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# zbMATH — the first resource for mathematics

Counting $$(k,l)$$-sumsets in groups of a prime order. (English) Zbl 1354.11021
From the text: A subset $$A$$ of a group $$G$$ is called $$(k,l)$$-sumset, if $$A=kB-lB$$ for some $$B\subseteq G$$, where $$kB-lB=x_1+\ldots+x_k-x_{k+1}-\ldots-x_{k+l}: x_1,\ldots,x_{k+l}\in B$$. Upper and lower bounds for the number $$(k,l)$$-sumsets in groups of prime order are provided.
Write $$SS_{k,l}(\mathbb Z_p)$$ for the collection of $$(k, l)$$-sumsets in $$\mathbb Z_p$$. B. Green and I. Ruzsa in [Stud. Sci. Math. Hung. 41, No. 3, 285–293 (2004; Zbl 1064.11020)] proved
$p^22^{p/3} \ll | SS_{2,0}(\mathbb Z_p)| \leq 2^{p/3+\theta(p)}$
where $$\theta(p)/p\to 0$$ as $$p\to\infty$$ and $$\theta(p)\ll p(\log \log p)^{2/3}(\log p)^{-1/9}$$.
The aim of this work is to obtain bounds for the number $$| SS_{k,l}(\mathbb Z_p)|$$. We prove
Theorem 1. Let $$p$$ be a prime number and $$k,l$$ be nonnegative integers with $$k + l \geq 2$$. Then there exists a positive constant $$C_{k,l}$$ such that
$C_{k,l}2^{p/(2(k+l)-1)} \leq | SS_{k,l}(\mathbb Z_p)| \leq 2^{(p/(k+l+1))+(k+l-2)+o(p)}. \tag{1}$

##### MSC:
 11B75 Other combinatorial number theory 68R05 Combinatorics in computer science 20D60 Arithmetic and combinatorial problems involving abstract finite groups
##### Keywords:
characteristic function; sumsets; granular set


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# Does this inequality hold true, in general?

Let

$$N = \prod_{i=1}^{\omega(N)}{{p_i}^{\alpha_i}}$$

be the prime factorization of the positive integer $N$.

Does the following inequality hold true in general?

$$\prod_{i=1}^{\omega(N)}{\frac{p_i}{p_i - 1}} < \frac{2\prod_{i=1}^{\omega(N)}{p_i}}{1 + \prod_{i=1}^{\omega(N)}{p_i}}$$

If not, for what particular values of $N$ does this inequality hold?

Note that $\omega(N)$ denotes the number of distinct prime factors of $N$.

It turns out that that inequality is true about 95% of the time.

As in my previous answer, cancel the numerator of each side to get the equivalent inequality

$\prod_{i=1}^{\omega(N)}{\dfrac{1}{p_i - 1}} < \dfrac{2}{1 + \prod_{i=1}^{\omega(N)}{p_i}}$ or $1 + \prod_{i=1}^{\omega(N)}{p_i} < 2\prod_{i=1}^{\omega(N)}(p_i - 1)$.

Dividing by $\prod_{i=1}^{\omega(N)}{p_i}$, and, using the fact that $\dfrac{\phi(n)}{n} =\prod_{i=1}^{\omega(N)}\dfrac{p_i-1}{p_i}$, this becomes $1+\dfrac{1}{\phi(n)} <2\dfrac{\phi(n)}{n}$.

Since $\dfrac{1}{\phi(n)}$ is small for large $n$, the proportion of $n$ satisfying this is the same as the proportion satisfying $\frac{1}{2} <\dfrac{\phi(n)}{n}$, or $2 >\dfrac{n}{\phi(n)}$.

At this point, I did a Google search for "density of euler phi function". The second link is http://www.ams.org/journals/proc/2007-135-09/S0002-9939-07-08771-0/S0002-9939-07-08771-0.pdf. This paper, by ANDREAS WEINGARTNER, is titled "THE DISTRIBUTION FUNCTIONS OF σ(n)/n AND n/ϕ(n)". Here is its abstract:

"Let σ(n) be the sum of the positive divisors of n. We show that the natural density of the set of integers n satisfying σ(n)/n ≥ t is given by $\exp\big(−e^{t e^{−γ}(1 + O(t^{−2}))}\big)$ , where γ denotes Euler’s constant. The same result holds when σ(n)/n is replaced by n/ϕ(n), where ϕ is Euler’s totient function."

This paper has clearly done the heavy lifting. If we put $t = 2$, and use $\gamma \approx 0.5772156649$ (I show each stage in the computation for checkability), $te^{-\gamma} = 1.1229189671$, $e^{te^{-\gamma}}=3.0738134815$, $\exp(-e^{te^{-\gamma}}) =0.0462444658$.

This is the density of $n$ for which $\dfrac{n}{\phi(n)} > 2$. The density for which $\dfrac{n}{\phi(n)} < 2$ is one minus this or $0.9537555342$.

• Thank you for your very comprehensive (and interesting!) answer, @martycohen. =) – Jose Arnaldo Bebita-Dris Jul 30 '13 at 1:31
• I apologize @martycohen, but I just spotted (today) what seems to be an "error" from the inequality $$1 + \prod_{i=1}^{\omega(N)}{p_i} < 2\prod_{i=1}^{\omega(N)}{(p_i - 1)},$$ from which we obtain, upon dividing through by $\prod_{i=1}^{\omega(N)}{p_i}$, $$1 + \frac{1}{N} < 2\frac{\phi(N)}{N}.$$ Thus, we get the inequality $$\frac{N}{\phi(N)} < \frac{2N}{N + 1}.$$ If $N > 1$, then $$I(N) = \frac{\sigma(N)}{N} < \frac{N}{\phi(N)} < \frac{2N}{N + 1},$$ where $I(N)$ is the abundancy index of $x$ and $\sigma(N)$ is the sum of divisors of $N$. – Jose Arnaldo Bebita-Dris Aug 7 '13 at 20:02
• This earlier MSE question of mine then appears relevant. In particular, MSE user Tharsis conjectures that ${\it all}$ odd deficient numbers satisfies $$I(N) = \frac{\sigma(N)}{N} < \frac{2N}{N + 1}.$$ – Jose Arnaldo Bebita-Dris Aug 7 '13 at 20:07
• Looking at your post beginning at the fourth sentence @martycohen, I can see that, since the inequality $$I(N) = \frac{\sigma(N)}{N} < \frac{N}{\phi(N)} < \frac{2N}{N + 1} < 2$$ from my earlier comment here results to the same inequality $$\frac{1}{2} < \frac{\phi(N)}{N}$$ from your fourth sentence in this post, the succeeding claims still hold true, I'd suppose? =) – Jose Arnaldo Bebita-Dris Aug 7 '13 at 22:25
• Please refer to this newer MSE question for more details. – Jose Arnaldo Bebita-Dris Aug 7 '13 at 22:27

We can cancel the numerator of each side to get the equivalent inequality

$\prod_{i=1}^{\omega(N)}{\dfrac{1}{p_i - 1}} < \dfrac{2}{1 + \prod_{i=1}^{\omega(N)}{p_i}}$ or $1 + \prod_{i=1}^{\omega(N)}{p_i} < 2\prod_{i=1}^{\omega(N)}(p_i - 1)$.

Let's look at the sum over primes $f(x) = \sum_{p < x} \ln (p-1)$. Since $\sum_{p < x} \ln p \approx x$ (from the prime number theorem),

\begin{align} f(x) &=\sum_{p < x} \ln (p-1)\\ &=\sum_{p < x} (\ln p+\ln(1-1/p))\\ &=\sum_{p < x} \ln p+\sum_{p < x}\ln(1-1/p)\\ &\approx x-\sum_{p < x}(1/p+1/(2p^2)+...)\\ &\approx x-\ln \ln x+C\\ \end{align}

From this, $e^{f(x)} \approx e^{x-\ln \ln x+C} = C_1 e^x/\ln x$.

If $N(x) = \prod_{p < x} p$ and $M(x) = \prod_{p < x} (p-1)$, $N(x) \approx e^x$ and $M(x) \approx C_1 e^x/\ln x$, $1+N(x) < 2M(x)$ means, for large enough $x$, $e^x < C_1 e^x/\ln x$, which is false.

So that inequality is often false.

• Thank you for your answer @martycohen! I think that your answer is somewhat inconsistent with what I was expecting, since the density of the integers that I "expect" to satisfy that inequality is a bit high - at least $\approx 0.831$, but then again, this is subject of course to an "underlying hypothesis". Again, my profuse thanks for taking the time to answer my MSE question. =) – Jose Arnaldo Bebita-Dris Jul 17 '13 at 7:22
• BTW, it can be shown that the inequality holds for odd primes and (odd) prime powers. It does ${\it not}$ hold for the (even) prime $2$. – Jose Arnaldo Bebita-Dris Jul 17 '13 at 7:25
• Please refer to this newer MSE question for more details. – Jose Arnaldo Bebita-Dris Aug 7 '13 at 22:28

format: plain


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# “n-partite n-clique” with added conditions

We are given a $n$-partite weighted graph $G$. Each partition has $n$ vertices, some of which may be isolated. Each partition must contain at least one non-isolated vertex. Let us number the vertices in some $i^{th}$ partition as $V_{i1},V_{i2},...,V_{in}$. Now each non-isolated vertex $V_{ij}$ has a set of $n-1$ neighbors (one in each of the remaining $n-1$ partitions) that form a permutation with $j$. Vertex $V_{ij}$ can have other neighbors as well. In fact it can have several such sets of neighbors.

Every non-isolated vertex has a positive weight $w_{ij}$ and every edge $(V_{ij},V_{kl})$ has a positive weight $e_{ijkl}$. For every non-isolated vertex $V_{ij}$, the following must be true $w_{ij}=\sum_{l}e_{ijkl}\forall k(\neq i)=\sum_{k}e_{ijkl}\forall l(\neq j)$. As a consequence of this condition, if a vertex has only one neighbor at position $l$ (of some partition $k$) among all partitions, then it cannot have any other neighbor in that partition $k$. Also, if a vertex has only one neighbor in some partition $k$ (at some position $l$), then it cannot have any neighbor at that position in any other partition.

We conjecture that the graph $G$ will always contain a $n$-clique. Is it true?

-
Why would you conjecture this after mathoverflow.net/questions/65952/n-partite-n-clique failed? –  Douglas Zare May 27 '11 at 14:59
The permutation condition is actually a consequence of the new set of conditions. Although it limits the number of legal edges these edges should fall at the right places –  Pawan Aurora May 27 '11 at 16:50
Erm. When $k=i$, the first sum is surely $0$ (there are no edges to its own part), so you'd better check what you wrote. By the way, the word "conjecture" doesn't mean "something I would like to be true". It means "something for which I have a lot of evidence but no proof" ;). –  fedja May 27 '11 at 18:58
Now, check what you are posting. I really mean it. Look at the condition you wrote and sum all edge weights looking columnwise. You'll get (n-1) times the sum of $w_{ij}$. Now do it rowwise. You'll get $n$ times the sum of $w_{ij}$. Nonsense, isn't it? I suspect that $l\le j$ is missing but I'm too lazy to make guesses. –  fedja May 28 '11 at 14:52
I meant $l\ne j$, of course. –  fedja May 28 '11 at 14:53

Take $n$ prime and connect $(i,j)$ with $(k,l)$ if $(i-k)(j-l)\equiv 1\mod n$. Clearly, each vertex has one neighbor in each row and column except its own. Also, if $xy=1$ and $zt=1$ in $\mathbb Z_n$, then there is no reason to expect that $(x-z)(y-t)=1$ (actually, if $-3$ is not a quadratic residue modulo $n$ (say, $n=5$), it is not merely unlikely but plainly impossible, so this graph contains no triangles, leave alone $n$-cliques.
Consider $n=3$. So we have the following edges $(V_{11},V_{22}),(V_{11},V_{33})$ as well as the edge $(V_{22},V_{33})$ apart from some other edges. Clearly we have a triangle (a $3$-clique here) among $V_{11},V_{22},V_{33}$. –  Pawan Aurora May 29 '11 at 3:49
I am totally lost here. As long as every vertex has exactly one neighbor in each row and column except its own, we always have a $n$-clique. In fact every vertex is part of some $n$-clique. What am i missing here? –  Pawan Aurora May 29 '11 at 18:39
OK, you have your proof and I offered my counterexample. Now, we can either check your proof (which may be long and hard) or refute my $n=5$ construction (which should be easy if it is wrong). So, you claim that you can find a triangle in it or a vertex with a wrong set of neighbors. Do it and I'll retract my claim. –  fedja May 29 '11 at 21:20


text:
# American Institute of Mathematical Sciences

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May  2019, 18(3): 1091-1115. doi: 10.3934/cpaa.2019053

## A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group

 Universidad Nacional de Cuyo-CONICET, Parque Gral. San Martin. M5502JMA Mendoza. Argentina

* Corresponding author

Received  December 2017 Revised  July 2018 Published  November 2018

Fund Project: The authors are supported by grants PICT 2015-1701 AGENCIA, and SECTyP UNCuyo B051

In this manuscript, we are interested in the study of existence, uniqueness and comparison of viscosity and weak solutions for quasilinear equations in the Heisenberg group. In particular, we highlight the limitation of applying the Euclidean theory of viscosity solutions to get comparison of solutions of sub-elliptic equations in the Heisenberg group. Moreover, we will be concerned with the equivalence of different notions of weak solutions under appropriate assumptions for the operators under analysis. We end the paper with an application to a Radó property.

Citation: Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053
##### References:

show all references

##### References:
 [1] L. Brandolini, M. Rigoli and A. G. Setti. On the existence of positive solutions of Yamabe-type equations on the Heisenberg group. Electronic Research Announcements, 1996, 2: 101-107. [2] Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747 [3] Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113 [4] Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 [5] Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 [6] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [7] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 [8] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 [9] Elder Jesús Villamizar-Roa, Henry Lamos-Díaz, Gilberto Arenas-Díaz. Very weak solutions for the magnetohydrodynamic type equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 957-972. doi: 10.3934/dcdsb.2008.10.957 [10] Yu-Zhu Wang, Yin-Xia Wang. Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 851-866. doi: 10.3934/cpaa.2013.12.851 [11] Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031 [12] Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221 [13] Guozhen Lu and Juncheng Wei. On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups. Electronic Research Announcements, 1997, 3: 83-89. [14] Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461 [15] Zhen Lei, Yi Zhou. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 575-583. doi: 10.3934/dcds.2009.25.575 [16] Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176 [17] Josef Diblík, Zdeněk Svoboda. Existence of strictly decreasing positive solutions of linear differential equations of neutral type. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 67-84. doi: 10.3934/dcdss.2020004 [18] Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025 [19] Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117 [20] Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897

2018 Impact Factor: 0.925


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Bits and bytes

# Bytes

AP.CSP:
DAT‑1 (EU)
,
DAT‑1.A (LO)
,
DAT‑1.A.4 (EK)
,
DAT‑1.A.6 (EK)
A bit is the smallest piece of information in a computer, a single value storing either start text, 0, end text or start text, 1, end text.
A byte is a unit of digital information that consists of 8 of those bits.
Here's a single byte of information:
start text, 11110110, end text
Here are three more bytes of information:
start text, 0, end text, start text, 0, end text, start text, 0, end text, start text, 0, end text, start text, 1010, end text, start text, 0, end text, start text, 101010, end text, start text, 0, end text, start text, 11011011, end text

### From bits to bytes

Conversion between bits and bytes is a simple calculation: divide by 8 to convert from bits to bytes or multiply by 8 to convert from bytes to bits. Try it yourself!
How many bytes long is this binary sequence?
start text, 101110, end text, start text, 01, end text, start text, 11011011, end text, start text, 011010, end text, start text, 01, end text, start text, 0, end text, start text, 0110101, end text, start text, 01111111, end text

How many bits are in 8 bytes of information?

### Why bytes?

What is so special about 8 bits that it deserves its own name?
Computers do process all data as bits, but they prefer to process bits in byte-sized groupings. Or to put it another way: a byte is how much a computer likes to "bite" at once.
A computer with a thought bubble with three bytes inside
The byte is also the smallest addressable unit of memory in most modern computers. A computer with byte-addressable memory can not store an individual piece of data that is smaller than a byte.
Memory
111110000
200001010
300100110
411000001

### What's in a byte?

A byte represents different types of information depending on the context. It might represent a number, a letter, or a program instruction. It might even represent part of an audio recording or a pixel in an image.
We'll explore how computers can use bits and bytes to represent all types of information in this unit.

## Want to join the conversation?

• What if we have a number of bits that are non-divisible by 8? Is that possible?
• You can always add zeroes to create 8 bit chunks.
• "At , what does the concept of byte-addressable imply? A computer should be able to store an individual piece of information, being made of wires?"
• A byte is a grouping of eight bits. Byte-addressable means that the computer stores data in bytes instead of single bits for example.

Wires are used for transporting information, computers generally don't use them as a storage device.
• How do quantum computers work with bits and bytes?
• if bits were a thing you can hold what would it look like?
(1 vote)
• We sort of already have that. Take a coin for example, it could either have heads facing up, or tails. So it has two possible states, which is what a bit is. However, a coin stores a bit, rather than actually being a bit. I don't exactly know what bits themself would look like, since they're really just an idea, rather than an actual object.
• Why bytes?Like why is it 8 and not like like 10 wouldn't that make it easier.
(1 vote)
• In our base 10 system, 10 is a place value represented by 10^1. In binary, however, it's in base 2, so 8 is a place value represented by 2^3.

It's 8, because 8 is close to our friend 10.
• How would a computer store a piece of information that only requires a bit like if a lightbulb is on or off in byte form?
• The type of information you are inquiring about is represented by a data type known as a Boolean. A Boolean data type is stored in memory as either a 0 (an equivalent to the lightbulb being in the off state) or 1 (an equivalent to the lightbulb being in the on state).

Using C++ as a reference, Boolean values are stored as 8-bit (1-byte) values so that they are still addressable. Memory is byte-addressable, so referencing a single bit becomes prohibitive. You can still represent one Boolean per bit, but that would require extra work on the part of the developer as they must now understand and work with bit manipulation.
• So if I'm understanding it correctly, a byte doesn't necessarily represent a number using 8 digits - for example the byte 10110110 can mean that:
First process = 1 = On
Second process = 0 = Off
Third = 1 = On
Fourth = 1 = On
Fifth = 0 = Off
Etc...

Is this correct? Or if a byte is representing separate processes it has to be 00000001 for the first process, 00000000 for the second, etc
• A byte can represent a number using 8 binary digits; 10110110 would be equivalent to the base-10 number 182 (if we are not considering two's complement representation) or -74 (if we are considering two's complement representation).

However, a byte (or a multitude of bytes) can represent different types of information depending on the context (as mentioned in the article). A multitude of bytes can represent an image, a text file, a video file, etc.

Using the byte to represent whether a process is "On" or "Off" in process counts of 8 is not a conventional use that I have come across. However, if you decided to formulate your own unique problem such that a byte represents processes which are on and off then it could be used in that manner (it makes for an intriguing thought experiment). One of the amazing properties of binary representation is how many different types of information can be represented by simple 0s and 1s.

Do note, though, that on the machine itself processes have process IDs (PIDs) and their metadata is managed by the operating system. Bytes will not be used to represent whether or not processes on a computer are running by flipping bits of the byte to 0 or 1, there are different mechanisms at play.
• how do you tell the difference between a 0 and a 1
• You have protocols to define the timing and so on (when does the message start, end).
And then you use different voltages ("high" and "low") to differentiate between 1 and 0.
• are there any patterns that one can pic up on in binary.


text:
Chapter 5, Problem 12P

### Fundamentals of Financial Manageme...

9th Edition
Eugene F. Brigham + 1 other
ISBN: 9781305635937

Chapter
Section

### Fundamentals of Financial Manageme...

9th Edition
Eugene F. Brigham + 1 other
ISBN: 9781305635937
Textbook Problem
46 views

# EFFECTIVE RATE OF INTEREST Find the interest rates earned on each of the following: a. You borrow $720 and promise to pay back$792 at the end of 1 year. b. You lend $720 and the borrower promises to pay you$792 at the end of 1 year. c. You borrow $65,000 and promise to pay back$98,319 at the end of 14 years. d. You borrow $15,000 and promise to make payments of$4,058.60 at the end of each year for 5 years.

(a)

Summary Introduction

To calculate: Interest rate earned.

Interest Rate:

Interest rate is an amount of money that is depicted as a percentage of principal. The interest rate is an amount that a lender charges from borrower to lend the money for borrower’s use. The interest rate as well as the term interest varies as per the discussion between the lender and the borrower.

Explanation

The formula for calculating interest rate is,

Interestrate(R)=IPT×100 (I)

Here,

• R is the interest rate.
• I is the amount paid in interest for a particular time.
• P is the principal amount.
• T is the time period

(b)

Summary Introduction

To calculate: Interest rate earned.

Interest Rate:

Interest rate is an amount of money that is depicted as a percentage of principal. The interest rate is an amount that a lender charges from borrower to lend the money for borrower’s use. The interest rate as well as the term interest varies as per the discussion between the lender and the borrower.

(c)

Summary Introduction

To calculate: Interest rate earned.

Interest Rate:

Interest rate is an amount of money that is depicted as a percentage of principal. The interest rate is an amount that a lender charges from borrower to lend the money for borrower’s use. The interest rate as well as the term interest varies as per the discussion between the lender and the borrower.

(d)

Summary Introduction

To calculate: Interest rate earned.

Interest Rate:

Interest rate is an amount of money that is depicted as a percentage of principal. The interest rate is an amount that a lender charges from borrower to lend the money for borrower’s use. The interest rate as well as the term interest varies as per the discussion between the lender and the borrower.

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text:
cumsum {base} R Documentation

Cumulative Sums, Products, and Extremes

Description

Returns a vector whose elements are the cumulative sums, products, minima or maxima of the elements of the argument.

Usage

cumsum(x)
cumprod(x)
cummax(x)
cummin(x)

Arguments

 x a numeric or complex (not cummin or cummax) object, or an object that can be coerced to one of these.

Details

These are generic functions: methods can be defined for them individually or via the Math group generic.

Value

A vector of the same length and type as x (after coercion), except that cumprod returns a numeric vector for integer input (for consistency with *). Names are preserved.

An NA value in x causes the corresponding and following elements of the return value to be NA, as does integer overflow in cumsum (with a warning).

S4 methods

cumsum and cumprod are S4 generic functions: methods can be defined for them individually or via the Math group generic. cummax and cummin are individually S4 generic functions.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. (cumsum only.)

Examples

cumsum(1:10)
cumprod(1:10)
cummin(c(3:1, 2:0, 4:2))
cummax(c(3:1, 2:0, 4:2))

[Package base version 4.2.0 Index]


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• ### First Data Release of the COSMOS Lyman-Alpha Mapping And Tomography Observations: 3D Lyman-$\alpha$ Forest Tomography at 2.05 < z < 2.55(1710.02894)

Aug. 3, 2018 astro-ph.CO, astro-ph.GA
Faint star-forming galaxies at $z\sim 2-3$ can be used as alternative background sources to probe the Lyman-$\alpha$ forest in addition to quasars, yielding high sightline densities that enable 3D tomographic reconstruction of the foreground absorption field. Here, we present the first data release from the COSMOS Lyman-Alpha Mapping And Mapping Observations (CLAMATO) Survey, which was conducted with the LRIS spectrograph on the Keck-I telescope. Over an observational footprint of 0.157$\mathrm{deg}^2$ within the COSMOS field, we used 240 galaxies and quasars at $2.17<z<3.00$, with a mean comoving transverse separation of $2.37\,h^{-1}\,\mathrm{Mpc^3}$, as background sources probing the foreground Lyman-$\alpha$ forest absorption at $2.05<z<2.55$. The Lyman-$\alpha$ forest data was then used to create a Wiener-filtered tomographic reconstruction over a comoving volume of $3.15\,\times 10^5\,h^{-3}\,\mathrm{Mpc^3}$ with an effective smoothing scale of $2.5\,h^{-1}\,\mathrm{Mpc}$. In addition to traditional figures, this map is also presented as a virtual-reality YouTube360 video visualization and manipulable interactive figure. We see large overdensities and underdensities that visually agree with the distribution of coeval galaxies from spectroscopic redshift surveys in the same field, including overdensities associated with several recently-discovered galaxy protoclusters in the volume. This data release includes the redshift catalog, reduced spectra, extracted Lyman-$\alpha$ forest pixel data, and tomographic map of the absorption.
• ### A Detection of $z$~2.3 Cosmic Voids from 3D Lyman-$\alpha$ Forest Tomography in the COSMOS Field(1710.02612)

June 26, 2018 astro-ph.CO
We present the most distant detection of cosmic voids ($z \sim 2.3$) and the first detection of three-dimensional voids in the Lyman-$\alpha$ forest. We used a 3D tomographic map of the absorption with effective comoving spatial resolution of $2.5\,h^{-1}\mathrm{Mpc}$ and volume of $3.15\times 10^5\,h^{-3}\mathrm{Mpc}^3$, which was reconstructed from moderate-resolution Keck-I/LRIS spectra of 240 background Lyman-break galaxies and quasars in a $0.16\,\mathrm{deg}^2$ footprint in the COSMOS field. Voids were detected using a spherical overdensity finder calibrated from hydrodynamical simulations of the intergalactic medium. This allows us to identify voids in the IGM corresponding to voids in the underlying matter density field, yielding a consistent volume fraction of voids in both data (19.5%) and simulations (18.2%). We fit excursion set models to the void radius function and compare the radially-averaged stacked profiles of large voids ($r > 5$ $h^{-1}$ Mpc) to stacked voids in mock observations and the simulated density field. Comparing with 432 coeval galaxies with spectroscopic redshifts in the same volume as the tomographic map, we find that the tomography-identified voids are underdense in galaxies by 5.95$\sigma$ compared to random cells.
• ### HST imaging of the brightest z~8-9 galaxies from UltraVISTA: the extreme bright end of the UV luminosity function(1706.04613)

Oct. 16, 2017 astro-ph.GA
We report on the discovery of three especially bright candidate $z_{phot} \gtrsim 8$ galaxies. Five sources were targeted for follow-up with HST/WFC3, selected from a larger sample of 16 bright ($24.8 \lesssim H\lesssim25.5$~mag) candidate $z\gtrsim 8$ LBGs identified over the 1.6 degrees$^2$ of the COSMOS/UltraVISTA field. These were identified as Y and J dropouts by leveraging the deep (Y-to-$K_{S} \sim 25.3-24.8$~mag, $5\sigma$) NIR data from the UltraVISTA DR3 release, deep ground based optical imaging from the CFHTLS and Subaru Suprime Cam programs and Spitzer/IRAC mosaics combining observations from the SMUVS and SPLASH programs. Through the refined spectral energy distributions, which now also include new HyperSuprime Cam g, r, i, z and Y band data, we confirm that 3/5 galaxies have robust $z_{phot}\sim8.0-8.7$, consistent with the initial selection. The remaining 2/5 galaxies have a nominal $z_{phot}\sim2$. However, if we use the HST data alone, these objects have increased probability of being at $z\sim9$. Furthermore, we measure mean UV continuum slopes $\beta=-1.91\pm0.26$ for the three $z\sim8-9$ galaxies, marginally bluer than similarly luminous $z\sim4-6$ in CANDELS but consistent with previous measurements of similarly luminous galaxies at $z\sim7$. The circularized effective radius for our brightest source is $0.9\pm0.2$ kpc, similar to previous measurements for a bright $z\sim11$ galaxy and bright $z\sim7$ galaxies. Finally, enlarging our sample to include the six brightest $z\sim8$ LBGs identified over UltraVISTA (i.e., including three other sources from Labbe et al. 2017, in prep.) we estimate for the first time the volume density of galaxies at the extreme bright ($M_{UV}\sim-22$~mag) end of the $z\sim8$ UV LF. Despite this exceptional result, the still large statistical uncertainties do not allow us to discriminate between a Schechter and a double power-law form.
• ### Neutral hydrogen (HI) gas content of galaxies at $z \approx 0.32$(1709.07596)

Sept. 22, 2017 astro-ph.GA
We use observations made with the Giant Metrewave Radio Telescope (GMRT) to probe the neutral hydrogen (HI) gas content of field galaxies in the VIMOS VLT Deep Survey (VVDS) 14h field at $z \approx 0.32$. Because the HI emission from individual galaxies is too faint to detect at this redshift, we use an HI spectral stacking technique using the known optical positions and redshifts of the 165 galaxies in our sample to co-add their HI spectra and thus obtain the average HI mass of the galaxies. Stacked HI measurements of 165 galaxies show that 95 per cent of the neutral gas is found in blue, star-forming galaxies. Among these galaxies, those having lower stellar mass are more gas-rich than more massive ones. We apply a volume correction to our HI measurement to evaluate the HI gas density at $z \approx 0.32$ as $\Omega_{HI}=(0.50\pm0.18) \times 10^{-3}$ in units of the cosmic critical density. This value is in good agreement with previous results at z < 0.4, suggesting no evolution in the neutral hydrogen gas density over the last $\sim 4$ Gyr. However the $z \approx 0.32$ gas density is lower than that at $z \sim 5$ by at least a factor of two.
• ### An ALMA survey of submillimeter galaxies in the COSMOS field: Multiwavelength counterparts and redshift distribution(1708.05748)

Sept. 7, 2017 astro-ph.GA
We carried out targeted ALMA observations of 129 fields in the COSMOS region at 1.25 mm, detecting 152 galaxies at S/N$\geq$5 with an average continuum RMS of 150 $\mu$Jy. These fields represent a S/N-limited sample of AzTEC / ASTE sources with 1.1 mm S/N$\geq$4 over an area of 0.72 square degrees. Given ALMA's fine resolution and the exceptional spectroscopic and multiwavelength photometric data available in COSMOS, this survey allows us unprecedented power in identifying submillimeter galaxy counterparts and determining their redshifts through spectroscopic or photometric means. In addition to 30 sources with prior spectroscopic redshifts, we identified redshifts for 113 galaxies through photometric methods and an additional nine sources with lower limits, which allowed a statistically robust determination of the redshift distribution. We have resolved 33 AzTEC sources into multi-component systems and our redshifts suggest that nine are likely to be physically associated. Our overall redshift distribution peaks at $z\sim$2.0 with a high redshift tail skewing the median redshift to $\tilde{z}$=2.48$\pm$0.05. We find that brighter millimeter sources are preferentially found at higher redshifts. Our faintest sources, with S$_{1.25 \rm mm}$<1.25 mJy, have a median redshift of $\tilde{z}$=2.18$\pm$0.09, while the brightest sources, S$_{1.25 \rm mm}$>1.8 mJy, have a median redshift of $\tilde{z}$=3.08$\pm$0.17. After accounting for spectral energy distribution shape and selection effects these results are consistent with several previous submillimeter galaxy surveys, and moreover, support the conclusion that the submillimeter galaxy redshift distribution is sensitive to survey depth.
• ### An ALMA survey of submillimetre galaxies in the COSMOS field: Physical properties derived from energy balance spectral energy distribution modelling(1707.00637)

July 3, 2017 astro-ph.GA
We determine the physical properties of a sample of SMGs in the COSMOS field that were pre-selected at the observed wavelength of $\lambda_{\rm obs}=1.1$ mm, and followed up at $\lambda_{\rm obs}=1.3$ mm with ALMA. We used MAGPHYS to fit the panchromatic (ultraviolet to radio) SEDs of 124 of the target SMGs, 19.4% of which are spectroscopically confirmed. The SED analysis was complemented by estimating the gas masses of the SMGs by using the $\lambda_{\rm obs}=1.3$ mm emission as a tracer of the molecular gas. The sample median and 16th-84th percentile ranges of the stellar masses, SFRs, dust temperatures, and dust and gas masses were derived to be $\log(M_{\star}/{\rm M}_{\odot})=11.09^{+0.41}_{-0.53}$, ${\rm SFR}=402^{+661}_{-233}$ ${\rm M}_{\odot}~{\rm yr}^{-1}$, $T_{\rm dust}=39.7^{+9.7}_{-7.4}$ K, $\log(M_{\rm dust}/{\rm M}_{\odot})=9.01^{+0.20}_{-0.31}$, and $\log(M_{\rm gas}/{\rm M}_{\odot})=11.34^{+0.20}_{-0.23}$, respectively. The median gas-to-dust ratio and gas fraction were found to be $120^{+73}_{-30}$ and $0.62^{+0.27}_{-0.23}$, respectively. We found that 57.3% of our SMGs populate the main sequence (MS) of star-forming galaxies, while 41.9% of the sources lie above the MS by a factor of >3 (one source lies below the MS). The largest 3 GHz radio sizes are found among the MS sources. Those SMGs that appear irregular in the rest-frame UV are predominantly starbursts, while the MS SMGs are mostly disk-like. The larger radio-emitting sizes of the MS SMGs compared to starbursts is a likely indication of their more widespread, less intense star formation. The irregular UV morphologies of the starburst SMGs are likely to echo their merger nature. Our results suggest that the transition from high-$z$ SMGs to local ellipticals via compact, quiescent galaxies (cQGs) at $z \sim 2$ might not be universal, and the latter population might also descend from the so-called blue nuggets.
• ### The Dust Attenuation of Star-forming Galaxies at $z\sim3$ and Beyond: New Insights from ALMA Observations(1705.01559)

May 25, 2017 astro-ph.GA
We present results on the dust attenuation of galaxies at redshift $\sim3-6$ by studying the relationship between the UV spectral slope ($\beta_{\rm UV}$) and the infrared excess (IRX; $L_{\rm IR}$/$L_{\rm UV}$) using ALMA far-infrared continuum observations. Our study is based on a sample of 67 massive, star-forming galaxies with a median mass of $M_{\ast}\sim 10^{10.7}\,M_{\rm \odot}$ spanning a redshift range $z=2.6-3.7$ (median $z=3.2$) that were observed with ALMA at $\lambda_{rest}=300\,{\rm \mu m}$. Both the individual ALMA detections (41 sources) and stacks including all galaxies show the IRX-$\beta_{\rm UV}$ relationship at $z\sim3$ is mostly consistent with that of local starburst galaxies on average. However, we find evidence for a large dispersion around the mean relationship by up to $\pm0.5$ dex. Nevertheless, the locally calibrated dust correction factors based on the IRX-$\beta_{\rm UV}$ relation are on average applicable to main-sequence $z\sim3$ galaxies. This does not appear to be the case at even higher redshifts, however. Using public ALMA observations of $z\sim4-6$ galaxies we find evidence for a significant evolution in the IRX-$\beta_{\rm UV}$ and the IRX-$M_{\ast}$ relations beyond $z\sim3$ toward lower IRX values. We discuss several caveats that could affect these results, including the assumed dust temperature. ALMA observations of larger $z>3$ galaxy samples will be required to confirm this intriguing redshift evolution.
• ### The FMOS-COSMOS survey of star-forming galaxies at $z\sim1.6$. V: Properties of dark matter halos containing H$\alpha$ emitting galaxies(1703.08326)

March 24, 2017 astro-ph.GA
We study the properties of dark matter halos that contain star-forming galaxies at $1.43 \le z \le 1.74$ using the FMOS-COSMOS survey. The sample consists of 516 objects with a detection of the H$\alpha$ emission line, that represent the star-forming population at this epoch having a stellar mass range of $10^{9.57}\le M_\ast/M_\odot \lesssim 10^{11.4}$ and a star formation rate range of $15\lesssim \mathrm{SFR}/(M_\odot \mathrm{yr^{-1}}) \lesssim 600$. We measure the projected two-point correlation function while carefully taking into account observational biases, and find a significant clustering amplitude at scales of $0.04$-$10~h^{-1}~\mathrm{cMpc}$, with a correlation length $r_0 = 5.21^{+0.70}_{-0.67}~h^{-1}~\mathrm{cMpc}$ and a bias $b=2.59^{+0.41}_{-0.34}$. We interpret our clustering measurement using a halo occupation distribution model. The sample galaxies appear to reside in halos with mass $M_\mathrm{h} = 4.6^{+1.1}_{-1.6}\times10^{12}~h^{-1}M_\odot$ on average that will likely become present-day halos of mass $M_\mathrm{h} (z=0) \sim2\times10^{13}~h^{-1}M_\odot$, equivalent to the typical halo mass scale of galaxy groups. We then confirm the decline of the stellar-to-halo mass ratio at $M_\mathrm{h}<10^{12}~M_\odot$, finding $M_\ast/M_\mathrm{h} \approx 5\times10^{-3}$ at $M_\mathrm{h}=10^{11.86}~M_\odot$, which is lower by a factor of 2-4 than those measured at higher masses. Finally, we use our results to illustrate the future capabilities of Subaru's Prime-Focus Spectrograph, a next-generation instrument that will provide strong constraints on the galaxy-formation scenario by obtaining precise measurements of galaxy clustering at $z>1$.
• ### An ALMA survey of submillimetre galaxies in the COSMOS field: The extent of the radio-emitting region revealed by 3 GHz imaging with the Very Large Array(1702.07527)

Feb. 24, 2017 astro-ph.GA
We determine the radio size distribution of a large sample of 152 SMGs in COSMOS that were detected with ALMA at 1.3 mm. For this purpose, we used the observations taken by the VLA-COSMOS 3 GHz Large Project. One hundred and fifteen of the 152 target SMGs were found to have a 3 GHz counterpart. The median value of the major axis FWHM at 3 GHz is derived to be $4.6\pm0.4$ kpc. The radio sizes show no evolutionary trend with redshift, or difference between different galaxy morphologies. We also derived the spectral indices between 1.4 and 3 GHz, and 3 GHz brightness temperatures for the sources, and the median values were found to be $\alpha=-0.67$ and $T_{\rm B}=12.6\pm2$ K. Three of the target SMGs, which are also detected with the VLBA, show clearly higher brightness temperatures than the typical values. Although the observed radio emission appears to be predominantly powered by star formation and supernova activity, our results provide a strong indication of the presence of an AGN in the VLBA and X-ray-detected SMG AzTEC/C61. The median radio-emitting size we have derived is 1.5-3 times larger than the typical FIR dust-emitting sizes of SMGs, but similar to that of the SMGs' molecular gas component traced through mid-$J$ line emission of CO. The physical conditions of SMGs probably render the diffusion of cosmic-ray electrons inefficient, and hence an unlikely process to lead to the observed extended radio sizes. Instead, our results point towards a scenario where SMGs are driven by galaxy interactions and mergers. Besides triggering vigorous starbursts, galaxy collisions can also pull out the magnetised fluids from the interacting disks, and give rise to a taffy-like synchrotron-emitting bridge. This provides an explanation for the spatially extended radio emission of SMGs, and can also cause a deviation from the well-known IR-radio correlation.
• ### Hidden Starbursts and Active Galactic Nuclei at 0<z<4 from the Herschel-VVDS-CFHTLS-D1 Field: Inferences on Coevolution and Feedback(1311.5228)

Dec. 24, 2016 astro-ph.CO
We investigate of the properties of $\sim$2000 Herschel/SPIRE-selected galaxies from $0<z<4$ using a combination of extensive spectroscopy, deep imaging from CFHT, VLA, Spitzer, XMM-Newton, and Herschel, and well-calibrated SED fitting. Herschel galaxies are observed to span a range of stellar masses, colors, and absolute magnitudes equivalent to galaxies undetected in SPIRE. Though many Herschel galaxies appear to be in transition, such galaxies are largely consistent with normal star-forming galaxies when rest-frame colors are utilized. The nature of the star-forming "main sequence" is studied and we warn against adopting this framework unless the main sequence is determined precisely. Herschel galaxies at different total infrared luminosities ($L_{TIR}$) are compared. Bluer colors, larger nebular extinctions, and larger contributions from younger stellar populations are observed for galaxies with larger $L_{TIR}$, suggesting that low-$L_{TIR}$ galaxies are undergoing rejuvenated starbursts while galaxies with higher $L_{TIR}$ are forming a larger percentage of their stellar mass. A variety of methods are used to select powerful active galactic nuclei (AGN). Galaxies hosting AGN are observed to be undergoing starbursts more commonly and vigorously than a matched sample of galaxies without powerful AGN and, additionally, the fraction of galaxies with an AGN increases with increasing star formation rate at all redshifts. At all redshifts ($0<z<4$) the most prodigious star-forming galaxies are found to contain the highest fraction of powerful AGN. For redshift bins that allow a comparison ($z>0.5$), the highest $L_{TIR}$ galaxies in a given redshift bin are unobserved by SPIRE at subsequently lower redshifts, a trend linked to downsizing. In conjunction with other results, this evidence is used to argue for prevalent AGN-driven quenching in starburst galaxies across cosmic time.
• ### Power law cosmology model comparison with CMB scale information(1610.03371)

Nov. 14, 2016 astro-ph.CO
Despite the ability of the cosmological concordance model ($\Lambda$CDM) to describe the cosmological observations exceedingly well, power law expansion of the Universe scale radius, $R(t)\propto t^n$, has been proposed as an alternative framework. We examine here these models, analyzing their ability to fit cosmological data using robust model comparison criteria. Type Ia supernovae (SNIa), baryonic acoustic oscillations (BAO) and acoustic scale information from the cosmic microwave background (CMB) have been used. We find that SNIa data either alone or combined with BAO can be well reproduced by both $\Lambda$CDM and power law expansion models with $n\sim 1.5$, while the constant expansion rate model $(n=1)$ is clearly disfavored. Allowing for some redshift evolution in the SNIa luminosity essentially removes any clear preference for a specific model. The CMB data are well known to provide the most stringent constraints on standard cosmological models, in particular, through the position of the first peak of the temperature angular power spectrum, corresponding to the sound horizon at recombination, a scale physically related to the BAO scale. Models with $n\geq 1$ lead to a divergence of the sound horizon and do not naturally provide the relevant scales for the BAO and the CMB. We retain an empirical footing to overcome this issue: we let the data choose the preferred values for these scales, while we recompute the ionization history in power law models, to obtain the distance to the CMB. In doing so, we find that the scale coming from the BAO data is not consistent with the observed position of the first peak of the CMB temperature angular power spectrum for any power law cosmology. Therefore, we conclude that when the three standard probes are combined, the $\Lambda$CDM model is very strongly favored over any of these alternative models, which are then essentially ruled out.
• ### ALMA spectroscopic survey in the Hubble Ultra Deep Field: CO luminosity functions and the evolution of the cosmic density of molecular gas(1607.06770)

Sept. 12, 2016 astro-ph.GA
In this paper we use ASPECS, the ALMA Spectroscopic Survey in the {\em Hubble} Ultra Deep Field (UDF) in band 3 and band 6, to place blind constraints on the CO luminosity function and the evolution of the cosmic molecular gas density as a function of redshift up to $z\sim 4.5$. This study is based on galaxies that have been solely selected through their CO emission and not through any other property. In all of the redshift bins the ASPECS measurements reach the predicted `knee' of the CO luminosity function (around $5\times10^{9}$ K km/s pc$^2$). We find clear evidence of an evolution in the CO luminosity function with respect to $z\sim 0$, with more CO luminous galaxies present at $z\sim 2$. The observed galaxies at $z\sim 2$ also appear more gas-rich than predicted by recent semi-analytical models. The comoving cosmic molecular gas density within galaxies as a function of redshift shows a factor 3-10 drop from $z \sim 2$ to $z \sim 0$ (with significant error bars), and possibly a decline at $z>3$. This trend is similar to the observed evolution of the cosmic star formation rate density. The latter therefore appears to be at least partly driven by the increased availability of molecular gas reservoirs at the peak of cosmic star formation ($z\sim2$).
• ### ALMA Spectroscopic Survey in the Hubble Ultra Deep Field: The Infrared Excess of UV-selected z=2-10 galaxies as a function of UV-continuum Slope and Stellar Mass(1606.05280)

Aug. 6, 2016 astro-ph.GA
We make use of deep 1.2mm-continuum observations (12.7microJy/beam RMS) of a 1 arcmin^2 region in the Hubble Ultra Deep Field to probe dust-enshrouded star formation from 330 Lyman-break galaxies spanning the redshift range z=2-10 (to ~2-3 Msol/yr at 1sigma over the entire range). Given the depth and area of ASPECS, we would expect to tentatively detect 35 galaxies extrapolating the Meurer z~0 IRX-beta relation to z>~2 (assuming T_d~35 K). However, only 6 tentative detections are found at z>~2 in ASPECS, with just three at >3sigma. Subdividing z=2-10 galaxies according to stellar mass, UV luminosity, and UV-continuum slope and stacking the results, we only find a significant detection in the most massive (>10^9.75 Msol) subsample, with an infrared excess (IRX=L_{IR}/L_{UV}) consistent with previous z~2 results. However, the infrared excess we measure from our large selection of sub-L* (<10^9.75 Msol) galaxies is 0.11(-0.42)(+0.32) and 0.14(-0.14)(+0.15) at z=2-3 and z=4-10, respectively, lying below even an SMC IRX-beta relation (95% confidence). These results demonstrate the relevance of stellar mass for predicting the IR luminosity of z>~2 galaxies. We furthermore find that the evolution of the IRX-stellar mass relationship depends on the evolution of the dust temperature. If the dust temperature increases monotonically with redshift (as (1+z)^0.32) such that T_d~44-50 K at z>=4, current results are suggestive of little evolution in this relationship to z~6. We use these results to revisit recent estimates of the z>~3 SFR density. One less obvious implication is in interpreting the high Halpha EWs seen in z~5 galaxies: our results imply that star-forming galaxies produce Lyman-continuum photons at twice the efficiency (per unit UV luminosity) as implied in conventional models. Star-forming galaxies can then reionize the Universe, even if the escape fraction is <10%.
• PFS (Prime Focus Spectrograph), a next generation facility instrument on the 8.2-meter Subaru Telescope, is a very wide-field, massively multiplexed, optical and near-infrared spectrograph. Exploiting the Subaru prime focus, 2394 reconfigurable fibers will be distributed over the 1.3 deg field of view. The spectrograph has been designed with 3 arms of blue, red, and near-infrared cameras to simultaneously observe spectra from 380nm to 1260nm in one exposure at a resolution of ~1.6-2.7A. An international collaboration is developing this instrument under the initiative of Kavli IPMU. The project is now going into the construction phase aiming at undertaking system integration in 2017-2018 and subsequently carrying out engineering operations in 2018-2019. This article gives an overview of the instrument, current project status and future paths forward.
• ### ALMA Spectroscopic Survey in the Hubble Ultra Deep Field: Survey Description(1607.06768)

July 22, 2016 astro-ph.GA
We present the rationale for and the observational description of ASPECS: The ALMA SPECtroscopic Survey in the Hubble Ultra-Deep Field (UDF), the cosmological deep field that has the deepest multi-wavelength data available. Our overarching goal is to obtain an unbiased census of molecular gas and dust continuum emission in high-redshift (z$>$0.5) galaxies. The $\sim$1$'$ region covered within the UDF was chosen to overlap with the deepest available imaging from HST. Our ALMA observations consist of full frequency scans in band 3 (84-115 GHz) and band 6 (212-272 GHz) at approximately uniform line sensitivity ($L'_{\rm CO}\sim$2$\times$10$^{9}$ K km/s pc$^2$), and continuum noise levels of 3.8 $\mu$Jy beam$^{-1}$ and 12.7 $\mu$Jy beam$^{-1}$, respectively. The molecular surveys cover the different rotational transitions of the CO molecule, leading to essentially full redshift coverage. The [CII] emission line is also covered at redshifts $6.0<z<8.0$. We present a customized algorithm to identify line candidates in the molecular line scans, and quantify our ability to recover artificial sources from our data. Based on whether multiple CO lines are detected, and whether optical spectroscopic redshifts as well as optical counterparts exist, we constrain the most likely line identification. We report 10 (11) CO line candidates in the 3mm (1mm) band, and our statistical analysis shows that $<$4 of these (in each band) are likely spurious. Less than 1/3 of the total CO flux in the low-J CO line candidates are from sources that are not associated with an optical/NIR counterpart. We also present continuum maps of both the band 3 and band 6 observations. The data presented here form the basis of a number of dedicated studies that are presented in subsequent papers.
• ### A New Constraint on the Ly$\alpha$ Fraction of UV Very Bright Galaxies at Redshift 7(1604.02214)

April 14, 2016 astro-ph.CO, astro-ph.GA
We study the extent to which very bright (-23.0 < MUV < -21.75) Lyman-break selected galaxies at redshifts z~7 display detectable Lya emission. To explore this issue, we have obtained follow-up optical spectroscopy of 9 z~7 galaxies from a parent sample of 24 z~7 galaxy candidates selected from the 1.65 sq.deg COSMOS-UltraVISTA and SXDS-UDS survey fields using the latest near-infrared public survey data, and new ultra-deep Subaru z'-band imaging (which we also present and describe in this paper). Our spectroscopy has yielded only one possible detection of Lya at z=7.168 with a rest-frame equivalent width EW_0 = 3.7 (+1.7/-1.1) Angstrom. The relative weakness of this line, combined with our failure to detect Lya emission from the other spectroscopic targets allows us to place a new upper limit on the prevalence of strong Lya emission at these redshifts. For conservative calculation and to facilitate comparison with previous studies at lower redshifts, we derive a 1-sigma upper limit on the fraction of UV bright galaxies at z~7 that display EW_0 > 50 Angstrom, which we estimate to be < 0.23. This result may indicate a weak trend where the fraction of strong Lya emitters ceases to rise, and possibly falls between z~6 and z~7. Our results also leave open the possibility that strong Lya may still be more prevalent in the brightest galaxies in the reionization era than their fainter counterparts. A larger spectroscopic sample of galaxies is required to derive a more reliable constraint on the neutral hydrogen fraction at z~7 based on the Lya fraction in the bright galaxies.
• ### The VIMOS Ultra Deep Survey: Lyman Alpha Emission and Stellar Populations of Star-Forming Galaxies at 2<z<6(1603.05992)

March 18, 2016 astro-ph.CO, astro-ph.GA
The extensive ground-based spectroscopy campaign from the VIMOS Ultra-Deep Survey (VUDS), and the deep multi-wavelength photometry in three very well observed extragalactic fields (ECDFS, COSMOS, VVDS), allow us to investigate physical properties of a large sample (~4000 galaxies) of spectroscopically confirmed faint (i_{AB}<~25 mag) star-forming galaxies, with and without Lyman alpha in emission, at z~2-6. The fraction of Lyman alpha emitters (LAEs; equivalent width (EW)=>20A) increases from ~10% at z~2 to ~40% at z~5-6, which is consistent with previous studies that employ higher Lyman alpha EW cut. This increase in the LAE fraction could be, in part, due to a decrease in the dust content of galaxies as redshift increases. When we compare best-fit SED estimated stellar parameters for LAEs and non-LAEs, we find that E(B-V) is smaller for LAEs at all redshifts and the difference in the median E(B-V) between LAEs and non-LAEs increases as redshift increases, from 0.05 at z~2 to 0.1 at z~3.5 to 0.2 at z~5-6. For the luminosities probed here (~L*), we find that star formation rates (SFRs) and stellar masses of galaxies, with and without Lyman alpha in emission, show small differences such that, LAEs have lower SFRs and stellar masses compared to non-LAEs. This result could be a direct consequence of the sample selection. Our sample of LAEs are selected based on their continuum magnitudes and they probe higher continuum luminosities compared to narrow-band/emission line selected LAEs. Based on our results, it is important to note that all LAEs are not universally similar and their properties are strongly dependent on the sample selection, and/or continuum luminosities.
• ### The Case for a James Webb Space Telescope Extragalactic Key Project(1512.04530)

Dec. 14, 2015 astro-ph.GA, astro-ph.IM
The upcoming launch of the James Webb Space Telescope (JWST) in less than three years is certain to bring a revolution in our understanding of many area of astrophysics, with one of the key goals being galaxy evolution. As the first proposals will be due in a little over two years, the time is ripe to take a holistic look at the science goals which the community would wish to accomplish with this observatory. Contrary to our experiences with the Hubble Space Telescope, which has now operated successfully for over two decades due to several timely servicing missions, the lifetime of JWST is finite and relatively short, with a lifetime requirement of five years, and a ten-year goal. Following the discussion session at the "Exploring the Universe with JWST" conference at ESA-ESTEC in October 2015, we highlight in this document the (non-local) extragalactic science goals for JWST. We describe how a concerted community effort could best address these, ensuring that the desired survey can be completed during the JWST mission.
• ### Mergers and Star Formation: The environment and Stellar Mass Growth of the Progenitors of Ultra-Massive Galaxies since z = 2(1509.00486)

Nov. 19, 2015 astro-ph.CO, astro-ph.GA
The growth of galaxies is a key problem in understanding the structure and evolution of the universe. Galaxies grow their stellar mass by a combination of star formation and mergers, with a relative importance that is redshift dependent. Theoretical models predict quantitatively different contributions from the two channels; measuring these from the data is a crucial constraint. Exploiting the UltraVISTA catalog and a unique sample of progenitors of local ultra massive galaxies selected with an abundance matching approach, we quantify the role of the two mechanisms from z=2 to 0. We also compare our results to two independent incarnations of semi-analytic models. At all redshifts, progenitors are found in a variety of environments, ranging from being isolated to having 5-10 companions with mass ratio at least 1:10 within a projected radius of 500 kpc. In models, progenitors have a systematically larger number of companions, entailing a larger mass growth for mergers than in observations, at all redshifts. Generally, in both observations and models, the inferred and the expected mass growth roughly agree, within the uncertainties. Overall, our analysis confirms the model predictions, showing how the growth history of massive galaxies is dominated by in situ star formation at z~2, both star-formation and mergers at 1<z<2, and by mergers alone at z<1. Nonetheless, detailed comparisons still point out to tensions between the expected mass growth and our results, which might be due to either an incorrect progenitors-descendants selection, uncertainties on star formation rate and mass estimates, or the adopted assumptions on merger rates.
• ### Shadow of a Colossus: A z=2.45 Galaxy Protocluster Detected in 3D Ly-a Forest Tomographic Mapping of the COSMOS Field(1509.02833)

Sept. 9, 2015 astro-ph.CO, astro-ph.GA
Using moderate-resolution optical spectra from 58 background Lyman-break galaxies and quasars at $z\sim 2.3-3$ within a $11.5'\times13.5'$ area of the COSMOS field ($\sim 1200\,\mathrm{deg}^2$ projected area density or $\sim 2.4\,h^{-1}\,\mathrm{Mpc}$ mean transverse separation), we reconstruct a 3D tomographic map of the foreground Ly$\alpha$ forest absorption at $2.2<z<2.5$ with an effective smoothing scale of $\sigma_{3d}\approx3.5\,h^{-1}\,\mathrm{Mpc}$ comoving. Comparing with 61 coeval galaxies with spectroscopic redshifts in the same volume, we find that the galaxy positions are clearly biased towards regions with enhanced IGM absorption in the tomographic map. We find an extended IGM overdensity with deep absorption troughs at $z=2.45$ associated with a recently-discovered galaxy protocluster at the same redshift. Based on simulations matched to our data, we estimate the enclosed dark matter mass within this IGM overdensity to be $M_{\rm dm} (z=2.45) = (9\pm4)\times 10^{13}\,h^{-1}\,\mathrm{M_\odot}$, and argue based on this mass and absorption strength that it will form at least one $z\sim0$ galaxy cluster with $M(z=0) = (3\pm 2) \times 10^{14}\,h^{-1}\mathrm{M_\odot}$, although its elongated nature suggests that it will likely collapse into two separate clusters. We also point out a compact overdensity of six MOSDEF galaxies at $z=2.30$ within a $r\sim 1\,h^{-1}\,\mathrm{Mpc}$ radius and $\Delta z\sim 0.006$, which does not appear to have a large associated IGM overdensity. These results demonstrate the potential of Ly$\alpha$ forest tomography on larger volumes to study galaxy properties as a function of environment, as well as revealing the large-scale IGM overdensities associated with protoclusters and other features of large-scale structure.
• ### Discovery of Massive, Mostly Star-formation Quenched Galaxies with Extremely Large Lyman-alpha Equivalent Widths at z ~ 3(1507.04439)

July 16, 2015 astro-ph.GA
We report a discovery of 6 massive galaxies with both extremely large Lya equivalent width and evolved stellar population at z ~ 3. These MAssive Extremely STrong Lya emitting Objects (MAESTLOs) have been discovered in our large-volume systematic survey for strong Lya emitters (LAEs) with twelve optical intermediate-band data taken with Subaru/Suprime-Cam in the COSMOS field. Based on the SED fitting analysis for these LAEs, it is found that these MAESTLOs have (1) large rest-frame equivalent width of EW_0(Lya) ~ 100--300 A, (2) M_star ~ 10^10.5--10^11.1 M_sun, and (3) relatively low specific star formation rates of SFR/M_star ~ 0.03--1 Gyr^-1. Three of the 6 MAESTLOs have extended Ly$\alpha$ emission with a radius of several kpc although they show very compact morphology in the HST/ACS images, which correspond to the rest-frame UV continuum. Since the MAESTLOs do not show any evidence for AGNs, the observed extended Lya emission is likely to be caused by star formation process including the superwind activity. We suggest that this new class of LAEs, MAESTLOs, provides a missing link from star-forming to passively evolving galaxies at the peak era of the cosmic star-formation history.
• ### Prime Focus Spectrograph for the Subaru telescope: massively multiplexed optical and near-infrared fiber spectrograph(1507.00725)

July 2, 2015 astro-ph.IM
The Prime Focus Spectrograph (PFS) is an optical/near-infrared multifiber spectrograph with 2394 science fibers distributed across a 1.3-deg diameter field of view at the Subaru 8.2-m telescope. The wide wavelength coverage from 0.38 {\mu}m to 1.26 {\mu}m, with a resolving power of 3000, simultaneously strengthens its ability to target three main survey programs: cosmology, galactic archaeology and galaxy/AGN evolution. A medium resolution mode with a resolving power of 5000 for 0.71 {\mu}m to 0.89 {\mu}m will also be available by simply exchanging dispersers. We highlight some of the technological aspects of the design. To transform the telescope focal ratio, a broad-band coated microlens is glued to each fiber tip. A higher transmission fiber is selected for the longest part of the cable system, optimizing overall throughput; a fiber with low focal ratio degradation is selected for the fiber-positioner and fiber-slit components, minimizing the effects of fiber movements and fiber bending. Fiber positioning will be performed by a positioner consisting of two stages of piezo-electric rotary motors. The positions of these motors are measured by taking an image of artificially back-illuminated fibers with the metrology camera located in the Cassegrain container; the fibers are placed in the proper location by iteratively measuring and then adjusting the positions of the motors. Target light reaches one of the four identical fast-Schmidt spectrograph modules, each with three arms. The PFS project has passed several project-wide design reviews and is now in the construction phase.
• ### The Science Case for Multi-Object Spectroscopy on the European ELT(1501.04726)

This White Paper presents the scientific motivations for a multi-object spectrograph (MOS) on the European Extremely Large Telescope (E-ELT). The MOS case draws on all fields of contemporary astronomy, from extra-solar planets, to the study of the halo of the Milky Way and its satellites, and from resolved stellar populations in nearby galaxies out to observations of the earliest 'first-light' structures in the partially-reionised Universe. The material presented here results from thorough discussions within the community over the past four years, building on the past competitive studies to agree a common strategy toward realising a MOS capability on the E-ELT. The cases have been distilled to a set of common requirements which will be used to define the MOSAIC instrument, entailing two observational modes ('high multiplex' and 'high definition'). When combined with the unprecedented sensitivity of the E-ELT, MOSAIC will be the world's leading MOS facility. In analysing the requirements we also identify a high-multiplex MOS for the longer-term plans for the E-ELT, with an even greater multiplex (>1000 targets) to enable studies of large-scale structures in the high-redshift Universe. Following the green light for the construction of the E-ELT the MOS community, structured through the MOSAIC consortium, is eager to realise a MOS on the E-ELT as soon as possible. We argue that several of the most compelling cases for ELT science, in highly competitive areas of modern astronomy, demand such a capability. For example, MOS observations in the early stages of E-ELT operations will be essential for follow-up of sources identified by the James Webb Space Telescope (JWST). In particular, multi-object adaptive optics and accurate sky subtraction with fibres have both recently been demonstrated on sky, making fast-track development of MOSAIC feasible.
• ### Lyman-alpha Forest Tomography from Background Galaxies: The First Megaparsec-Resolution Large-Scale Structure Map at z>2(1409.5632)

Sept. 19, 2014 astro-ph.CO, astro-ph.GA
We present the first observations of foreground Lyman-$\alpha$ forest absorption from high-redshift galaxies, targeting 24 star-forming galaxies (SFGs) with $z\sim 2.3-2.8$ within a $5' \times 15'$ region of the COSMOS field. The transverse sightline separation is $\sim 2\,h^{-1}\mathrm{Mpc}$ comoving, allowing us to create a tomographic reconstruction of the 3D Ly$\alpha$ forest absorption field over the redshift range $2.20\leq z\leq 2.45$. The resulting map covers $6\,h^{-1}\mathrm{Mpc} \times 14\,h^{-1}\mathrm{Mpc}$ in the transverse plane and $230\,h^{-1}\mathrm{Mpc}$ along the line-of-sight with a spatial resolution of $\approx 3.5\,h^{-1}\mathrm{Mpc}$, and is the first high-fidelity map of large-scale structure on $\sim\mathrm{Mpc}$ scales at $z>2$. Our map reveals significant structures with $\gtrsim 10\,h^{-1}\mathrm{Mpc}$ extent, including several spanning the entire transverse breadth, providing qualitative evidence for the filamentary structures predicted to exist in the high-redshift cosmic web. Simulated reconstructions with the same sightline sampling, spectral resolution, and signal-to-noise ratio recover the salient structures present in the underlying 3D absorption fields. Using data from other surveys, we identified 18 galaxies with known redshifts coeval with our map volume enabling a direct comparison to our tomographic map. This shows that galaxies preferentially occupy high-density regions, in qualitative agreement with the same comparison applied to simulations. Our results establishes the feasibility of the CLAMATO survey, which aims to obtain Ly$\alpha$ forest spectra for $\sim 1000$ SFGs over $\sim 1 \,\mathrm{deg}^2$ of the COSMOS field, in order to map out IGM large-scale structure at $\langle z \rangle \sim 2.3$ over a large volume $(100\,h^{-1}\mathrm{Mpc})^3$.
• ### Progress with the Prime Focus Spectrograph for the Subaru Telescope: a massively multiplexed optical and near-infrared fiber spectrograph(1408.2825)

Aug. 12, 2014 astro-ph.IM
The Prime Focus Spectrograph (PFS) is an optical/near-infrared multi-fiber spectrograph with 2394 science fibers, which are distributed in 1.3 degree diameter field of view at Subaru 8.2-meter telescope. The simultaneous wide wavelength coverage from 0.38 um to 1.26 um, with the resolving power of 3000, strengthens its ability to target three main survey programs: cosmology, Galactic archaeology, and galaxy/AGN evolution. A medium resolution mode with resolving power of 5000 for 0.71 um to 0.89 um also will be available by simply exchanging dispersers. PFS takes the role for the spectroscopic part of the Subaru Measurement of Images and Redshifts project, while Hyper Suprime-Cam works on the imaging part. To transform the telescope plus WFC focal ratio, a 3-mm thick broad-band coated glass-molded microlens is glued to each fiber tip. A higher transmission fiber is selected for the longest part of cable system, while one with a better FRD performance is selected for the fiber-positioner and fiber-slit components, given the more frequent fiber movements and tightly curved structure. Each Fiber positioner consists of two stages of piezo-electric rotary motors. Its engineering model has been produced and tested. Fiber positioning will be performed iteratively by taking an image of artificially back-illuminated fibers with the Metrology camera located in the Cassegrain container. The camera is carefully designed so that fiber position measurements are unaffected by small amounts of high special-frequency inaccuracies in WFC lens surface shapes. Target light carried through the fiber system reaches one of four identical fast-Schmidt spectrograph modules, each with three arms. Prototype VPH gratings have been optically tested. CCD production is complete, with standard fully-depleted CCDs for red arms and more-challenging thinner fully-depleted CCDs with blue-optimized coating for blue arms.