisms that Tannakian categories are supposed to satisfy, they don't commute with the trick, because the "(at least up to isomorphism)" remark comes back to haunt you. This is my understanding of the explicit point where the subtlety is buried. I should say however that I figured all this out myself once and I might be wrong. If I'm wrong I will no doubt shortly be put right!

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Thanks to both of you! – Timo Keller Jan 10 '10 at 11:26
Great answer. Do you know of any situation where "the trick" is used to prove sth. about a representation of a gp $G$ through a corresponding property of a representation of another group? – Maxime Bourrigan Feb 5 at 17:21

It might be worthwhile to supplement Kevin Buzzard's answer by describing exactly what goes wrong in the case of $Q$ and $D_4$. Let $V$ and $W$ be the two dimensional irreducible representations of $Q$ and $D_4$ respectively.

The representation $V \otimes V$ is isomorphic to $1 \oplus \chi_1 \oplus \chi_2 \oplus \chi_3$, where the $\chi_i$ are the nontrivial (one-dimensional) characters of $Q$. Similarly, $W \otimes W$ is isomorphic to $1 \oplus \eta_1 \oplus \eta_2 \oplus \eta_3$ where the $\eta_i$ are the nontrivial (one-dimensional) characters of $D_4$. The reason these formulas look the same is that the representation rings of $Q$ and $D_4$ are isomorphic.

However, we cannot build this isomorphism out of maps of vector spaces which commute with tensor product. Specifically, we cannot find an isomorphism $\alpha : V \to W$ of vector spaces so that the induced map $\alpha \otimes \alpha : V \otimes V \to W \otimes W$ carries $1$ to $1$ and $\chi_i$ to $\eta_i$. (Of course, I haven't told you how to label the $\chi_i$ and $\eta_i$. My statement is that there is no choice of labeling for which you can do this.)

This is very easy to see. The map $\alpha \otimes \alpha$ will carry the anti-symmetric elements of $V \otimes V$ to the anti-symmetric elements of $W \otimes W$. But $\wedge^2 V$ is the trivial representation, and $\wedge^2 W$ is not! So isomorphisms of the form $\alpha \otimes \alpha$ can't carry $1$ to $1$.

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Interesting. Out of curiosity, is the structure of R(G) as a lambda-ring enough to distinguish finite groups? – Qiaochu Yuan Jan 10 '10 at 16:38
I was wondering that myself! I don't know. – David Speyer Jan 10 '10 at 16:54
Thanks David. I knew from the Tannakian formalism that "something like this had to be true", but seeing the explicit example is really great. I feel like I should steal from your answer and Theo's answer to make my answer "even better" with edits, but on the other hand I could just save myself a job and do nothing because your answers won't go away. I think this is great: it seems to me that we now have completely concrete illustrations of how the categories are similar and how they differ in this case. – Kevin Buzzard Jan 10 '10 at 20:58
Qiaochu's question about Lambda rings is answered below by Mariano; the answer is no. – David Speyer Jan 10 '10 at 21:46
The Lambda ring structure is enough to show that R(D_4) is different from R(Q) via Adams operations built from the Lambda ring structure. – Paul Pearson Feb 20 '13 at 2:02

Regarding a question which arose in comments to another answer: The lambda ring structure is on $RG$ is not enough to reconstruct the group. Dade has given examples (MathSciNet review here; paper here) of pairs of groups which have the same character table with the same power maps, and from this it follows that the whole lambda ring structure is the same.

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Nice! I don't think I would have found that. I've added the link for you. – David Speyer Jan 10 '10 at 21:44
Here's a link to the paper: dx.doi.org/10.1016/0021-8693(64)90002-X – Steven Sam Jan 10 '10 at 22:10
Thanks Steven! I've edited that in the main answer. – David Speyer Jan 11 '10 at 16:46
In the positive direction, I recently encountered a result of Hoehnke and Johnson, arxiv.org/abs/math/9210219 : A group is determined by the collection of functions $G^k \to \mathbb{C}$ for $k=1$, $2$ and $3$ given by $g \mapsto \chi(g)$, $(g,h) \mapsto \chi(g)\chi(h)- \chi(gh)$ and $(f,g,h) \mapsto \chi(f) \chi(g) \chi(h) - \chi(fg) \chi(h) - \chi(gh) \chi(f) - \chi(h) \chi(fg) + \chi(fgh) + \chi(fhg)$ for all characters $\chi$. – David Speyer Dec 20 '14 at 16:58

Edit: Somehow I totally misread the question. I talked about the group algebra $\mathbb C[G]$, which is not at all the same as the character ring $R(G)$. Over $\mathbb C$ (or any other field of characteristic 0), $R(G)$ is naturally a subalgebra of $(\mathbb C[G])^*$, which is the algebra of functions on $G$ with pointwise multiplication, and now the comultiplication encodes the group structure. On the other hand, it is not a subbialgebra: the coproduct of a class function is not a class function.

Anyway, original post below, with the obviously wrong things struck out. So it's really an answer to Kevin, rather than anything else.

Well, it depends on what you mean by "$R(G)$". I won't address TK duality, and most of what I'll say is essentially a follow-up to Kevin's answer, rather than an answer in its own right. Also, I'm only going to address finite groups and their finite-dimensional representations. Also, for me the word "ring" means (associative, unital, noncommutative) "$\mathbb C$-algebra".

Recall that a complex representation of $G$ is the same as an algebra representation of $\mathbb C[G]$. Let $R$ be a ring. As Kevin says, it's in general impossible to define an $R$-module structure on $M\otimes N$ when $M,N$ are $R$-modules. (When $R$ is abelian, which is not the case here, one can define a tensor product $M \otimes_R N$, but that's not the tensor product of representations anyway.) What would a tensor product of modules require? It would require a rule that assigns to each $r\in R$ and each pair $M,N$ of $R$-modules an endomorphism of $M\otimes N$, of course, and we should impose all sorts of axioms that force the tensor product to be well-behaved. Among other things, it's much easier if the endomorphism is an element of the tensor product $\text{End}(M) \otimes \text{End}(N) \subseteq \text{End}(M\otimes N)$. And we already have some distinguished elements of $\text{End}(M)$ and $\text{End}(N)$, namely the action of $R$.

So one way to try to construct a well-behaved tensor product on the category of $R$-modules is to find a nice map $\Delta: R \to R\otimes R$. Then the axioms for this map that assure that the tensor product is good are that $\Delta$ be an algebra homomorphism, and that it be "coassociative": $(\text{id}\otimes \Delta)\circ \Delta = (\Delta \otimes \text{id})\circ \Delta)$. Let's suppose that there's also a distinguished "trivial" representation $\epsilon: R \to \text{End}(\mathbb C) = \mathbb C$; if this is to be the monoidal unit, then we'd need $(\text{id}\otimes \epsilon) \circ \Delta = \text{id} = (\epsilon \otimes \text{id})\circ \Delta$. The maps $\Delta, \epsilon$ satisfying these axioms define on $R$ the structure of a bialgebra.

By the way, the map is called "$\Delta$" because if $G$ is a group (or monoid) and $R = \mathbb C[G]$, then the map $R \to R\otimes R$ given on the basis $G$ by the diagonal map $\Delta: g \mapsto g\otimes g$ is such a structure.

Then here's a cool fact. Define an element $r\in R$ to be grouplike if $\Delta(r) = r\otimes r$. Then the grouplike elements are a multiplicative submonoid of $R$. And when $R = C[G]$, the grouplike elements are precisely $G$.

So my answer to your question is that "the additional information contained in $R(G)$ as opposed to the character table" is its bialgebra structure.

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Thanks Theo, that's a great comment. You've elucidated a point which I only had a much vaguer appreciation of when I wrote my answer. – Kevin Buzzard Jan 10 '10 at 20:54
The representation ring $R(G)$ from the question is not the group algebra, as far as I can see, and it is the latter which is a bialgebra. – Mariano Suárez-Alvarez Jan 11 '10 at 6:44
oops! I wasn't thinking. – Theo Johnson-Freyd Jan 11 '10 at 16:23
An important point to make here is that the tensor product structure on R(G) comes from the comultiplication on C[G], which means you then have to ask yourself: where did the multiplication go? – Qiaochu Yuan Jan 11 '10 at 16:42

It might be worth explaining why you shouldn't expect $R(G)$ to tell you everything about a group. $R(G)$ is naturally isomorphic to the ring of class functions $G \to \mathbb{C}$ (the functions constant on conjugacy classes) under pointwise addition and multiplication, and as such the information it contains is precisely the multiset of sizes of each the number of conjugacy classes of $G$. That's it! No other information. (Note that $D_4$ and $Q$ both have conjugacy classes of sizes $1, 1, 2, 2, 2$ five conjugacy classes.)

In other words, the abstract structure of the representation ring actually gives you less information than the character table; the character table at least hands you a distinguished basis of $R(G)$. Without this basis, $R(G)$ can't even tell you what tensor products of representations look like.

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How do you get the class sizes from the abstract ring $R(G)\subseteq\mathbb C^G$? It is a semisimple commmutative $\mathbb C$-algebra, so its only invariant is the dimension. – Mariano Suárez-Alvarez Jan 11 '10 at 16:59
Whoops. I seem to have been secretly assuming that R(G) came with the dual basis on Hom(G, C), which is obviously wrong. Thanks! – Qiaochu Yuan Jan 11 '10 at 17:19
That's what R(G) \otimes C can tell you. If you don't tensor with C, you can get a little more. For example, R(Z/2 x Z/2) is not isomorphic to R(Z/4) because, after tensoring with a field k of characteristic 2, the former becomes k[u,v]/<u^2, v^2> and the latter becomes k[u]/u^4. But, morally, I agree with this answer. – David Speyer Jan 11 '10 at 17:37

An RWTH Aachen thesis by Helena Skrzipczyk [1992] is referred to by Eick and J. Mu"ller J. ALg 304 (2006) On Brauer pairs. She gives several examples of non-isomorphic groups with bijections between the character tables and power maps. They are the smallest order Brauer pairs assuming such to be p-groups.

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## Right Angle Example

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Introduce the other angles by referring to and comparing them to the right angle. 02mN-399860. The latest example is a piece of writing in Vanity Fair that slams Melania Trump even for things she hasn't done — and even for her good work. A Right angle measures exactly 90 degrees. The angle CAT is a right angle. The Low Angle. You can either play on your own or challenge a friend. The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse. Writing paper, text books, newspaper, magazines. Community See All. Other examples include the angle at the base of a capital L or the top of a capital T. 4 ultra-wide camera and a 26mm f/1. The following figure shows you an example. We show a right triangle below. When an equilateral triangle. An example of a straight. Note: The lines that surround the letter C are the formal way of denoting a right angle. The name hypotenuse is given to the longest edge in a right-angled triangle. NIH Sample Documents The National Institutes of Health (NIH): NIAID has recently provided some resources for investigators. Right Triangle Trigonometry and SOHCAHTOA 1 This entry was posted on September 4, 2016 by Todd Helmenstine (updated on September 6, 2016 ) Sohcahtoa isn’t actually an Egyptian god, but if it helps to remember him that way, you’ll have an easier time recalling right angle trig relationships. Using the ratios that come from the right triangle, and understanding the application of the unit circle, you can solve a wide variety of problems involving angles and lengths. label the three sides: the hypotenuse. WOW,This whole thing is made of right traingles. Organized by task and by each textbox within a task, each example includes a stronger and weaker written response that you can use to compare against the rubric and your own work in order to. Third graders can also learn to distinguish right angles from other. In spherical geometry, there can be as many as three right angles, such as on the surface of a sphere, e. Examples of obtuse angles are 96°, 110°, 138°, 175°, 189° Straight line. We therefore say that BC is the opposite side to angle A and AB is the adjacent side to angle A. If the radius is a length of 1, then that means that every Reference Triangle that we create has a hypotenuse of 1, which makes it so much easier to compare one angle to another. The first is to use a support e. Find the lengths of the sides and hypotenuse of the triangle. In this tutorial, you'll see how to find the tangent of a particular angle in a right triangle. Circular hole becomes ellipse Distorted width Accurately presents object’s details, i. Notice how the tips of each slice make a corner that is a right angle. Angles having magnitudes 220°, 250°, 310° are all reflex angles. Thus the complement of 60° is 30°. For example, the craftsmen measures a distance of four feet along an established base line from a given point and then measures a distance of three feet along a reference line extending from said point at approximately a right angle. To understand how ω and φ act together to cause the shift in time, think of sin ( ) as a function machine. A ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. right angle n. So we have to pay attention to this arc to make sure that the tool is looking at the same angle that we are. According to the Pythagorean theorem,. the angle between two radii of a circle that. It can be shown, for example, that for masonry having a density of 3, water being 1, the figure of minimum section is a right-angled triangle, with the water against its vertical face; while for a greater density the water face must lean towards the water, and for a less density away from the water, so that the water may lie upon it. Also learn the facts to easily understand math glossary with fun math worksheet online at Splash Math. Interactive, animated maths dictionary for kids with over 600 common math terms explained in simple language. Other examples include the angle at the base of a capital L or the top of a capital T. 3 SOLID ANGLE 4 units of solid angle are called steradians, abbreviated sr. The first one is done. Prove that the length of the bisector (of the right angle) is equal to \frac{a\cdot b\cdot \sqrt{2}}{a+b Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. A very simple introduction into right angles. Approaches To Find Random events; What Everybody Dislikes About Dealer’s strategy And Why; Describing Uncomplicated Russian Brides “Professor” will su. Right triangles. The base could be on the equator, from the 0 meridian to the 90th meridian. Template library PoweredTemplate. A right angle is an internal angle which is equal to 90 degrees. You may have noticed that the side opposite the right angle is always the triangle's longest side. I then provide each student with one the index cards used in the previous activity. right angle definition: The definition of a right angle is the meeting point between two perpendicular straight lines. We can use: tan / B = opp/adj = b/40 tan 65 = b/40 40 tan 65 = b 85. Example: A searchlight located 200 meters from a weather office is turned on. Example shows that if the object is above the level of the observer, then the angle between the horizontal and the observer's line of sight is the angle of elevation. If we know one of these angles, we can easily substitute that value and find the missing one. The relation between the sides and angles of a right triangle is the basis for trigonometry. It is used, for example, in gradients and in some transform functions. Alcalde; Impresos; Equipo; Contacto. Non-example: right trapezoid with only two right angles instead of four Rhombus Definition: a quadrilateral with both pairs of opposite sides parallel and congruent (parallelogram) with four congruent sides and opposite angles congruent. Right Angles. For example, the craftsmen measures a distance of four feet along an established base line from a given point and then measures a distance of three feet along a reference line extending from said point at approximately a right angle. This demo shows you the right triangle trig function calculations for several angles, explanations below. Based on the measurement, the angles can be broadly classified as acute, right, obtuse and reflex. angle; the hypotenuse is always the longest side of the right triangle. Given the right triangle ABC with height h (CD) to the hypotenuse, h = ( x y ), whereas a = ( c x ), and b = ( c y ). An acute angle has a measure, or it's smaller, than a right angle. If the opposite vertex is (3, 2), then find the equations to its sides. Right angle detectors were issued and the children were shown how to use them to find right angles. 90 degree angles are inefficient and so are rare in nature. The tangent ratio is just one of these ratios. Example: A child is photographed while they sit scared in a corner. Find the length of segment BD. Example: The angle ABC measures 40 degrees. We can use: tan / B = opp/adj = b/40 tan 65 = b/40 40 tan 65 = b 85. The other two sides of the triangle, AC and CB are referred to as the 'legs'. Also, in this example, the angles are supplementary, which mean that each of the two angles combined equals 180 degrees (one of those straight lines that intersected to form the four angles). There are four important types of angle: acute, right, obtuse, and reflex. EXAMPLE: Setting out a right angle Step 1. Under this configuration, a deviation of 90° still occurs, but without changing parity. In this quick tutorial I will let you know how to create right angel in PHP. If the radius is a length of 1, then that means that every Reference Triangle that we create has a hypotenuse of 1, which makes it so much easier to compare one angle to another. Now place the protractor on the horizontal line. 2 yard line. Vectors and Angles. An angle measuring between 0° and 90° is called an acute angle. We know ∠ACB is an inscribed angle intercepting arc AB, so finding m∠ACB shouldn't be a problem. (i) Find the value of ∠c and ∠a if ∠b is 88°. Angle C is always 90 degrees; angle 3 is either angle B or angle A, whichever is NOT entered. Using an example, point out the right angle and tell students that this square tells you that the type of angle is a right angle and the number tells you the angle size (90). an angle whose sides lie in opposite directions from the vertex in the same straight line and which equals two right angles…. I personally prefer the second method. ⭐⭐⭐⭐⭐ If you seeking special discount you'll need to searching when special time come or holidays. Example: The angle ABC measures 40 degrees. A Straight angle measures exactly 180 degrees. Right angles are one of the most common and interesting angles in all of geometry. Solution: We are given the value of one of the angles, so we can find the value of the other. The other two sides of the triangle, AC and CB are referred to as the 'legs'. Angle of Elevation: Angle of Depression: Angle of elevation is measured when the object is viewed upward. One of its angles is 45°. Example 1: Figure 1 shows a triangle with angles of different measures. It includes a fun video. What is the Resultant Force and How to Find it (with Examples) In this article, you will learn what the resultant force (also known as net force) is, and how to find it when an object is subject to parallel forces as well as non-parallel forces with the help of examples. • Hold your arms so they form a right angle and are parallel to each other. These type question use to judge your logical ability in code. (noun) An example of a right angle is a 90º angle. Measure 90˚ and mark it with a point. 1) 13 12 B A C θ 2) 4 13 A B C θ 3) 9 6 A B C θ 4) 11. Right Triangle: A three-sided polygon with one right angle is called right triangle. To understand how the most popular prisms work and how each can best be used in light reflection and refraction applications, consider right angle prisms, roof prisms, and combination prisms. • Theorem #4: In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. Straight angle Right angle Acute angle Obtuse angle A B C A B C A B C B A C Types Of Angles Right angle: An angle whose measure is 90 degrees. Below is the proof that two triangles are congruent by Side Angle Side. What is m∠ACB? By definition of arc measure, the measure of arc AB is 55°. Examples are shown below. For example, the angle at the corner of a square or rectangle is a right angle. Example 4 You can walk across the Sydney Harbour Bridge and take a photo of the Opera House from about the same height as top of the highest sail. The Pythagorean Theorem, named after the Greek mathematician Pythagoras, is a formula used to calculate the length of the sides of a right triangle. Let A and B denote the other two angles, and a and b the sides opposite them, respectively. Download Right Angle Word templates designs today. Improve your math knowledge with free questions in "Types of angles" and thousands of other math skills. Sure, our industry can be. Right-Angle Weave 101. Definitions: A miter joint joins the cut ends of two boards. It will then calculate the hypotenuse and print the result along with the input out to screen. Magnetic Force. From the Isosceles Triangle Theorem if follows that the angles of an equilateral triangle are all the same. We looked at how when two straight lines or segments meet, this creates an angle. Types of Triangles - right triangles, acute triangles, obtuse triangles, oblique triangles, equilateral triangles, equiangular triangles, isosceles triangles and scalene triangles, examples and step by step solutions, how to solve problems involving angles and sides of triangles, examples with step by step solutions. Trigonometric functions of an acute angle are ratios of different pairs of sides of a right-angled triangle ( Fig. Right triangles. The angles are classified according to their sizes. Page 2 of 5 Unknown sides and angles in right angled triangles can be found using these ratios.

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• ### The mass-size relation of LRGs from BOSS and DECaLS(1802.01596)

Feb. 5, 2018 astro-ph.GA
We use the DECaLS DR3 survey photometry matched to the SDSS-III/BOSS DR12 spectroscopic catalog to investigate the morphology and stellar mass-size relation of luminous red galaxies (LRGs) within the CMASS and LOWZ galaxy samples in the redshift range $0.2<z<0.7$. The large majority of both samples is composed of early-type galaxies with De Vaucouleurs profiles, while only less than 20% are late-type exponentials. We calibrate DECaLS effective radii using the higher resolution CFHT/MegaCam observations and optimise the correction for each morphological type. By cross-matching the photometric properties of the early-type population with the Portsmouth stellar mass catalog, we are able to explore the high-mass end of the distribution using a large sample of 313,026 galaxies over 4380 deg$^{2}$. We find a clear correlation between the sizes and the stellar masses of these galaxies, which appears flatter than previous estimates at lower masses. The sizes of these early-type galaxies do not exhibit significant evolution within the BOSS redshift range, but a slightly declining redshift trend is found when these results are combined with $z\sim0.1$ SDSS measurements at the high-mass end. The synergy between BOSS and DECaLS has important applications in other fields, including galaxy clustering and weak lensing.
• ### The Sloan Lens ACS Survey. XIII. Discovery of 40 New Galaxy-Scale Strong Lenses(1711.00072)

Jan. 12, 2018 astro-ph.GA
We present the full sample of 118 galaxy-scale strong-lens candidates in the Sloan Lens ACS (SLACS) Survey for the Masses (S4TM) Survey, which are spectroscopically selected from the final data release of the Sloan Digital Sky Survey. Follow-up Hubble Space Telescope (HST) imaging observations confirm that 40 candidates are definite strong lenses with multiple lensed images. The foreground lens galaxies are found to be early-type galaxies (ETGs) at redshifts 0.06 to 0.44, and background sources are emission-line galaxies at redshifts 0.22 to 1.29. As an extension of the SLACS Survey, the S4TM Survey is the first attempt to preferentially search for strong-lens systems with relatively lower lens masses than those in the pre-existing strong-lens samples. By fitting HST data with a singular isothermal ellipsoid model, we find total projected mass within the Einstein radius of the S4TM strong-lens sample ranges from $3 \times10^{10} M_{\odot}$ to $2 \times10^{11} M_{\odot}$. In [Shu15], we have derived the total stellar mass of the S4TM lenses to be $5 \times10^{10} M_{\odot}$ to $1 \times10^{12} M_{\odot}$. Both total enclosed mass and stellar mass of the S4TM lenses are on average almost a factor of 2 smaller than those of the SLACS lenses, which also represent typical mass scales of the current strong-lens samples. The extended mass coverage provided by the S4TM sample can enable a direct test, with the aid of strong lensing, for transitions in scaling relations, kinematic properties, mass structure, and dark-matter content trends of ETGs at intermediate-mass scales as noted in previous studies.
• The fourth generation of the Sloan Digital Sky Survey (SDSS-IV) began observations in July 2014. It pursues three core programs: APOGEE-2, MaNGA, and eBOSS. In addition, eBOSS contains two major subprograms: TDSS and SPIDERS. This paper describes the first data release from SDSS-IV, Data Release 13 (DR13), which contains new data, reanalysis of existing data sets and, like all SDSS data releases, is inclusive of previously released data. DR13 makes publicly available 1390 spatially resolved integral field unit observations of nearby galaxies from MaNGA, the first data released from this survey. It includes new observations from eBOSS, completing SEQUELS. In addition to targeting galaxies and quasars, SEQUELS also targeted variability-selected objects from TDSS and X-ray selected objects from SPIDERS. DR13 includes new reductions of the SDSS-III BOSS data, improving the spectrophotometric calibration and redshift classification. DR13 releases new reductions of the APOGEE-1 data from SDSS-III, with abundances of elements not previously included and improved stellar parameters for dwarf stars and cooler stars. For the SDSS imaging data, DR13 provides new, more robust and precise photometric calibrations. Several value-added catalogs are being released in tandem with DR13, in particular target catalogs relevant for eBOSS, TDSS, and SPIDERS, and an updated red-clump catalog for APOGEE. This paper describes the location and format of the data now publicly available, as well as providing references to the important technical papers that describe the targeting, observing, and data reduction. The SDSS website, http://www.sdss.org, provides links to the data, tutorials and examples of data access, and extensive documentation of the reduction and analysis procedures. DR13 is the first of a scheduled set that will contain new data and analyses from the planned ~6-year operations of SDSS-IV.
• ### The BOSS Emission-Line Lens Survey V. Morphology and Substructure of Lensed Lyman-$\alpha$ Emitters at redshift $z\approx2.5$ in the BELLS GALLERY(1708.08854)

Aug. 29, 2017 astro-ph.GA
We present a morphological study of the 17 lensed Lyman-$\alpha$ emitter (LAE) galaxies of the Baryon Oscillation Spectroscopic Survey Emission-Line Lens Survey (BELLS) for the GALaxy-Ly$\alpha$ EmitteR sYstems (BELLS GALLERY) sample. This analysis combines the magnification effect of strong galaxy-galaxy lensing with the high resolution of the Hubble Space Telescope ($HST$) to achieve a physical resolution of $\sim$80 pc for this $2<z<3$ LAE sample, allowing a detailed characterization of the LAE rest-frame ultraviolet continuum surface brightness profiles and substructure. We use lens-model reconstructions of the LAEs to identify and model individual clumps, which we subsequently use to constrain the parameters of a generative statistical model of the LAE population. Since the BELLS GALLERY sample is selected primarily on the basis of Lyman-$\alpha$ emission, the LAEs that we study here are likely to be directly comparable to those selected in wide-field narrow-band LAE surveys, in contrast with the lensed LAEs identified in cluster lensing fields. We find an LAE clumpiness fraction of approximately 88%, significantly higher than found in previous (non-lensing) studies. We find a well-resolved characteristic clump half-light radii of $\sim$350 pc, a scale comparable to the largest H II regions seen in the local universe. This statistical characterization of LAE surface-brightness profiles will be incorporated into future lensing analyses using the BELLS GALLERY sample to constrain the incidence of dark-matter substructure in the foreground lensing galaxies.
• ### Galaxy clustering dependence on the $\left[\mathrm{O\scriptsize{II}}\right]$ emission line luminosity in the local Universe(1611.05457)

Aug. 2, 2017 astro-ph.GA
We study the galaxy clustering dependence on the $\left[\mathrm{O\scriptsize{II}}\right]$ emission line luminosity in the SDSS DR7 Main galaxy sample at mean redshift $z\sim0.1$. We select volume-limited samples of galaxies with different $\left[\mathrm{O\scriptsize{II}}\right]$ luminosity thresholds and measure their projected, monopole and quadrupole two-point correlation functions. We model these observations using the 1$h^{-1}\rm{Gpc}$ MultiDark Planck cosmological simulation and generate light-cones with the SUrvey GenerAtoR algorithm. To interpret our results, we adopt a modified (Sub)Halo Abundance Matching scheme, accounting for the stellar mass incompleteness of the emission line galaxies. The satellite fraction constitutes an extra parameter in this model and allows to optimize the clustering fit on both small and intermediate scales (i.e. $r_p\lesssim 30h^{-1}\rm{Mpc})$, with no need of any velocity bias correction. We find that, in the local Universe, the $\left[\mathrm{O\scriptsize{II}}\right]$ luminosity correlates with all the clustering statistics explored and with the galaxy bias. This latter quantity correlates more strongly with the SDSS $r$-band magnitude than $\left[\mathrm{O\scriptsize{II}}\right]$ luminosity. In conclusion, we propose a straightforward method to produce reliable clustering models, entirely built on the simulation products, which provides robust predictions of the typical ELG host halo masses and satellite fraction values. The SDSS galaxy data, MultiDark mock catalogues and clustering results are made publicly available.
• ### Observational evidence of galaxy assembly bias(1705.00013)

April 28, 2017 astro-ph.CO, astro-ph.GA
We analyze the spectra of 300,000 luminous red galaxies (LRGs) with stellar masses $M_* \gtrsim 10^{11} M_{\odot}$ from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). By studying their star-formation histories, we find two main evolutionary paths converging into the same quiescent galaxy population at $z\sim0.55$. Fast-growing LRGs assemble $80\%$ of their stellar mass very early on ($z\sim5$), whereas slow-growing LRGs reach the same evolutionary state at $z\sim1.5$. Further investigation reveals that their clustering properties on scales of $\sim$1-30 Mpc are, at a high level of significance, also different. Fast-growing LRGs are found to be more strongly clustered and reside in overall denser large-scale structure environments than slow-growing systems, for a given stellar-mass threshold. Our results imply a dependence of clustering on stellar-mass assembly history (naturally connected to the mass-formation history of the corresponding halos) for a homogeneous population of similar mass and color, which constitutes a strong observational evidence of galaxy assembly bias.
• ### A Direct Measurement of the High-Mass End of the Velocity Dispersion Function at $z\sim0.55$ from SDSS-III/BOSS(1607.06820)

April 6, 2017 astro-ph.GA
We report the first direct spectroscopic measurement of the velocity dispersion function (VDF) for the high-mass red sequence (RS) galaxy population at redshift $z\sim0.55$. We achieve high precision by using a sample of 600,000 massive galaxies with spectra from the Baryon Oscillation Spectroscopic Survey (BOSS) of the third Sloan Digital Sky Survey (SDSS-III), covering stellar masses $M_*\gtrsim10^{11}~M_{\odot}$. We determine the VDF by projecting the joint probability-density function (PDF) of luminosity $L$ and velocity dispersion $\sigma$, i.e. $p(L,\sigma)$, defined by our previous measurements of the RS luminosity function and $L-\sigma$ relation for this sample. These measurements were corrected from red--blue galaxy population confusion, photometric blurring, incompleteness and selection effects within a forward-modeling framework that furthermore correctly accommodates the low spectroscopic signal-to-noise ratio of individual BOSS spectra. Our $z\sim0.55$ RS VDF is in overall agreement with the $z\sim0$ early-type galaxy (ETG) VDF at $\log_{10}\sigma\gtrsim2.47$, however the number density of $z=0.55$ RS galaxies that we report is larger than that of $z=0$ ETG galaxies at $2.35\gtrsim\log_{10}\sigma\gtrsim 2.47$. The extrapolation of an intermediate-mass L-$\sigma$ relation towards the high-mass end in previous low-z works may be responsible for this disagreement. Evolutionary interpretation of this comparison is also subject to differences in the way the respective samples are selected; these differences can be mitigated in future work by analyzing $z=0$ SDSS data using the same framework presented in this paper. We also provide the sample PDF for the RS population (i.e. uncorrected for incompleteness), which is a key ingredient for gravitational lensing analyses using BOSS.
• ### BOSS Great Wall: morphology, luminosity, and mass(1703.08444)

March 24, 2017 astro-ph.CO
We study the morphology, luminosity and mass of the superclusters from the BOSS Great Wall (BGW), a recently discovered very rich supercluster complex at the redshift $z = 0.47$. We have employed the Minkowski functionals to quantify supercluster morphology. We calculate supercluster luminosities and masses using two methods. Firstly, we used data about the luminosities and stellar masses of high stellar mass galaxies with $\log(M_*/h^{-1}M_\odot) \geq 11.3$. Secondly, we applied a scaling relation that combines morphological and physical parameters of superclusters to obtain supercluster luminosities, and obtained supercluster masses using the mass-to-light ratios found for local rich superclusters. We find that the BGW superclusters are very elongated systems, with shape parameter values of less than $0.2$. This value is lower than that found for the most elongated local superclusters. The values of the fourth Minkowski functional $V_3$ for the richer BGW superclusters ($V_3 = 7$ and $10$) show that they have a complicated and rich inner structure. We identify two Planck SZ clusters in the BGW superclusters, one in the richest BGW supercluster, and another in one of the poor BGW superclusters. The luminosities of the BGW superclusters are in the range of $1 - 8\times~10^{13}h^{-2}L_\odot$, and masses in the range of $0.4 - 2.1\times~10^{16}h^{-1}M_\odot$. Supercluster luminosities and masses obtained with two methods agree well. We conclude that the BGW is a complex of massive, luminous and large superclusters with very elongated shape. The search and detailed study, including the morphology analysis of the richest superclusters and their complexes from observations and simulations can help us to understand formation and evolution of the cosmic web.
• ### Discovery of a Very Bright and Intrinsically Very Luminous, Strongly Lensed Ly{\alpha} Emitting Galaxy at z = 2.82 in the BOSS Emission-Line Lens Survey(1702.00989)

Feb. 6, 2017 astro-ph.GA
We report the discovery of a very bright (r = 20.16), highly magnified, and yet intrinsically very luminous Ly{\alpha} emitter (LAE) at z = 2.82. This system comprises four images in the observer plane with a maximum separation of ~ 6" and it is lensed by a z = 0.55 massive early-type galaxy. It was initially identified in the Baryon Oscillation Spectroscopic Survey (BOSS) Emission-Line Lens Survey for GALaxy-Ly{\alpha} EmitteR sYstems (BELLS GALLERY) survey, and follow-up imaging and spectroscopic observations using the Gran Telescopio Canarias (GTC) and William Herschel Telescope (WHT) confirmed the lensing nature of this system. A lens model using a singular isothermal ellipsoid in an external shear field reproduces quite well the main features of the system, yielding an Einstein radius of 2.95" +/- 0.10", and a total magnification factor for the LAE of 8.8 +/- 0.4. This LAE is one of the brightest and most luminous galaxy-galaxy strong lenses known. We present initial imaging and spectroscopy showing the basic physical and morphological properties of this lensed system.
• ### The BOSS Emission-Line Lens Survey. IV. : Smooth Lens Models for the BELLS GALLERY Sample(1608.08707)

Dec. 27, 2016 astro-ph.GA
We present \textsl{Hubble Space Telescope} (\textsl{HST}) F606W-band imaging observations of 21 galaxy-Ly$\alpha$ emitter lens candidates in the Baryon Oscillation Spectroscopic Survey (BOSS) Emission-Line Lens Survey (BELLS) for GALaxy-Ly$\alpha$ EmitteR sYstems (BELLS GALLERY) survey. 17 systems are confirmed to be definite lenses with unambiguous evidence of multiple imaging. The lenses are primarily massive early-type galaxies (ETGs) at redshifts of approximately $0.55$, while the lensed sources are Ly$\alpha$ emitters (LAEs) at redshifts from 2 to 3. Although the \textsl{HST} imaging data are well fit by smooth lens models consisting of singular isothermal ellipsoids in an external shear field, a thorough exploration of dark substructures in the lens galaxies is required. The Einstein radii of the BELLS GALLERY lenses are on average $60\%$ larger than those of the BELLS lenses because of the much higher source redshifts which will allow a detailed investigation of the radius evolution of the mass profile in ETGs. With the aid of the average $\sim 13 \times$ lensing magnification, the LAEs are resolved to comprise individual star-forming knots of a wide range of properties with characteristic sizes from less than 100 pc to several kpc, rest-frame far UV apparent AB magnitudes from 29.6 to 24.2, and typical projected separations of 500 pc to 2 kpc.
• ### The BOSS Emission-Line Lens Survey. III. : Strong Lensing of Ly$\alpha$ Emitters by Individual Galaxies(1604.01842)

July 25, 2016 astro-ph.GA
We introduce the Baryon Oscillation Spectroscopic Survey (BOSS) Emission-Line Lens Survey (BELLS) for GALaxy-Ly$\alpha$ EmitteR sYstems (BELLS GALLERY) Survey, which is a Hubble Space Telescope program to image a sample of galaxy-scale strong gravitational lens candidate systems with high-redshift Ly$\alpha$ emitters (LAEs) as the background sources. The goal of the BELLS GALLERY Survey is to illuminate dark substructures in galaxy-scale halos by exploiting the small-scale clumpiness of rest-frame far-UV emission in lensed LAEs, and to thereby constrain the slope and normalization of the substructure-mass function. In this paper, we describe in detail the spectroscopic strong-lens selection technique, which is based on methods adopted in the previous Sloan Lens ACS (SLACS) Survey, BELLS, and SLACS for the Masses Survey. We present the BELLS GALLERY sample of the 21 highest-quality galaxy--LAE candidates selected from $\approx 1.4 \times 10^6$ galaxy spectra in the BOSS of the Sloan Digital Sky Survey III. These systems consist of massive galaxies at redshifts of approximately 0.5 strongly lensing LAEs at redshifts from 2--3. The compact nature of LAEs makes them an ideal probe of dark substructures, with a substructure-mass sensitivity that is unprecedented in other optical strong-lens samples. The magnification effect from lensing will also reveal the structure of LAEs below 100 pc scales, providing a detailed look at the sites of the most concentrated unobscured star formation in the universe. The source code used for candidate selection is available for download as a part of this release.
• ### The Correlation Between Halo Mass and Stellar Mass for the Most Massive Galaxies in the Universe(1607.04678)

July 15, 2016 astro-ph.CO, astro-ph.GA
We present measurements of the clustering of galaxies as a function of their stellar mass in the Baryon Oscillation Spectroscopic Survey. We compare the clustering of samples using 12 different methods for estimating stellar mass, isolating the method that has the smallest scatter at fixed halo mass. In this test, the stellar mass estimate with the smallest errors yields the highest amplitude of clustering at fixed number density. We find that the PCA stellar masses of Chen etal (2012) clearly have the tightest correlation with halo mass. The PCA masses use the full galaxy spectrum, differentiating them from other estimates that only use optical photometric information. Using the PCA masses, we measure the large-scale bias as a function of Mgal for galaxies with logMgal>=11.4, correcting for incompleteness at the low-mass end of our measurements. Using the abundance-matching ansatz to connect dark matter halo mass to stellar mass, we construct theoretical models of b(Mgal) that match the same stellar mass function but have different amounts of scatter in stellar mass at fixed halo mass, sigma_logM. Using this approach, we find sigma_logM=0.18^{+0.01}_{-0.02}. This value includes both intrinsic scatter as well as random errors in the stellar masses. To partially remove the latter, we use repeated spectra to estimate statistical errors on the stellar masses, yielding an upper limit to the intrinsic scatter of 0.16 dex.
• ### The High-Mass End of the Red Sequence at z~0.55 from SDSS-III/BOSS: completeness, bimodality and luminosity function(1410.5854)

May 26, 2016 astro-ph.GA
We have developed an analytical method based on forward-modeling techniques to characterize the high-mass end of the red sequence (RS) galaxy population at redshift $z\sim0.55$, from the DR10 BOSS CMASS spectroscopic sample, which comprises $\sim600,000$ galaxies. The method, which follows an unbinned maximum likelihood approach, allows the deconvolution of the intrinsic CMASS colour-colour-magnitude distributions from photometric errors and selection effects. This procedure requires modeling the covariance matrix for the i-band magnitude, g-r colour and r-i colour using Stripe 82 multi-epoch data. Our results indicate that the error-deconvolved intrinsic RS distribution is consistent, within the photometric uncertainties, with a single point ($<0.05~{\rm{mag}}$) in the colour-colour plane at fixed magnitude, for a narrow redshift slice. We have computed the high-mass end ($^{0.55}M_i \lesssim -22$) of the $^{0.55}i$-band RS Luminosity Function (RS LF) in several redshift slices within the redshift range $0.52<z<0.63$. In this narrow redshift range, the evolution of the RS LF is consistent, within the uncertainties in the modeling, with a passively-evolving model with $\Phi_* = (7.248 \pm 0.204) \times10^{-4}$ Mpc$^{-3}$ mag$^{-1}$, fading at a rate of $1.5\pm0.4$ mag per unit redshift. We report RS completeness as a function of magnitude and redshift in the CMASS sample, which will facilitate a variety of galaxy-evolution and clustering studies using BOSS. Our forward-modeling method lays the foundations for future studies using other dark-energy surveys like eBOSS or DESI, which are affected by the same type of photometric blurring/selection effects.
• ### The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Modeling the clustering and halo occupation distribution of BOSS-CMASS galaxies in the Final Data Release(1509.06404)

May 3, 2016 astro-ph.CO, astro-ph.GA
We present a study of the clustering and halo occupation distribution of BOSS CMASS galaxies in the redshift range 0.43 < z < 0.7 drawn from the Final SDSS-III Data Release. We compare the BOSS results with the predictions of a Halo Abundance Matching (HAM) clustering model that assigns galaxies to dark matter halos selected from the large BigMultiDark $N$-body simulation of a flat $\Lambda$CDM Planck cosmology. We compare the observational data with the simulated ones on a light-cone constructed from 20 subsequent outputs of the simulation. Observational effects such as incompleteness, geometry, veto masks and fiber collisions are included in the model, which reproduces within 1-$\sigma$ errors the observed monopole of the 2-point correlation function at all relevant scales: from the smallest scales, 0.5 $h^{-1}$ Mpc, up to scales beyond the Baryonic Acoustic Oscillation feature. This model also agrees remarkably well with the BOSS galaxy power spectrum (up to $k\sim1$ $h$ Mpc$^{-1}$), and the Three-point correlation function. The quadrupole of the correlation function presents some tensions with observations. We discuss possible causes that can explain this disagreement, including target selection effects. Overall, the standard HAM model describes remarkably well the clustering statistics of the CMASS sample. We compare the stellar to halo mass relation for the CMASS sample measured using weak lensing in the CFHT Stripe 82 Survey with the prediction of our clustering model, and find a good agreement within 1-$\sigma$. The BigMD-BOSS light-cone including properties of BOSS galaxies and halo properties is made publicly available.
• The Extended Baryon Oscillation Spectroscopic Survey (eBOSS) will conduct novel cosmological observations using the BOSS spectrograph at Apache Point Observatory. Observations will be simultaneous with the Time Domain Spectroscopic Survey (TDSS) designed for variability studies and the Spectroscopic Identification of eROSITA Sources (SPIDERS) program designed for studies of X-ray sources. eBOSS will use four different tracers to measure the distance-redshift relation with baryon acoustic oscillations (BAO). Using more than 250,000 new, spectroscopically confirmed luminous red galaxies at a median redshift z=0.72, we project that eBOSS will yield measurements of $d_A(z)$ to an accuracy of 1.2% and measurements of H(z) to 2.1% when combined with the z>0.6 sample of BOSS galaxies. With ~195,000 new emission line galaxy redshifts, we expect BAO measurements of $d_A(z)$ to an accuracy of 3.1% and H(z) to 4.7% at an effective redshift of z= 0.87. A sample of more than 500,000 spectroscopically-confirmed quasars will provide the first BAO distance measurements over the redshift range 0.9<z<2.2, with expected precision of 2.8% and 4.2% on $d_A(z)$ and H(z), respectively. Finally, with 60,000 new quasars and re-observation of 60,000 quasars known from BOSS, we will obtain new Lyman-alpha forest measurements at redshifts z>2.1; these new data will enhance the precision of $d_A(z)$ and H(z) by a factor of 1.44 relative to BOSS. Furthermore, eBOSS will provide improved tests of General Relativity on cosmological scales through redshift-space distortion measurements, improved tests for non-Gaussianity in the primordial density field, and new constraints on the summed mass of all neutrino species. Here, we provide an overview of the cosmological goals, spectroscopic target sample, demonstration of spectral quality from early data, and projected cosmological constraints from eBOSS.
• ### A Steep Slope and Small Scatter for the High-Mass End of the L-$\sigma$ Relation at $z\sim0.55$(1505.03866)

Dec. 11, 2015 astro-ph.GA
We measure the intrinsic relation between velocity dispersion ($\sigma$) and luminosity ($L$) for massive, luminous red galaxies (LRGs) at redshift $z \sim 0.55$. We achieve unprecedented precision by using a sample of 600,000 galaxies with spectra from the Baryon Oscillation Spectroscopic Survey (BOSS) of the third Sloan Digital Sky Survey (SDSS-III), covering a range of stellar masses $M_* \gtrsim 10^{11} M_{\odot}$. We deconvolve the effects of photometric errors, limited spectroscopic signal-to-noise ratio, and red--blue galaxy confusion using a novel hierarchical Bayesian formalism that is generally applicable to any combination of photometric and spectroscopic observables. For an L-$\sigma$ relation of the form $L \propto \sigma^{\beta}$, we find $\beta = 7.8 \pm 1.1$ for $\sigma$ corrected to the effective radius, and a very small intrinsic scatter of $s = 0.047 \pm 0.004$ in $\log_{10} \sigma$ at fixed $L$. No significant redshift evolution is found for these parameters. The evolution of the zero-point within the redshift range considered is consistent with the passive evolution of a galaxy population that formed at redshift $z=2-3$, assuming single stellar populations. An analysis of previously reported results seems to indicate that the passively-evolved high-mass L-$\sigma$ relation at $z\sim0.55$ is consistent with the one measured at $z=0.1$. Our results, in combination with those presented in Montero-Dorta et al. (2014), provide a detailed description of the high-mass end of the red sequence (RS) at $z\sim0.55$. This characterization, in the light of previous literature, suggest that the high-mass RS distribution corresponds to the "core" elliptical population.
• The third generation of the Sloan Digital Sky Survey (SDSS-III) took data from 2008 to 2014 using the original SDSS wide-field imager, the original and an upgraded multi-object fiber-fed optical spectrograph, a new near-infrared high-resolution spectrograph, and a novel optical interferometer. All the data from SDSS-III are now made public. In particular, this paper describes Data Release 11 (DR11) including all data acquired through 2013 July, and Data Release 12 (DR12) adding data acquired through 2014 July (including all data included in previous data releases), marking the end of SDSS-III observing. Relative to our previous public release (DR10), DR12 adds one million new spectra of galaxies and quasars from the Baryon Oscillation Spectroscopic Survey (BOSS) over an additional 3000 sq. deg of sky, more than triples the number of H-band spectra of stars as part of the Apache Point Observatory (APO) Galactic Evolution Experiment (APOGEE), and includes repeated accurate radial velocity measurements of 5500 stars from the Multi-Object APO Radial Velocity Exoplanet Large-area Survey (MARVELS). The APOGEE outputs now include measured abundances of 15 different elements for each star. In total, SDSS-III added 2350 sq. deg of ugriz imaging; 155,520 spectra of 138,099 stars as part of the Sloan Exploration of Galactic Understanding and Evolution 2 (SEGUE-2) survey; 2,497,484 BOSS spectra of 1,372,737 galaxies, 294,512 quasars, and 247,216 stars over 9376 sq. deg; 618,080 APOGEE spectra of 156,593 stars; and 197,040 MARVELS spectra of 5,513 stars. Since its first light in 1998, SDSS has imaged over 1/3 of the Celestial sphere in five bands and obtained over five million astronomical spectra.
• ### The Sloan Lens ACS Survey. XII. Extending Strong Lensing to Lower Masses(1407.2240)

Feb. 10, 2015 astro-ph.GA
We present observational results from a new Hubble Space Telescope (HST) Snapshot program to extend the methods of the Sloan Lens ACS (SLACS) Survey to lower lens-galaxy masses. We discover 40 new galaxy-scale strong lenses, which we supplement with 58 previously discovered SLACS lenses. In addition, we determine the posterior PDFs of the Einstein radius for 33 galaxies (18 new and 15 from legacy SLACS data) based on single lensed images. We find a less-than-unity slope of $0.64\pm0.06$ for the $\log_{10} {\sigma}_*$-$\log_{10} {\sigma}_{\rm SIE}$ relation, which corresponds to a 6-$\sigma$ evidence that the total mass-density profile of early-type galaxies varies systematically in the sense of being shallower at higher lens-galaxy velocity dispersions. The trend is only significant when single-image systems are considered, highlighting the importance of including both "lenses" and "non-lenses" for an unbiased treatment of the lens population when extending to lower mass ranges. By scaling simple stellar population models to the HST I-band data, we identify a strong trend of increasing dark-matter fraction at higher velocity dispersions, which can be alternatively interpreted as a trend in the stellar initial mass function (IMF) normalization. Consistent with previous findings and the suggestion of a non-universal IMF, we find that a Salpeter IMF is ruled out for galaxies with velocity dispersion less than $180$ km/s. Considered together, our mass-profile and dark-matter-fraction trends with increasing galaxy mass could both be explained by an increasing relative contribution on kiloparsec scales from a dark-matter halo with a spatial profile more extended than that of the stellar component.
• The Sloan Digital Sky Survey (SDSS) has been in operation since 2000 April. This paper presents the tenth public data release (DR10) from its current incarnation, SDSS-III. This data release includes the first spectroscopic data from the Apache Point Observatory Galaxy Evolution Experiment (APOGEE), along with spectroscopic data from the Baryon Oscillation Spectroscopic Survey (BOSS) taken through 2012 July. The APOGEE instrument is a near-infrared R~22,500 300-fiber spectrograph covering 1.514--1.696 microns. The APOGEE survey is studying the chemical abundances and radial velocities of roughly 100,000 red giant star candidates in the bulge, bar, disk, and halo of the Milky Way. DR10 includes 178,397 spectra of 57,454 stars, each typically observed three or more times, from APOGEE. Derived quantities from these spectra (radial velocities, effective temperatures, surface gravities, and metallicities) are also included.DR10 also roughly doubles the number of BOSS spectra over those included in the ninth data release. DR10 includes a total of 1,507,954 BOSS spectra, comprising 927,844 galaxy spectra; 182,009 quasar spectra; and 159,327 stellar spectra, selected over 6373.2 square degrees.
• ### Host Galaxy Spectra and Consequences for SN Typing From The SDSS SN Survey(1308.6818)

Aug. 30, 2013 astro-ph.CO
We present the spectroscopy from 5254 galaxies that hosted supernovae (SNe) or other transient events in the Sloan Digital Sky Survey II (SDSS-II). Obtained during SDSS-I, SDSS-II, and the Baryon Oscillation Spectroscopic Survey (BOSS), this sample represents the largest systematic, unbiased, magnitude limited spectroscopic survey of supernova (SN) host galaxies. Using the host galaxy redshifts, we test the impact of photometric SN classification based on SDSS imaging data with and without using spectroscopic redshifts of the host galaxies. Following our suggested scheme, there are a total of 1166 photometrically classified SNe Ia when using a flat redshift prior and 1126 SNe Ia when the host spectroscopic redshift is assumed. For 1024 (87.8%) candidates classified as likely SNe Ia without redshift information, we find that the classification is unchanged when adding the host galaxy redshift. Using photometry from SDSS imaging data and the host galaxy spectra, we also report host galaxy properties for use in future nalysis of SN astrophysics. Finally, we investigate the differences in the interpretation of the light curve properties with and without knowledge of the redshift. When using the SALT2 light curve fitter, we find a 21% increase in the number of fits that converge when using the spectroscopic redshift. Without host galaxy redshifts, we find that SALT2 light curve fits are systematically biased towards lower photometric redshift estimates and redder colors in the limit of low signal-to-noise data. The general improvements in performance of the light curve fitter and the increased diversity of the host galaxy sample highlights the importance of host galaxy spectroscopy for current photometric SN surveys such as the Dark Energy Survey and future surveys such as the Large Synoptic Survey Telescope.
• ### The clustering of galaxies at z~0.5 in the SDSS-III Data Release 9 BOSS-CMASS sample: a test for the LCDM cosmology(1202.6057)

April 22, 2013 astro-ph.CO
We present results on the clustering of 282,068 galaxies in the Baryon Oscillation Spectroscopic Survey (BOSS) sample of massive galaxies with redshifts 0.4<z<0.7 which is part of the Sloan Digital Sky Survey III project. Our results cover a large range of scales from ~0.5 to ~90 Mpc/h. We compare these estimates with the expectations of the flat LCDM cosmological model with parameters compatible with WMAP7 data. We use the MultiDark cosmological simulation together with a simple halo abundance matching technique, to estimate galaxy correlation functions, power spectra, abundance of subhaloes and galaxy biases. We find that the LCDM model gives a reasonable description to the observed correlation functions at z~0.5, which is a remarkably good agreement considering that the model, once matched to the observed abundance of BOSS galaxies, does not have any free parameters. However, we find a deviation (>~10%) in the correlation functions for scales less than ~1 Mpc/h and ~10-40 Mpc/h. A more realistic abundance matching model and better statistics from upcoming observations are needed to clarify the situation. We also estimate that about 12% of the "galaxies" in the abundance-matched sample are satellites inhabiting central haloes with mass M>~1e14 M_sun/h. Using the MultiDark simulation we also study the real space halo bias b(r) of the matched catalogue finding that b=2.00+/-0.07 at large scales, consistent with the one obtained using the measured BOSS projected correlation function. Furthermore, the linear large-scale bias depends on the number density n of the abundance-matched sample as b=-0.048-(0.594+/-0.02)*log(n/(h/Mpc)^3). Extrapolating these results to BAO scales we measure a scale-dependent damping of the acoustic signal produced by non-linear evolution that leads to ~2-4% dips at ~3 sigma level for wavenumbers k>~0.1 h/Mpc in the linear large-scale bias.
• The Baryon Oscillation Spectroscopic Survey (BOSS) is designed to measure the scale of baryon acoustic oscillations (BAO) in the clustering of matter over a larger volume than the combined efforts of all previous spectroscopic surveys of large scale structure. BOSS uses 1.5 million luminous galaxies as faint as i=19.9 over 10,000 square degrees to measure BAO to redshifts z<0.7. Observations of neutral hydrogen in the Lyman alpha forest in more than 150,000 quasar spectra (g<22) will constrain BAO over the redshift range 2.15<z<3.5. Early results from BOSS include the first detection of the large-scale three-dimensional clustering of the Lyman alpha forest and a strong detection from the Data Release 9 data set of the BAO in the clustering of massive galaxies at an effective redshift z = 0.57. We project that BOSS will yield measurements of the angular diameter distance D_A to an accuracy of 1.0% at redshifts z=0.3 and z=0.57 and measurements of H(z) to 1.8% and 1.7% at the same redshifts. Forecasts for Lyman alpha forest constraints predict a measurement of an overall dilation factor that scales the highly degenerate D_A(z) and H^{-1}(z) parameters to an accuracy of 1.9% at z~2.5 when the survey is complete. Here, we provide an overview of the selection of spectroscopic targets, planning of observations, and analysis of data and data quality of BOSS.
• The Sloan Digital Sky Survey III (SDSS-III) presents the first spectroscopic data from the Baryon Oscillation Spectroscopic Survey (BOSS). This ninth data release (DR9) of the SDSS project includes 535,995 new galaxy spectra (median z=0.52), 102,100 new quasar spectra (median z=2.32), and 90,897 new stellar spectra, along with the data presented in previous data releases. These spectra were obtained with the new BOSS spectrograph and were taken between 2009 December and 2011 July. In addition, the stellar parameters pipeline, which determines radial velocities, surface temperatures, surface gravities, and metallicities of stars, has been updated and refined with improvements in temperature estimates for stars with T_eff<5000 K and in metallicity estimates for stars with [Fe/H]>-0.5. DR9 includes new stellar parameters for all stars presented in DR8, including stars from SDSS-I and II, as well as those observed as part of the SDSS-III Sloan Extension for Galactic Understanding and Exploration-2 (SEGUE-2). The astrometry error introduced in the DR8 imaging catalogs has been corrected in the DR9 data products. The next data release for SDSS-III will be in Summer 2013, which will present the first data from the Apache Point Observatory Galactic Evolution Experiment (APOGEE) along with another year of data from BOSS, followed by the final SDSS-III data release in December 2014.
• ### The SDSS DR6 Luminosity Functions of Galaxies(0806.4930)

June 30, 2008 astro-ph
We present number counts, luminosity functions (LFs) and luminosity densities of galaxies obtained using the Sloan Digital Sky Survey Sixth Data Release in all SDSS photometric bands. Thanks to the SDSS DR6, galaxy statistics have increased by a factor of ~9 in the u-band and by a factor of ~4-5 in the rest of the SDSS bands with respect to the previous work of Blanton et al. (2003b). In addition, we have achieved a high redshift completeness in our galaxy samples. Firstly, by making use of the survey masks, provided by the NYU-VAGC DR6, we have been able to define an area on the sky of high angular redshift completeness. Secondly, we guarantee that brightness-dependent redshift incompleteness is small within the magnitude ranges that define our galaxy samples. With these advances, we have estimated very accurate SDSS DR6 LFs in both the bright and the faint end. In the {0.1}^r-band, our SDSS DR6 luminosity function is well fitted by a Schechter LF with parameters Phi_{*}=0.90 +/- 0.07\$, M_{*}-5log_{10}h=-20.73 +/- 0.04 and alpha=-1.23 +/- 0.02. As compared with previous results, we find some notable differences. In the bright end of the {0.1}^u-band luminosity function we find a remarkable excess, of ~1.7 dex at M_{{0.1}^u}=-20.5, with respect to the best-fit Schechter LF. This excess weakens in the {0.1}^g-band, fading away towards the very red {0.1}^z-band. A preliminary analysis on the nature of this bright-end bump reveals that it is mostly comprised of active galaxies and QSOs. It seems, therefore, that an important fraction of this exceeding luminosity may come from nuclear activity. In the faint end of the SDSS DR6 luminosity functions, where we can reach 1-1.5 magnitudes deeper than the previous SDSS LF estimation, we obtain a steeper slope [ABRIDGED].
• ### The DEEP2 Galaxy Redshift Survey: The Red Sequence AGN Fraction and its Environment and Redshift Dependence(0801.2769)

Jan. 17, 2008 astro-ph
We measure the dependence of the AGN fraction on local environment at z~1, using spectroscopic data taken from the DEEP2 Galaxy Redshift Survey, and Chandra X-ray data from the All-Wavelength Extended Groth Strip International Survey (AEGIS). To provide a clean sample of AGN we restrict our analysis to the red sequence population; this also reduces additional colour-environment correlations. We find evidence that high redshift LINERs in DEEP2 tend to favour higher density environments relative to the red population from which they are drawn. In contrast, Seyferts and X-ray selected AGN at z~1 show little (or no) environmental dependencies within the same underlying population. We compare these results with a sample of local AGN drawn from the SDSS. Contrary to the high redshift behaviour, we find that both LINERs and Seyferts in the SDSS show a slowly declining red sequence AGN fraction towards high density environments. Interestingly, at z~1 red sequence Seyferts and LINERs are approximately equally abundant. By z~0, however, the red Seyfert population has declined relative to the LINER population by over a factor of 7. We speculate on possible interpretations of our results.


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# Three Fast Methods to Compute the DCT-2

After doing my "proof thing" of the fast DFT based implementation of the DCT-4 (which does actually come in useful in my line of work), I thought I would have a fiddle with some of the other DCT variants. The DCT-2 is a transform which I doubt I will ever use, but I managed to find three ways (excluding the slow and obvious one) to build it. Here is the definition:

Like in the DCT-4 post, I hate trig functions so the first thing to do is transform $\cos$ into a complex exponential. I'll use this as the starting point for the methods.

## Method 1: Recursion and the DCT-4

All of the basis functions of the DCT-2 have a symmetry about $n=\frac{N}{2}$; the symmetry is even for even $k$ and odd for odd $k$. This is a big hint that we can benefit from splitting the input sequence into two half-length sequences where one sequence is reversed:

As on the DCT-4 page, we can conjugate any terms we like in the summation as we are only interested in the real part of the result. This allows us to rewrite the above as:

Split $X_k$ into even and odd terms:

The end. It can be seen that the $X_{2k}$ terms can be obtained by recursively calling the function again. The $X_{2k+1}$ terms can be obtained via a DCT-4. Writing out the steps explicitly:

• Create a sequence $y_n = x_n + x_{N-n-1}$ of length $\frac{N}{2}$
• Create a sequence $z_n = x_n - x_{N-n-1}$ of length $\frac{N}{2}$
• The resulting $X_{2k+1}$ terms are given by the DCT-4 of the sequence $z_n$
• The resulting $X_{2k}$ terms are given by performing this same process on the sequence $y_n$

## Method 2: A real DFT and a DCT-4

This method is probably not particularly useful given how the next method works - but I thought it was interesting so I'm listing it. Starting from the beginning again:

Rearrange the above into two halves by interleaving the input:

Again, we conjugate terms where we know it won't impact the result:

For the $X_{2k}$ case, we can write this as:

So for the even output indices, we can use a DFT on some rearranged data followed by a twiddle. For the $X_{2k+1}$ outputs, this particular form gets ugly - but recalling that in the first method, the $X_{2k+1}$ outputs were not recursive and simply based on a DCT-4, we just steal that result:

I won't bother explicitly listing the steps for this one. They are simple to figure out.

## Method 3: A single real DFT

This is almost certainly the method you should use to compute the DCT-2. Recall the third equation in the second method:

I didn't spot this immediately, but the summation term also happens to be the first stage of a decimation in frequency transform. Recall the DIF FFT stage:

So we create a new vector $y_n$:

Then substitute back into $X_k$:

This actually becomes an $N/2$ length DFT if you use a real transform.

# Derivation of fast DCT-4 algorithm based on DFT

It's well known that an $N$ point DCT-4 can be computed using an $N/2$ point complex FFT. Although the algorithm is widespread, the texts which I have read on the subject have not provided the details as to how it works. I've been trying to understand this for a while (I tend not to use algorithms until I understand them) and finally figured it out and thought I would share (please provide comments/links if you can find a better or shorter explanation - this is as good as I could get).

Trig functions are hard to manipulate in expressions (for me anyway), so knowing that $x_n$ is real, we change the $\cos$ into a complex exponential and obtain the following:

It is worth noting that the sign of the exponent is irrelevant.

We then split the expression up to operate over two half-length sequences composed of $x_{2n}$ and $x_{N-1-2n}$. The second sequence is reversed and decimated because when the $n$ is replaced by $N-1-2n$ in the $n + \frac{1}{2}$ term of the exponential, the expression is negated rather than the offset being modified. We move the $N$ term into another exponential which becomes trivial as $N$ cancels the denominator. i.e.:

Or:

The exponentials still contain no term which resembles an DFT (over length $N/2$ anyway). To get one, we break up $X_k$ into the terms $X_{2k}$ and $X_{N-1-2k}$. Again we choose to reverse the second sequence because it will keep the exponential terms in a similar but negated form.

Now because we are only interested in the real terms, we can ignore or conjugate anything which contributes to the imaginary term of the above expressions. This means that we can conjugate the exponentials when they are only modulating a real term. Doing this, we obtain:

Almost there, but to obtain the full benefit we need the inner terms to be the same. If we now multiply the $X_{N-1-2k}$ term by $- \mathrm{j}$, we get:

Done. If we expand out the $X_{2k}$ and $X_{N-1-2k}$ terms completely we get:

Which give us the steps for a DFT based DCT-4 algorithm:

• Transform the $N$ point real sequence $x_n$ into the $N/2$ point complex sequence $y_n = x_{2n} + \mathrm{j} x_{N-1-2n}$.
• Multiply each element of the sequence $y_n$ by $\mathrm{e}^{- \frac{\mathrm{j} \pi n}{N}}$.
• Find $Y$, the DFT of the sequence $y$.
• Multiply each element of $Y_k$ by $\mathrm{e}^{- \frac{\mathrm{j} \pi \left( 2k + \frac{1}{2} \right) }{2N}}$.
• The DCT-4 outputs are given by:


text:
# Closed 1-form on a Riemann surface

Let $R$ be a Riemann surface of genus $g\ge 2$ and $p\in R$ a point. I need to find a way to produce a closed 1-form $\omega$ on $R$ which satisfies the following conditions:

• $\omega$ is not exact, not holomorphic and not anti-holomorphic
• $\omega$ is zero in $p$
• $\int_R\omega\wedge \overline{\omega}>0$

Can you help me?

The answer of this question Constructing one-forms on a Riemann surface using the uniformization theorem suggests to use the uniformization theorem, but they do it for meromorphic quadratic forms and I don't know how to apply it to my case.

Here is what I suggest. First, start with a holomorphic (1,0)-form $\alpha$ which is not exact. We know it exists, since the Dolbeault group $H^{(1,0)}(R)$ is $g$-dimensional. $\alpha$ is automatically closed since $R$ has complex dimension $1$. Moreover, we immediately get that

$i \int_R \alpha \wedge \bar{\alpha} > 0$.

Also, since the Euler characteristic of $R$ is non-zero (it is $2-2g < 0$), it follows that $\alpha$ vanishes somewhere in $R$. If it vanishes precisely at $p$, then this is great. Otherwise, pull back $\alpha$ using a smooth diffeomorphism $f: R \to R$ which maps $p$ to a point where $\alpha$ vanishes (does there exist such a diffeomorphism?). $f^*(\alpha)$ is automatically closed, and not exact, and also vanishes at $p$. Also the positivity condition of the integral still holds if $f$ is assumed to be orientation-preserving.

If $f^*(\alpha)$ happens to be either holomorphic or anti-holomorphic, then just add to it $\epsilon$ times the complex conjugate of $f^*(\alpha)$, where $\epsilon$ is some small enough positive real number, so as to preserve positivity of the integral.

Thus I have solved your problem, provided the answer to the following is yes: given any two points $p_1$ and $p_2$ in $R$, does there exists an orientation-preserving diffeomorphism which maps $p_1$ to $p_2$?

Edit: I am surprised I wrote the answer above. There is no $\omega$ satisfying all the requirements in the original post. First of all, the integral in the inequality is pure imaginary, but I suppose that it is just because of a typo, and the author forgot to multiply by $i$. However, $i$ times the integral is negative, by the Gauss-Bonnet theorem, since $g \geq 2$. But let us assume that the author had asked the same questions, but with the inequality being $i \int_R \omega \wedge \bar{\omega} < 0$ which is actually automatically satisfied, by the Gauss-Bonnet theorem.

We start with a holomorphic $(1,0)$-form $\alpha$ which is not exact, as I did in the beginning of my answer. If at $p$ we have that $i \alpha \wedge \bar{\alpha} \neq 0$, then we just add $df$ to $\alpha$, where $f$ is a smooth complex-valued function, chosen so that $i df \wedge d\bar{f}$ is minus $i \alpha \wedge \bar{\alpha}$ at $p$. This will make sure that $\omega = \alpha + df$ is still closed, and vanishes at $p$. It remains to destroy holomorphicity of $\omega$, in case $\omega$ happens to be holomorphic, but this can be done by replacing $\omega$ with a suitable linear combination of $\omega$ and $\bar{\omega}$. This should fix my wrong answer above.

• I understand your reasoning, but to me your last hypothesis seems a little too strong to be verified for every couple of points on every Riemann surface.. – user00169 Aug 27 '16 at 22:17
• @user00169: see mathoverflow.net/questions/104104/… – Malkoun Aug 27 '16 at 22:40
• in particular, I quote Liviu Nicolaescu's comment: The group of diffeomorphisms of a smooth connected manifold acts transitively on that manifold. – Malkoun Aug 27 '16 at 22:42
• See the answer here – user99914 Aug 28 '16 at 4:18
• @Arctic Char: thank you! It is a nice proof via the isotopy extension theorem. – Malkoun Aug 28 '16 at 4:24