In the fall of 2017, The Cooperage Project began offering a Pop Up Club for Middle School age students at Wayne Highlands School District. The after-school program provides a wide range of activities in and around the community, conquering topics from coding and robotics to pizza making at a local restaurant.
The youngsters help formulate their own adventures, helping to engage their interest in the various subjects be th ey academic or mundane. The effort has been a great success, and the Wallenpaupack Area School District has expressed an interest in expanding the club to serve that population as well.
The Cooperage Project recently received a two-year, $50,000 grant from the Harry and Jeanette Weinberg Foundation to support the free after-school program that puts an emphasis on STEAM subjects. The grant allowed the Pop Up Club to expand to three days a week come fall.
Pennsylvania Statewide Afterschool/Youth Development Network (PSAYDN) honored Arrah Fisher, as an Afterschool Champion in the category of Quality for outstanding work in developing, supporting and promoting meaningful high-quality afterschool/out-of- school time programs to benefit children, youth and families across Pennsylvania. The awards took place on March 14, 2019 during PSAYDN's annual reception, which brought together elected officials, program providers, youth, faith-based leaders, parents and business leaders from across the commonwealth to emphasize the importance of afterschool programs in the state.
Arrah works every day to provide a fun, engaging, safe space for middle school students to learn something new, make friends and challenge themselves during The Cooperage Project's Pop-Up Club, a free STEAM-focused afterschool program for students from Wayne Highlands Middle School as well as cyber and home school students. This program has grown from just once a week to a three-day a week program that is an integral part of the middle school students' lives. They look forward to seeing Arrah at Pop-Up Club and are eager to learn each day when they come to the club after school.
Arrah is dedicated to providing high quality experiences that embody the mission of The Cooperage Project to embrace diversity and celebrate shared experiences. She brings her varied background and perspective to the program that seeks to provide unique real-world learning experiences to her community's youth.


text:
Home » TV Actors » Celebrity Wiki
Matt Blashaw Wiki: Facts about the HGTV and DIY Star
HGTV's Matt Blashaw attends as DIRECTV partners with Scripps Networks Interactive (Photo by Joe Scarnici/Getty Images for DIRECTV)
About Matthew Blashaw
Birth November 6, 1975 Orange County
Spouse Teresa Blashaw,
Lindsay Archer 2017 - now
Children Jace William Blashaw
Parents Deborah Klee Blashaw, Lawrence Blashaw
Owns One Call Fixes It All
TV Shows Professional Grade
Yard Crashers
Vacation House for Free
Matt Blashaw is one of the biggest stars HGTV has produced for good reason. With hit shows like Yard Crashers to his name, he has become one of the most engaging personalities in the home improvement reality show space. Blashaw's fans are always dying to know to more about the guy behind "One Call Fixes It All." So, we've got Matt Blashaw's wiki, with everything to know about him all in one place.
Matt Blashaw's Family
Matthew Blashaw was born on November 6, 1975 in California. He is the son of Deborah Klee Blashaw and Lawrence Blashaw. Most of Matt's family is based in Orange County where he is from.
Matt's grandfather, Lawrence Blashaw Sr., fought in the Battle of Bulge during World War II. He met Matt's grandmother, Nita, in England. Lawrence Blashaw Sr. passed away in 2017.
Matt's mother Deborah is active in several charities. She's also dabbled in theater, acting in a production of The Emperor's New Clothes at the Gallery Theatre in Anaheim.
Matt Blashaw's Career
Having grown up on a lumberyard, Matt Blashaw was well-versed in the know-how of home improvement. A licensed contractor, he has been in the business of renovating homes for more than two decades.
His company slogan, "One Call Fixes It All," made him popular when he became a reality TV star. He's the popular host of shows on two major networks. Matt hosts HGTV's Professional Grade and is the star of Yard Crashers and Vacation House for Free, while also hosting DIY Network's Money Hunters, Deconstruction, and Project Xtreme.
Blashaw has also appeared in shows like DIY Insider, Ellen's Design Challenge, and Matt Blashaw: A Day in the Life.
Matt Blashaw went from getting jobs through family referrals to filming over 400 episodes of home renovation shows collectively, and running a thriving contractor business.
Matt Blashaw's Wife
While filming his long-running show Yard Crashers, Matt Blashaw met Lindsay Archer. She is a landscape architect and expert in container gardening from Kansas City, and her job introduced her to her future husband.
Archer had no interest in appearing on camera, but when Yard Crashers was following her client's home renovation, she had no choice but to make a few appearances. It was while working together that Archer and Blashaw grew closer.
With the help of the project's contractor, Blashaw and Archer went on their first date a few days after they met. They worked through their long-distance relationship and got engaged.
Matt Blashaw and Lindsay Archer married in 2017. On July 31, 2018, they welcomed their son, Jace William Blashaw.
This is Matt's second marriage. He reportedly was previously married to one Teresa Bradshaw, but split with her before he met Lindsay.

redpajama_set_name: RedPajamaCommonCrawl


As cosy as sleeping bags are, nowadays they're designed so that you often feel like you're being buried alive. Lovely to be so warm, not so lovely to feel quite so incapacitated. Introducing the most revolutionary, and definitely the coolest (though very warm of course) sleeping bag ever invented. The SELK is the sleeping bag you wear, so if it's still freezing outside (or inside if the heating's packed in) when you get up, you don't need to get out of your sleeping bag! You can walk around quite happily in temperatures as ludicrously low as -8.4 degrees C!
It has an integral hood, and you can unzip the ends of your arms to get your mitts out to spark up the primus on Everest - or pop some bread in the toaster if your flat's dropped below a civilized temperature. The suit lining is made from soft-touch nylon, is padded with a Polyester Hollow Fibre Bonded filling, and comes in four colours and two sizes, small-medium, and large - extra-large. The SELK makes winter something to look forward to, and of course, it's the ultimate answer in how to be cool when you're warm.


text:
Why you should vote YES…
by Bill Cash · Published 2 Nov 2005 · Updated 4 Jan 2016
Reading the editorial page of the Columbus Dispatch is like being in an abusive relationship. Just when you think you're safe, you wake up, and out of nowhere you get brained. Unfortunately, unlike a battered wife, I really have to admit that it's all my fault and I don't know why I keep going back to him.
This Sunday the Dispatch opposed Reform Ohio Now's issues 2 through 5. With all the thoughtful writing they've done on local city issues and on abuses of power, plus with the series of articles we've seen on Ohio's lopsidedly gerrymandered legislative districts, I was sure they would back at least some of the issues. Unfortunately, I'll have to be lying to my friends that "I fell down the stairs," because this hurt and I sure didn't see it coming.
I was not whole-heartedly behind each issue. In very brief, the issues are:
2: absentee voting in advance
3: campaign contribution limits
4: end party-based gerrymandering of districts
5: pass statewide control of elections to a board rather than the Secretary of State
I have been leaning toward opposing 2 and 3 and supporting 4 and 5.
Issue 2 allows people to vote up to 35 days before the date of the actual election. I think this is a terrible idea. While the idea of absentee voting seems nice, a lot can happen in the five weeks before an election. As we allow voting farther and farther ahead, people are more likely to miss defining events on the campaign trail, and once your vote is cast, it can't be changed. Consider what happened last year in the 26th District where I lived: the Democratic primary was won by Mike Mitchell, who defeated his opponent by only 173 votes. But (according to the paper on March 9, 2004) Mitchell had gotten a DUI citation just two days before the election. He was also charged with going 70 in a 55 and driving on a suspended license. (Unfortunately, this didn't hit the news until it was too late.) If people had known about this before they voted, I think he would have lost the election. Early voting costs you the chance to make important last-minute decisions.
Issue 3 is campaign contribution limits; I honestly don't have a strong position here and I don't want to talk about it. I think they are a mistake.
Issue 4, however, is the real jewel of this year's reform package. Under issue 4, anyone could submit a plan for the state's legislative districts, and that plan would be scored for its political competitiveness. The formula — which is insanely complicated (go read it) — gives higher points if a district is more evenly balanced between the two parties. The highest overall point-getting plan generally wins and is chosen by an impartial board. It's more complicated than that, but it would end the system where last year Democrats won their Ohio House seats by an average of some 60 percent and Republicans by some 40 percent. Gerrymandering used to be more of an art, but in the last go-round, computer mapping technology had advanced so much that these crazy unwinnable districts have been much easier to draw. (I know, because I write mapping software.)
The gerrymandering in our state is really costing us, because the people who get elected just have to win their party primary, not appeal to a broad electorate. In the case of the 26th District, by the way, Mike Mitchell went on to run unopposed in the general election, and he won. (The Dispatch said that the "solution would be worse than the problem" and that "Democrats' weak challenges to GOP officeholders have played a role in this problem." But it cuts both ways when Republicans don't field any candidate at all.)
Issue 5 would take election oversight duties away from the Secretary of State and give them to a board of election officials. Although a board might be less responsive, I think it would be a worthwhile step. In Ohio in 2004, as in Florida in 2000, the Secretary of State was at the center of a heated, disputatious election, and the Secretary of State was also the chairman of the Bush campaign. It presented a ridiculous conflict of interest. I understand Florida has already stripped their SOS of election duties, by the way.
I think the Dispatch does have a good point to make about the complexity and specificity of the laws, saying that "they seek to write highly detailed, prescriptive language and untested policy into the Ohio Constitution. Once there, fixing any deficiencies would require another statewide vote." While this is true, I am not exactly holding my breath for either party to voluntarily level the playing field. Also, while issues 2 and 3 generally deal with regulation of elections and are probably better suited to being acts of the legislature, issues 4 and 5 are structural in nature and therefore do belong in our state's constitution. As an alternative, the broad outlines of issue 4's formula might have been put into the constitution, along with the requirement of a more detailed enabling act to be passed by the legislature, so that the variables could be tinkered with, but I'm not comfortable with anything coming out of the Statehouse these days.
Our electoral system is fundamental to our functioning democracy. Vote yes on issues 4 and 5!
Tags: Ohiopoliticsvoting
Next story …and who NOT to vote for!
Previous story From the Bureau
What's the matter with Ohio?
by Bill Cash · Published 30 Jan 2005 · Last modified 4 Jan 2016
by Bill Cash · Published 17 Mar 2005 · Last modified 4 Jan 2016
Election night update
by Bill Cash · Published 5 Nov 2002 · Last modified 21 Aug 2014
2 Nov 2005 at 9:05pm
Your far from a battered wife. Your more like A spoiled princess. A.B.


The 'Third annual EU conference on EAFRD financial instruments for agriculture and rural development in 2014-2020' took place on 10 October 2017 in Paris.
informed stakeholders about the fi-compass advisory platform support to Member States and managing authorities and about further support available from cooperation between the European Commission and EIB Group for agriculture and rural development.
This high-level conference was opened by Phil Hogan, Commissioner for Agriculture and Rural Development. He was joined by Ms Catherine Geslain-Lanéelle, Director-General for the Economic and Environmental Performance of Enterprises, Ministry for Agriculture, France, Ms Carole Delga, President of the Occitanie region, France, Pim van Ballekom, Vice-President of the European Investment Bank (EIB) and Pier Luigi Gilibert, Chief Executive of the European Investment Fund (EIF).
They were joined by further distinguished speakers from the European Commission, the EIB Group, EAFRD managing authorities from France and other Member States, farm unions, and the French banking sector.
Organised by the European Commission's Directorate General for Agriculture and Rural Development (DG AGRI) this flagship fi-compass conference welcomed participants from 25 EU Member States to the 'Third annual EU conference on EAFRD financial instruments for agriculture and rural development in 2014-2020' on 10 October 2017 in Paris.
In his opening statement, Phil Hogan, European Commissioner for Agriculture and Rural Development, highlighted the importance of innovation, modernisation and sustainability for European agriculture. He reminded the audience that "We need to provide farmers, agri-food operators and rural entrepreneurs with the full range of financing possibilities existing under the EAFRD so that they are in a position to compete fairly with all other economic sectors which already benefit from cohesion policy or other EU-level financing."
Catherine Geslain-Lanéelle, Director-General for the Economic and Environmental Performance of Enterprises, Ministry for Agriculture, France, emphasised the exemplary role of the Occitanie and Nouvelle Aquitaine regions in pioneering financial instruments for agriculture and rural development. She also underlined that the Ministry is fully supporting the regions in their pioneering role.
Occitanie region's support for agriculture in the form of financial instruments was also the focus of the speech of Carole Delga, President of the Occitanie region, France. She highlighted the significance of agriculture for the regional economy and the key role played by regions in implementing financial instruments.
Pim van Ballekom, Vice-President of the European Investment Bank (EIB) reviewed the merits of financial instruments in improving the efficiency of public policies for the agricultural sector by "doing more with less". He also called for improvements in the use of financial instruments in terms of rationalisation, simplification and better complementarity between centralised and decentralised instruments. He was followed by Pier Luigi Gilibert, European Investment Fund (EIF), who also highlighted the usefulness of financial instruments as a highly effective complement to grants and subsidies and gave an overview of recent EIF involvement in setting up EAFRD financial instruments in France and Italy.
Keynote speakers were joined by representatives from the farm unions and the French banking sector who gave insights into investment needs in the agricultural sector in France and the other EU Member States. The European Commission, together with the EIB Group, presented financing opportunities offered by the European Fund for Strategic Investments (EFSI) and options for blending EFSI with EAFRD resources. The role of Omnibus in offering further possibilities of combining EFSI with EAFRD resources was highlighted.
Furthermore, the conference featured recent developments in the use of EAFRD financial instruments in France and the other Member States. Managing authorities also reported on their positive experience with targeted coaching on financial instruments for EAFRD managing authorities.
This fi-compass conference followed previous macro-regional and annual fi-compass EAFRD events that took place in Dublin, Vienna, Riga, Madrid, Rome, Warsaw, Brussels and Tallin. The next EAFRD Conference will be in Sofia.
The conference in Paris was graphically recorded and drawings from the event are included in the photo gallery. We encourage you to look at the video recordings and presentations from the speakers. A selection of additional fi-compass resources can be also found below.
The press release of the European Commission about the conference can be found here.


text:
Q: Using MutationObserver correctly I'm trying to monitor updates to a page using MutationObserver. I've read through the documentation (https://developer.mozilla.org/en-US/docs/Web/API/MutationObserver & https://developers.google.com/web/updates/2012/02/Detect-DOM-changes-with-Mutation-Observers) but can't seem to get this right.
What am I doing wrong?
var observer = new MutationObserver((mutations) => {
    mutations.forEach((mutation) => {
        console.log(mutation)
    })
})

var config = {childList: true}

observer.observe(document, config)

A: Use the DOM element instead
observer.observe(document.body, config)

A: // observer instance
var observer = new MutationObserver(function(mutations) {
  mutations.forEach(function(mutation) {
    console.log(mutation);

    // include your code for reacting to each mutation here

  });
});

// target node
var target = document.getElementById('my-id');

// configuration
var config = { attributes: true, childList: true, characterData: true };

// start observer
observer.observe(target, config);

// stop observer -- use this when you want to stop observing
// observer.disconnect();

redpajama_set_name: RedPajamaStackExchange


text:
Sharp Says It's Shipping Screens For The New iPhone This Month
Jamie Condliffe
Aug 3, 2012, 4:00am
The next iPhone is on the way, and a handful of rumours point to a launch on September 12. Now, Sharp has announced that it's shipping screens for the new device this month — so it looks like those dates could be believable.
Sharp's new president, Takashi Okuda, said at a press conference this morning that "shipments will start in August". If rumours are to be believed, the screens are going to be four inches on the diagonal, with an aspect ratio of 16:9, and pack technology to make them thinner and improve battery life. Let's wait and see. [Reuters]


The building code of Canada states that a yearly inspection is required on all propane, natural gas and wood burning fireplaces and stoves. This is an opportunity to take advantage of our worry free solution to that annual inspection, cleaning and maintenance check. Our fast and efficient forward scheduling system will ensure your unit is inspected and serviced each spring well before the next heating season.


We are leading in manufacturer and wholesaler of ladies bridal dresses, lehenga, etc.
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Location: San Polo dei Cavalieri, Italy.
Notes: Castle Orsini-Cesi-Borghese is located in San Polo dei Cavalieri in the province of Rome.The first news of the castle date from the second half of the tenth century when it is mentioned as the land belonging to a place unidentified small church of San Paolo (now destroyed), which for various misspellings dialect, over the centuries it became San Polo (from which, perhaps the name of the village), located in the area. At first it was the Knights belonging to the Sovereign Military Order of Malta, then to the Roman monastery of St. Paul Outside the Walls, but only in the twelfth century is referred to a Castrum Sancti Pauli. In this period the castle was simply a stronghold in the shelter of a small bell. The Castrum was a focal point and intersection between the Sabina and the Aniene Valley, in fact, around the eleventh century castle was the dispute of the feudal lords of Marcellina, Montecelio, Palombara Sabina, Tivoli and the abbey of San Paul outside the walls. By Pope, in the 2nd half of '300, the castrum was leased to the Orsini, who bought it in the fifteenth century and turned it into a military stronghold. By a statute of the Orsini of 1479 it shows that the castle was contracted out to a temporary boarding. In 1558 they sold it to the Orsini Cesi castle which turned it into a summer residence frequented by aristocrats. In 1603 it was transformed by Cesi in the Accademia dei Lincei, which was founded by Federico Cesi, who hosted the castle Galileo Galilei. After the plague of 1656 the castle passed to the Borghese becoming part of the Papal States but then became autonomous with the unification of Italy. In the eighteenth century it was sold to a certain Peter Trusiani, but returned as a property of the Borghese in the early twentieth century, but in 1946 the court of Rome was auctioned and was bought by Dr. Chiappini who in 1958 sold it to the architect Luca Brasini which it began the restoration which was later completed by Andrea Berardi jurist. After countless sales and disposals is private property now.
Info sul sito di Tibursuperbum, tibursuperbum.it.
Info sul portale di San Polo dei Cavalieri, sanpolodeicavalieri.com.


text:
Home Latest News Int'l Democracy Day: Tambuwal warns against rigging Edo, Ondo governorship polls
Int'l Democracy Day: Tambuwal warns against rigging Edo, Ondo governorship polls
The Sokoto State Governor Aminu Tambuwal who is also Chairman of the Peoples Democratic Party (PDP) Governors Forum, has declared that any electoral victory in the Edo and Ondo governorship elections, obtained through rigging, amounts to a coup d'etat.
Tambuwal handed the warning while speaking on the commemoration of International Day of Democracy marked on15 September of every year by the United Nation since 2007. The UN had set aside the day for the purpose of promoting and upholding the principles of democracy while inviting all member states and organisations to commemorate the day in an appropriate manner that contributes to raising public awareness.
But in a statement, Tambuwal noted that strengthening of Nigerians electoral infrastructure has become very urgent, if the country is to restore the hope and aspiration of Nigerians that democracy still counts. He also called on the National Assembly to immediately conclude the long overdue amendments to the Electoral Act to entrench provisions that make it impossible for rigging and manipulation of elections.
"Indeed, the upcoming Gubernatorial elections in Edo and Ondo States is a test case of our preparedness for a rule based and consent enabled governance. "It has become imperative to rethink the role of not just the electoral umpire, but also the activities of various security agencies in our electrical process in such a manner as to instill in them the culture of inflexible support for democracy and its practice.
"Any electoral victory obtained other than through the legitimate votes of the people is nothing short of a COUP D'ETAT. "It completely violates our Constitution which envisaged that governmental authority must be obtained only through the will of the people." He therefore called on the National Assembly to immediately conclude the long overdue amendments to the Electoral Act to entrench provisions that make it impossible for rigging and manipulation of elections. "In the same vein, any further amendments to the Constitution must take cognisance of our experiences in previous elections.
"It may be that time has come to introduce aspects of technology in our elections that are fool proof and such should be test run with other elections before the general elections in 2023. "Finally, it bears repeating to say that good governance anchored on a sound economy and corrupt free progressive social, cultural, and infrastructural development is a sine qua non for sustainable democracy. "It behoves on us all, particularly the political and economic actors on stage to do our utmost to ensure a stable Nigeria.
"Finally, I call on the Civil Society and the Press to continue to play its watchdog role of keeping government in check in order to ensure the survival of our Democracy." He added that to mark this day in Nigeria, it is important to note that the struggle for Democracy has involved sweat, blood, diplomacy, compromises and negotiations. "This had traversed the whole period of pre and post-independence struggle, the First, Second, Third Republics. And our current experiment since 1999.
Tambuwal noted that In contemporary Nigeria that many young people who did not participate in the various struggles may not understand the intensity of the price paid by the nation's forebears. "It is with this in mind that I wish to flag various dangers lurking around and threatening to derail our present Democratic experiment,"
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text:
Biografia
Dal 1992 vive ad Ascoli Piceno, è sposato e ha due figli.

Carriera

Club
Con la Dinamo Tirana, club nel quale ha trascorso l'intera carriera, ha vinto 5 campionati e 3 coppe albanesi. È stato altresì capocannoniere del campionato albanese per 6 stagioni consecutive, dal 1971 al 1976.

Nazionale
Ha esordito con la Nazionale albanese nel 1971, giocando 15 partite e segnando 5 gol sino al 1981.

Palmarès

Club

Dinamo Tirana: 1966-67, 1974-75, 1975-76, 1976-77, 1979-80

Dinamo Tirana: 1970-71, 1973-74, 1977-78

Individuale
 Capocannoniere del campionato albanese: 6
1970-71, 1971-72, 1972-73, 1973-74, 1974-75, 1975-76

Note

Collegamenti esterni

Calciatori della Nazionale albanese

redpajama_set_name: RedPajamaWikipedia


text:
Q: Proving inequality by induction only Suppose $a_1,\dots,a_n$ are real positive numbers s.t. $\prod_{i=1}^n a_i=1$. My book claims that by induction only (i.e. the use of AM-GM is forbidden), one can prove that $$\sum a_i\ge n$$ and that equiality exists if and only if $\forall i,a_i=1$. I tried to prove it:
for $n=2,$ denote $a_1=a,a_2=\frac 1 a$ then: $$0\le \frac{(a-1)^2}{a}=a+\frac{1}{a}-2\Rightarrow a_1+a_2\ge 2.$$
Now we need to prove it for $n+1$. Denote by $b_n=a_na_{n+1}$, By hypothesis $$a_1\cdots a_{n-1} b_n=1\Rightarrow a_1+\dots a_{n-1}+b_n\ge n$$ which means $$a_1+\dots+a_{n-1}+a_n +a_{n+1} \ge n-b_n+a_n+a_{n+1}.$$ That means I need to prove $$a_n+a_{n+1}-a_na_{n+1}\ge 1$$ but I don't know how.
How can I complete the proof?

A: The inequality you need to prove (I think you should change the $n-1$ to $n+1$ if you follow what you have on the previous line),
$$
a_n+a_{n+1}-a_na_{n+1}\geq 1
$$
is not in general true. Take $a_n=a_{n+1}=2$ (which is perfectly valid). This gives zero in the left-hand side.
So, maybe what you can do, is to assume that you, in your product $b_n=a_na_{n+1}$ choose the elements $a_n$ and $a_{n+1}$ that has a certain property?

A: $$a_n+a_{n+1}-a_na_{n+1}\ge 1 \Leftrightarrow 1 - a_n -a_{n+1} + a_na_{n+1} \le 0. $$
Factorize the LHS and you will see which $a_n$ and $a_{n+1}$ you should pick.

A: What we want to show is that
$\prod_{i=1}^n a_i=1
\implies \sum a_i\ge n
$.
This is certainly true for $n=1$.
Suppose it is true for $n$.
Suppose we have
$\prod_{i=1}^{n+1} a_i=1
$.
We want to show that
$ \sum_{i=1}^{n+1} a_i\ge n+1
$.
As a first try,
let's look at
$\prod_{i=1}^{n} a_i
$.
If
$A
=(\prod_{i=1}^{n} a_i)^{1/n}
$
and
$b_i
=a_i/A
$,
then
$\prod_{i=1}^{n} b_i
= 1
$,
so
$\sum_{i=1}^{n} b_i
\ge n
$,
or
$\sum_{i=1}^{n} a_i
\ge An
$.
Therefore
$\sum_{i=1}^{n+1} a_i
\ge An+a_{n+1}
$.
If we can show that
$ An+a_{n+1}
\ge n+1
$, we are done.
This is the same as
$\frac{An+a_{n+1}}{n+1}
\ge 1
$.
But
$1
=\prod_{i=1}^{n+1} a_i
=a_{n+1}\prod_{i=1}^{n} a_i
=a_{n+1}A^n
$,
so this is the same as
$\frac{An+a_{n+1}}{n+1}
\ge a_{n+1}A^n
$
or
$An+a_{n+1}
\ge (n+1)a_{n+1}A^n
$
or
$An+1/A^n
\ge (n+1)a_{n+1}A^n
$
By the AM-GM means inequality,
since $nA$ is $n$ copies of $A$,
$\begin{array}\\
\frac{ An+a_{n+1}}{n+1}
&\ge \sqrt[n]{A^n a_{n+1}}\\
&= \sqrt[n]{(\prod_{i=1}^n a_i) a_{n+1}}\\
&= \sqrt[n]{\prod_{i=1}^{n+1} a_i}\\
&= 1\\
\end{array}
$
and,
to my pleasant surprise,
we are done.
(added even later)
We are not allowed to use
the AM-GM inequality.
However,
I will now show,
based on some earlier work of my own,
that this particular use
on the AM-GM inequality,
in which all but one of the values
are equal,
is a consequence of
Bernoulli's inequality.
Bernoulli's inequality,
which is easily proved by induction,
states that
$(1+x)^m \ge 1+mx
$
if $x \ge 0$ and
$m$ is a positive integer
with equality if and only if
$m = 1$
or
$x = 0$.
Therefore
$(1+\frac{v/u-1}{m})^m
\ge 1+m\frac{v/u-1}{m}
= \frac{v}{u}
$ with
equality only if
$m=1$ or $v/u-1 = 0
$,
which is the same as
$v = u$.
Multiplying by $u^m$,
this becomes
$\begin{array}\\
u^{m-1}v
&\le u^m(1+\frac{v/u-1}{m})^m \\
&= (u+\frac{v-u}{m})^m \\
&= (\frac{v+(m-1)u}{m})^m \\
\text{or}\\
(u^{m-1}v)^{1/m}
&\le \frac{v+(m-1)u}{m}\\
\end{array}
$
But this is just
the AM-GM inequality
with all but one of the values equal!
Therefore,
this use of the
AM-GM inequality
is OK to use,
since it is a consequence
of Bernoulli's inequality,
not the full
AM-GM inequality.
(added later)
If there is equality,
then
$A = a_{n+1}
$.
Since
$1
=A^n a_{n+1}
$,
$1 = a^{n+1}_{n+1}
$,
so
$a_{n+1} = 1$.
This implies that
there is equality for
$n$,
so that
$a_n = 1$.
This allows us to
work our way back down
in an odd kind of
inverse induction
to show that
equality at any step
shows that
at that step
and all preceding steps
$a_i = 1$.
This is weird,
but I think that
it is correct.


text:
\section{Introduction}
\label{sec:intro}

Much of the effort to study infrared-luminous galaxies has centered on
wavelengths greater than 50 $\mu$m.  Modeling work is focused on the
near-IR (e.g. \markcite{chok94}Chokshi et al. 1994) and far-IR (e.g.
\markcite{hac87}Hacking et al. 1987; \markcite{rrbinson96}Rowan-Robinson
et al. 1996) portions of the galaxian spectrum.
However, the mid-infrared is well-suited for studying starburst and
ultraluminous galaxies.  About 40\% of the luminosity from starburst
galaxies is radiated from 8-40 $\mu$m (\markcite{soi87}Soifer et al.
1987).  Extinction effects are small, and problems due to infrared
cirrus are minimized.  Most importantly for space astrophysics, for
a fixed telescope aperture, the spatial resolution is higher at shorter
wavelengths, and the confusion limit lies at higher redshifts.  Recent
work using the {\it Infrared Space Observatory (ISO)} (e.g.
\markcite{knapp96}Knapp et al.  1996; \markcite{boul96}Boulade et al.
1996; \markcite{rrobinson96}Rowan-Robinson et al. 1996)
shows the relative importance of the 7 $\mu$m and 15 $\mu$m bands
for galaxy studies.  The {\it Wide-Field Infrared Explorer (WIRE)},
a Small Explorer mission due to launch in late 1998
(\markcite{hac96}Hacking et al.\ 1996;
\markcite{schemb96}Schember et al. 1996), will conduct a very deep survey
at 24 $\mu$m to study
starburst galaxy evolution.  The {\it Space Infrared Telescope Facility
(SIRTF)} is also expected to conduct surveys in mid-infrared bands.
To interpret the results of these surveys now in
progress or soon to commence, it is necessary to better understand
the mid-infrared properties of galaxies in the local Universe.

The 25 $\mu$m luminosity function provides the basis
for predicting the
faint source counts in the mid-infrared.  The empirical model of
\markcite{hac91}Hacking \& Soifer (1991) uses an analytic fit to the
luminosity function derived by Soifer \& Neugebauer \markcite{soi91}
(1991).  This function was estimated from a complete subsample of the
Bright Galaxy Sample (\markcite{soi87}Soifer et al.\ 1987) containing
135 galaxies to a flux density limit of 1.26 Jy.  The availability of
many more redshifts of IRAS galaxies (principally from the 1.2 Jy Survey
(\markcite{str90}Strauss et al.\ 1990; \markcite{fis95} Fisher et al.\
1995)) enables a much larger sample to be studied, reducing uncertainties
at high and low luminosities.

In this paper we present the selection of a large galaxy sample that is
flux-limited at 25 $\mu$m, and derive the local luminosity function
based on this sample.  The sample selection is described in the next
section.  In Section \ref{sec:lfresults} we describe the $1/V_{max}$ and
the maximum-likelihood estimators for deriving the local luminosity
function, and present the results. The completeness of the sample is
discussed in Section \ref{sec:compdisc}. In Section \ref{sec:correct} we
calculate the radial density distribution of the sample using a
maximum-likelihood method. The radial density fit is used to correct the
$1/V_{max}$ estimate of the local luminosity function, as well as the
redshift distribution with which the luminosity function can be
compared.  Section 6 includes discussions of the different luminosity
function estimators, a comparison of our newly derived luminosity
function with previous estimates, the implications for mid-infrared
backgrounds and number counts, and the effects of evolution on the
derivation of the luminosity function. The color properties of the
sample are treated in another paper (\markcite{fang98}Fang et al.\ 1998).

\section{Sample Selection}
\label{sec:sample}

Our sample is based on a selection from the IRAS Faint Source Survey
(FSS;\markcite{mos92} Moshir et al.\ 1992).  The principal data product
of the FSS is the Faint Source Catalog (FSC).  The FSC was produced
by coadding IRAS detector scans before extracting sources, and
is roughly one magnitude deeper than the Point Source Catalog.
Another FSS database is the Faint Source Reject File (FSCR), which contains
possible detections that were not included in the FSC for assorted
quality-control reasons.  This section treats the selection of sources
from the FSC and the FSCR in turn.

To minimize the effects of cirrus and contamination from stars, we chose a
sky coverage limit of $|b| \ge 30^\circ$.  In addition,
the same excluded areas used by Strauss et al.\ (1990) were excluded
from our sky coverage region.  At high Galactic latitudes, the
excluded regions consist almost wholly of gaps not covered by IRAS,
and the Large and Small Magellanic Clouds.  Our sample covers
47.7\% of the sky.

The 90\% completeness limit for the FSC lies at a 25 $\mu$m flux
density of 210 mJy for sky covered by 2 HCONs, and at 170 mJy for
coverage of 3 HCONs, for $|b| > 10^\circ$ (Moshir et al.\ 1992).  Our
preliminary searches of redshift databases indicated, however, that
for such faint flux density limits, many of the faintest objects
would have no redshift available.  A flux density limit of 250 mJy
was found to have a small redshift incompleteness (see below) and was
selected for our sample.  Furthermore, sources with moderate (SNR=3
to 5) or high-quality (SNR $> 5$) flux densities at 25 $\mu$m are
included in the sample.

A set of weak color criteria were chosen to further discriminate against
stars.  The great majority of galaxies have larger flux density at 60
$\mu$m than at 25 $\mu$m.  For these sources, no constraint was placed
on the 12 $\mu$m -- 25 $\mu$m color.  A small percentage of galaxies
will have a smaller flux density at 60 $\mu$m than at 25 $\mu$m.  For
these sources, we have limited the ratio of $F_{\nu}(25)/F_{\nu}(60)$
to lie between 1 and 1.6, and have further constrained
$F_{\nu}(12)/F_{\nu}(25)$ to be less than 1 to avoid admitting
large numbers of stars into the sample.  Color-color diagrams of
the sources in the sample that have been identified as galaxies with
redshifts (see below) are displayed in Figure \ref{fig:colorcolor}.
These diagrams show that only a handful of galaxies
satisfy $F_{\nu}(25)/F_{\nu}(60) \ge 1$, and that our color criteria
are fairly liberal for sources identified as galaxies.

To summarize, our sample selection criteria are:
\[ \begin{array}{l}
    F_{\nu}(25) \ge 250  {\rm mJy;} \\
    \mbox{moderate or good quality detection at}~25\mu{\rm m;} \\
    |b| \ge 30^\circ, \mbox{ and not in Strauss et al.\ excluded zone}; \\
    F_{\nu}(25)/F_{\nu}(60) < 1 \mbox{ and no constraint on } F_{\nu}(12)/F_{\nu}(25), \\
     {\rm or} ~ 1  < F_{\nu}(25)/F_{\nu}(60) < 1.6 ~{\rm and}
                               ~F_{\nu}(12)/F_{\nu}(25) < 1.
\end{array} \]

1619 sources in the FSC meet the above criteria.  To identify galaxies and
non-galaxies (stars, infrared cirrus, etc.), and to obtain redshifts for
identified galaxies, a positional match of these 1619 sources was made with
the 1.2 Jy Survey catalog, the November 1993 public version of J.P. Huchra's
ZCAT, the NED database, and the SIMBAD database.  A matching radius of 1
arcminute was used to match the FSC sources with these catalogs.  Additional
redshifts were culled from J.P. Huchra's private ZCAT catalog, and from
additional observations made at the Fred L. Whipple Observatory (FLWO) 1.5m.
This matching process resulted in 1438 galaxies with redshifts.  Of the
remainder, 166 sources were identified as Galactic objects or H{\sc II}
regions in nearby galaxies, leaving 15 sources identified as galaxies without
redshifts.

The selection of sources from the FSCR is more complicated due to the
low reliability of sources in that database.  Applying our selection
criteria to the Faint Source Reject File results in 851 sources.  These
were matched with the IRAS OPTical IDentification (OPTID) database at
IPAC.  The OPTID database is a special version of the FSC and FSCR with
optical identifications added from the Guide Star Catalog (GSC:
\markcite{lasker90}Lasker et al.\ 1990), the Tycho Input Catalog (TIC:
\markcite{egr92}Egret et al.\ 1992)), and the COSMOS/UKST Southern Sky
Object Catalogue (COSCAT: \markcite{yen92}Yentis et al.\ 1992).  Sources
matching with stars within $1\sigma$ error ellipses on the OPTID Plots
were excluded. We examined those sources which have no match with either
a star or a galaxy within $1\sigma$ error ellipses by studying the OPTID
Reports and by comparing with the optical Digitized Sky Survey images.
We found only a few of these sources to be possible galaxies which,
along with the reject sources that match with galaxies within the
$1\sigma$ error ellipses, gives a small sample of 27 sources.  Among
these, there are 5 sources which are extremely blue, and have null SNRs
at 60 $\mu$m.  ADDSCANS (one-dimensional coadds of IRAS scans) of
these sources show no obvious detections at their locations at all four
IRAS bands.  No redshifts for these sources were found when matching
them with the redshift catalogs (see above), and we suspect that they
are not real galaxies.  The final yield from the Reject File is 22
sources.

Matching the 22 FSCR sources with the same redshift catalogs gives 20
sources with redshifts.  To verify that these matches are not random, a
sample of 851 sources randomly distributed in our sampling regions was
generated.  The number of matches with galaxies is only 2.  This
indicates that these 22 reject sources are not likely to be just random
matches with galaxies.  Furthermore, it shows that the number of
coincidental matches in our total sample should also be very small.

The breakdown of sources for redshifts for our entire sample (FSC plus
FSCR sources) is listed in Table \ref{tab:redsrc}.  The previously
unpublished redshifts obtained at FLWO are tabulated in Table
\ref{tab:redshifts}. The total number of galaxies with redshifts is
1458. There are 17 galaxies remaining without redshifts.

The 1.2 Jy Survey catalog also provides the ADDSCAN flux densities for the
sources extended at 60 $\mu$m.  For those sources which have no ADDSCAN
flux densities in the 1.2 Jy Survey but are extended at 25 $\mu$m, we have
obtained new ADDSCAN flux density measurements and substituted these for the
FSC values.

Figure \ref{fig:ncounts} shows the $\log(N)$ versus $\log(F_{\nu}(25))$ plot
for the sample of galaxies with redshifts.  The points follow a slope of -1.5
all the way to our flux limit, demonstrating the high completeness of our
sample as a function of flux.  The apparent excess at high flux values is most
likely due to the Local Supercluster.  Further checks of completeness are
presented in Section \ref{sec:compdisc}, including an investigation of
possible systematic effects of flux errors on derived luminosity functions.

For each galaxy, secondary distances are used when available.
Otherwise, distances have been assigned using a linear Virgocentric
inflow model (\markcite{aar82}Aaronson et al.\ 1982) and a Hubble
constant of 75 km~s$^{-1}$ Mpc$^{-1}$ (see Appendix \ref{sec:h0disc} for
the case of a different Hubble constant).  The same criteria as Soifer
et al. \markcite{soi87} (1987) have been used to identify Virgo cluster
galaxies (with a distance of 17.6 Mpc).  The monochromatic luminosity of
each source is computed from the distance, using a $k$-correction that
assumes a power-law slope of the SED between 12 $\mu$m  and 25 $\mu$m.
The monochromatic luminosities are expressed as $\nu L_\nu$ and have
units of solar luminosities.

\section{Luminosity function results}
\label{sec:lfresults}

\subsection{1/$V_{max}$ and parametric maximum likelihood}
\label{sec:parametric}

The classic method of estimating luminosity functions is via
the $1/V_{max}$ estimator (Schmidt 1968).  This technique is
non-parametric, so no analytic form of the luminosity function
need be assumed, but the data must be binned.  The infrared
luminosity functions presented by Soifer et al.\ (1987) and
Soifer \& Neugebauer (1991) were derived via this method.
The space density $\rho_L$ and its error $\sigma_{\rho}$ are
computed from the following quantities:
\begin{equation}
\rho_L = \left ( {{4\pi} \over \Omega} \right )
        \left ( {\sum {1 \over {V_{max}}}} \right ) ,
\end{equation}
\begin{equation}
\sigma_{\rho} = \left ( {{4\pi} \over \Omega} \right )
        {\left ( {\sum {1 \over {V_{max}^2}}} \right ) }^2 ,
\end{equation}
where $\Omega$ is the solid angle of the survey, $V_{max}$ is the
maximum volume to which the object could have been detected, and the
sum is over all galaxies in each luminosity bin.  The assumption
that galaxies are
distributed uniformly in space may be tested by checking that
$V/V_{max}$ is 0.5 in each luminosity bin.

The luminosity function may also be derived via maximum likelihood
techniques that are independent of density variations (e.g.\markcite{san79}
Sandage, Tammann, \& Yahil 1979). The parametric method has the
advantage that no binning of the data is required, but an analytic form
of the luminosity function must be assumed. For all of the parametric
maximum likelihood fits in this paper, we have assumed a cumulative luminosity
function of the form used by Yahil et al.\ \markcite{yah91} (1991) in
the analysis of the 1.2 Jy Survey:
\begin{equation}
\Psi(L)=C{\left({{L}\over{L_*}}\right)}^{-\alpha}{\left( 1+{{L}\over{L_*}}
                                                 \right)}^{-\beta}
\end{equation}
with its corresponding differential luminosity function
\begin{equation}
\Phi(L)={\left({\alpha \over L} + {\beta \over
                        {L_* + L}}\right)}\Psi(L).
\end{equation}

A flux-limited sample contains only a small number of very sub-luminous
galaxies, making it difficult to determine the luminosity function at
these luminosities.  We have followed Yahil et al.\
\markcite{yah91}(1991)
in imposing a lower limit on luminosity $L_s = 4 \pi r_s^2 \nu
f_m$, where $r_s$ is the distance corresponding to a velocity of 500 km
s$^{-1}$, and $f_m$ is the survey flux limit.  For $f_m$ = 250 mJy,
$L_s$ is $4.0\times 10^7 L_\odot$. Then the minimum
detectable luminosity at distance $r$ is $L_{min}(r) = {\rm Max} (L_s,
4\pi r^2 \nu f_m)$.  The probability of detecting a galaxy of
luminosity $L_i$ at distance $r_i$ is then
\begin{equation}
\label{eqn:condprob}
f(L_i | r_i) = \left \{ \begin{array}{ll}
        {{\Phi(L_i)}/{\Psi(L_{min})}} & \mbox{if $L_i \ge L_{min}$} \\
        0 & \mbox{otherwise}
         \end{array}
         \right .
\end{equation}
The likelihood function is the product of all of these probabilities:
\begin{equation}
\Lambda = - 2 \sum_i \ln f(L_i | r_i).
\end{equation}
This (reciprocal) likelihood function is minimized to determine
$\alpha$, $\beta$, and
$L_*$.  The variance of these parameters is estimated by making
use of the asymptotic
normality of the maximum likelihood estimators for a large sample
(\markcite{kendall87}Kendall, Stuart \& Ord 1987).

The likelihood function contains ratios of the differential and cumulative
luminosity functions, so another calculation is required to find
the normalization $C$.
One method is to use
the $n_1$ estimator of Davis \& Huchra \markcite{dav82}(1982) after
the shape parameters have been determined.  The quantity $n_1$ is the
number density of galaxies with $L \ge L_s$.  (See Equations 11 and
13 of \markcite{yah91}Yahil et al.\ (1991) for the formulas appropriate
to our choice of the parametric luminosity function.) An upper limit
to the redshift must be chosen to fix the sample volume used in this
density estimator.  Based on a calculation of $n_1$ in radial shells,
we have
chosen an upper limit on redshift of 20,000 km~s$^{-1}$ ($z \sim 0.067$)
for the normalization of the maximum likelihood fit.  (However, this
upper redshift limit is {\it not} applied in determination of the
shape of the luminosity function.)  Effects of  incompleteness or
evolution may become significant beyond this redshift.
A maximum-likelihood fit to the density
variations of the sample as a function of redshift (see Section
\ref{sec:correct}) further indicates that the density is well-behaved
to this limit.  A second means of determining the normalization is to
require that the number of sources predicted by the luminosity function
over some redshift range equals the number in the sample.  Computing
the normalization from source counts in this way from 500 km s$^{-1}$
to 20,000 km~s$^{-1}$ gives essentially the same normalization as
the $n_1$ estimator.

The derived parameters for the maximum likelihood
fit are listed in Table \ref{tab:fitpars}.
Luminosity functions derived by the $1/V_{max}$ method and the
parametric maximum likelihood method are shown in Figure
\ref{fig:newvis}.  We show the luminosity
functions in the form of the visibility $\Theta = \Phi L^{2.5}$.
The visibility is proportional to the number of galaxies visible
per log luminosity bin,
and it has the advantage of showing the relative numbers of galaxies
in a flux-limited sample.  The open triangles are from the $1/V_{max}$
estimate of the entire sample, and the solid curve is the parametric
maximum-likelihood fit to galaxies with redshifts greater than
500 km~s$^{-1}$, with normalization from a volume-limited sample
as described above.    The solid curve is truncated at
$L_s = 4.0\times 10^7 L_\odot$.  The $1/V_{max}$ estimate (made in bins
of $\log(\nu L_\nu) = 0.4$) was converted to visibility by
multiplying $\rho_L$ by  $L^{1.5}{2.5 \over {\ln 10}}$.

A plot of $V/V_{max}$ as a function of luminosity for our sample is
included in Figure \ref{fig:vratio1}.  The variations in this
statistic for the full sample and northern sample at luminosities
less than $10^{10} L_\odot$ are most likely due to the Local
Supercluster.  The southern sample is much more uniform at these
luminosities, supporting this interpretation.

The general agreement between the $1/V_{max}$ luminosity function and
the shape of the parametric fit in Figure \ref{fig:newvis}, together
with the uniformity of the $V/V_{max}$ statistic, confirm that the
parametric fit is a good description of the shape of the local
luminosity function.    The $1/V_{max}$ point at $\log \nu L_\nu = 8.6$
corresponds to the luminosity at which the flux limit samples the
Virgo cluster, and hence lies well above the solid curve.

The number of galaxies without redshifts (17) is slightly more than one
percent of our sample.  One method for estimating the effects of this known
incompleteness on the luminosity function is to assign to each of these
galaxies the median redshift of their log flux bin, and recalculate the
luminosity function.  When the median redshifts are computed for the whole
sample, all the assigned redshifts are around 7,000 km sec$^{-1}$.  When the
median redshifts are calculated from those galaxies whose redshifts were not
obtained from public sources (a more realistic estimate), the assigned
redshifts range from 12,000 to 18,000 km sec$^{-1}$.  Since all these galaxies
have a 25 $\mu$m flux density within a factor 1.5 of the flux limit, their
luminosities all fall near $L_*$.  When the luminosity function is
recalculated and plotted as a visibility, we have found that the asymptotes are
unchanged, and the peak visibility is increased by a few percent.

\subsection{Nonparametric maximum likelihood}
\label{sec:lfnonpar}

A non-parametric maximum-likelihood method (a.k.a ``Stepwise Maximum
Likelihood'') has been used to calculate the galaxy luminosity function in
various surveys (\markcite{lyn71}Lynden-Bell 1971;
\markcite{cho86}Choloniewski 1986; \markcite{efs88}Efstathiou et al.\ 1988;
\markcite{sau90}Saunders et al.\ 1990; \markcite{marzke94}Marzke et al. 1994).
This method does not assume a specific functional form for the luminosity
function, but rather describes it by a non-negative step function
(i.e. requires binning) which makes the observed galaxy luminosity
distribution as likely as possible.  We have followed the prescriptions
set out by \markcite{efs88}Efstathiou et al.\ (1988) and
\markcite{sau90}Saunders et al.\ (1990) for calculating the non-parametric
luminosity function.

As in the parametric maximum-likelihood method, the non-parametric approach
also eliminates the effect of any density inhomogeneities, but loses the
overall normalization by assuming that the galaxy spatial and luminosity
distributions are independent.  The
normalization of the non-parametric function has been chosen to give the
same source-counts as the parametric function over the same bins.

Figure \ref{fig:lf_swml} shows the visibility function obtained by using
the non-parametric maximum-likelihood method (circles), which agrees within
the error bars with our parametric results (solid line, from Figure
\ref{fig:newvis}).
The non-parametric method is commonly used to assess whether the parametric
functional form is acceptable.  Figure \ref{fig:lf_swml}
shows that the parametric form of Figure \ref{fig:newvis} gives a reasonable
description of the density-independent 25 $\mu$m luminosity function.
We have found that the non-parametric function is sensitive to the bin
width, however.  Figure \ref{fig:lf_swml} shows the case for bins in
$\log L$ = 0.2, but for a bin width of 0.4, the non-parametric results is
systematically higher than the parametric curve for high luminosities.
The coarser binning probably results in a loss of information and a less
reliable luminosity function.  The choice of bin size is a compromise
between this effect and tolerating the statistical uncertainty from
having too few sources in each bin.

\section{Completeness Checks}
\label{sec:compdisc}

Systematic variations of noise across the sky may change the
detection quality and cause incompleteness problems for the sample.
In the 25 $\mu$m passband of IRAS, noise is dominated by photon shot noise
from the zodiacal background, and hence is a strong function of the
ecliptic latitude.  We have examined the noise variation in the
sample and in the corresponding FSS plates and found two regions that
contain higher noise at $25\mu$m.  They are essentially the two
quadrants defined by $(0^\circ < l < 180^\circ, b < -30^\circ)$ and
$(180^\circ < l < 360^\circ, b > 30^\circ)$, which contain the ecliptic
plane.  Most of the moderate-quality detection sources (generally,
those with SNRs between 3 and 5) in our
sample are in these two regions, indicating the systematic effect.

To estimate this systematic effect on our sample completeness, we
compared luminosity functions estimated for two subsamples of our
250-mJy-limited full sample.  The first subsample is defined by raising
the flux-density limit to 400 mJy in the two quadrants of the sky
$(0^\circ < l < 180^\circ, b < -30^\circ)$ and $(180^\circ < l <
360^\circ, b > 30^\circ)$ where the noise is systematically higher, and
keeping the 250 mJy limit in the other two quadrants.  Most of the
moderate quality detection sources are left out of this subsample, so we
call these 1050 sources the high-quality subsample.  The second
subsample consists of those 408 sources in the full sample that are left out
of the high-quality subsample, and is therefore a low-quality subsample.
Figure \ref{fig:comphighlowQ} compares the luminosity functions of the
high-quality and the low-quality subsamples, using the $1/V_{max}$
estimator since it accomodates the different flux limits more easily
than either of the maximum-likelihood estimators.  There is no
systematic deficiency of galaxies in the low-quality subsample.  The
luminosity functions agree within the $1\sigma$ error bars, with the
exception of the point at $\log \nu L_\nu = 8.6$.  The low-quality
subsample includes galaxies in the Virgo cluster with 25 $\mu$m
flux densities between 250 and 400 mJy, most of which fall into this
luminosity bin.

In Figure \ref{fig:comphighQwhole} we compare the $1/V_{max}$ luminosity
functions of the high-quality subsample and the full 250 mJy-limited sample.
They agree very well, indicating that the full sample is complete at 250 mJy
flux-density across the entire sampling sky.  Therefore we are confident
that the full 250 mJy-limited sample is well-defined.

As another check of the completeness of our full sample, we have computed
the cumulative probability of finding a galaxy of luminosity $L$
at distance $r$, following the method of Yahil et al.\
\markcite{yah91} (1991).  This probability is the integral of
Equation \ref{eqn:condprob} and is $F(L|r)=\Psi(L)/\Psi(L_{min}(r))$.
This probability should be uniformly distributed on the interval
[0,1] independent of distance.  If the sample were incomplete at
the faint end, we would expect a decrease in the relative number
of galaxies as $F\rightarrow 1$.

We have computed $F(L_i | r_i )$ for each galaxy in the sample, then
computed the histogram of this statistic in different distance ranges,
normalized to 1 in each range.  The result is shown in Figure
\ref{fig:compcheck}.
There is no systematic decrease in the relative galaxy counts as
$F\rightarrow 1$ (albeit the statistics are poor beyond 400 Mpc).
This result provides further confirmation that our 250 mJy-limited sample
is complete to these distances.  Since this method relies upon ratios
of the luminosity function, it should be insensitive to inhomogeneities
in the galaxy distribution.

\section{Density Variations and the Corrections for $V/V_{max}$}
\label{sec:correct}

Just as the galaxy luminosity function independent of the
inhomogeneities of the density field can be obtained, the spatial
density variations independent of the galaxy luminosity function can
also be calculated using the maximum-likelihood method
(\markcite{sau90}Saunders et al.\ 1990).  Here we only investigate the
radial density distribution.  The probability for detecting a source of
luminosity $L_{i}$ at radial distance $r_{i}$ is
\begin{equation}
p_{i}=\frac{\rho(r_{i})dV_{i}}{\int_{0}^{r_{max,i}}\rho(r)(dV/dr)dr},
\end{equation}
where $\rho(r_{i})$ is the density at $r_{i}$, and $r_{max,i}$ is the
maximum distance for the source to be included in the sample.  The
likelihood function for all the detected sources would be $\cal L = \prod
\it p_{i}$, \rm given the density distribution $\rho(r)$ we are trying to
calculate.  In the non-parametric approach, this distribution can be found
by maximizing $ln \cal L$ with respect to the binned density field
$\rho(r_{j})$, which gives
\begin{equation}
\label{eqn:bindensity}
\rho(r_{j})=N_{j}(\sum_{i}\frac{f_{ij}V_{j}}{V_{eff,i}})^{-1},
\end{equation}
where $V_{eff,i} = \sum_{k=0}^{r_{max,i}} \rho(r_{k})V_{k}$,
$N_{j}$ is the number of sources in the $j$-th distance bin with volume
$V_{j}$, $f_{ij}$ is the fraction
of the bin contained in $[0, r_{max,i}]$ where the $i$-th source is observable,
and the sum is over all sources.

Figure \ref{fig:den_swml} illustrates the non-parametric maximum-likelihood
estimate of the radial density variation (solid line) as compared with the
flux-limited observed distribution (dashed line).  Here the radial distance
bins are in logarithmic scale, and the density variations are normalized to
the average density over all bins of interest.
The method correctly recovers the density enhancement at $\sim
17$ Mpc caused by Virgo.

We do not find a systematic increase in the radial density at large
distances.  If our sample is assumed to be complete, this result means
that there is no indication of density evolution in this sample.
We return to this point in section \ref{sec:evolution}.

The benefits of calculating the radial density variations in our sample
are two-fold.  First, it may be used to correct the $V/V_{max}$
statistic and the traditional $1/V_{max}$ estimate of the
density-dependent luminosity function (\markcite{sch68}Schmidt 1968;
\markcite{fel77} Felton 1977).  For the $V/V_{max}$ test, since
\begin{equation}
\frac{\int_{0}^{r_{max}}\frac{\int_{0}^{r}\rho(r')D^{2}dr'}
{\int_{0}^{r_{max}}\rho(r')D^{2}dr'}\rho(r)D^{2}dr}
{\int_{0}^{r_{max}}\rho(r)D^{2}dr} =
\frac{\int_{0}^{r_{max}}(\int_{0}^{r}\rho(r')D^{2}dr')d(\int_{0}^{r}\rho(r)
D^{2}dr)}{(\int_{0}^{r_{max}}\rho(r)D^{2}dr)^{2}} = \frac{1}{2},
\end{equation} where $\rho(r)$ is the radial density field, $r_{max}$ is
the maximum comoving distance for a source to be included in a sample,
and $D$ is the effective distance (\markcite{lon78}Longair 1978), we can
modify the $V/V_{max}$ test so that \begin{equation}
\label{eqn:modvovmax}
\frac{V}{V_{max}}=\frac{\int_{0}^{r}\rho(r)D^{2}dr}
{\int_{0}^{r_{max}}\rho(r)D^{2}dr}, \end{equation} which gives
$\overline{V/V_{max}}=1/2$ for an inhomogeneous radial distribution.
This also suggests that we use the $V_{eff,i}$ in equation
\ref{eqn:bindensity} instead of $V_{max}$ in the $1/V_{max}$ estimator
to calculate a density-corrected luminosity function.  The $\rho(r)$
calculated from the maximum-likelihood method can provide these
corrections.  The results are shown in Figure \ref{fig:vmax2} for the
new $V/V_{max}$ values and in Figure \ref{fig:lfvmax} for the
$1/V_{max}$ luminosity function.  Compared with the results without the
inhomogeneity correction in Figure \ref{fig:vratio1}, the $V/V_{max}$ values
are closer to 0.5 for those bins most strongly affected by
the Local Supercluster.  (However, the
corrected $V/V_{max}$ values are not suitable as a test of the sample
completeness, since the density correction will mask any incompleteness
in the sample.) The new $1/V_{max}$ luminosity function is smoother and
closer to the ones estimated by the maximum-likelihood method.  As
mentioned in Saunders et al. (1990), the binning of the distance in
Figure \ref{fig:den_swml} may account for the residual effects of the
density inhomogeneities in these Figures.

Second, the ``true'' density variations in Figure \ref{fig:den_swml}
also correct the observed redshift distribution. The
density-variation-corrected redshift distribution should be predicted by
a density-independent luminosity function estimated in the same sample.
This provides an independent test to verify the luminosity function.
Figure \ref{fig:srccnt} shows our results.  Here the dashed histogram is
the observed redshift distribution. (Here we show the source-counts in
distance bins instead of redshift bins since some galaxies have
secondary distances.) It is divided by the density of the corresponding
distance bins in Figure \ref{fig:den_swml} which gives the solid-line
histogram.  The source-counts affected by large-scale structures, such
as the ones near Virgo, are corrected by this procedure. The dotted line
shows the prediction by the parametric density-independent luminosity
function of Figure \ref{fig:newvis} and Table \ref{tab:fitpars}.  This
prediction fits the solid-line histogram well, demonstrating the
reliability of our luminosity function.  Both the solid-line histogram
and the dotted line are normalized so that they contain the same total
number of sources as observed within 20,000 km~s$^{-1}$.

\section{Discussion}
\label{sec:disc}

\subsection{Limitations of the methods}
\label{sec:limits}

The preceding sections have presented the 25 $\mu$m luminosity function
calculated by a number of methods.  In this section, the limitations of
the various methods are revisited in greater detail.

The $1/V_{max}$ estimate of the luminosity function varies the most from
the other estimators, particularly at luminosities less than $L_*$, where
our sample is most strongly influenced by the Local Supercluster.  As
noted above, this estimator assumes that the 3-dimensional galaxy
distribution is homogeneous.  The error bars in all of our $1/V_{max}$
estimates similarly depend on the assumption of uniformity, and do not
reflect the additional uncertainties that arise from an inhomogeneous
distribution.  We consider the $1/V_{max}$ estimates to be the least
reliable descriptions of the 25 $\mu$m luminosity function out of those
we have presented.

The parametric and non-parametric maximum-likelihood estimates should
be independent of structures in the galaxy distribution.  However,
these estimators assume that the luminosity function is independent
of density.  This assumption is difficult to check.  In the case of a
large 60 $\mu$m-selected sample, \markcite{yah91} Yahil et al. (1991)
demonstrated that their similarly-derived luminosity function
provided a good description of the luminosity distribution of
galaxies even in regions of high density.

The non-parametric maximum likelihood method applied to our sample is
sensitive to the choice of binning, as described in section
\ref{sec:lfnonpar}. We therefore consider the parametric
maximum-likelihood description of the 25 $\mu$m luminosity function with
parameters in Table \ref{tab:fitpars} to be the best out of those we
present in this paper.

\subsection{Comparison with other luminosity functions}
\label{sec:compare}

As mentioned in the Introduction,
\markcite{soi91}Soifer \& Neugebauer (1991) derived the local luminosity
function at 25 $\mu$m from a complete subsample of the 60-$\mu$m-selected
Bright Galaxy
Sample (\markcite{soi87}Soifer et al.\ 1987).  This subsample contained
135 galaxies to a flux density limit of 1.26 Jy, and the luminosity function
was determined using the classical $1/V_{max}$ estimator. Their luminosity
function is plotted as a space density in Figure \ref{fig:spacedens},
together with our parametric and $1/V_{max}$ estimates.

Our flux density limit of 250 mJy samples the Virgo cluster at $\log \nu
L_\nu = 8.6$ (section \ref{sec:parametric}), and our $1/V_{max}$ estimate
is much higher than the parametric curve in that bin. The Soifer \&
Neugebauer points in Figure \ref{fig:newvis} in $\log \nu L_\nu = 9.4$
bin is also much higher than the curve, as expected since their flux
density limit is five times higher than ours.  We conclude that the
Soifer \& Neugebauer luminosity function is strongly affected by the
Local Supercluster, particularly because the Bright Galaxy Sample  was
selected to be visible from Palomar Observatory (Soifer et al.\ 1984)
and does not include the south Galactic cap.

\subsection{Implications}
\label{sec:implications}

The improved luminosity function that is the main result of this paper
has implications the luminosity density of the local Universe, infrared
number counts predictions at 25 $\mu$m, and the infrared background
expected at this wavelength.

\markcite{soi91}Soifer \& Neugebauer (1991) calculated the local luminosity
density in the four IRAS passbands based on their derived luminosity
functions.  At 25 $\mu$m, they found that the log of the luminosity density in
units of $L_\odot$ Mpc$^{-3}$ is 7.2.  Integrating our parametric function
gives a log luminosity density of 6.9, which is a factor of 2 lower in linear
terms than the Soifer \& Neugebauer result. The differences in the sub-$L_*$
portion of the luminosity functions account for the different integrated
luminosity densities.

\markcite{hac91} Hacking \& Soifer (1991) presented number count
distributions at 25 $\mu$m, using a fit to the \markcite{soi91} Soifer
\& Neugebauer (1991) luminosity function.  (Those authors fit a
hyperbola to the  visibility function, which has the power-law
asymptotes at high and low luminosities like our parametric form, except
that it forces the absolute values of the slopes of the asymptotes to be
identical.) As they noted, the Soifer \& Neugebauer functions at other
wavelengths are 20\%-30\% higher than those derived from other samples
(\markcite{sau90}Saunders et al. 1990).  Our improved 25 $\mu$m function
is similarly about 15\% lower in normalization than the Soifer \&
Neugebauer estimate.  Hence, models based on this luminosity function
and its normalization will produce number counts about 15\% lower at the
bright end than those in Figure 1(c) of Hacking \& Soifer (1991).  Such
a change would bring the number counts in better agreement with the IRAS
number counts presented in their same figure.

Finally, we have substituted our luminosity function into the
backgrounds model of Hacking \& Soifer (1991).  The modified model
yields estimates of the infrared background at 25 $\mu$m that are
factors of 2.5 to 2.7 times smaller than the original Hacking \& Soifer
estimates.  Our results indicate that the detection of an IR
background at this wavelength is likely to be even more difficult
than previously predicted.

\subsection{Evolution}
\label{sec:evolution}

To this point, we have neglected any effects of galaxy evolution on our
derivation of the luminosity function.  Since the maximum-likelihood
estimators are independent of density variations, the shape of the
luminosity function will be unchanged by density evolution of the
galaxy population.  Our normalization considers galaxies out to 20,000
km~s$^{-1}$, a redshift where evolution effects should be small.

More importantly, our maximum likelihood fit to the radial density
(section \ref{sec:correct} and Figure \ref{fig:den_swml}) does not show
a systematic increase with redshift and hence no indication of density
evolution.  This result is in direct contrast to that of
\markcite{sau90}Saunders et al.\ (1990).  From their 60-$\mu$m-selected
sample, those authors found that density evolution of $(1+z)^{6.7}$
matched their radial density fit.  However, we note that the existence
of evolution from IRAS samples is not yet settled --- for the deepest
IRAS sample, \markcite{ashby96}
Ashby et al.\ (1996) did not find the high-redshift tail expected from
strong evolution.

Nevertheless, we have computed the shape of the parametric luminosity function
for the case of exponential luminosity evolution corresponding to $(1+z)^3$ at
low redshifts (cf. Saunders et al.\ (1990)), by correcting the luminosity of
each galaxy to the present epoch.  The values of the shape parameters of this
function are $\alpha = 0.437 \pm 0.032$, $\beta = 1.769 \pm 0.067$, and $L_* =
3.98 \pm 0.34 \times 10^9 L_\circ$.  The only change from the function
described in Table \ref{tab:fitpars} is that the slope of the luminosity
function at the high-luminosity end is slightly lowered.

\section{Summary and conclusions}
\label{sec:conclusions}

The following are the results of this paper:

1. We have selected a sample of 1458 galaxies with redshifts
from the IRAS Faint Source
Survey with a flux density limit of 250 mJy at 25 $\mu$m.  An additional
17 galaxies do not have redshifts available.

2. The local luminosity function is derived using the $1/V_{max}$ estimator
and both parametric and non-parametric maximum likelihood methods.
The $1/V_{max}$ estimate
is significantly affected by the Local Supercluster.  The maximum likelihood
methods are independent of density variations, and we consider the parametric
fit with parameters in Table \ref{tab:fitpars} to be the best estimate of the
local luminosity function at 25 $\mu$m.

3. A maximum likelihood fit to the radial density in this sample is used
to correct the $1/V_{max}$ estimate.  The fit shows no sign of a
systematic increase with redshift of the density, as would result from
density evolution of the galaxy population.

4. The $1/V_{max}$ luminosity function derived from a smaller sample by
Soifer \& Neugebauer (1991) is significantly contaminated by the Local
Supercluster.  Predictions of number counts and local luminosity density
based on that function are 15-20\% higher than those indicated by our
improved luminosity function.  The new function also leads to lower
predictions of the mid-infrared background due to galaxies.

\acknowledgments

We thank Susan Tokarz, Perry Berlind, and Jim Peters for observations
made at F.L.\ Whipple Observatory.
We also thank Iffat Khan and Joe Mazzarella for assistance with database
searches, and Carol Lonsdale and Tom Soifer for helpful discussions.
We are grateful to Cong Xu for a critical reading of this manuscript.
We thank the anonymous referee and our Scientific Editor, Steven Willner,
for numerous comments that have improved this paper.
The SIMBAD database is maintained by CDS in Strasbourg, France.  The
NED database is supported at IPAC by NASA.

redpajama_set_name: RedPajamaArXiv


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Home > News & reports > News > Nikau Foundation celebration
Nikau Foundation celebration
The Malaghan Institute hosted the Nikau Foundation and their supporters recently and unveiled the new "Nikau Foundation Immunotherapy Laboratory", named in recognition of their loyal support to the Malaghan Institute.
Nikau is part of the world's fastest growing movement in philanthropy – Community Foundations where people give to invest in the future to make an enduring impact on society.
One of our staff, PhD Student Karmella Naidoo, is a recipient of a Nikau Foundation Scholarship. While Professor Graham Le Gros gave the group an overview of our recent discoveries Karmella shared with the group progress on the first year of her research into eczema and its associated immune cell responses. One in five New Zealand children suffers from eczema; an allergic disease that typically starts in infancy and results from an inappropriate immune response to innocuous substances such as house dust mites or pollen.
Karmella's PhD project is supervised by Malaghan Institute's Gut Immunology Team Leader Dr Elizabeth Forbes-Blom. Karmella's investigations are connected both nationally and internationally; including research collaborations with academic institutions and industry (partners include Fonterra, University of Otago, University of Queensland, the Mayo Clinic, Danone and the University of Utrecht).
Chris Milne, Chair of the Nikau Foundation said, "I was impressed by the promise of immunotherapy for our community. The dedication of scientists at the Malaghan Institute makes me optimistic for the future of medical research in New Zealand."
The Nikau Foundation, on behalf of their donors, have funded medical research at the Malaghan Institute for over a decade, and we are grateful for their continuing support.