text:
Mathieu Bisson est un acteur français. Il est le fils de Jean-Pierre Bisson.

Filmographie

Cinéma
 1991 : Le Voleur d'enfants de Christian de Chalonge : Simon
 2004 : Entre vivir y soñar d'Alfonso Albacete et David Menkes : Philippe
 2006 : Un ticket pour l'espace de Éric Lartigau : le technicien 3
 2007 : Jean de la Fontaine - Le Défi de Daniel Vigne : Nicolas Boileau
 2016 : Lutine d'Isabelle Broué : Gaël
 2018 : Les Animaux fantastiques : Les Crimes de Grindelwald (Fantastic Beats : The Crimes of Grindelwald) de David Yates

Court métrage
 2002 : Bien fait de Jeanne Biras
 2004 : Mon homme de Stéphanie Cotta : le jeune homme
 2004 : La Vache qui pleure de Stanislas Carré de Malberg : le père de Jean
 2012 : L'Équation de Mathieu Hippeau : Serge
 2014 : Le Grand Jeu d'Agnès Vialleton : Mathias
 2016 : La Femme et le TGV de Timo Von Gunten : Pierre

Télévision

 2002 : Jean Moulin d'Yves Boisset : l'employé à la réception
 2002 : Nestor Burma, épisode Noblesse désoblige de Philippe Venault : Bertrand
 2003 : Les Enfants du miracle de Sébastien Grall : Paul Daumier
 2003 : Et toc ! de Samuel Tasinaje et Martin Valente
 2003 : Joséphine, ange gardien, épisode Sens dessus dessous de David Delrieux
 2004 : La Nourrice de Renaud Bertrand : Monsieur de Lignac
 2004 : Sauveur Giordano, épisode Disparitions : Hervé Guillot
 2004 : Petits Mythes urbains : épisode Scalpel illégitime de Óskar Santos Gómez et Stéphane Gateau : un passager
 2005 : Clara Sheller de Renaud Bertrand, saison 1 : Mathieu, le frère de Clara
 2005 : Commissaire Moulin, épisode Un coupable trop parfait de Jean-Luc Breitenstein : Lefrère
 2006 : Les Secrets du volcan de Michaëla Watteaux : Christophe
 2006 : L'Oncle de Russie de Francis Girod : François
 2007 : L'Affaire Ben Barka de Jean-Pierre Sinapi
 2007 : H.B. Human Bomb de Patrick Poubel : Michel Marie
 2007 : Notable, donc coupable de Dominique Baron et Francis Girod :  François Calmels
 2008 : À droite toute de Marcel Bluwal : Caron
 2008 : Mitterrand à Vichy de Serge Moati : François Mitterrand
 2009 : Alice Nevers, le juge est une femme : "Faute d'Adn" de Denis Amar : Luc Sarron
 2009 : Les Vivants et les Morts de Gérard Mordillat : Docteur Kopps
 2009 : Marion Mazzano, épisode Le Contrat de Marc Angelo : le juge d'instruction
 2009 : Sœur Thérèse.com, épisode Pardon ma sœur de Bertrand Van Effenterre : Alexandre Mercier
 2010 : Sélection naturelle de Régis Musset
 2010 : Affaires étrangères, saison 1 épisode 2 Maroc de Vincenzo Marano : Monsieur Lascombe
 2011 : Pour Djamila de Caroline Huppert : le directeur de cabinet
 2011 : Clem, saison 2 épisode 2 La Mutation de Joyce Bunuel : Didier
 2013 : RIS police scientifique, épisode Chute Libre de Hervé Brami
 2014 : Section de recherches, épisode Sexy à mort de Delphine Lemoine
 2014 : Nicolas Le Floch, saison 6 épisode 1 Le Cadavre anglais de Philippe Bérenger : Armand Desplat
 2017 : Prof T , six épisodes de Nicolas Cuche et Jean-Christophe Delpias : Prof T.
 2021 : Laval, le collaborateur de Laurent Heynemann : Jacques Baraduc

Théâtre

 2000 : La Bonne Mère, mise en scène Sophie Lorotte
 2001 : Fragments d'une lettre d'adieu lue par des géologues, mise en scène Camille Chamoux, Lavoir Moderne Parisien
 2004 : La Quatrième Sœur de Janusz Głowacki, mise en scène Camille Chamoux, Théâtre Silvia Monfort
 2004 : Trois Jours de pluie de Richard Greenberg, mise en scène Jean-Marie Besset et Gilbert Desveaux, Théâtre de l'Atelier
 2005 : L'Autre de Florian Zeller, mise en scène Annick Blancheteau, Théâtre des Mathurins
 2006 : Romance de David Mamet, mise en scène Pierre Laville, Théâtre Tristan Bernard
 2007 : Cité radieuse d'après Conor McPherson, mise en espace Gilbert Désveaux, Festival NAVA
 2011 : Hitch de Alain Riou et Stéphane Boulan, mise en scène Sebastien Grall, Théâtre du Lucernaire
 2012 : Le Vicaire de Rolf Hochhuth, mise en scène Jean-Paul Tribout, Théâtre 14 Jean-Marie Serreau
 2013/14 : L'Importance d'être sérieux d'Oscar Wilde, traduction Jean-Marie Besset, mise en scène Gilbert Désveaux, Théâtre des 13 vents,  Théâtre Montparnasse

Distinctions
 2011 : Prix d'interprétation au Festival d'Anjou pour la pièce Hitch.

Notes et références

Liens externes

Acteur français de cinéma
Acteur français de télévision
Acteur français de théâtre
Acteur ayant incarné François Mitterrand
Talents Cannes
Date de naissance non renseignée (XXe siècle)


text:
# 2452 Believe This?
Home/# 2452 Believe This?
Q. Recently the Government of Canada turned down the tax-exempt status for the Atheistic Church.
If they decide to sell properties, can one buy them or are they considered in Halacha as having been used for avoda zarah?
A. Last week as reported by the press, (N. P. Dec 7), the Federal Court of Appeal upheld Revenue Canada's rejection of an application for charitable tax- exempt status made by a "Church of Atheism" in Ontario's Lanark Highlands.
The Halacha viewpoint in regard to this institution depends on what they actually belief. The "Church of Atheism" tried to make the argument that it should qualify for a tax exemption as a teacher and promoter of atheism per se. Using the premise that "advancement of religion" has been recognized for centuries as a charitable purpose in itself. If that would be their only claim, it likely would not be defined as avoda zara.
However, they also proposed to treat its interpretation of "Atheism" as a religion, rather than the absence or rejection of religion. The "Church" maintains the worship of "mainstream science" and claims to possess a "Ten Commandments of Energy" which were "created by a wise human being who consists of pure, invisible Energy and has acknowledged Energy's existence."
Revenue Canada had demanded evidence that the CoA believes "in a higher unseen power such as a G-d, Supreme Being, or entity," but that was not forthcoming, thus the rejection.
One may argue that atheism, although prohibited for Bnai Noach, does not qualify as avoda zara and therefore the resulting restrictions do not apply. However, it has been suggested that they may be believers on the divine power of man and his natural energy, and that may indeed qualify as avoda zara.
Rabbi A, Bartfeld as revised by Horabonim Dovid Pam and Aaron Miller Shlit'a

redpajama_set_name: RedPajamaCommonCrawl


text:
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redpajama_set_name: RedPajamaGithub


text:
Home / National News / National Guardsman missing after flash flooding devastates Ellicott City, Maryland: 'I'm hopeful that he'll be OK'
National Guardsman missing after flash flooding devastates Ellicott City, Maryland: 'I'm hopeful that he'll be OK'
(ELLICOT, CITY, Md.) — A National Guardsman is missing after devastating flash flooding swept through Ellicott City, Maryland, turning streets into rivers and carrying away cars.
The missing man, 39-year-old Eddison Hermond of Severn, Maryland, was last seen at about 5:20 p.m. Sunday, Howard County police said.
Hermond, a member of the Maryland National Guard, was at an Ellicott City restaurant with his friend from the Air Force, Joseph Lopez, when they noticed the parking lot starting to flood because of the heavy rain, Lopez told ABC News.
"It happened so fast," Lopez said. "The main street became a river in less than 20 minutes."
The friends were helping to block the restaurant doors to prevent flooding when one woman with a cat carrier wanted to leave, Lopez said.
Eddison was helping her when he lost his balance and was swept away into a man-made river and then under a bridge, Lopez said.
Eddison, who was wearing a white T-shirt and black shorts, hasn't been seen since, Lopez said.
"Eddison is a great swimmer so I'm hopeful that he will be OK," Lopez said.
The governor of Maryland declared a state of emergency in the wake of the flash flooding in the historic mill town and other areas Sunday. Ellicott City saw 8.4 inches of rain in just a few hours. Nearby Catonsville, Maryland, saw over 10 inches of rain.
Howard County County Executive Allan Kittleman this morning called the flooding worse than the one that hit Ellicott City in 2016, killing two people and causing millions of dollars of damage.
"My heart's broken when I walk through the town and see it," Kittleman told "Good Morning America." "All I'm thinking about is the folks whose lives have been devastated for a second time in two years."
There were about 30 water rescues and about 250 to 300 people were helped out of buildings amid the flash flooding, officials said.
Cheryl Nitz came to Ellicott City for a trip with her family Sunday when "it started raining … and all of a sudden it just started raining harder and harder," she told ABC News. "Then we noticed the water was coming up over the curbs. Then all of a sudden the water was actually running down the middle of the street."
Nitz and her family raced up to higher ground to their car.
"We were able to get the car out of the parking spot and we moved toward Main Street," she said, but "there were other cars stuck in the way. We couldn't move."
The water rose so rapidly that her family had to be rescued from their car, she added.
"It rose so quickly probably within 5 or 10 minutes it just started to kind of cover the road… it was rushing," she said. "It was unbelievable."
Andrew Kolozsvary of New Hampshire was also visiting Ellicott City, enjoying a Sunday drive, when it started "raining pretty hard, so we decided to park the truck in front of the county welcome center," he told ABC News. "The water came and it came fast."
While in the truck, the quickly rising water was "inching the car along," he said.
After his rescue, the truck ended up washed down the street.
"I was glad to get out of it," he said. "The second deluge was the really bad one — that took the truck."
Amber Twait was in Ellicott City as a band member for her friend's wedding Sunday when the venue flooded.
The couple ended up getting married at a nearby Mexican restaurant that was on higher ground, she told ABC News.
Eventually, Twait added, the restaurant started to flood as well, but not before they finished the ceremony.
"Obviously not ideal, but they were still able to say they're married at the end of the day," she said.
"It was a little scary, just because the lights were flickering but it was just really sweet and they both had such a good attitude about it," Twait added. "Hell or high water, they still were going to get married."


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Blackrock Slips as Firm Likely to Get into Investment Mode
ByInvesting.com
By Dhirendra Tripathi
Investing.com – BlackRock stock (NYSE:) fell 2.4% on Friday on concerns the fund giant is likely to get into investment mode again, a move that could keep its profit margin under pressure.
"Looking forward, they are going into investment mode again," Reuters quoted Edward Jones analyst Kyle Sanders as saying. "They are going to spend a lot, and that is probably going to make it tough for them to really have meaningful profit margin and EPS growth in 2022," he said, as quoted by the agency.
For the fourth quarter ended December 31, Blackrock's 14% gain in revenue, to $5.10 billion, just met expectations while adjusted profit per share of $10.42 topped estimates.
Performance fees, the sole revenue stream to erode in the fourth quarter, fell by $90 million, primarily reflecting lower revenue from liquid alternative products.
Assets under management at the world's largest money manager scaled a new peak of $10 trillion by the end of December, up from $8.68 trillion a year earlier.
"Our business is more diversified than ever before – active strategies, including alternatives, contributed over 60% of 2021 organic base fee growth," Chief Executive Officer Larry Fink said in a statement.
Net inflows for the quarter were at $212 billion, of which long-term net flows accounted for $169 billion.
ETFs attracted a net $104 billion, 11% higher than a year ago and a record for the company, Blackrock said.
BlackRock now manages $2.6 trillion in actively-managed funds that include ETFs and mutual funds. The firm's alternatives business, which includes hedge funds, saw $5.5 billion of inflows, taking total assets to $265 billion, according to Bloomberg.
Jamie Dimon Takes High View of Fed Hikes: 'It Could Be Six or Seven'
Disney Shares Tumble After Guggenheim Downgrade
U.S. shares lower at close of trade; Dow Jones Industrial Average down 1.30%
Jan 22, 2022 Investing.com
Canada shares lower at close of trade; S&P/TSX Composite down 2.29%


I came a cross an article from the first of the year that I thought would be interesting. The Center for Medicare and Medicare Services (CMS) launched two new models in January. One is a new payment model and the other has updated an existing model which is aimed at lowering drug prices, and better serving patients of Medicare Part D and Medicare Advantage plans.
The new Part D plan, called the Part D Payment Modernization Model, aims at creating incentives to lower drug prices, as well as out-of-pocket costs.
Currently in Part D, when a patient's prescription drug spending is high enough for the patient to enter the highest phase of the benefit, known as the "catastrophic phase," Medicare is responsible for 80 percent of the drug costs.
With this rule in place, plans have little reason to negotiate lower costs for the highest-spending patients.
The new model would make it to where participating Part D plans would share in the savings if they stay below the target, but would be responsible for losses if they did. It would also provide tools to Part D plans to control drug costs and help enrollees choose more affordable drugs.
The update to the Medicare Advantage Value-Based Insurance Design model will allow plans to provide reduced cost sharing and other benefits to enrollees, in a much more targeted way.
The new model allows plans to boost the rewards and incentives programs that can help beneficiaries improve their health and increases access to telehealth services by allowing plans to use virtual services instead of in-person visits.
I think this is good news across both forms of Medicare. The main complaint about Medicare Advantage has been drug costs and limited networks. Think these new models could be better? What else should be implemented and reworked?
Interesting. Not sure on your question. Was going to throw it back to you actually.. which do you think got the better updates in your estimation? Did one improve to make it the better option, traditional + D vs Advantage? I know there are pros and cons to both..


I'm back. So I guess I'll just post whatever I want whenever I feel like it on here. I've been gone mostly due to some personal projects and stuff, but it's good to be back and not just stalking or ignoring the tab I have open on this site. I've made quite a lot of characters and creatures for stories and I figured why not describe them here and actually use them for something other than storytelling. So I decided to start with Ajac, who is from a weird collaboration story thing that me and my buds did a while back. Ajac has a wide knowledge about plants and animals. He only kept one animal he helped, a shadow creature named Kezo. Kezo is usually in the form of a wolf, which has black fur and pale blue eyes. A shadow creature is basically a shapeshifter that can turn into different types of animals. Shadow creatures are as the name implies, made of shadows that were given life and a physical form by magic, so whatever animal they take the form of will have black fur/scales/skin. A shadow creature's eyes are a pale color.
Hello. I am Ajac and this is my friend, Kezo. I suppose that Scar has already spoke about me, so I'll just be over here.
Really, man? You're just gonna stand there.
What else would I do? I have already fed the animals and given the medical bay animals their treatment.
Can't you just talk about yourself or one of those animals you helped?
Fine, I suppose I could do that. I, Ajac, spent two years on the run for minorly injuring a poacher, who had injured a young shadow creature, which are hard to find due to their elusive nature. I took care of the creature until I was able to release it back into the wild, but it refused to leave. So I continued to care for it and gave it the name Kezo. Eventually I ran into a group of outlaws, who too were running from the law for mediocre things. I joined the group and continued my work of helping animals and ended up having to care of them as well. They have a knack for getting into trouble and getting on my nerves. We ended up getting caught, but the judge allowed us to tell our sides of the story. Most of us were pardoned or let off with a minor punishment. After that, I settled down and opened up an animal hospital and sanctuary, where I care for injured animals I find and those that people have brought to me. Certain animals are unfit to be released into the wild usually end up living with me for the rest of their life or are adopted into families. Of course I do take time to check up on those adopted whenever I can. I often give my thanks to the volunteers and the few workers I have, since they allow me to have more free time. I believe that is all, unless you wish for me to tell a much more detailed version.
That's fine. I feel like I might do this sort of thing with other characters.(I totally didn't already start another one) I'd add more, but this post is getting long. I'll just make another post once this goes up. That one will probably be random stuff like most of my posts.


Do you have a new home loan and can't wait to get debt free as soon as possible?
You try to pay a little extra each month; but life gets in the way at times. It could be an unexpected car repair or a much-needed weekend getaway or a little too many drinks, more than a couple of times, at your favourite bar – these small overheads can add to your monthly expenditure considerably.
So, what is the key to financial freedom?
Discipline – as everyone would tell you. But, where's the motivation?
Lisa borrowed $350,000 to purchase a house worth $500,000. She borrowed the money for 25 years at a rate of 4.00 per cent.
Using HashChing's online repayment calculator, you'd find that the loan, at that rate, would cost Lisa $554,228.68 over it's full term. Thus, Lisa would be paying $204,228.68 in interest, in addition to the $500,000 that she paid for her house! That's a whopping $704,228 of her hard-earned money!
So, now that you have done the math, you must be wondering how to avoid throwing your money away to the banks?
Let's start by telling you a secret. Right now, is probably the cheapest time to pay off your home loan. With talks about the next movement in interest rates being upward, there couldn't be a better time to pay down your mortgage than now.
Of course, you could choose to stick to the bank's schedule and wait 25 years to own your home while you pay a 'tub' full of interest. Or, you throw every extra dollar at your mortgage to bring down the principal amount – which means, savings in interest, lower repayment amount, and early financial freedom from debt.
But, wait. Isn't that the crux of the problem? Having extra dollars to pay down your mortgage each month?
Creating a good household budget, and sticking to it, is an obvious solution to that problem. But, there's something else you could do about that, too. By merely shrugging off your laziness and checking the rates you are paying on everything from your insurance to utilities to mortgage, you might find that you are spending much higher than the market average. By comparing rates from different service providers, it is possible to land yourself a better deal and save thousands of dollars each month.
Lisa's home loan is a case in point. At 4.00 per cent, she is probably paying too much interest to her lender. Especially for an over 80 per cent LVR loan, it is possible to lock in a rate as low as 3.39 per cent with HashChing – where you can compare broker pre-negotiated home loan deals from over 60 lenders in Australia.
If Lisa switches to a rate of 3.39 per cent from her current rate of 4.00 per cent, her monthly repayment amount will reduce by $115.83, saving her $34,747.97 over the life of the loan. But what if Lisa switches to this rate and continues paying the same repayment amount?
That's an additional repayment of $115 each month!
Assuming that Lisa makes the switch in the third year of her home loan, she will save an additional $13,000 and also reduce two years from her loan term. Use our handy additional repayment calculator to crunch the numbers for yourself.
With several competitive home loan deals available in the market currently, if you are paying anything more than 3% on your home loan, you are probably paying too much interest.
According to experts, reviewing your home loan every six months or so helps you stay on top of your mortgage, by confirming whether you are on the best deal in the market or not.
Size up your mortgage against the most competitive home loan deals in the market by comparing home loan rates. Just a few seconds of your time could put hundreds of dollars in your pocket each month. Or, if you don't feel up to it, fill up this short contact form so that we can put a verified mortgage broker in touch with you to do all the legwork on your behalf and help you achieve the financial freedom you deserve!
Remember, money saved is money earned. And every dollar counts….


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Per Enflo
Enflo in 1972
(1944-05-20) 20 May 1944 (age 76)
Approximation problem
Schauder basis
Hilbert's fifth problem (infinite-dimensional)
uniformly convex renorms of super-reflexive Banach spaces
embedding metric spaces (unbounded distortion of cube)
"Concentration" of polynomials at low degree
Invariant subspace problem
Mazur's "live goose" for solving "Scottish Book" Problem 153
Analytic number theory
École Polytechnique, Paris
The Royal Institute of Technology, Stockholm
Doctoral advisor
Hans Rådström
Angela Spalsbury
Bruce Reznick
Joram Lindenstrauss
Bernard Beauzamy
Per H. Enflo (Swedish: [ˈpæːr ˈěːnfluː]; born 20 May 1944) is a Swedish mathematician working primarily in functional analysis, a field in which he solved problems that had been considered fundamental. Three of these problems had been open for more than forty years:[1]
The basis problem and the approximation problem[2] and later
the invariant subspace problem for Banach spaces.[3]
In solving these problems, Enflo developed new techniques which were then used by other researchers in functional analysis and operator theory for years. Some of Enflo's research has been important also in other mathematical fields, such as number theory, and in computer science, especially computer algebra and approximation algorithms.
Enflo works at Kent State University, where he holds the title of University Professor. Enflo has earlier held positions at the Miller Institute for Basic Research in Science at the University of California, Berkeley, Stanford University, École Polytechnique, (Paris) and The Royal Institute of Technology, Stockholm.
Enflo is also a concert pianist.
1 Enflo's contributions to functional analysis and operator theory
1.1 Hilbert's fifth problem and embeddings
1.1.1 Applications in computer science
1.2 Geometry of Banach spaces
1.3 The basis problem and Mazur's goose
1.3.1 Basis problem of Banach
1.3.2 Problem 153 in the Scottish Book: Mazur's goose
1.3.3 Grothendieck's formulation of the approximation problem
1.3.4 Enflo's solution
1.4 Invariant subspace problem and polynomials
1.4.1 Multiplicative inequalities for homogeneous polynomials
1.4.1.1 Bombieri norm
1.4.2 Applications
2 Mathematical biology: Population dynamics
2.1 Human evolution
5 External sources
5.1 Databases
Enflo's contributions to functional analysis and operator theory[edit]
In mathematics, Functional analysis is concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. In functional analysis, an important class of vector spaces consists of the complete normed vector spaces over the real or complex numbers, which are called Banach spaces. An important example of a Banach space is a Hilbert space, where the norm arises from an inner product. Hilbert spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics, stochastic processes, and time-series analysis. Besides studying spaces of functions, functional analysis also studies the continuous linear operators on spaces of functions.
Hilbert's fifth problem and embeddings[edit]
See also: Hilbert's fifth problem
At Stockholm University, Hans Rådström suggested that Enflo consider Hilbert's fifth problem in the spirit of functional analysis.[4] In two years, 1969–1970, Enflo published five papers on Hilbert's fifth problem; these papers are collected in Enflo (1970), along with a short summary. Some of the results of these papers are described in Enflo (1976) and in the last chapter of Benyamini and Lindenstrauss.
Applications in computer science[edit]
See also: Approximation algorithm
Enflo's techniques have found application in computer science. Algorithm theorists derive approximation algorithms that embed finite metric spaces into low-dimensional Euclidean spaces with low "distortion" (in Gromov's terminology for the Lipschitz category; c.f. Banach–Mazur distance). Low-dimensional problems have lower computational complexity, of course. More importantly, if the problems embed well in either the Euclidean plane or the three-dimensional Euclidean space, then geometric algorithms become exceptionally fast.
However, such embedding techniques have limitations, as shown by Enflo's (1969) theorem:[5]
For every m ≥ 2 {\displaystyle m\geq 2} , the Hamming cube C m {\displaystyle C_{m}} cannot be embedded with "distortion D {\displaystyle D} " (or less) into 2 m {\displaystyle 2^{m}} -dimensional Euclidean space if D < m {\displaystyle D<{\sqrt {m}}} . Consequently, the optimal embedding is the natural embedding, which realizes { 0 , 1 } m {\displaystyle \{0,1\}^{m}} as a subspace of m {\displaystyle m} -dimensional Euclidean space.[6]
This theorem, "found by Enflo [1969], is probably the first result showing an unbounded distortion for embeddings into Euclidean spaces. Enflo considered the problem of uniform embeddability among Banach spaces, and the distortion was an auxiliary device in his proof."[7]
Geometry of Banach spaces[edit]
See also: Uniformly convex space
A uniformly convex space is a Banach space so that, for every ϵ > 0 {\displaystyle \epsilon >0} there is some δ > 0 {\displaystyle \delta >0} so that for any two vectors with ‖ x ‖ ≤ 1 {\displaystyle \|x\|\leq 1} and ‖ y ‖ ≤ 1 , {\displaystyle \|y\|\leq 1,}
‖ x + y ‖ > 2 − δ {\displaystyle \|x+y\|>2-\delta }
implies that
‖ x − y ‖ < ϵ . {\displaystyle \|x-y\|<\epsilon .}
Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short.
In 1972 Enflo proved that "every super-reflexive Banach space admits an equivalent uniformly convex norm".[8][9]
The basis problem and Mazur's goose[edit]
See also: Schauder basis and Approximation problem
With one paper, which was published in 1973, Per Enflo solved three problems that had stumped functional analysts for decades: The basis problem of Stefan Banach, the "Goose problem" of Stanislaw Mazur, and the approximation problem of Alexander Grothendieck. Grothendieck had shown that his approximation problem was the central problem in the theory of Banach spaces and continuous linear operators.
Basis problem of Banach[edit]
Main article: Schauder basis
See also: Stefan Banach, Banach space, and Separable space
The basis problem was posed by Stefan Banach in his book, Theory of Linear Operators. Banach asked whether every separable Banach space has a Schauder basis.
A Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that for Hamel bases we use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces.
Schauder bases were described by Juliusz Schauder in 1927.[10][11] Let V denote a Banach space over the field F. A Schauder basis is a sequence (bn) of elements of V such that for every element v ∈ V there exists a unique sequence (αn) of elements in F so that
v = ∑ n ∈ N α n b n {\displaystyle v=\sum _{n\in \mathbb {N} }\alpha _{n}b_{n}\,}
where the convergence is understood with respect to the norm topology. Schauder bases can also be defined analogously in a general topological vector space.
In 1937, Polish mathematician Stanislaw Mazur promised a "live goose" as the prize for solving problem 153 in the Scottish Book. In 1972, Mazur presented the goose to Per Enflo.
Problem 153 in the Scottish Book: Mazur's goose[edit]
In 1972 Stanislaw Mazur awarded Enflo the promised live goose for solving a problem in the Scottish book.
See also: Scottish Café, Scottish Book, and Stanislaw Mazur
Banach and other Polish mathematicians would work on mathematical problems at the Scottish Café. When a problem was especially interesting and when its solution seemed difficult, the problem would be written down in the book of problems, which soon became known as the Scottish Book. For problems that seemed especially important or difficult or both, the problem's proposer would often pledge to award a prize for its solution.
On 6 November 1936, Stanislaw Mazur posed a problem on representing continuous functions. Formally writing down problem 153 in the Scottish Book, Mazur promised as the reward a "live goose", an especially rich price during the Great Depression and on the eve of World War II.
Fairly soon afterwards, it was realized that Mazur's problem was closely related to Banach's problem on the existence of Schauder bases in separable Banach spaces. Most of the other problems in the Scottish Book were solved regularly. However, there was little progress on Mazur's problem and a few other problems, which became famous open problems to mathematicians around the world.[12]
Grothendieck's formulation of the approximation problem[edit]
Main articles: Approximation property and Approximation problem
See also: Compact operator and Alexander Grothendieck
Grothendieck's work on the theory of Banach spaces and continuous linear operators introduced the approximation property. A Banach space is said to have the approximation property, if every compact operator is a limit of finite-rank operators. The converse is always true.[13]
In a long monograph, Grothendieck proved that if every Banach space had the approximation property, then every Banach space would have a Schauder basis. Grothendieck thus focused the attention of functional analysts on deciding whether every Banach space have the approximation property.[13]
Enflo's solution[edit]
In 1972, Per Enflo constructed a separable Banach space that lacks the approximation property and a Schauder basis.[14] In 1972, Mazur awarded a live goose to Enflo in a ceremony at the Stefan Banach Center in Warsaw; the "goose reward" ceremony was broadcast throughout Poland.[15]
Invariant subspace problem and polynomials[edit]
Main article: Invariant subspace problem
See also: Invariant subspace
In functional analysis, one of the most prominent problems was the invariant subspace problem, which required the evaluation of the truth of the following proposition:
Given a complex Banach space H of dimension > 1 and a bounded linear operator T : H → H, then H has a non-trivial closed T-invariant subspace, i.e. there exists a closed linear subspace W of H which is different from {0} and H such that T(W) ⊆ W.
For Banach spaces, the first example of an operator without an invariant subspace was constructed by Enflo. (For Hilbert spaces, the invariant subspace problem remains open.)
Enflo proposed a solution to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987[16] Enflo's long "manuscript had a world-wide circulation among mathematicians"[17] and some of its ideas were described in publications besides Enflo (1976).[18][19] Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Beauzamy, who acknowledged Enflo's ideas.[16]
In the 1990s, Enflo developed a "constructive" approach to the invariant subspace problem on Hilbert spaces.[20]
Multiplicative inequalities for homogeneous polynomials[edit]
See also: homogeneous polynomial
An essential idea in Enflo's construction was "concentration of polynomials at low degrees": For all positive integers m {\displaystyle m} and n {\displaystyle n} , there exists C ( m , n ) > 0 {\displaystyle C(m,n)>0} such that for all homogeneous polynomials P {\displaystyle P} and Q {\displaystyle Q} of degrees m {\displaystyle m} and n {\displaystyle n} (in k {\displaystyle k} variables), then
| P Q | ≥ C ( m , n ) | P | | Q | , {\displaystyle |PQ|\geq C(m,n)|P|\,|Q|,}
where | P | {\displaystyle |P|} denotes the sum of the absolute values of the coefficients of P {\displaystyle P} . Enflo proved that C ( m , n ) {\displaystyle C(m,n)} does not depend on the number of variables k {\displaystyle k} . Enflo's original proof was simplified by Montgomery.[21]
This result was generalized to other norms on the vector space of homogeneous polynomials. Of these norms, the most used has been the Bombieri norm.
Bombieri norm[edit]
Main article: Bombieri norm
The Bombieri norm is defined in terms of the following scalar product: For all α , β ∈ N N {\displaystyle \alpha ,\beta \in \mathbb {N} ^{N}} we have
⟨ X α | X β ⟩ = 0 {\displaystyle \langle X^{\alpha }|X^{\beta }\rangle =0} if α ≠ β {\displaystyle \alpha \neq \beta }
For every α ∈ N N {\displaystyle \alpha \in \mathbb {N} ^{N}} we define | | X α | | 2 = | α | ! α ! , {\displaystyle ||X^{\alpha }||^{2}={\frac {|\alpha |!}{\alpha !}},}
where we use the following notation: if α = ( α 1 , … , α N ) ∈ N N {\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{N})\in \mathbb {N} ^{N}} , we write | α | = Σ i = 1 N α i {\displaystyle |\alpha |=\Sigma _{i=1}^{N}\alpha _{i}} and α ! = Π i = 1 N ( α i ! ) {\displaystyle \alpha !=\Pi _{i=1}^{N}(\alpha _{i}!)} and X α = Π i = 1 N X i α i . {\displaystyle X^{\alpha }=\Pi _{i=1}^{N}X_{i}^{\alpha _{i}}.}
The most remarkable property of this norm is the Bombieri inequality:
Let P , Q {\displaystyle P,Q} be two homogeneous polynomials respectively of degree d ∘ ( P ) {\displaystyle d^{\circ }(P)} and d ∘ ( Q ) {\displaystyle d^{\circ }(Q)} with N {\displaystyle N} variables, then, the following inequality holds:
d ∘ ( P ) ! d ∘ ( Q ) ! ( d ∘ ( P ) + d ∘ ( Q ) ) ! | | P | | 2 | | Q | | 2 ≤ | | P ⋅ Q | | 2 ≤ | | P | | 2 | | Q | | 2 . {\displaystyle {\frac {d^{\circ }(P)!d^{\circ }(Q)!}{(d^{\circ }(P)+d^{\circ }(Q))!}}||P||^{2}\,||Q||^{2}\leq ||P\cdot Q||^{2}\leq ||P||^{2}\,||Q||^{2}.}
In the above statement, the Bombieri inequality is the left-hand side inequality; the right-hand side inequality means that the Bombieri norm is a norm of the algebra of polynomials under multiplication.
The Bombieri inequality implies that the product of two polynomials cannot be arbitrarily small, and this lower-bound is fundamental in applications like polynomial factorization (or in Enflo's construction of an operator without an invariant subspace).
Applications[edit]
See also: Polynomial factorization
Enflo's idea of "concentration of polynomials at low degrees" has led to important publications in number theory[22] algebraic and Diophantine geometry,[23] and polynomial factorization.[24]
Mathematical biology: Population dynamics[edit]
See also: Mathematical biology and Population dynamics
In applied mathematics, Per Enflo has published several papers in mathematical biology, specifically in population dynamics.
Human evolution[edit]
Enflo has also published in population genetics and paleoanthropology.[25]
Today, all humans belong to one population of Homo sapiens sapiens, which is individed by species barrier. However, according to the "Out of Africa" model this is not the first species of hominids: the first species of genus Homo, Homo habilis, evolved in East Africa at least 2 Ma, and members of this species populated different parts of Africa in a relatively short time. Homo erectus evolved more than 1.8 Ma, and by 1.5 Ma had spread throughout the Old World.
Anthropologists have been divided as to whether current human population evolved as one interconnected population (as postulated by the Multiregional Evolution hypothesis), or evolved only in East Africa, speciated, and then migrating out of Africa and replaced human populations in Eurasia (called the "Out of Africa" Model or the "Complete Replacement" Model).
Neanderthals and modern humans coexisted in Europe for several thousand years, but the duration of this period is uncertain.[26] Modern humans may have first migrated to Europe 40–43,000 years ago.[27] Neanderthals may have lived as recently as 24,000 years ago in refugia on the south coast of the Iberian peninsula such as Gorham's Cave.[28][29] Inter-stratification of Neanderthal and modern human remains has been suggested,[30] but is disputed.[31]
With Hawks and Wolpoff, Enflo published an explanation of fossil evidence on the DNA of Neanderthal and modern humans. This article tries to resolve a debate in the evolution of modern humans between theories suggesting either multiregional and single African origins. In particular, the extinction of Neanderthals could have happened due to waves of modern humans entered Europe – in technical terms, due to "the continuous influx of modern human DNA into the Neandertal gene pool."[32][33][34]
Enflo has also written about the population dynamics of zebra mussels in Lake Erie.[35]
A concert pianist, Per Enflo debuted at the Stockholm Concert Hall in 1963.[36]
Piano[edit]
Per Enflo is also a concert pianist.
A child prodigy in both music and mathematics, Enflo won the Swedish competition for young pianists at age 11 in 1956, and he won the same competition in 1961.[37] At age 12, Enflo appeared as a soloist with the Royal Opera Orchestra of Sweden. He debuted in the Stockholm Concert Hall in 1963. Enflo's teachers included Bruno Seidlhofer, Géza Anda, and Gottfried Boon (who himself was a student of Arthur Schnabel).[36]
In 1999 Enflo competed in the first annual Van Cliburn Foundation's International Piano Competition for Outstanding Amateurs.[38]
Enflo performs regularly around Kent and in a Mozart series in Columbus, Ohio (with the Triune Festival Orchestra). His solo piano recitals have appeared on the Classics Network of the radio station WOSU, which is sponsored by Ohio State University.[36]
^ Page 586 in Halmos 1990.
^ Per Enflo: A counterexample to the approximation problem in Banach spaces. Acta Mathematica vol. 130, no. 1, Juli 1973
^ * Enflo, Per (1976). "On the invariant subspace problem in Banach spaces". Séminaire Maurey--Schwartz (1975--1976) Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14-15. Centre Math., École Polytech., Palaiseau. p. 7. MR 0473871.
Enflo, Per (1987). "On the invariant subspace problem for Banach spaces". Acta Mathematica. 158 (3): 213–313. doi:10.1007/BF02392260. ISSN 0001-5962. MR 0892591.
^ Rådström had himself published several articles on Hilbert's fifth problem from the point of view of semigroup theory. Rådström was also the (initial) advisor of Martin Ribe, who wrote a thesis on metric linear spaces that need not be locally convex; Ribe also used a few of Enflo's ideas on metric geometry, especially "roundness", in obtaining independent results on uniform and Lipschitz embeddings (Benyamini and Lindenstrauss). This reference also describes results of Enflo and his students on such embeddings.
^ Theorem 15.4.1 in Matoušek.
^ Matoušek 370.
^ Beauzamy 1985, page 298.
^ Pisier.
^ Schauder J (1927). "Zur Theorie stetiger Abbildungen in Funktionalraumen" (PDF). Mathematische Zeitschrift. 26: 47–65. doi:10.1007/BF01475440. hdl:10338.dmlcz/104881. S2CID 123042807.
^ Schauder J (1928). "Eine Eigenschaft des Haarschen Orthogonalsystems". Mathematische Zeitschrift. 28: 317–320. doi:10.1007/BF01181164. S2CID 120228356.
^ Mauldin
^ a b Joram Lindenstrauss and L. Tzafriri.
^ Enflo's "sensation" is discussed on page 287 in Pietsch, Albrecht (2007). History of Banach spaces and linear operators. Boston, MA: Birkhäuser Boston, Inc. pp. xxiv+855 pp. ISBN 978-0-8176-4367-6. MR 2300779. Introductions to Enflo's solution were written by Halmos, by Johnson, by Kwapień, by Lindenstrauss and Tzafriri, by Nedevski and Trojanski, and by Singer.
^ Kałuża, Saxe, Eggleton, Mauldin.
^ a b Beauzamy 1988; Yadav.
^ Yadav, page 292.
^ For example, Radjavi and Rosenthal (1982).
^ Heydar Radjavi & Peter Rosenthal (March 1982). "The invariant subspace problem". The Mathematical Intelligencer. 4 (1): 33–37. doi:10.1007/BF03022994. S2CID 122811130.
^ Page 401 in Foiaş, Ciprian; Jung, Il Bong; Ko, Eungil; Pearcy, Carl (2005). "On quasinilpotent operators. III". Journal of Operator Theory. 54 (2): 401–414. . Enflo's method of ("forward") "minimal vectors" is also noted in the review of this research article by Gilles Cassier in Mathematical Reviews: MR2186363 Enflo's method of minimal vector is described in greater detail in a survey article on the invariant subspace problem by Enflo and Victor Lomonosov, which appears in the Handbook of the Geometry of Banach Spaces (2001).
^ Schmidt, page 257.
^ Montgomery. Schmidt. Beauzamy and Enflo. Beauzamy, Bombieri, Enflo, and Montgomery
^ Bombieri and Gubler
^ Knuth. Beauzamy, Enflo, and Wang.
^ The model for the evolution of human population genetics (developed by Enflo and his coauthors) was reported on the cover page of a major Swedish newspaper. Jensfelt, Annika (14 January 2001). Svenska Dagbladet: 1. Missing or empty |title= (help)
^ Mellars, P. (2006). "A new radiocarbon revolution and the dispersal of modern humans in Eurasia". Nature. 439 (7079): 931–935. Bibcode:2006Natur.439..931M. doi:10.1038/nature04521. PMID 16495989. S2CID 4416359.
^ Banks, William E.; Francesco d'Errico; A. Townsend Peterson; Masa Kageyama; Adriana Sima; Maria-Fernanda Sánchez-Goñi (24 December 2008). Harpending, Henry (ed.). "Neanderthal Extinction by Competitive Exclusion". PLOS ONE. Public Library of Science. 3 (12): e3972. Bibcode:2008PLoSO...3.3972B. doi:10.1371/journal.pone.0003972. ISSN 1932-6203. PMC 2600607. PMID 19107186.
^ Rincon, Paul (13 September 2006). "Neanderthals' 'last rock refuge'". BBC News. Retrieved 2009-10-11.
^ Finlayson, C., F. G. Pacheco, J. Rodriguez-Vidal, D. A. Fa, J. M. G. Lopez, A. S. Perez, G. Finlayson, E. Allue, J. B. Preysler, I. Caceres, J. S. Carrion, Y. F. Jalvo, C. P. Gleed-Owen, F. J. J. Espejo, P. Lopez, J. A. L. Saez, J. A. R. Cantal, A. S. Marco, F. G. Guzman, K. Brown, N. Fuentes, C. A. Valarino, A. Villalpando, C. B. Stringer, F. M. Ruiz, and T. Sakamoto. 2006. Late survival of Neanderthals at the southernmost extreme of Europe. Nature advanced online publication.
^ Gravina, B.; Mellars, P.; Ramsey, C. B. (2005). "Radiocarbon dating of interstratified Neanderthal and early modern human occupations at the Chatelperronian type-site". Nature. 438 (7064): 51–56. Bibcode:2005Natur.438...51G. doi:10.1038/nature04006. PMID 16136079. S2CID 4335868.
^ Zilhão, João; Francesco d'Errico; Jean-Guillaume Bordes; Arnaud Lenoble; Jean-Pierre Texier; Jean-Philippe Rigaud (2006). "Analysis of Aurignacian interstratification at the Châtelperronian-type site and implications for the behavioral modernity of Neandertals". PNAS. 103 (33): 12643–12648. Bibcode:2006PNAS..10312643Z. doi:10.1073/pnas.0605128103. PMC 1567932. PMID 16894152.
^ Page 665:
Pääbo, Svante et alia. "Genetic analyses from ancient DNA." Annu. Rev. Genet. 38, 645–679 (2004).
^ Jensfelt, Annika (14 January 2001). Svenska Dagbladet: 1. Missing or empty |title= (help)
^ "'Per Enflo's theory is extremely well thought-out and of the highest significance'...said American anthropologist Milford Wolpoff, professor at the University of Michigan." (Page 14 in Jensfelt, Annika (14 January 2001). "Ny brandfackla tänder debatten om manniskans ursprung (Swedish)". Svenska Dagbladet: 14–15. )
^ Saxe
^ a b c * Chagrin Valley Chamber Music Concert Series 2009-2010 Archived 2012-11-11 at the Wayback Machine.
^ Saxe.
^ Michael Kimmelman (August 8, 1999). "Prodigy's Return". The New York Times Magazine. Section 6, p. 30.
"Recipients of 2005 Distinguished Scholar Award at Kent State University Announced", eInside, 2005-4-11. Retrieved on February 4, 2007.
Bibliography[edit]
Enflo, Per. (1970) Investigations on Hilbert's fifth problem for non locally compact groups (Stockholm University). Enflo's thesis contains reprints of exactly five papers:
Enflo, Per (1969a). "Topological groups in which multiplication on one side is differentiable or linear". Math. Scand. 24: 195–197. doi:10.7146/math.scand.a-10930.
Per Enflo (1969). "On the nonexistence of uniform homeomorphisms between Lp spaces". Ark. Mat. 8 (2): 103–5. Bibcode:1970ArM.....8..103E. doi:10.1007/BF02589549.
Enflo, Per (1969b). "On a problem of Smirnov". Ark. Math. 8 (2): 107–109. Bibcode:1970ArM.....8..107E. doi:10.1007/bf02589550.
Enflo, Per (1970a). "Uniform structures and square roots in topological groups I". Israel J. Math. 8 (3): 230–252. doi:10.1007/BF02771560. S2CID 189773170.
Enflo, Per (1970b). "Uniform structures and square roots in topological groups II". Israel J. Math. 8 (3): 253–272. doi:10.1007/BF02771561. S2CID 121193430.
Enflo, Per. 1976. Uniform homeomorphisms between Banach spaces. Séminaire Maurey-Schwartz (1975—1976), Espaces, L p {\displaystyle L^{p}} , applications radonifiantes et géométrie des espaces de Banach, Exp. No. 18, 7 pp. Centre Math., École Polytech., Palaiseau. MR0477709 (57 #17222) [Highlights of papers on Hilbert's fifth problem and on independent results of Martin Ribe, another student of Hans Rådström]
Enflo, Per (1972). "Banach spaces which can be given an equivalent uniformly convex norm". Israel Journal of Mathematics. 13 (3–4): 281–288. doi:10.1007/BF02762802. MR 0336297. S2CID 120895135.
Enflo, Per (1973). "A counterexample to the approximation problem in Banach spaces". Acta Mathematica. 130: 309–317. doi:10.1007/BF02392270. MR 0402468.
Enflo, Per (1976). "On the invariant subspace problem in Banach spaces" (PDF). Séminaire Maurey--Schwartz (1975--1976) Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14–15. Centre Math., École Polytech., Palaiseau. pp. 1–7. MR 0473871.
Beauzamy, Bernard; Bombieri, Enrico; Enflo, Per; Montgomery, Hugh L. (1990). "Products of polynomials in many variables". Journal of Number Theory. 36 (2): 219–245. doi:10.1016/0022-314X(90)90075-3. hdl:2027.42/28840. MR 1072467.
Beauzamy, Bernard; Enflo, Per; Wang, Paul (October 1994). "Quantitative Estimates for Polynomials in One or Several Variables: From Analysis and Number Theory to Symbolic and Massively Parallel Computation". Mathematics Magazine. 67 (4): 243–257. JSTOR 2690843. (accessible to readers with undergraduate mathematics)
P. Enflo, John D. Hawks, M. Wolpoff. "A simple reason why Neanderthal ancestry can be consistent with current DNA information". American Journal Physical Anthropology, 2001
Enflo, Per; Lomonosov, Victor (2001). "Some aspects of the invariant subspace problem". Handbook of the geometry of Banach spaces. I. Amsterdam: North-Holland. pp. 533–559.
Bartle, R. G. (1977). "Review of Per Enflo's "A counterexample to the approximation problem in Banach spaces" Acta Mathematica 130 (1973), 309–317". Mathematical Reviews. 130: 309–317. doi:10.1007/BF02392270. MR 0402468.
Beauzamy, Bernard (1985) [1982]. Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. ISBN 0-444-86416-4. MR 0889253.
Beauzamy, Bernard (1988). Introduction to Operator Theory and Invariant Subspaces. North Holland. ISBN 0-444-70521-X. MR 0967989.
Enrico Bombieri and Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P. ISBN 0-521-84615-3.
Roger B. Eggleton (1984). "Review of Mauldin's The Scottish Book: Mathematics from the Scottish Café". Mathematical Reviews. MR 0666400.
Grothendieck, A.: Produits tensoriels topologiques et espaces nucleaires. Memo. Amer. Math. Soc. 16 (1955).
Halmos, Paul R. (1978). "Schauder bases". American Mathematical Monthly. 85 (4): 256–257. doi:10.2307/2321165. JSTOR 2321165. MR 0488901.
Paul R. Halmos, "Has progress in mathematics slowed down?" Amer. Math. Monthly 97 (1990), no. 7, 561—588. MR1066321
William B. Johnson "Complementably universal separable Banach spaces" in Robert G. Bartle (ed.), 1980 Studies in functional analysis, Mathematical Association of America.
Kałuża, Roman (1996). Ann Kostant and Wojbor Woyczyński (ed.). Through a Reporter's Eyes: The Life of Stefan Banach. Birkhäuser. ISBN 0-8176-3772-9. MR 1392949.
Knuth, Donald E (1997). "4.6.2 Factorization of Polynomials". Seminumerical Algorithms. The Art of Computer Programming. 2 (Third ed.). Reading, Massachusetts: Addison-Wesley. pp. 439–461, 678–691. ISBN 0-201-89684-2.
Kwapień, S. "On Enflo's example of a Banach space without the approximation property". Séminaire Goulaouic-Schwartz 1972—1973: Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 8, 9 pp. Centre de Math., École Polytech., Paris, 1973. MR407569
Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
Lindenstrauss, J.; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977. Springer-Verlag.
Matoušek, Jiří (2002). Lectures on Discrete Geometry. Graduate Texts in Mathematics. Springer-Verlag. ISBN 978-0-387-95373-1. .
R. Daniel Mauldin, ed. (1981). The Scottish Book: Mathematics from the Scottish Café (Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979). Boston, Mass.: Birkhäuser. pp. xiii+268 pp. (2 plates). ISBN 3-7643-3045-7. MR 0666400.
Nedevski, P.; Trojanski, S. (1973). "P. Enflo solved in the negative Banach's problem on the existence of a basis for every separable Banach space". Fiz.-Mat. Spis. Bulgar. Akad. Nauk. 16 (49): 134–138. MR 0458132.
Pietsch, Albrecht (2007). History of Banach spaces and linear operators]. Boston, MA: Birkhäuser Boston, Inc. pp. xxiv+855 pp. ISBN 978-0-8176-4367-6. MR 2300779.
Pisier, Gilles (1975). "Martingales with values in uniformly convex spaces". Israel J. Math. 20 (3–4): 326–350. doi:10.1007/BF02760337. MR 0394135. S2CID 120947324.
Heydar Radjavi & Peter Rosenthal (March 1982). "The invariant subspace problem". The Mathematical Intelligencer. 4 (1): 33–37. doi:10.1007/BF03022994. S2CID 122811130.
Karen Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics, 2002 Springer-Verlag, New York. (Pages 122–123 sketch a biography of Per Enflo.)
Schmidt, Wolfgang M. (1980 [1996 with minor corrections]) Diophantine approximation. Lecture Notes in Mathematics 785. Springer.
Singer, Ivan. Bases in Banach spaces. II. Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. viii+880 pp. ISBN 3-540-10394-5. MR610799
Yadav, B. S. (2005). "The present state and heritages of the invariant subspace problem". Milan Journal of Mathematics. 73: 289–316. doi:10.1007/s00032-005-0048-7. ISSN 1424-9286. MR 2175046. S2CID 121068326.
External sources[edit]
Biography of Per Enflo at Canisius College
Homepage of Per Enflo at Kent State University
Enflo, Per (25 April 2011). "Personal notes, in my own words". perenflo.com. Archived from the original on 26 April 2012. Retrieved 13 December 2011.
Databases[edit]
Per Enflo at the Mathematics Genealogy Project
Google Scholar. "Per Enflo". Retrieved 2010-05-15.
Mathematical Reviews. "Per Enflo". Retrieved 2010-05-14.
MGP: 20609
WorldCat Identities: viaf-4913716
Retrieved from "https://en.wikipedia.org/w/index.php?title=Per_Enflo&oldid=997358198"
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Надёжность

Понятия
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Enjoy a variety of cources to learn on the go. Each snack-size course is divided into chapters concise enough to make your learning experience fun and snappy. Before proceeding to a new chapter, you will be prompted with a quiz to recap and test your best.
This introduction to Hatha yoga will teach you the benefits, poses and terms to begin or improve a safe, enjoyable yoga practice. Here you will find an overview of the history of yoga, follow clear instructions to guide you through poses and find tips on nutritious eating to improve your practice.
This course is based on the classic business book from 1937, Think and Grow Rich. Based on interviews of some of the wealthiest men at the time, the book explores their secret to making money and shares practical tips on how to apply these formulas for personal success.
People go through different levels of confidence - it's perfectly natural. But there are techniques to help gain a stronger sense of self and build confidence. In this course, you'll learn how to remove insecurities that will help you excel at work, social situations and personal relationships.


Phoenix, AZ – (October 11, 2007) – Businesses and entrepreneurs looking for funding, talent, real estate plays for the future, green initiatives or innovative ideas will find them all at The Second Annual Arizona Entrepreneurship conference November 8 at the Phoenix Airport Marriott. The day-long event will connect participants with local and national entrepreneurs and investors.
"While Arizona will never be Silicon Valley, this conference is designed to connect the two and help entrepreneurs here build their businesses," said Francine Hardaway, organizer of the event.
Keynote speakers include a collection of heavyweights sharing their in-the-trenches experiences in building, funding and selling their companies.
• Michael Gerber, Best Selling Author of "The E-Myth"
The First Annual Arizona Entrepreneurship Conference [photos | video] was held in Fall 2006 and was standing-room only.


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Serum magnesium in adult patients with idiopathic and symptomatic epilepsy in Maiduguri, Northeast Nigeria
I Abdullahi1, MM Watila2, N Shahi3, YW Nyandaiti4, SA Bwala5
1 Federal Medical Center Azare, Bauchi State, Nigeria
2 Neurology Unit, Department of Medicine, University of Maiduguri Teaching Hospital, Maiduguri, Borno State; Department of Clinical and Experimental Epilepsy, UCL Institute of Neurology, London, UK
3 Neurosurgery Department, Kings College Hospital, London, UK
4 Neurology Unit, Department of Medicine, University of Maiduguri Teaching Hospital, Maiduguri, Borno State, Nigeria
5 Department of Medicine, National Hospital Abuja, Abuja, Nigeria
Dr. M M Watila
Neurology Unit, Department of Medicine, University of Maiduguri Teaching Hospital, PMB 1414, Maiduguri, Borno State
DOI: 10.4103/njcp.njcp_252_18
Background: Alteration in the homeostasis of trace elements such as magnesium may play a role in the development of epileptic seizures. This study aims to investigate the levels of serum magnesium in people with idiopathic generalized epileptic (IGE) seizures and symptomatic seizures in Northeast Nigeria. Materials and Methods: Serum magnesium level was measured using atomic absorption spectrometry among 40 adults with IGE, 20 adults with symptomatic epileptic seizures, and 30 healthy controls. Serum calcium, potassium, phosphate, and albumin were also measured. Results: The mean serum magnesium level was significantly lower among people with epilepsy compared with the controls [0.79 ± 0.18 mmol/L vs 0.90 mmol/L ± 0.17, P = 0.007, 95% confidence interval (CI): (−0.189 to −0.031)]. People with IGE had significantly lower levels of magnesium compared with those with symptomatic seizures [0.74 ± 0.17 mmol/L vs 0.9 ± 0.16 mmol/L, P < 0.001 95% CI: (−0.251 to −0.069)]. The mean magnesium level for all groups was in the reference range, but the lowest levels were observed in those with IGE. There is no significant correlation between the level of serum magnesium and the severity of seizure attacks. There was significantly lower level of calcium in people with IGE compared with those with symptomatic seizures [2.3 ± 0.13 mmol/L vs 2.4 ± 0.16 mmol/L, P = 0.012, 95% CI: (−0.177 to 0.023)] or controls [2.3 ± 0.13 mmol/L vs 2.4 ± 0.12 mmol/L, P < 0.01, 95% CI: (−0.156 to −0.044)]. No significant differences were observed in the levels of potassium, phosphate, and albumin. Conclusion: This study suggests that low serum magnesium and calcium may play a role in IGE, and supplementation may be useful in reducing seizures in Black patients with epilepsy.


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Who Is Seth Rogen's Character, Rand Gauthier?
By: Alex jack
Seth Rogen stars in Hulu's Pam & Tommy as Rand Gauthier, a real-life figure who was integral in releasing Anderson and Lee's infamous sex tape.
Seth Rogen stars in the cast of Hulu's series Pam & Tommy as Rand Gauthier, but who is the real-life figure Rogen plays? Taking a break from his comedy film roles, Seth Rogen returns to television for Pam & Tommy, Hulu's miniseries that tells the true story of Pamela Anderson and Tommy Lee's infamous sex tape. Not only does Rogen star in a main role, but he was also instrumental in developing the series and getting it to Hulu.
Rogen began his career as a television actor in the short-lived '90s series Freaks & Geeks. Since then, Seth Rogen has produced and starred in a variety of successful comedy movies like Pineapple Express, Superbad, and Neighbors. Although Pam & Tommy's true story has been a divisively-received project for Rogen, the actor is no stranger to controversy. In 2014, Seth Rogen and frequent collaborator James Franco got into international hot water after the release of their comedy The Interview, where the duo's characters are sent to assassinate North Korean leader Kim Jong-un. While the responses to Anderson and Lee's Pam & Tommy portrayals have varied, they at least haven't been subject to intense international scrutiny.
Related: How Much Is Tommy Lee Worth? Why He Wouldn't Pay Rand
Pam & Tommy stars Lily James and Sebastian Stan as model Pamela Anderson and Mötley Crüe drummer Tommy Lee, documenting their turbulent marriage and media frenzy after a disgruntled former employee publicly releases their sex tape. Set in the mid-1990s, Pam & Tommy's true story begins when Seth Rogen's character Rand Gauthier, working as a carpenter for the couple, is abruptly and unfairly fired without pay. As revenge, both Pam & Tommy's character and the real-life Gauthier stole Lee's safe, finding a sex tape between the drummer and actress locked inside. Seth Rogen's character, a former porn actor, then works with a former colleague in the porn industry to widely and publicly distribute the tape via the newly-popular internet in hopes of profit.
Pam & Tommy: The Real-Life Rand Gauthier Explained
Rand Gauthier (played by Seth Rogen in Pam & Tommy) was an electrician for Tommy Lee and Pamela Anderson at the time of their marriage. After Lee fired him and never paid for his services in real life, Gauthier decided to humiliate the rocker and make some fast cash – cue the theft of the infamous sex tape that drives the narrative of Pam & Tommy. The true Rand Gauthier designed a heist where, under the guise of returning to collect his tools, he stole Lee's refrigerator-sized safe holding the couple's jewels and guns, later discovering it also contained their sex tape. While the heist is portrayed accurately, Seth Rogen's real-life Pam & Tommy character was originally at the mansion to install a studio and sound system, whereas the show depicts him as a carpenter.
As is portrayed in Pam & Tommy, Gauthier then used his contacts from the porn industry to make several copies of the sex tape and, with help from friend Milton Ingley, devised to sell them online at about $60 apiece. Gauthier never actually made any money from the distribution, however, and the true story of Seth Rogen's Pam & Tommy character only got weirder from there. Since Gauthier never found a proper distributor for the sex tape, he ended up owing money to the mob instead to pay off his debts. Since Pam & Tommy's true story takes place long before the majority of proper internet privacy laws were set, the real Gauthier never faced any jail time for publicly releasing the sex tape. He now lives further up the California coast and remains an electrician to this day.
Although Pam & Tommy primarily utilized Seth Rogen's more dramatic depths, his stoner reputation and comedy talents were still tapped into for Gauthier's failed heist. Rogen has played some diverse roles in the past, but a former porn-star-turned-electrician-turned-blackmailer for Pam & Tommy is one of the strangest in his portfolio. While Seth Rogen's Rand Gauthier isn't the primary focus of Pam & Tommy, his instigation of their tumultuous media scandal and personal troubles makes for a riveting plot.
Next: How Rand Broke Into Pam & Tommy's House: Was He Really Disguised As A Dog?
The Office Season 6 Spent $45,000 on the Rights to Use One Song
Jordan Williams
(799 Articles Published)
Jordan Williams is a Movie/TV Feature Senior Staff Writer for Screen Rant. She graduated from the University of Oregon in 2020 with a B.A. in Psychology and a minor in Media Studies.
Jordan is based in Seattle, Washington and enjoys exploring the natural beauty the PNW has to offer. She runs on coffee and classic movies, taking pride in having watched every film on AFI's 100 Greatest Films list and every Best Picture Oscar winner.
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The Fever Syndrome review – a family home, and its dramas and rivalries, dissected | Stage
Alex jack


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Josie Long: Cara Josephine (at SoHo Playhouse)
Posted by Sean L. McCarthy | Oct 16, 2015 | Reviews | 0 |
Josie Long has mounted six one-person shows since 2006 at the Edinburgh Fringe Festival, with three of them nominated for Best Comedy Show in consecutive years (2010-2011-2012), and her first extended New York City run comes courtesy of SoHo Playhouse's Fringe Encores series — which had the theater's director audibly over the moon introducing her Thursday night. "She said yes!"
Long will soon have you smitten, too.
Despite the fact that her show here, "Cara Josephine," revolves around her heartbreak over the course of six months in 2013, when she was 31, and not even knowing she's 33 now ("it's true, because that's how time works") changes the universality of processing heartbreak for her or for us.
Long's charm is quickly apparent. Even before she appears, in fact, as she delivers a pre-show announcement from just behind the stage. "I'm going to entertain the living shit out of you!" Long proclaims. "If you have seat belts…I don't know why you would…" And then silence.
Once onstage, she fills the audience in on some administrative details. Yes, this is her 2014 Edinburgh show (which also sold out runs in London, throughout the UK, and for two weeks in Australia at the Melbourne International Comedy Festival). No, you won't hear a specific hated British politician's name within the following 70 minutes. Long changed the reference to Donald Trump. "It's the same point," she says. "But also, what a treat. What a gift!" And even just since last summer, speaking of gifts, could she or any of us imagine — and now, please try not to imagine — the prime minister "not fucking a pig?!" Sorry. Now you know how we feel about it!
But back to Long and her personal longing for love once more.
Heartbreak feels much like war, she has learned over the past two years. In the throes of a relationship in decline, denial reigns. Only afterward do you realize how grave your mistake.
Each heartbreak, Long acknowledges "is a cunty snowflake," unique to your own experiences. For her, having a mother who despises all men, yet also must have a man in her life at all times, may not represent the best role model. Then there's her sister, engaged so many times she wears the rings of past would-be suitors as a necklace.
And yet. Her sister found love, and gave birth to a daughter — Josie's niece, so named Cara Josephine — and with it, hope for her yet.
In the meantime, Josie Long will maintain her affinity for saying modern phrases and lyrics in the style of 1920s/1930s film dialogue, singing the hits of 2011 in the style of Bob Dylan, and in between, imagining herself as a love poet with catchphrases aplenty.
Take that Trisha Goddard. Whoever she is.
And take that, too, Walt Whitman. Josie Long will be our captain. O captain, our captain! Rise up and ring your bells! At least for the next 10 nights. And then "she's had enough" of us New Yorkers. For now. We can only hope "she's forgotten something" and shall return again.
Josie Long: Cara Josephine plays through Oct. 25, 2015, at SoHo Playhouse, 15 Vandam St., New York, NY. Running time: 70 minutes. Tickets: $25
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Episode #27: Josie Long | The Comic's Comic - […] for Best Comedy Show at the Fringe, and she's making her biggest splash yet in the United States with…


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The Forties Field, Blocks 21/10 and 22/6a, UK North Sea
View ORCID ProfileP. T. S. Rose, G. W. Byerley, E. College, J. R. Pyle, D. J. Ralph, P. S. Rowbotham, L. A. van Oorschot, J. Towart, O. Vaughan and M. Vermaas
Geological Society, London, Memoirs, 52, 454-467, 30 October 2020, https://doi.org/10.1144/M52-2017-41
P. T. S. Rose
Apache North Sea Ltd, Caledonia House, Prime Four Business Park, Kingswells Causeway, Kingswells, Aberdeen AB15 8PU, UK
Roles: [Investigation (Lead)], [Writing - Original Draft (Lead)]
ORCID record for P. T. S. Rose
For correspondence: phil.rose@apachecorp.com
G. W. Byerley
Roles: [Formal analysis (Equal)], [Investigation (Equal)], [Writing - Review & Editing (Equal)]
E. College
J. R. Pyle
D. J. Ralph
P. S. Rowbotham
L. A. van Oorschot
J. Towart
Roles: [Supervision (Equal)], [Validation (Equal)], [Writing - Review & Editing (Equal)]
O. Vaughan
M. Vermaas
https://doi.org/10.1144/M52-2017-41
Geological Society of London
Received 13 December 2017
Revision received 14 March 2018
Accepted 19 March 2018
Published 30 October 2020.
© 2020 The Author(s). Published by The Geological Society of London. All rights reserved
P. T. S. Rose, [Investigation (Lead)], [Writing - Original Draft (Lead)]*,
G. W. Byerley, [Formal analysis (Equal)], [Investigation (Equal)], [Writing - Review & Editing (Equal)],
E. College,
J. R. Pyle, [Formal analysis (Equal)], [Investigation (Equal)], [Writing - Review & Editing (Equal)],
D. J. Ralph, [Formal analysis (Equal)], [Investigation (Equal)], [Writing - Review & Editing (Equal)],
P. S. Rowbotham, [Formal analysis (Equal)], [Investigation (Equal)], [Writing - Review & Editing (Equal)],
L. A. van Oorschot, [Formal analysis (Equal)], [Investigation (Equal)], [Writing - Review & Editing (Equal)],
J. Towart, [Supervision (Equal)], [Validation (Equal)], [Writing - Review & Editing (Equal)],
O. Vaughan, [Formal analysis (Equal)], [Investigation (Equal)], [Writing - Review & Editing (Equal)] and
M. Vermaas, [Formal analysis (Equal)], [Investigation (Equal)], [Writing - Review & Editing (Equal)]
Apache North Sea Ltd, Caledonia House, , Prime Four Business Park, Kingswells Causeway, Kingswells, Aberdeen AB15 8PU, UK
↵*Correspondence: phil.rose{at}apachecorp.com
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Geological Society, London, Memoirs
P. T. S. Rose, G. W. Byerley, E. College, J. R. Pyle, D. J. Ralph, P. S. Rowbotham, L. A. van Oorschot, J. Towart, O. Vaughan and M. Vermaas
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History of exploration, appraisal and development
Regional context
Data acquisition and drilling history
Production history and reserves
Forties Field data summary
UK oil and gas fields: an overview
Chapter 1 Precambrian basins of India: stratigraphic and tectonic context
Chapter 1 Introduction to the geology of Myanmar
Chapter 1 Introduction to the Jurassic Arabian Intrashelf Basin
Chapter 13 Anatomy of the Andaman–Nicobar subduction system from seismic reflection data


Innovation, as a great impetus for the development of the society, has been widely discussed at the national strategic level. Accordingly, as effective ways of innovation, entrepreneurial activities are emerging worldwide.
As such, entrepreneurial teams are receiving extensive attention. At present, some scholars have begun to explore entrepreneurial team innovative capabilities from the perspective of social net-work. However, most of them either pay much attention to the impact that the external social net-work brings to entrepreneurial performance or focus merely on one of the dimensions of social network.
This research, based on the perspective of internal social network, focused on the impact of team heterogeneity and tie strength on entrepreneurial team innovative capabilities. Specifically, this research explored the following issues: what is the relationship between team heterogeneity and innovative capabilities? What is the relationship between tie strength and innovative capabilities? Is there a nonlinear relationship between the variables such as inverted U-shaped relationship? How do network heterogeneity and tie strength interact with each other?
(1) Entrepreneurial team heterogeneity significantly influences the innovative capabilities of the team; individual characteristic heterogeneity and information heterogeneity have positive impact on innovative capabilities while heterogeneity of values has no significant influence.
(2) Tie strength lays a significant impact on the innovative capabilities of entrepreneurial teams, and there is an inverted U-shaped relationship between the two, implying that there is an optimal level of tie strength which enhances innovative capabilities the most.
(3) There is a positive moderating effect of team heterogeneity on the relationship between tie strength and team innovative capabilities.
After the conclusions were drawn, limitations and inadequacies of this study, as well as its man-agerial implications were discussed. Finally, future research directions of social network and en-trepreneurial team innovative capabilities were discussed.


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NGC 1354 é uma galáxia  localizada na direcção da constelação de Eridanus. Possui uma declinação de -15° 13' 15" e uma ascensão recta de 3 horas, 32 minutos e 29,2 segundos.

A galáxia NGC 1354 foi descoberta em 30 de Dezembro de 1785 por William Herschel.

Ver também
Astronomia extragaláctica
Lista de galáxias
Lista de objectos NGC
New General Catalogue

Ligações externas

NGC 1354
Constelação de Eridanus


5 year manufacturer backed warranty, including print heads.
With its rugged construction and dual print heads in a staggered formation, the XT-640 is built to meet the demanding production requirements of the textile industry, providing stability, exceptional print quality and impressive high-speed performance.
The XT-640 features dual print heads in a staggered formation to achieve high-speed performance. Its reinforced rail, grid roller, and frame provide a firm foundation for smooth operation and exceptional print quality. To capture every gradation and detail in your design, each head fires precise droplets in seven different sizes to accommodate media characteristics and desired print quality.
The XT-640 Roland Ink System ensures stable, long-run printing with one litre airtight sealed pouches. In four colour mode inks are available in two pouches (two litres) per colour, while the proprietary Roland Ink Switching System automatically switches to the backup ink pouch when the primary pouch runs out – meaning you can replace an empty pouch without stopping the printer. And while you're away from the printer, the Roland Printer Assist iPad* app allows remote management of common printer functions.
The XT-640 ensures precise media tracking for high volume production. An advanced vacuum system keeps transfer paper flat for printing, while a new convenient Feed Adjuster at the front and rear of the printer provides even tensioning and prevents skewing of the media, resulting in a precisely wound roll on the take-up reel for smooth transition to the heat transfer process.
The Texart XT-640 yields a marked difference to the colours and quality of your sublimated fabrics. The XT-640's advanced print control technology in combination with Texart ink produces beautifully sharp images quality with brilliant colour. For maximum production, the 4 colour ink configuration delivers fine detail and superb contrast at impressive print speeds. To optimise the already wide colour gamut, the 8 colour configuration includes orange and violet ink to give your prints maximum impact for flawless photographic reproduction, smooth gradations and delicate skin tones.
ErgoSoft Roland Edition is developed especially for Roland DG by ErgoSoft, the textile industry's premier RIP software developer. This feature-packed RIP delivers outstanding productivity and image quality, enabling powerful RIP processing and high density printing with minimal ink usage. Optimising digital textile production with useful features such as Step and Repeat, Colour Replace and Scale and Shrinkage Correction.
Roland DG's Texart ink used in the RT-640 dye sublimation transfer printer meets the requirements of Oeko-Tex® Standard 100, class 1, which is the most difficult certification in the class. Oeko-Tex® Standard 100 is a global uniform testing and certification system for textile raw materials, intermediate and end products at all stages of production with the aim of completely eliminating harmful substances.


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The flooding of the Zambezi to form Lake Kariba created a whole new ecosystem, changing a river into a huge lake. Existing river fish were unable to survive in deep water so kept to the shallows, leaving two-thirds of the lake devoid of fish. But the fishery potential of this new resource was obviously considerable, so in the early 1960s research was undertaken to find a suitable fish to introduce. Lake Tanganyika supports a thriving population of pelagic freshwater sardine, Limnothrissa miodon, so attempts were made to introduce this to Kariba. After a slow start the sardine (locally known as kapenta) gradually built up a sufficiently large population from which to start a viable fishing industry. Once this critical mass of fish had been reached by the turn of the decade, populations soared and a wonderful new food resource became available, not to mention a great employment opportunity.
(Photo: Crayfish known as 'yabby' © Paul Murray)
Kapenta fishing is done at night with a specially adapted netting technique. The large-framed scoop net, hinged out from the front of the boat, is lowered into the water. Shoals are lured to the surface using powerful lights, while the net is manoeuvred below them. The lights are then turned off and the fish descend into the net. Once on land the kapenta, which only grow to 6–7cm, are salted and sun-dried to form an extremely important protein-rich addition to the local diet.
A very disturbing discovery has recently been made in Lake Kariba which could have disastrous effects on the local environment and economy. A completely alien species of freshwater crayfish has been found to be breeding in increasingly alarming numbers. It's a voracious feeder and with no effective predators it's literally threatening to wipe out Kariba's entire fish stocks including the kapenta fishing industry on which countless people rely as a source of protein. And as with virtually all alien introductions the wider effect on the local environment can be immense – for instance what will happen to the fish eagles if the fry of their prey species get consumed by crayfish? So what to do? Well this cray is very popular in Australia where it is known as the 'Yabby' and is eaten by humans in very large quantities, which is the very reason why it was introduced deliberately into Kariba by some thoughtless person who really should have known better. So now we're stuck with it the best short term solution is for us to consume them at every opportunity. Try to order them even if they don't appear on the menu because it's the duty of every visitor to Zimbabwe to join the locals and do their bit to help eat this delicious terrorist out of existence.
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I received the following report from the master of all things Maine. Eleazar Peabody has the inside and outside scope on everything that happens "naturally" in Maine…probably unnaturally as well. He's better than any farmer's almanac and certainly no weather scientist can hold a candle to his weather and climate predictions.
Earlier this week I spotted the first Mud Runts of the season. They looked quite healthy and seemed in good spirits. They were cavorting in a wet spot just off the Patch Mountain Road, near Greenwood City, Maine. They disappeared quite quickly when they saw my broad smile and caught sight of Ole Brinker…Cheers and Happy Spring!!


Glenlake South is built by Ashton Woods and conveniently located in the heart of Raleigh, overlooking the beautiful Crabtree Valley. Glenlake is just minutes from premier shopping and dining at Crabtree Valley and North Hills Malls, Rex healthcare facilities, the RBC sports arena and many cultural and fine arts venues. In addition, the location is easily accessible to the 440 beltline, downtown Raleigh, and RDU International airport.
The architecture of Glenlake epitomizes old world elegance, with stone, brick and shake accents that enhance outdoor living on porches and within the fenced rear yards. Enjoy maintenance free living, surrounded by 12 acres of open space, including lakes, numerous sidewalks and a rose garden pergola.
Homes start in the Low $400's! Also, Ashton Woods is the only builder with Power House Green and Environments for Living™; two programs designed with healthy and comfortable living in mind that enhances the performance of your home. From high levels of efficiency and comfort to meeting the Environments for Living™ performance standards, your Ashton Woods Home results in enhanced performance.
Ready for Your Personal Tour of Glenlake?
Below are a few of Glenlake spotlight properties for sale. Visit the Glenlake Community Page to see all homes for sale in Glenlake.


We hired a nanny. She started full time today. She's been working for us on an as needed basis for the last month. She's been coming over once a week to clean. We had her over on occation to watch the kids. But its been really sporadic.
Having her start full time just feels strange, but in a good way. She spent the day with the kids. They went up to the park to play and took the dogs with them. They seem to really enjoy having her around. She also managed to make homemade burritos for everyone for lunch. Anna came down to my office and took my "lunch order" on a post-it note! Then she made bannana bread. And for dinner she made homemade lasagna. Oh yeah, she also placed a grocery order on safeway.com. She planned a menu for the whole week. And she called several carpet cleaners to get some quotes. And when she left, the place was tidy and the kitchen was clean, the laundry was put away, and I was happy and felt less stressed than I normally do.
Even in just one day, I can see the positive effect this will have on all of our lives. The kids won't be bored as much. We're going to eat homemade meals and healthier instead of eating at fast food and/or restaurants 5 or 6 nights a week. Rachele and I will feel less stressed with household things and we'll be able to spend our free time with the kids instead of doing chores. There will be some structure added to the kids lives. With Rachele starting her new job, its really a good time for us to be doing this.