text:
Hinkley Point near melt-down as French socialist party calls for freeze. Britain's Hinkley Point nuclear project is close to unravelling after France's ruling socialist party threw its support behind dissident trade union leaders and called for a fundamental review of the high-cost venture. The whole saga has now become freighted with politics and misunderstandings in a three-way jostle between France, Britain, and China, with no outcome in sight that can please everybody. The French socialists warned that Hinkley threatens the financial viability of EDF, the state-owned energy giant responsible for two thirds of the £18bn funding and for limitless liabilities if it all goes wrong. Tong Zhao from the Carnegie-Tsinghua Centre for Global Policy in Beijing said Chinese leaders were flabbergasted by the sudden decision to freeze the plan and they doubt assurances that the dispute is over costs. Tong Zhao said Hinkley was a flagship project for China and was hailed at the time as a break-through into the Western nuclear market. "President Xi Jinping himself promoted the project when he was in London and it became bigger than a mere contract. It has taken on symbolic meaning at a political level," he said. Socialist senator Yannick Vaugrenard warns that Hinkley Point risks becoming the "Concord of French energy", a view widely-shared since EDF's chief financial officer resigned in protest alleging that the company was gambling its future on an unknown risk. Nuclear power cannot easily be switched on and off. It is ill-adapted for use as a back-up source to cover lulls in renewable power. "In a world moving towards cheaper, flexible, decentralized power systems, investing in eye-wateringly expensive, always-on 'base-load' plants increasingly looks like a 20th Century solution for a 21st Century problem," said Richard Black from the Energy and Climate Intelligence Unit.
Telegraph 9th Aug 2016 read more »
France's ruling Socialist Party has said it has "cause for reservation" over the government's continued pursuit of a much-protracted deal to develop nuclear reactors at the Hinkley Point power plant. Socialist president François Hollande has consistently supported the deal, but the leadership of his party released a communiqué citing "structural concerns" over a plan from France's EDF to build two new nuclear reactors at the Somerset site, citing financial risks and "contextual concerns" stemming from the Brexit vote.
FT 9th Aug 2016 read more »
Fresh evidence of links between the government and EDF Energy has led to concerns over the firm possibly receiving "preferential treatment" for its flagship nuclear project planned for Hinkley Point in Somerset. Ten advisers and civil servants who worked at the now defunct Department for Energy Climate Change (DECC) in the last five years had ties to EDF, according to an analysis of online professional networking accounts by Energydesk. Among the 10 EDF-linked government employees is a regulatory and licensing officer currently working for the French company — recently employed by DECC and previously an operating reactors programme manager at the Office for Nuclear Regulation. There also features an EDF strategy manager, who has been working for the company since 2014 following a 13-month secondment to DECC's commercial team (from October 2011) while at previous employer KPMG. DECC's commercial team played a crucial role in deciding to press ahead with the Hinkley project and NNB Generation Company Limited, an EDF subsidiary, submitted a proposal to the National Planning Inspectorate for a new nuclear power plant at Hinkley on October 31st 2011.
Energydesk 10th Aug 2016 read more »
Theresa May's first test as British premier has come from the Chinese in the form of a remarkable article in the Financial Times. The article is puzzling. What sort of diplomat negotiates on serious issues through the media? Wouldn't they normally work discreetly to identify the cause of the problem — if there is one — and then seek to find a quiet solution? Issuing threats is not very diplomatic. Indeed, the article reads as if it had been written by a PR firm instructed to put pressure on ministers. One wonders how Beijing would react if the British ambassador there were to write an article demanding that Hong Kong be allowed to choose its own leaders. The point that seems to be missed is that Mrs May as a new prime minister cannot afford to give in to the first threat she faces. To blink is to fail and to be seen to have failed. How could she go into a tough European negotiation on the terms of Brexit having bent to the first gust of wind from the east? The challenge, therefore, is to find a way out of the problem that saves face on all sides. Here is a suggestion. The development of any new nuclear station at Bradwell is some years away and will remain so whatever happens to Hinkley Point. The Bradwell development will no doubt be tendered openly. Whatever half promises have been made to the Chinese are irrelevant. No British government can bind its successors. Beijing, which may actually be pleased if the ill-fated Hinkley project is dropped or postponed indefinitely, should be told it would be welcome to bid for Bradwell or other projects provided it can be demonstrated that all cyber espionage against UK commercial and industrial targets has been stopped and has remained stopped for five years. Of course, some espionage can go undetected. That is why real trust will only come when a comprehensive cyber security treaty is signed that binds all sides.
The SNP has today urged Prime Minister Theresa May to bring an end to the Hinkley Point nuclear power project after trade unions in France called for the EDF investment in the £18 billion plant to be declared invalid. Three French trade unions have stated that information on the delay was withheld from board members by EDF ahead of a vote approving investment – further showing the utter chaos at the very heart of the nuclear project. The SNP have repeated calls for the UK government to not simply delay the beleaguered project further, but bring an end to the proposals as soon as possible with the cost to the taxpayer continuing to run out of control. SNP MSP Gillian Martin, a member of Holyrood's Economy & Jobs Committee, said: "There seems to be no end to the confusion and chaos running at the very heart of the Tory government's nuclear power project – from ongoing uncertainty and delays, to problems emerging from France. Theresa May must pull the plug on this costly white elephant project immediately. Experts have already stated that Hinkley is yesterday's answer to today's energy needs – and I urge Theresa May to listen to expert opinion and take the long overdue decision to end the project for good."
Scottish Energy News 9th Aug 2016 read more »
The Chinese authorities' claim to legitimacy rests upon their ability to provide economic wellbeing, which is proving increasingly tough, and this idea of renewed national pride, seized and amplified by Xi Jinping since he took power. It is one reason he wanted his people to see his red carpet welcome in London last year: the former aggressor was now bowing its knee. It also increases Beijing's sensitivity to the 11th-hour halt on Hinkley Point and the public airing of suspicions about China's behaviour and intentions. The undiplomatic handling will be almost as unwelcome as the decision. But the Chinese are pragmatic and the real lesson from history is that remembering the days when Britannia ruled the waves accentuates its decline. The UK is a waning power, especially since voting for Brexit, which "show[ed] a losing mindset", in the words of one populist state-run Chinese newspaper. Beijing might, in fact, describe the UK somewhat as Lord Macartney, the first British envoy to China, described China: "An old, crazy, first-rate man of war, which a fortunate succession of able and vigilant officers have contrived to keep afloat …" What the UK and other western countries want most, as the former chancellor George Osborne's strategy highlighted, is a stable China with which they can do business. Whether or not Theresa May decides Hinkley Point should go ahead, that is unlikely to change.
Guardian 9th Aug 2016 read more »
The relationship between the British and Chinese governments is under strain after the prime minister, Theresa May, decided to delay approval of the planned £18bn Hinkley Point C nuclear power station because of concerns over China's 30% stake in the project. The move has prompted a warning from the Chinese ambassador to the UK, Liu Xiaoming, who said in a pointed Financial Times comment piece that relations with Britain were at a "crucial historical juncture". Here are some of the key questions regarding the Sino-British relationship after the Hinkley Point delay.
The Conservative Government has defended its handling of an £18bn Chinese-backed deal to build a new nuclear power plant, following criticism from Beijing. Prime Minister Theresa May stunned the energy sector and the international community when she ordered an eleventh hour pause in the project last month.
Herald 9th Aug 2016 read more »
Belfast Telegraph 9th Aug 2016 read more »
Reuters 9th Aug 2016 read more »
The relationship between the UK and China was at a "crucial historical juncture" over the development of the Hinkley Point nuclear power plant, Beijing´s top diplomat in the UK said. China's ambassador to the UK, Liu Xiaoming, wrote in the Financial Times that there was a link between the government's decision to delay going ahead with the nuclear reactor and the future of bilateral ties between the two nations.
Share Cast 9th Aug 2016 read more »
The Chinese Government has waded into the row surrounding plans to build a new nuclear power station at Hinkley Point.
Bristol Post 9th Aug 2016 read more »
Theresa May was today urged to speak out about Chinese human rights abuses as the row grew over Hinkley Point nuclear power station. Tory MP Fiona Bruce, chair of the Conservative Party's Human Rights Commission, said the Government should review its policy on the country. "Early indications are that this may be more likely under Mrs May," she said. "Of course we must continue to engage and trade with China, but we can and also should speak out more on its appalling human rights abuses."
Evening Standard 9th Aug 2016 read more »
The High Court of Paris on 5 August rejected a request by EDF's Central Works Council to suspend the final investment decision for the Hinkley Point C nuclear power plant project in the UK. Meanwhile, China's ambassador to the UK has called for the UK government to make a prompt decision on the project. In a statement, EDF said it welcomed the court's decision. The Central Works Council submitted its request to the court on 28 July – the same day as the EDF board of directors agreed to go ahead with the £18 billion ($23.6 billion) project to construct two EPR reactors at Hinkley Point in Somerset. In addition, EDF said its chairman and CEO, Jean-Bernard Lévy, plans to bring legal action against the Sud Energie union which claimed he was aware ahead of the board meeting that the UK government planned to conduct a further review of the Hinkley Point C project. EDF said the union issued a statement on 5 August which "inaccurately claimed that he had lied to journalists when he confirmed he had not known before the meeting".
World Nuclear News 9th Aug 2016 read more »
Sinn Féin MP Mickey Brady has described the pending news that the Nuclear Plant in Hinkley Point, Somerset is supposedly set for final approval as unwelcome and unwanted, saying it added 'another deadly danger off our coastline'. Brady said the £18 billion nuclear plant, if approved, would join the 'discredited Sellafield' as another source of deadly danger and potential contamination to the island of Ireland.
Newry Times 9th Aug 2016 read more »
Almost half of Engineer readers think that a UK-developed small modular reactor would be a better option than the current deal to build and operate a new nuclear power station at Hinkley Point. With over 700 responses to our poll and a lively comments section with over 5 contributions, the decision to allow EDF to build a new twin-reactor nuclear power station at Hinkley Point in Somerset with China supplying a significant proportion of the finance is clearly one that still arouses strong feelings among Engineer readers. A very large majority of respondents agreed that the government's last-minute decision to delay committing to the deals agreed by former Chancellor George Osborne represented a good chance to rethink the project, with only 11 per cent saying that the deal should go ahead as per the existing agreements. The largest group, 48 per cent, thought that UK-developed small modular reactor would be the best option for the UK's nuclear future (presumably abandoning the entire EPR strategy in the process) while 19 per cent though the deals could be reworked to give more opportunities for UK industry. The next group, 13 per cent, thought new nuclear build should be abandoned altogether, while 8 per cent thought China's roe in the project needed to be renegotiated. Just 2 per cent declined to choose an option.
The Engineer 9th Aug 2016 read more »
Daily Top Story | New Nuclear | News | News 2016 | News August 2016


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Симметрии в квантовой механике — преобразования пространства-времени и частиц, которые оставляют неизменными уравнения квантовой механики. Рассматриваются во многих разделах квантовой механики, которые включают релятивистскую квантовую механику, квантовую теорию поля, стандартную модель и физику конденсированного состояния. В целом, симметрия в физике, законы инвариантности и сохранения являются основополагающими ограничениями для формулирования физических теорий и моделей. На практике это мощные методы решения задач и прогнозирования того, что может случиться. Хотя законы сохранения не всегда дают конечное решение проблемы, но они формируют правильные ограничения и наметки к решению множества задач.

В этой статье описывается связь между классической формой непрерывных симметрий, а также их квантовыми операторами, которые связывают их с группами Ли и релятивистскими преобразованиями в группе Лоренца и группе Пуанкаре.

Обозначения
Условные обозначения, используемые в этой статье, следующие. Жирным шрифтом обозначены векторы, 4-векторы, матрицы и векторные операторы, в то время как квантовые состояния обозначаются скобками (бра и кет нотация). Широкие шляпки предназначены для операторов, узкие — для единичных векторов (включая их компоненты в тензорных индексах). Если не указано иное, используется соглашение о суммировании повторяющихся тензорных индексов. Метрическая сигнатура пространства Минковского (+ −−−).

Преобразования симметрии волновой функции в нерелятивистской квантовой механике

Непрерывные симметрии
Как правило, соответствие между непрерывными симметриями и законами сохранения дается квантовым аналогом теоремы Нётер.

Форма фундаментальных квантовых операторов, например энергии как частной производной по времени и импульса как градиента (от пространственных координат), становится ясной, если рассмотреть начальное состояние, а затем немного изменить один из его параметров. Такой подход работает для смещения (длины), продолжительности (времени) и углов (вращения). Кроме того, инвариантность некоторых величин можно увидеть, проделав преобразования длин и углов, что свидетельствует о сохранении этих величин.

В дальнейшем рассмотрим преобразования только для одночастичных волновых функций вида:

где  обозначает унитарный оператор. Унитарность обычно требуется для операторов, представляющих преобразования пространства, времени и спина, поскольку норма состояния (представляющая полную вероятность нахождения частицы с некоторым спином в некотором объёме пространства) должна быть инвариантной относительно этих преобразований. Обратное преобразование задаётся эрмитовым сопряжением  . Эти результаты можно распространить на многочастичные волновые функции. В нотации Дирака преобразования квантовых состояний:

Тогда действие оператора  преобразует волновую функцию ψ (r, t) в ψ (r ′, t ′), так что обратный оператор  заменяет ψ (r ′, t ′) на ψ (r, t), поэтому какой-либо оператор  будет инвариантен относительно преобразования  при условии

и поэтому:

для любых состояний ψ. Квантовые операторы, соответствующие наблюдаемым, также должны быть эрмитовыми, чтобы их собственные значения были действительными числами, то есть оператор равен своему эрмитово сопряженному, .

Обзор теории групп Ли
Ниже приведены ключевые положения теории групп, относящиеся к квантовой теории, а примеры приводятся на протяжении всей статьи. В альтернативном подходе используются группы матриц (см. Книги Холла)

Пусть G — группа Ли, которая локально параметризуется конечным числом N действительных непрерывно меняющихся параметров ξ 1, ξ 2 ,. . . ξ N. Или на другом языке это означает, что G — гладкое многообразие, которое также является группой, с гладкими групповыми операциями.

 Размерность группы равна N — то есть равно количеству параметров группы.
 элементы группы g в G — это функции параметров:

 и когда все параметры обнулены, то это соответствует нейтральному элементу группы:

 Элементы группы часто представляются в виде матриц, действующих на векторы, или преобразований, действующих на функции.

 Генераторы группы — это частные производные элементов группы по параметрам группы в точке, когда параметр равен нулю, а именно

 На языке многообразий генераторы группы — это элементы касательного пространства к G в единице. Генераторы также известны как элементы инфинитезимальной группы или как элементы алгебры Ли группы G. (См. Обсуждение коммутатора ниже.)
 Одно из преимуществ генераторов в теоретической физике заключается в том, что этим операторам соответствуют симметрии, которые можно записать в виде матриц или как дифференциальных операторов. В квантовой теории для унитарных представлений группы, генераторы умножаются на i :

 Генераторы группы образуют векторное пространство, что означает, что линейные комбинации генераторов также образуют генератор.

 Генераторы (матрицы или дифференциальные операторы) удовлетворяют коммутационным соотношениям :

 где f abc — (зависящие от базиса) структурные константы группы. Вместе со свойствами векторного пространства генераторы группы задают базис алгебры Ли. Из-за антисимметрии скобок (коммутатора) структурные константы группы антисимметричны по первым двум индексам.

 Представления группы описывают способы, которыми группа G (или её алгебра Ли) действует в векторном пространстве. (В качестве векторного пространства можно взять, например пространство собственных векторов гамильтониана, имеющего G в качестве группы симметрии.) Обозначим представления через заглавную D. Затем можно дифференцировать D, чтобы получить представление алгебры Ли, часто также обозначаемое D. Эти два представления связаны следующим образом:

 без суммирования по повторяющемуся индексу j. Представления группы — это линейные операторы, поставленные в соотствие каждому элементу группы и для которых выполняется правило композиции:

Представление, которое не может быть разложено в прямую сумму других представлений, называется неприводимым . Неприводимые представления принято помечать верхним индексом n в скобках, как например в D (n), или, если имеется более одного числа, то записывают D (n, m,. . . ) .

В квантовой теории возникает дополнительная тонкость: два вектора, различающиеся скалярным множителем, задают одно и то же физическое состояние. Тогда подходящим понятием представления выбирают проективное представление, которое удовлетворяет закону композиции только с точностью до скалярного множителя. В контексте квантово-механического спина такие представления называются спинорным.

Импульс и энергия как генераторы переноса, эволюции во времени и вращения
Оператор пространственных трансляций  действует на волновую функцию, сдвигая пространственные координаты на бесконечно малое смещение Δ r. Явное выражение для оператора  можно получить с помощью разложения в ряд Тейлора ψ (r + Δ r, t) в окрестности r, а затем (сохраняя член первого порядка и пренебрегая членами второго и более высоких порядков), заменить пространственные производные (градиент) оператором импульса . Точно так же для оператора сдвига по времени, действующего на параметр времени, в разложении в ряд Тейлора для ψ (r, t + Δ t) в окрестности t, заменяют производную по времени оператором энергии  .

{|class="wikitable"
|-
!Название
!Оператор трансляций
!Оператор эволюции во времени
|-
!Действие на волновую функцию
|
|
|-
!Инфинитезимальный оператор
|
|
|-
!конечный оператор
|
|
|-
!Генератор
|Оператор импульса
|Оператор энергии
|-
|}

Экспоненциальные функции возникают по определению данному Эйлером, и их физический и математический смысл понимается следующим образом. Чистый перенос состоит из множества небольших переносов, поэтому, чтобы получить оператор сдвига для конечного приращения, нужно заменить Δ r на Δ r / N и Δ t на Δ t / N, где N — положительное ненулевое целое число. Затем с увеличением N величина Δ r и Δ t становится ещё меньше, при этом их величины остаются неизменными. Действие инфинитезимальных операторов на волновую функцию N раз и переход к пределу, когда N стремится к бесконечности, приводит к виду конечных операторов.

Трансляции пространства и времени коммутируют, что также означает коммутацию их операторов и генераторов.

{|class="wikitable"
|+Коммутаторы
!Операторы
|
|
|-
!Генераторы
|
|
|-
|}

Для гамильтониана, не зависящего явно от времени, энергия сохраняется во времени, а квантовые состояния называются стационарными состояниями: собственные состояния гамильтониана — это собственные значения энергии E:

и все стационарные состояния принимают вид

где t 0 — начальный момент времени, обычно равны нулю, поскольку при выборе начального времени непрерывность не нарушается.

В других обозначениях можно записать  .

Угловой момент как генератор вращения

Орбитальный угловой момент
Оператор вращения действует на волновую функцию таким образом, что пространственные координаты частицы поворачиваются на постоянный угол Δ θ :

где r ′ обозначают повернутые вокруг оси координаты. Ось задаётся единичным вектором  а поворот — через угловое приращение Δ θ, определяемое по формуле :

где  матрица вращения, зависящая от оси и угла. На языке групп, матрицы вращения — это элементы группы, а углы и ось  — параметры трехмерной специальной ортогональной группы SO (3). Матрицы вращения вокруг стандартного базиса декартовой системы  на угол , и соответствующие образующие поворотов  :

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В более общем смысле для вращений вокруг оси, определяемой вектором , элемены матрицы вращения задаются

где δ ij — символ Кронекера, а ε ijk — символ Леви-Чивиты .

Неочевидно как определить оператор вращения по сравнению с трансляциями пространства и времени. Можно рассмотреть частный случай (вращение вокруг оси x, y или z), а затем вывести общий результат или напрямую использовать общую матрицу вращения и тензорные индексы с δij и εijk . Чтобы вывести оператор бесконечно малого вращения, который соответствует малому Δ θ, мы используем приближения малых углов sin (Δ θ) ≈ Δ θ и cos (Δ θ) ≈ 1 и тейлоровское разложение в окрестности r или r i при сохранении только первого порядка и в конце подставим компоненты оператора углового момента.

{| class="wikitable"
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Z -компоненту оператора углового момента можно заменить на проекцию вдоль оси, определяемой вектором , используя скалярное произведение  .

Опять же, конечное вращение можно выполнить используя множество небольших вращений, заменяя Δ θ на  и переходя к пределу, когда N стремится к бесконечности. Это приводит к оператору вращения для конечного вращения.

Вращения вокруг одной и той же оси коммутируют, например, поворот на углы θ 1 и θ 2 вокруг оси i можно записать

Однако вращения вокруг разных осей не коммутируют. Общие правила коммутации операторов углового момента

В этом смысле орбитальный угловой момент описывает вращения. Каждый из вышеперечисленных коммутаторов можно легко представить, взяв повседневный объект и повернув его последовательно на один и тот же угол вокруг оси 1 и оси 2 или наоборот вокруг оси 2 и оси 1 — окончательные положения тела окажутся разными.

В квантовой механике есть ещё одна форма вращения, которая математически кажется похожей на орбитальный случай, но имеет другие свойства, описанные ниже.

Спин
Все предыдущие величины имеют классические аналоги. Спин — это величина, которой обладают частицы в квантовой механике без какого-либо классического аналога, имеющая размерность единицы углового момента. Оператор вектора спина обозначается . Собственные значения его компонент — это возможные величины (в единицах ) измерения спина, спроецированного на базисные вектора.

Вращения (обычного пространства) вокруг оси  на угол θ относительно единичного вектора  в пространстве, действующее на многокомпонентную волновую функцию (спинор) в точке пространства, представляется в видеОднако, в отличие от орбитального углового момента, в котором квантовое число l может принимать только положительные или отрицательные целые значения (включая ноль), спиновое квантовое число s может принимать все положительные и отрицательные полуцелые значения. Для каждого спинового квантового числа существуют матрицы вращения.

Вычисление экспоненты для z-проекции с заданным спиновым квантовым числом s дает (2 s + 1) -мерную матрицу спина. Что можно использовать для определения спинора как вектора-столбца из 2 s + 1 компонент, который преобразуется повороте системы координат в соответствии со спиновой матрицей в фиксированной точке пространства.

Для простейшего нетривиального случая для состояния с s = 1/2 оператор спина имеет вид

где матрицы Паули в стандартном представлении:

Полный угловой момент
Оператор полного углового момента представляет собой сумму орбитального и спинового моментов

и имеет важное значение для многочастичных систем, особенно в ядерной физике и квантовой химии многоэлектронных атомов и молекул.

Аналогичная матрица вращения

Сохраняющиеся величины для квантового гармонического осциллятора
Группа динамической симметрии n- мерного квантового гармонического осциллятора — это специальная унитарная группа SU (n). Например, число инфинитезимальных генераторов соответствующих алгебр Ли для групп SU (2) и SU (3) равно трем и восьми соответственно. Это приводит ровно к трем и восьми независимым сохраняющимся величинам (кроме гамильтониана) в этих системах.

Двумерный квантовый гармонический осциллятор обладает ожидаемыми сохраняющимися величинами как гамильтониан и угловой момент, но также имеет дополнительные скрытые сохраняющиеся величины такие как разности уровней энергии и другую форму углового момента.

Группа Лоренца в релятивистской квантовой механике
Ниже рассматривается группа Лоренца (бусты и вращения в пространстве-времени). В этом разделе см.

Преобразования Лоренца можно параметризовать быстротой  для буста в направлении трехмерного единичного вектора , и уголом поворота  вокруг трехмерного единичного вектора , который определяя направление оси. Тогда  а также  вместе задают шесть параметров группы Лоренца (три для вращений и три буста). Группа Лоренца имеет шесть измерений.

Чистые вращения в пространстве-времени
Рассмотренные выше матрицы вращения и генераторы вращения образуют пространственноподобную часть четырёхмерной матрицы, представляющую собой чистое вращение. Три элемента группы Лоренца  а генераторы  для чистых вращений:

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Матрицы вращения действуют на любые 4-вектора A = (A 0, A 1, A 2, A 3) и вращают пространственно-подобные компоненты в соответствии с формулой

оставив временную координату неизменной. В матричном представлении вектор A рассматривается как вектор-столбец.

Чистый буст в пространстве-времени
Буст со скоростью  в направлениях x, y или z, заданных декартовой системе коорднинат с базисом , — матрицы преобразования буста. Эти матрицы  а соответствующие генераторы  — это оставшиеся три элемента группы и генераторы группы Лоренца:

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Матрицы буста действуют на любые 4-векторы A = (A 0, A 1, A 2, A 3) и смешивают временные и пространственные компоненты в соответствии с формулой

Термин «буст» относится к относительной скорости между двумя системами отсчёта, и его не следует объединять с импульсом как генератором трансляций, как объясняется ниже.

Комбинирование бустов и вращений
Произведение вращений дают другое вращение (частый пример подгруппы), в то время как произведения бустов или бустов и вращений нельзя выразить через чистые бусты или чистые вращения. В общем случае, любое преобразование Лоренца можно выразить как произведение чистого вращения и чистого буста. Для получения дополнительной информации см. и ссылки в них.

Представления генераторов бустов и вращений обозначаются  и  соответственно, где заглавная  в этом контексте указывает на представление группы.

Для группы Лоренца представления  и  генераторов  и  удовлетворяют следующим правилам коммутации.

{|class="wikitable"
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Во всех коммутаторах бусты смешиваются с вращениями, хотя коммутаторы только вращений приводят к другому вращению. Экспоненциальное отображение генераторов группы даёт операторы бустов и вращений, которые объединяются в общее преобразование Лоренца, при котором пространственно-временные координаты преобразуются из покоящейся системы отсчёта в другую посредствлом бустов и/или вращений. Точно так же экспоненциальное отображение представлений генераторов дает представления операторов бустов и вращений, в соответствии с которыми преобразуется спинорное поле частицы.

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В литературе генераторы бустов  и генераторы вращения  иногда объединяются в один генератор для преобразований Лоренца , антисимметричную четырёхмерную матрицу с элементами:

и, соответственно, параметры бустов и вращений собираются в другую антисимметричную четырёхмерную матрицу  с элементами:

Таким образом, общее преобразование Лоренца:

с суммированием по повторяющимся матричным индексам α и β . Матрицы Λ действуют на любые 4-векторы A = (A 0, A 1, A 2, A 3) и смешивают времениподобные и пространственно-подобные компоненты в соответствии с формулой

Преобразования спинорных волновых функций в релятивистской квантовой механике
В релятивистской квантовой механике волновые функции больше не являются однокомпонентными скалярными полями, а представляют собой спинорные поля состоящие из 2 (2 s + 1) компонент, где s — спин частицы. Ниже приведены преобразования этих функций в пространстве-времени.

При правильном ортохронном преобразовании Лоренца  в пространстве Минковского все одночастичные квантовые состояния  локально преобразуются в некотором представлении  для группы Лоренца по формуле

где  — конечномерное представление, другими словами, квадратная матрица размерности , а  рассматривается как вектор-столбец, содержащий компоненты с  допустимые значения спина  :

Неприводимые вещественные представления и спин
Неприводимые представления  и  можно использовать для построения спиновых представлений группы Лоренца. Определение новых операторов:

так что  и  комплексно сопряженные друг другу. Из этого следует, что они удовлетворяют симметрично записанным коммутаторам:

и это по существу коммутаторы, которым удовлетворяют операторы орбитального и спинового углового момента. Следовательно,  и  образуют операторные алгебры, аналогичные угловому моменту; одни и те же лестничные операторы, z -проекции и т. д. независимо друг от друга, поскольку каждый из их компонент коммутирует друг с другом. По аналогии со спиновым квантовым числом введём положительные целые или полуцелые числа  с соответствующими наборами собственных значений  и  . Матрицы, удовлетворяющие указанным выше коммутационным соотношениям, такие же, как для спинов a и b, имеют компоненты, заданные умножением значений дельты Кронекера на матричные элементы углового момента:

где в каждом случае номер строки m ′ n ′ и номер столбца mn разделены запятой. Тогда

и аналогично для J (n)
. Три квадратные матрицы J (m) — каждая размерности , а три J (п) с размерностью . Целые или полуцелые числа m и n нумеруют все неприводимые представления с помощью эквивалентных обозначений, используемых здесь: , каждое из которых имеет вид квадратных матриц размерности .

Применим это рассуждение к частицам со спином  ;

 левосторонние  -компонентные спиноры преобразуются относительно непреводимых вещественных представлений  ,
 правые  -компонентные спиноры преобразуются относительно непреводимых вещественных представлений  ,
 взяв прямые суммы, обозначенные символом  (см. более простую концепуию для матриц прямую сумму матриц), получаем представления, при которых преобразуются  -компонентные спиноры:  где  . Это тоже непреводимые вещественные представления, но, как показано выше, они распадаются на комплексно-сопряжённые.

В этих случаях  относится к любому из ,  или к полному преобразованию Лоренца  .

Релятивистские волновые уравнения
В контексте уравнения Дирака и уравнения Вейля спиноры Вейля, удовлетворяющие уравнению Вейля, преобразуются при простейших неприводимых спиновых представлениях группы Лоренца, поскольку спиновое квантовое число в этом случае является наименьшим допустимым ненулевым числом: 1/2 . 2-компонентный левый спинор Вейля преобразуется посредством , а 2-компонентный правый спинор Вейля преобразуется по  . Спиноры Дирака, удовлетворяющие уравнению Дирака, преобразуются по представлению  — прямой сумме непреводимых вещественных представлений спиноров Вейля.

Группа Пуанкаре в релятивистской квантовой механике и теории поля
Пространственные трансляции, временные трансляции, вращения и бусты, вместе взятые, образуют группу Пуанкаре. Элементами группы являются три матрицы вращения и три матрицы бустов (как в группе Лоренца), одна для трансляций времени и три для пространственных трансляций в пространстве-времени. Для каждого элемента существует генератор. Следовательно, группа Пуанкаре 10-мерна.

В специальной теории относительности пространство и время могут быть собраны в 4-вектор , и аналогично энергия и импульс объединяются в четырехмерный вектор импульса  . С учётом релятивистской квантовой механики параметры временной интервала и пространственного смещения (всего четыре параметра, один для времени и три для пространства) объединяются в пространственно-временное смещение , а операторы энергии и импульса подставляются в четырёхмерный импульс, чтобы получить четырёхмерный оператор

которые представляют собой генераторы трансляций пространства-времени (всего четыре генератора, один для времени и три для пространства):

Запишем коммутационные соотношения между компонентами 4-импульса P (генераторы пространственно-временных трансляций) и угловым моментом M (генераторы преобразований Лоренца), которые определяют алгебру Пуанкаре:

где η — метрический тензор Минковского. (Обычно для операторов 4-импульсов в коммутационных соотношениях снимаются шляпы). Эти уравнения содержат фундаментальные свойства пространства и времени, насколько они известны сегодня. У этих соотношений есть классический аналог, в котором коммутаторы заменены скобками Пуассона.

Для описания спина в релятивистской квантовой механике используют псевдовектор Паули – Любанского

оператор Казимира, представляет собой постоянный спиновой вклад в полный угловой момент. Коммутационные соотношения между P и W и между M и W запишутся в виде

Инварианты, построенные из W, инварианты Казимира, можно использовать для классификации неприводимых представлений группы Лоренца.

Симметрии в квантовой теории поля и физике частиц

Унитарные группы в квантовой теории поля
Теория групп — это абстрактный способ математического анализа симметрий. Унитарные операторы обладают первостепенным значением в квантовой теории, поэтому унитарные группы важны в физике элементарных частиц. Группа N- мерных унитарных квадратных матриц обозначается U (N). Унитарные операторы сохраняют скалярное произведение, что означает, что вероятности также сохраняются, поэтому квантовая механика какой-либо системы должна быть инвариантной относительно унитарных преобразований. Пусть  — унитарный оператор, а обратный к неиу — эрмитово сопряженный , который коммутирует с гамильтонианом:

Тогда наблюдаемая величина, соответствующая оператору  сохраняется, а гамильтониан инвариантен относительно преобразования  .

Поскольку предсказания квантовой механики должны быть инвариантными под действием группы, учёные ищут унитарные преобразования для представления группы.

Важными подгруппами каждой группы U (N) являются те унитарные матрицы, которые имеют единичный определитель (или являются «унимодулярными»): они также называются специальными унитарными группами и обозначаются SU (N).

U (1)
Простейшая унитарная группа — это U (1), которая представляет собой просто комплексные числа по модулю 1. Этот элемент одномерной матрицы записывают в виде

в которой θ — параметр группы. Эта группа абелева, поскольку одномерные матрицы всегда коммутируют при матричном умножении. Лагранжианы в квантовой теории поля для комплексных скалярных полей часто инвариантны относительно преобразований U (1). Если есть квантовое число a, связанное с симметрией U (1), например барион и три лептонных числа в электромагнитных взаимодействиях, то

U (2) и SU (2)
Общий вид элемента группы U (2) параметризуется двумя комплексными числами a и b :

а для SU (2) определитель равен 1:

На языке теории групп, матрицы Паули являются генераторами специальной унитарной группы в двух измерениях, обозначаемой SU (2). Их коммутатор такой же, как и для орбитального углового момента, за исключением коэффициента 2:

Элемент группы SU (2) можно записать:

где σ j — матрица Паули, а параметры группы — углы поворота вокруг оси заданной вектором .

Двумерный изотропный квантовый гармонический осциллятор обладает группой симметрии SU (2), в то время как алгебра симметрий анизотропного осциллятора является нелинейным расширением u (2).

U (3) и SU (3)
Восемь матриц Гелл-Манна  (см. Статью о них и структурных константах) важны для квантовой хромодинамики. Первоначально они появились в теории SU (3) для аромата, которая до сих пор используется в ядерной физике. Они задают генераторы группы SU (3), поэтому элемент группы SU (3) можно записать аналогично элементу группы SU (2):

где θ n — восемь независимых параметров. Матрицы  удовлетворяют коммутатору:

где индексы , ,  принимают значения 1, 2, 3 … 8. Структурные константы f abc полностью антисимметричны по всем индексам, аналогично индексам SU (2). В стандартном базисе цветного заряда (r для красного, g для зелёного, b для синего):

цветовые состояния — это собственные состояния матриц  и , в то время как другие матрицы отвечают за смешивание цветовых состояний.

Состояния восьми глюонов (8-мерные векторы-столбцы) — это собственные состояниями присоединенного представления группы , 8-мерного представления, действующего на его собственной алгебре Ли , для матриц  и  . Формируя тензорные произведения представлений (стандартное представление и двойственное ему) и принимая соответствующие отношения, протоны, нейтроны и другие адроны представляются собственными состояниями различных представлений  цвета. Представления SU (3) можно описать «теоремой максимального веса».

Вещество и антивещество
В релятивистской квантовой механике релятивистские волновые уравнения обладают замечательной симметрией природы: каждая частица имеет соответствующую античастицу. Математически это выражается спинорными полями, которые являются решениями релятивистских волновых уравнений.

Зарядовое сопряжение переставляет частицы и античастицы. Физические законы и взаимодействия, неизменные в результате этой операции, обладают C-симметрией.

Дискретные симметрии пространства-времени
 Четность отражает ориентацию пространственных координат как левой и правой. Неформально пространство «отражается» в зеркале. Физические законы и взаимодействия, неизменные в результате этой операции, обладают P-симметрией.
 Обращение времени переворачивает временную координату, то есть возникает время, бегущее из будущего в прошлое. Любопытное свойство времени, которого нет в пространстве, заключается в том, что оно однонаправленно: частицы, движущиеся вперед во времени, эквивалентны античастицам, движущимся назад во времени. Физические законы и взаимодействия, неизменямые этой операцией, обладают T-симметрией.

C, P, T симметрии

 CPT теорема
 Нарушение CP
 PT-симметрия
 Нарушение Лоренца

Калибровочная теория
В квантовой электродинамике обладает группой симметрии U (1), которая абелева. В квантовой хромодинамике соответствующая группа симметрии SU (3) является неабелевой.

Электромагнитное взаимодействие осуществляется фотонами, у которых нет электрического заряда. Тензор электромагнитного поля задаётся через 4-потенциальное электромагнитное поле, обладающее калибровочной симметрией.

Сильное (цветное) взаимодействие обеспечивается глюонами, которые различаются восемью цветовыми зарядами. Имеется восемь тензоров напряженности глюонного поля с соответствующими полями 4-глюонных потенциалов, каждый из которых обладает калибровочной симметрией.

Сильное (цветное) взаимодействие

Цветовой заряд
По аналогии с оператором спина существуют операторы цветового заряда в терминах матриц Гелл-Манна λ j :

и поскольку цветной заряд сохраняется, все операторы цветного заряда должны коммутировать с гамильтонианом:

Изоспин
Изоспин сохраняется при сильных взаимодействиях.

Слабые и электромагнитные взаимодействия

Преобразование двойственности
Магнитные монополи могут существовать теоретически, хотя текущие наблюдения и теория согласуются с обоими исходами существования монополей или несуществования. Электрические и магнитные заряды можно эффективно «преобразовывать друг в друга» с помощью преобразования дуальности.

Электрослабая симметрия
 Электрослабая симметрия
 Нарушение электрослабой симметрии

Суперсимметрия
Супералгебра Ли — это алгебра, в которой (подходящие) базисные элементы либо подчиняются правилам коммутации, либо антикоммутации. В суперсимметрии предполагается, что все фермионные частицы имеют бозонные аналоги, и наоборот. Эта симметрия теоретически привлекательна, поскольку не делается никаких дополнительных предположений (например, о существовании струн), препятствующих симметрии. Кроме того, допуская суперсимметрию, можно решить ряд загадочных проблем. Эти симметрии, которые представлены супералгебрами Ли, экспериментально не подтверждены. Сейчас считается, если она существуют, то эта симметрия нарушена. Предполагается, что темная материя представляет собой гравитино, частицу со спином 3/2 (фермион) и массой, а её суперсимметричным партнером является гравитон со спином 2 (бозон).

Перестановочная симметрия
Концепция перестановочной симметрии выводится из фундаментального постулата квантовой статистики, который гласит, что никакая наблюдаемая физическая величина не должна изменяться после замены двух идентичных частиц друг на друга. В ней говорится, что, поскольку все наблюдаемые пропорциональны квадрату волновой функции  для системы одинаковых частиц, то волновая функция  должна либо остаться прежней, либо изменить свой знак при таком обмене. В более общем смысле, для системы из n одинаковых частиц волновая функция  должна преобразоваться как неприводимое представление конечной симметрической группы S n. Согласно теореме Паули о статистике, фермионные состояния преобразуются как антисимметричное неприводимое представление S n, а бозонные состояния — как симметричное неприводимое представление. Для классификации симметрии ровибронных состояний молекул Лонге-Хиггинс ввел группу молекулярной симметрии как группу соответствующих перестановок неразличимых ядер и перестановок с пространственной инверсией.

Поскольку обмен двух неразличимых частиц математически эквивалентен повороту каждой частицы на 180 градусов (и, следовательно, повороту системы отсчёта одной частицы на 360 градусов) симметричный характер волновой функции зависит от спина частицы после применения к нему оператора вращения . Частицы с целым спином не меняют знака своей волновой функции при повороте на 360 градусов, поэтому знак волновой функции всей системы не меняется. Частицы с полуцелым спином меняют знак своей волновой функции при повороте на 360 градусов (подробнее см. Теорема Паули).

Частицы, у которых волновая функция не меняет знак при обмене, называются бозонами или частицами с симметричной волновой функцией. Частицы, у которых волновая функция системы меняет знак при перестановке, называются фермионами, или частицами с антисимметричной волновой функцией.

Таким образом, фермионы подчиняются другой статистике (называемой статистикой Ферми — Дирака), чем бозоны (которые подчиняются статистике Бозе — Эйнштейна). Одним из следствий статистики Ферми — Дирака является принцип Паули для фермионов: никакие два идентичных фермиона не могут иметь одно и то же квантовое состояние (другими словами, волновая функция двух одинаковых фермионов в одном и том же состоянии равна нулю). Это, в свою очередь, приводит к давлению вырождения для фермионов — сильному сопротивлению фермионов сжатию. Это сопротивление приводит к «жесткости» или «твёрдости» обычной атомной материи (поскольку атомы содержат электроны, которые являются фермионами).

Комментарий

Примечания

Дальнейшее чтение

Теория групп
Квантовая механика
Специальная теория относительности
Страницы с непроверенными переводами

redpajama_set_name: RedPajamaWikipedia


Members Present: – Keith Cragg, Alan Garside, Dan Jones and Alex King.
Matthew Milburn has decided to call it a day with our club due to work, family and other commitments. Matthew was a quiet lad who always had a good eye for scenery and buildings. He will be sadly missed and the door will be always being open if he decides to return.
We were sorry to learn of the news that our first chairman Harry Dennis passed away recently after suffering from dementia. It was Harry's idea back in 1978 to put together the club with others although in his later years he wasn't able to attend as much as he would of linked due to poor health, he will be missed and remembered by all.
Ramsbottom – Not really much to report except that we put all tools down over two weeks to run the program. It highlighted a few errors in the track work, program diagrams but all in all was a success and enjoyed by all. We hope to do this more often. The layout on the whole could do with a good clean up and the tracks giving a good scrub.
Bacup Shed – A couple of weeks ago it was decided that we should possibly head in a different direction with Bacup Shed. This was due to many factors such as damp, mdf base board, continuous problems with the peco turntable (whole other story) and continous faults with trackwork and wiring. It was decided that the layout couldn't really offer much in terms of interest. So with great regret it has now been stripped and a new layout will take its place but we have no name yet. Scrapped 27-02-14.
Unnamed Layout – So the fun begins all over again. The new layout will be a countryside layout similar to a GWR branch line…..but LMS/BR region with DCC and hopefully sound. It will be an end to end layout with a fiddle yard at one end. We are hoping to have a small/medium sized station, cattle dock, goods shed, small engine shed and sidings but with lots of scenery and plenty to do. We realized that Bacup shed would only give you so much to do due to it only being a shed. This way you can have 2/3 coach trains, wagons, tankers, box vans, light engine movements and more happening even at once due to it being dcc. The reason for LMS/BR region is so that way we can use existing stock between Ramsbottom and the new layout. Just have to remember to remove or add the chips first. We also did look at some real life locations such as Ewood Bridge, Summerseat and Holcombe Brook but decided to go free lance.
The four sections were stripped down for part recovery which included wiring, track, points, point motors, pits, turntable, components and ballast. Most of this will be reusable and with other bits probably go for sale.
MDF top boards were removed and donated to a person who has a wood burner.
2x large chip board sections that Keith brought down when we were originally doing Bacup have been put on top of the frame work to see above measurements etc. Although a little warped these will be brought into place once screwed down. When Roger was alive he had brought down some old kitchen units and work tops well some of these are still around and will be used as baseboards for the new layout. These have been brought out of store to see about sizes.
The current thickness ranges from 2cm down to 1 ½cm so there will be some packing to do to get a continuous level.
Alan has been evicted from his table in the corner due to extending this layout into the corner. Alan has now found a new home on the table next to it which was originally a computer desk. The table in the corner is one of these where it extends and has been in the extended position well now it's been reduced and pushed right into the corner to give more room and a solid base to mount the extension on.
Alan has had to move all his materials and equipment from the corner. These will be mounted to the wall and he has said he will have a good sort out to reduce the clutter.
A damp proof course membrane will be fitted to run the length of the wall where the layout will be. This will prevent the damp from touching the layout.
Our neighbors at No3 are in the process of moving house so we thought it would only be right to get them a little going away presents to show our appreciation for the help and support they have shown us over the last few years (after all they do supply us with water). A bottle of wine and a £10 gift card for next were the order of the day.
A new dehumidifier has been ordered from Amazon due to the other playing up from time to time. This will hopefully be here within 3-5 working days so by this time next week we will have it. This will allow us when we get time to have a look at ours and maybe have two working one day. This should help the building get a little dryer. The dehumidifier also has a tube which we can place into the sink for continuous running so no need to empty anymore.
The website will be getting a general make over with a new template, decluttered and reworked once my current work on www.burystandard4group.org.uk has been completed which should be too long now.
We are also looking into the possibility of getting rid of our gas heaters with the replacements of either electric heaters or storage heaters. We will need to investigate and do our homework first. This is due to the rising cost of gas over the last 3 years. This has risen from £83 for 6 bottles to £87 just for 3 bottles!! If anyone has had any storage heaters fitted with then we would like you to get in touch with regards to costing, suppliers etc.
I received a phone call last night around 7.30pm from Roger who is Harrys son to inform me and the club of Harrys passing earlier that morning.
Harry was one who first came up with the idea of forming a club without him maybe we would never of existed. He was our first chairman who seemed to dedicate a large of his time to club and its aims.
In the later years of his life he couldnt attend as much as he would of liked due to his wife having dementia and needing full time care but when he did he would come down, run the layout and let us know of any faults.
Harry was a local postman serving the local town of Ramsbottom which led him to work with our chairman Alan Gartside and the late Roger Wilson.
Rest In Peace Harry, you will be missed by all of us from the club.


Being a bestseller isn't about any fame, glory or about money; its about the message, those words that the world needs to hear.
There is a voice that doesn't have words. Listen to it. Go for it. You might not get any other chance so take a glance.
It's not what you suppose to be but it's what you want to be that matters.
From her birthday cake to her special gift, from the venue to the invitation, from her dress to the tiara. They were taking care of each and everything. Everything was being done so perfectly, as it was just not only her special day it was their too. They try to illuminate her path with the brightest joy. Her friends took five weeks to plan her birthday. But those special three words from that special person (love of her life) took away all the thunder of their planing and attention of the day.
P. S. – A musing, hope you have enjoyed. You are free to share your words, ideas and thoughts in thecomment section.
Let the world shine by the sparkle of your smile.


Addicted to Agility, RI Toys for Tots, Young at Heart Run & Thank-You!!!
Thank you everyone who came out this past weekend! We could not do it without your support and help.
It means more to us that you could ever know. We love hosting these NADAC trials and are so happy that you come time and time again to trial with us. Each and everyone of you have become part of our "extended family".
What makes us even more appreciative is the support that you gave to Tots for Tots!
with the "do not take a Q and we will give $1 to Toys for Tots" throughout the year at various trials.
of you who stayed and helped us load the trailer.....all without us asking! Thank you again!
A special THANK YOU also to all of you for supporting our new "Young at Heart" run. It was great that you guys cheered on the dogs who ran in this class. It was especially touching for me to see the dogs who got us to where we are in the sport of agility happily running again.
Way to go to the following people who entered our first Young at Heart Run!
It may just seem like words but THANK YOU! THANK YOU! Words can not express how much we appreciate everything each and everyone of you do!
Also, be sure to check out (and "Like") Addicted to Agility's Facebook page for the latest information!
Hope to see you on June 23 & 24th!
Re: Addicted to Agility, RI Toys for Tots, Young at Heart Run & Thank-You!!!
Thank you go Addicted for the Young at Heart Run! I wasn't sure what she would do, but Chloe looked so happy to be out in the ring again, even if the jump bars were on the ground. She was always one to enjoy people cheering for her.


Every company has a "Contact Us" page on their website. But with Sento Bene, we really do want to know about you. This is your invitation to connect with us, and we can't wait to hear what you have to say.
Tell us about your last celebration with Sento Bene, and what you were toasting on that occasion. Or, if you have any questions or comments about our wines, please let us know.


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By damian Events & Programs, Fundraising, The Panther Prowl
Week of January 10, 2022 Mark your calendar: Jan. 14 – Noon dismissal-Parent/Teacher Conferences Jan. 17 – No school MLK Jan. 24 – Chick-Fil-A Family Night *Jan. 30 – School…
Week of January 3, 2022 Mark your calendar: Jan 3 – No School for students– Teacher Retreat/Professional day Jan. 4 – Classes resume Jan. 14 – Noon dismissal-Parent/Teacher Conferences Jan….
By damian Counselor's Corner, Events & Programs, Fundraising, The Panther Prowl
Week of December 13, 2021 Mark your calendar: Dec. 13-16 Ongoing Advent food drive (see below) Dec. 13 – Reconciliation – middle school Dec. 14 – Reconciliation – grades 3-5…
The Panther Prowl | Nov 21, 2021
Week of November 22, 2021 Many thanks for all the generous donations to our Food Collection for St. Matthew's House! Since Nov. 3, we collected over 1000 lbs. of food…
The Panther Prowl | Oct 24, 2021
Week of October 25, 2021 Fall picture day is Wed. 10/27! Students will be taking individual and classroom pictures in their school uniform. Ordering envelopes will be sent home on…
Week of October 11, 2021 A Salute to Veterans! Wed., Nov. 10, 2021 @ 9:00am We will have a Veteran's Day mass beginning at 9:00am followed by a Tribute to…
The Panther Prowl | Oct 3, 2021
By damian Events & Programs, Fundraising, HASA, The Panther Prowl, Volunteering
Week of October 4, 2021 Used Uniform Closet – Open Monday, 10/4 – 3:30-4:30 p.m. In front of the SES Gym – Open for clothes turn-in or purchase. Cash or…
The Panther Prowl | Sep 26, 2021
By damian Events & Programs, Fundraising, Holiday Events, The Panther Prowl
Week of September 27, 2021 Message from Chef Dan School Lunches resume Tuesday, September 28. Anyone who has open orders will have food supplied on the weekday as requested. Please…
Week of September 13, 2021 Used Uniform Closet Please note: The Used Uniform Closet is open Monday, Sept. 13 from 3:30-4:30pm. The closet will be open for item donation and/or…
The Panther Prowl | Sep 5, 2021
By damian Events & Programs, Fundraising, HASA, The Panther Prowl
Week of September 6, 2021 Reminders! Sept. 6 – Labor Day, NO School – We wish everyone a safe and happy weekend! Sept. 7 –Parents are invited to join us…

redpajama_set_name: RedPajamaCommonCrawl


The chassis was constructed of special laminated plywood on the monocoque principle and joined using the synthetic Aerolite 300 adhesive. Small sections are assembled into a whole by gluing and stapling. Localized stress areas were reinforced. There are not many photographs of the chassis but a diagram appears in Frank Costin's book. A good verbal description interestingly appeared in a review published by "Sports Car and Lotus Owner" (April 1960).
Diameter of inlet valves: 32.2 - 32.3 m.m.
Diameter of exhaust valves: 30.18 - 30.32 m.m.
Overall lenght of car: 384 cm.
Overall width of car: 144 cm.
Overhall height of car: 122 cm.


This shoot started with Casey from Tharpe Social Media & Events asking me to be the photographer of her styled shoot! She said there was going to be an old truck- so naturally I was hooked right off the bat. It wasn't until we were all warming up after the portraits, that I found out the real reason behind the everything!
Cassidy and Austin tied the knot back in October at a venue in Powhatan. They hired a company to do their wedding photography and videography. Not only was their photography experience awful, but so were their images. No one hiring a professional should receive images like they did of their WEDDING day. I beg of you.. if you are getting married and looking for a photographer.. Please make sure you know who you are hiring. The images are the only thing you have left after your big day! They're so important!!
The couple is now completely in love with their images, and they can't wait to hang them up in their home! I'm SO grateful I was able to give them the portraits that they deserve! And I am so thankful that this team of vendors put this on for them!
STATIONARY | Kara Anne & Co.


Human invasive breast cancer: immunohistochemical staining for cytokeratin 14 using NCL-LCK14. Note intense membrane and cytoplasmic staining of a proportion of tumor cells. Paraffin section.
Cytokeratins 14 and 5 are useful to distinguish stratified epithelial cell types from simple epithelial cell types. Cytokeratin 14 has been reported to be expressed in neoplasms of squamous cell origin.
Clone LL002 reacts with the human cytokeratin intermediate filament protein (50 kD) identified as cytokeratin 14.


In the aftermath of the financial crisis financial stability considerations have become more prominent in central banks' policies and require substantial discretion in their implementation. We argue that transparency can serve as an instrument for central banks to balance the conflicting interests of different stakeholders and manage market participants' expectations, explain their strategy and report on its implementation progress and receive comments and feedback by stakeholders. In view of the increasingly contested concept of central banks' independence, we show that increasing and institutionalizing ex ante and ex post transparency on policy decisions and their implementation strengthen central banks' accountability as a key ingredient and prerequisite for their independence. Finally, we hold that central banks' new responsibilities for financial stability call for the development of a more comprehensive and inclusive approach given their strong impact on private actors and their linkages with other policy fields and governmental agencies.


text:
Malaysia: More dengue cases, but no report of Zika virus yet
The total number of localities, which are still affected by dengue increased from 647 to 660 for Week 26.
July 9, 2016 19:23 +08
Dengue fever is on the rise again in Malaysia, with 2,017 more cases reported between June 26 and July 2 (Week 26). But Zika virus has so far not been reported in the country.
The National Public Health Laboratory and the Institute for Medical Research had tested nearly 507 blood samples until July 2. Health screenings were also organised at several immigration checkpoints in the country. But there were no reports of Zika virus during the exercise.
The Director-General, Health, Dr. Noor Hisham Abdullah, said that until now nine states have recorded a rise in the instance of dengue cases, namely Selangor with the maximum of 1,145 cases, Johor (165), Perak (75), Sarawak (68), Kelantan (51), Kedah (42), Pahang (39), Penang (31) and Perlis (5).
"The cumulative cases of dengue fever from January up to July 2 stood at 57,782 cases compared to 59,365 cases during the corresponding period last year, a decline of 2.7 per cent (1,583 cases)," he told Bernama on Saturday.
Dr. Noor Hisham also said that 10 fatalities related to dengue were reported during the last week.
The total number of localities which are still affected by dengue increased from 647 to 660 for Week 26. But the outbreak hotspots have declined from 74 to 73 areas as compared to the previous week.
Dr. Noor Hisham advised citizens to be alert to signs and symptoms of the disease, such as fever, muscle and joint pain, rashes, pain behind the eyes and headache, as they expect the number of dengue cases to increase after the festive season.
"This includes dengue fever pre-existing signs such as stomach pain, frequent vomiting, restlessnes, feeling weak and bleeding from the gums and nose," he added.
He also warned saying, "If there are such signs and symptoms, get treatment at the nearest clinic or hospital immediately as early detection and treatment can save lives."
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text:
Home >> Priyanka's Another International Hit
Cine Buzz
Priyanka's Another International Hit
Updated: October 17, 2015 5:15 am
The 33-year-old, Dil Dhadkane Do actress Priyanka Chopra who is currently enjoying the success of her International Television series 'Quantico' is all set make to make it big once again in the music industry too. The actress turned singer, who has given international hits in the past like In My City, Exotic & I Can't Make You Love Me, will feature in the N.A.S.A's new track titled 'Meltdown', which also features rapper DMX. This song is a part of 'We Are Blood' film's soundtrack and was released recently. Priyanka Chopra took Twitter to share this news with her fans, the 'Mary Kom' actress tweeted, "Big news everyone!! My next song!! Check it out!! Goooo Punjabis!! #MeltDown @nasaofficial @DMX Out Friday."
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text:
/*
 *
 */
package org.jspare.vertx.samples.route;

import java.util.Arrays;
import java.util.List;
import java.util.Map;

import org.jspare.vertx.samples.model.Car;
import org.jspare.vertx.web.annotation.handler.Handler;
import org.jspare.vertx.web.annotation.handling.ArrayModel;
import org.jspare.vertx.web.annotation.handling.MapModel;
import org.jspare.vertx.web.annotation.handling.Parameter;
import org.jspare.vertx.web.annotation.method.Get;
import org.jspare.vertx.web.annotation.method.Post;
import org.jspare.vertx.web.annotation.subrouter.SubRouter;
import org.jspare.vertx.web.handler.APIHandler;

@SubRouter("/cars")
public class CarRoute extends APIHandler {

	@Get("/:id")
	@Handler
	public void getById(@Parameter("id") Integer id) {

		success(new Car("Car " + id, 2000 + id));
	}

	@Get
	@Handler
	public void list() {

		success(Arrays.asList(new Car("Punto", 2010), new Car("Veloster", 2013)));
	}

	@Post
	@Handler
	public void save( Car car) {

		success(car);
	}

	@Post("/bulk")
	@Handler
	public void saveBulk(@ArrayModel(Car.class) List<Car> cars) {

		success(cars);
	}

	@Post("/type/bulk")
	@Handler
	public void saveTypeBulk(@MapModel(Car.class) Map<String, Car> cars) {

		success(cars);
	}

}

redpajama_set_name: RedPajamaGithub


text:
Q: Stronger result than bertrand's postulate It is well known that there is a prime number between $n$ and $2n$ for all $n$. I decided to go deeper: is there a lower bound on the number of primes between $n$ and $2n$ for "large enough" $n$?
For instance, I found empirically that $\pi(n)-\pi(n/2)\ge \sqrt n$ at around $n\ge 100$ (In particular, I also used a program to show $\pi(n^2)-\pi(n^2/2)\ge n$ for $n\ge 10$). Does there exist a nice elementary proof of this?
Obviously, we can show this result asymptotically for very great $n$. I ask to consider an elementary proof of the above bound.

A: From Theorem 1.2 and 1.3 of Axlar's 2017 paper, we have for  $x \ge 49$,
$$
\pi(x) < \frac{x}{\ln x - 1 - \frac{1}{\ln x} - \frac{3.15}{\ln^2 x} - \frac{12.85}{\ln^3 x} - \frac{71.3}{\ln^4 x} - \frac{463.2275}{\ln^5 x} - \frac{4584}{\ln^6 x} }
$$
and for $x \ge 19,033,744,403$,
$$
\pi(x) > \frac{x}{\ln x - 1 - \frac{1}{\ln x} - \frac{2.85}{\ln^2 x} - \frac{13.15}{\ln^3 x} - \frac{70.7}{\ln^4 x} - \frac{458.7275}{\ln^5 x} - \frac{3428.7225}{\ln^6 x} }
$$
Plug in these inequalities you will get the best currently known bounds on $\pi(n) - \pi(n/2)$.
Another way to look at a stronger form of Bertrand's postulate is found in Theorem 3.6 of Das and Paul's 2019 paper where they proved that

For any integer $n$ and $k$, if $n \ge 1.1\ln(2.5k)$, there are at least $k-1$ primes between $n$ and $kn$.

Both of these are relatively new results so I don't think much has been improved since their publication.

redpajama_set_name: RedPajamaStackExchange


Daylight Saving Time Proposition 7: Will California Abandon Time Changes?
The Golden State could be getting even more sunshine in winter evenings after voters in California backed a proposition that could see it making daylight saving time permanent.
On the ballot for the midterm elections Tuesday was Proposition 7, which asked California voters whether or not they agreed to stop changing their clocks twice a year. Just two days earlier, the state fell back to standard time, in which California, along with most of the U.S., gained an hour of sunlight in the morning but lost one in the evening.
With more than 3 million ballots counted, some 60.3 percent of voters voted yes, while 39.7 percent of people opposed the proposition, according to DesertSun.com.
The Daylight Saving Time Act came into force in 1949 in the U.S., requiring clocks to be set forward by an hour every spring and fall back an hour every fall.
Democratic Representative Kansen Chu of San Jose put forward the proposal, arguing that time changes are dangerous. Heart attacks and car accidents increase when people lose an hour of sleep by putting their clocks forward every March, he told the Associated Press, citing a 2009 Michigan State University study.
"It's a public safety measure, and I don't know anybody who really enjoys doing this adjustment of their schedule twice a year," he said.
He argued that keeping daylight saving time permanent would mean that children could spend more time outside during late afternoons in the winter, and people would go out for dinner more during those months, which would help businesses.
However, any change still requires the backing of a two-thirds majority in Congress, as well as a change in federal law.
Year-round daylight saving time would mean the sun won't rise until 8 a.m. on some winter months in California. Democratic state Senator Hannah-Beth Jackson, who represents Ventura County, told the AP this could lead to more accidents during the morning commute.
"I just don't share this belief that this is a major crisis," she said.
Republican state Senator Jim Nielsen said in June: "It's fixing something that is not broken. Our society has acculturated itself to daylight savings time. I think it would create too much confusion to change it again," The Mercury News reported.
Hawaii and Arizona (with the exception of Navajo Nation) do not recognize daylight saving time. In March, Florida became the first state to approve year-round daylight saving time, but the move still requires the approval of Congress.


text:
Foods that can boost your immunity
9 foods to help you eat healthy this winter.
Charmaine Yabsley
It tends to be more to common to catch a cold or flu during the colder months, so it's important to eat a wide range of foods to give your immune system the best chance to fight off any nasty bugs.
While it's tempting to reach for comfort food when it's cold outside and you're spending more time indoors, try and include some of the suggestions below to boost your immunity and help you to stay cold-free this winter.
1. Eat seasonally
Natasha Murray, an Accredited Practising Dietitian and spokesperson for the Dietitians Association of Australia, recommends eating seasonal fruit and vegetables, as they're more likely to be full of the nutrients you need.
"These pack a punch nutrient-wise. If you eat seasonally you're going to give your body a fantastic set-up to get through winter because seasonal fruits such as apples, pears, and citrus, including grapefruit, mandarins and oranges are at their peak of nutrition."
Winter veg includes broccoli, cauliflower, carrots, mushrooms and spinach.
2. Meat
"Lean poultry, such as chicken, turkey breast with skin removed and lean red meat are proteins that have a good amount of iron and zinc in them, which is important for the immune system," she says. "We find that when people have low zinc levels, their immune system suffers."
If you eat red meat, don't go for the supersized ribs. Keep your serving size in mind – ideally a 90-100g raw/65g cooked serving size, which is about the size of the palm of your hand. For white meat, like chicken and turkey, one serving is a little more – around 100g raw/80g cooked.
3. Oily fish
"Oily fish such as salmon, tuna and sardines have omega-3 fatty acids in them, which are essential for a healthy immune system, as well as Vitamins A, D, B6 and B12," says Murray. "And sardines have little bones which contain calcium."
If you're a fish eater, aim to have it 2–3 times a week. "Fresh, frozen or tinned are good," says Murray. "If you buy tinned fish then choose those in spring water as this will reduce your sodium [salt] intake."
"A fantastic snack or an addition to a meal, nuts contain protein and healthy fats and are a really good source of vitamin E, which is needed for a strong immune system," says Murray.
Murray says that it's best to choose raw and unsalted nuts, otherwise you may unknowingly be doubling your sodium intake.
"Roast nuts in the oven and add extra flavour with spices such as cumin, garlic or chilli."
Be careful not to overdo serving sizes, though. "You should aim to eat around 30g of nuts a day. A serve is a small handful, which is about 30 almonds, or 4 chestnuts, or 15 cashews."
5. Rolled oats
Oats contain beta-glucan, a type of soluble fibre, which has immune-boosting properties and can help your body to fight infection.
Add to a morning smoothie, or soak in water or apple juice overnight and heat in the morning. "Add some Greek yoghurt, berries for sweetness and cinnamon. Or make an oat crumble for a winter dessert with stewed fruit," she says.
6. Greek yoghurt
"Full of B vitamins and folate and calcium, there's also a lot of good bacteria in yoghurt," says Murray.
These nutrients, found in good quality yoghurts, may boost immune function, digestion and the absorption of nutrients, and help fight against unfriendly bacteria, according to Harvard Medical School.
Although plain yoghurt is also good for you, Greek yoghurt is likely to have less sugar and more protein.
Add to breakfasts or have as a dessert, with a teaspoon of honey and winter berries added for extra sweetness. "Or use as a replacement for sour cream, which has a lot of saturated fat," says Murray.
"Garlic is a little powerhouse," says Murray. "It contains the antioxidant allicin, which has antibacterial properties and may help immunity and reduce your risk of catching colds."
"Garlic has a beautiful flavour and is quite warming, so you can add it to most [savoury] meals," says Murray. "Add a couple of cloves as a base to sauces, dishes, or roast a whole garlic and squeeze it out on sourdough bread."
To ensure you get the most out of this health-boosting ingredient when cooking, leave it for 15 minutes at room temperature after crushing it before you use it.
"We often forget that what we drink can help boost our immune system," says Murray. "Green tea contains a whole lot of polyphenols – antioxidants – which [can help] protect the body's cells against damage."
Remember that green tea still contains caffeine, although it's much lower than coffee, so aim for no more than 3 cups a day. Decaf green tea is thought to have similar potential health benefits.
9. Chicken soup
It may be good for the soul, but it might also help if you have a cold. A study published in the journal Chest reported that chicken soup may have anti-inflammatory properties that can help to stave off upper-respiratory tract infections.
"[Almost] any type of homemade soup is a fantastic way to get a lot of vegetables into your diet, and keep your hydration levels up, since the fluid adds to your water intake."
Murray recommends adding a rainbow of vegetables into your soup to keep your immune system strong. "We get different nutrients and vitamins from each colour," she says.
"Try Thai pumpkin soup, minestrone with pasta and lentils, beans or pulses, chicken and corn soup, or potato and leek," she says. "Use winter veg you may not eat otherwise, such as turnips, swede and parsnips."
Food myths
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THE FUTURE OF MEDICINE: THE FLU AND ANTIBIOTICS
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DOCTORS PRESCRIBING VEGIES FOR BETTER HEALTH
Most of us still aren't eating enough vegetables. Now some doctors are driving the message home by writing vegie prescriptions.
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The labels can promise everything from heart health to weight loss, but are these claims research-based or just clever marketing?


Nowadays, so many of the products you find in the grocery stores are processed to the point that they can barely be considered food at all. Sometimes you can't even pronounce half of the things in the ingredients list! At Bob's Red Mill, we don't do it that way. When it comes to grains, beans and seeds, we think nature got it right. In their original state, these wholesome foods provide a magnificent combinations of nutrients that are essential to health. We honor that innate perfection by keeping our products simple, and our White Popcorn is an excellent example of that practice. Virtually unprocessed, chock full of fiber, and boasting a naturally sweet flavor, these precious seeds are as healthy as they are delicious.
Pure and simple. Those are the perfect words to describe Bob's Red Mill White Popcorn. Take a look at the ingredients - or rather, the ingredient. That's right. There's only one: whole grain corn. It doesn't get much simpler than that. And we think that's just how it should be.
Heat 1 tbsp vegetable oil in a medium pot over medium-high heat. Add 1/4 cup popcorn kernels and cover with a tight lid. Shake the pop frequently until most of the kernels have popped, or there are 3 seconds between pops. Makes about 4 cups popped popcorn.
Place 1/4 cup popcorn kernels in a small paper bag. Pop on 100% power for 1:30 to 2:30, until most of the kernels have popped, or there are 3 seconds between pops. Makes about 4 cups popped popcorn.


If you want to improve your home's energy-efficiency – and save money – one of the most impactful changes you can make is to replace your windows. This is because older windows are often responsible for a great deal of unwanted heat transfer. In summer, inefficient windows can let excessive heat into your home; in winter, inefficient windows allow heat to escape.
Today's high-efficiency windows are much better at managing heat transfer – reflecting away heat in the summer and trapping heat in the winter. When heat transfer is managed, you're more comfortable – and less heating or air conditioning is required to keep you that way. And when your heating and air system gets a rest, your pocketbook does, too.
Energy-efficient windows, such as these from Richmond's replacement window experts, feature advanced low-e (low-emissivity) glass coatings. These coatings are designed to emit low levels of heat, reflecting the sun's infrared heat rays away rather than allow them to pass through. These low-e coatings are quite effective; while standard glass absorbs more than 80% of infrared rays, low-e glass absorbs a third. Despite their effect on infrared rays, low-e coatings won't compromise your window's view or the natural light that you love.
Modern windows also employ two or sometimes three panes. Each extra pane provides an extra buffer, adding a layer of insulation between your home's interior and exterior. Gases such as argon or krypton are often injected between the panes; these gases are clear, non-toxic, and odorless. More importantly, argon and krypton conduct heat only half as well as regular air, providing a stronger barrier to heat transfer.
Even the spacers between the panes of glass play an important role in energy efficiency. Not so long ago, spacers were made of aluminum – a metal that conducts heat very easily. Instead of metal, the newest spacers are made of foam. Because foam can expand and contract without losing integrity, it offers longer-lasting seals. And because foam blocks heat transfer, it offers excellent insulation.
When glass coatings, extra panes, gas fills, and spacers work together, windows can be very efficient. This efficiency is most commonly measured along two metrics: U-factor and solar heat gain coefficient (or SHGC). U-factor refers to the heat transferring quality of the glass; lower U-factors mean less heat transfer. SHGC reflects the amount of sunlight that penetrates into the house through the panes; lower SHGC values mean less light.
Windows with the right U-factor and SHGC values for the local climate can be certified by ENERGY STAR as being energy-efficient. In the Richmond area, we live in what's called the North-Central Climate Zone – characterized by cold winters and hot summers. Here windows require a U-factor of 0.30 or less and an SHGC value of 0.40 or less qualify for ENERGY STAR certification.
So what can you expect if you replace old windows with new ones that have ENERGY STAR certification? In our South Atlantic region, we can expect to save about $450 per year when upgrading from single-pane glass and $59 per year when replacing double-pane glass. To learn more about your projected savings, visit ENERGY STAR.
Replacing your windows is an investment that provides excellent returns in comfort and energy savings. If you're still living with single-pane or older double-pane windows, now would be a great time to consider an upgrade.
If your allergies are just as bad in the summer as they are in the spring or fall, you're not alone. It simply means that you're more sensitive to summertime allergy causes than most. Thankfully, there's a lot you can do to alleviate summer allergy symptoms.
Let's look at the causes, starting with grass. When grass is growing rapidly and displaying seed pods, then it's pollinating. Grass tends to release pollen from mid-spring through early fall. The same can be said of weeds, although they tend to release pollen in earnest a few weeks after grass does. While they may lack the sheer volume of trees, grasses and weeds release pollen that can travel in the air for miles.
Despite the abundance of pollen – and potentially dust mites – you don't have to give into the idea of suffering from allergies. There are several practices you can follow to mitigate the effects of pollen and dust mites, both in and around your home. You can also mitigate their effects on you.
Start by keeping pollen out of your home. Keep your windows closed. If you've been outdoors for a few hours, change and wash your clothes after you re-enter your home. If you have pets, wash them more regularly to reduce the amount of pollen they bring into your home.
No matter what you do, some pollen will enter your home – but you can manage it. Start by using high-quality air-filters in the intakes for your heating and air system; be sure to change them regularly. Dust and vacuum regularly. Schedule a professional carpet and upholstery cleaning; Chem-Dry, for example, has cleaning processes that have been proven to be particularly effective to removing allergens from homes.
You can mitigate the effects of dust mites by keeping your household humidity low; dust mites can't survive when humidity levels are kept under 50%. Wash your bed linens regularly. And, again, use high-quality air filters to pull dust out of the air.
You can also manage the pollen in your yard. Keep your lawn short, cutting grasses and weeds before they have a chance to produce seed pods. Keep weeds or other pollinators from proliferating in your yard by spraying them or pulling them out before they can bloom. If yard work makes your allergies worse, wear a mask designed to pull pollen from the air while you work.
Don't forget to manage pollen's effects on you when you're not in your yard. Many over-the-counter anti-histamines and nasal sprays are effective when taken regularly. If these aren't enough, an allergist may be able to prescribe additional remedies that can provide relief. Check the daily pollen count in your area. On a dry, windy summer's day when the count is liable to be high, plan to minimize your time outdoors – and, again, keep your windows closed.
You can't control pollen or dust mites, but smart practices can minimize their effects on you. If you're suffering from itchy eyes, a runny nose, or incessant sneezing, do something about it.
No matter how often you clean your home, your furnishings, or your carpet, they can each play host to a range of impurities. Some of these impurities are airborne. Others are trapped in your upholstery, carpet fibers, or other surfaces in your house. Some impurities, such as dust or pollen, can become airborne when there is some disturbance.
These impurities aren't just unsightly – they can also cause allergies and induce asthma attacks. Furthermore, they can worsen conditions such as sinusitis. Since you don't want to develop any of these respiratory conditions, it's important you clean your home thoroughly on a regular basis. But simply cleaning more often isn't the answer; you also need the right equipment.
The best cleaning technicians have the ability to deep-clean carpets, rugs, and upholstery. They're able to apply cleaning solutions more deeply and remove them more thoroughly than is possible with typical household equipment. When combining these processes with the use of sanitizers, expert cleaners like my preferred carpet cleaner in Tacoma, are able to remove the vast majority of impurities from both surface and air.
But we can't rely on professionals to clean our carpets and upholstery all of the time. Effectively managing impurities requires us to deal with them in between visits from our local carpet cleaner. Vacuuming is a great start, but traditional vacuums aren't quite up to the task.
Along with air, vacuum cleaners inhale dust and other minute particles from the carpet, rugs, furnishings at other surfaces. This dirt- and allergen-filled air is passed through a filter before being released back into the room. These filters usually have tiny holes that trap most of the dirt while allowing air to pass through. Unfortunately, many types of impurities are also small enough to pass through a typical vacuum air filter.
In order to effectively capture the dust, allergens, and other impurities you vacuum, you need to employ a more efficient air filter. What you'll want is a vacuum that uses a HEPA filter.
HEPA stands for "high-efficiency particulate air." HEPA filters are mechanical, employing a fine mesh that can trap tiny airborne particles. These filters are so efficient that they can trap tobacco smoke, pet dander, pollen, and dust mites along with other tiny air contaminants. The most efficient HEPA filters are even capable of dealing with military-level biological, chemical, and radiological particles.
According to the US National Institute for Occupational Safety and Health, a true HEPA air filter should be able to trap 99.97% of particles that are 0.3 microns thick, or thicker. To put this into perspective, the human hair has a thickness of 50 to 150 microns; true HEPA filters can trap particles up to 450 times smaller than the thickness of your hair. This level of efficiency ensures that HEPA filters can trap all the common allergens found at home.
Not even the best HEPA filter can capture every allergen at once. You'll still want to to vacuum on a regular basis. But you can do so knowing that you're doing something that is helping to keep you and your family safe.
A standby generator automatically provides power in the event of an outage. These types of generators are permanently installed on your residence, usually in concrete, and are connected to the building's electrical circuits. During a blackout, an automatic transfer switch will turn the generator on within seconds, bringing power back to your home. When the blackout is over, the transfer switch will shut the generator down again.
Unlike standby generators, backup generators are not permanently installed and don't work automatically. These portable generators must be started manually by the user after a blackout has occurred. Many households elect to plug devices directly into the generator; others prefer to connect the generator to the house's electrical panel in order to power outlets as well as household systems.
Generators can be easily connected to an electrical panel by installing an interlock kit that functions like a manual transfer switch. While the interlock kit will not automatically engage or shut down power, it will isolate the household and utility systems to protect against unsafe overloads.
The type of generator that will work best for you depends entirely on the needs of your family and your home or business. A portable generator may work best for small homes with few electrical needs or in an area where blackouts are not common. However, for those considering the more reliable (and larger and more expensive) standby generators, there are still a few more things to take into consideration.
Different sizes of generators exist to suit different needs, and standby backup generators can be quite large. A large generator creates more power, making them ideal for homes with a lot of electrical needs. If you need to run your air conditioner during a blackout, for instance, you'll need to purchase a stronger generator. You'll need to know exactly how much power you require in order to select the right unit for you. Portable generators typically max out at 10,000-watt units, while standby generators can be as powerful as 30,000 watts.
The vast difference in power also results in a vast difference in price. Portable generators are a far more economical choice with units ranging from $500 to $1500. On the other hand, standby generators carry a heftier price tag. The permanent generators start at a price of $5000 but can get as pricey as $15,000. In addition to that price tag, the units will need to be installed.
Picking the right generator for your family really comes down your specific needs. Some families may not be able to handle the cost of a permanent standby generator, while others wouldn't be able to survive without it. When it comes to picking a backup generator, carefully weigh the pros against the cons.


Heero drew in a deep breath, the raspy sound of his breath the only sound in the still room, lacking the sleepy mumurs that Duo used to voice, the rustle of body and limbs moving gently within the fabrics of the bed. Or perhaps the phantom sounds of himself ... finding himself drawing closer to the boy lying at his side or his own sleepy mumurs in reply to the other's.
Duo Maxwell never lied. And Heero knew... no, felt... believed that the other boy spoke of the future as if it were already determined, Heero succumbing to Relena's offers and presence. What an irony then that Shinigami's own predictions had caused a stir within himself. A stir, an uneasy ripple, uneasy in that he was so unused to these sensations, for something more than which was offered. More than either boy was willing to offer, in these times, in these circumstances. But Heero wanted more, even if the mind had yet to register, to catalogue, these feelings as such. His body felt them in the way they were constantly drawn to touch, to watch beautiful Death. And his heart felt them in that last moment, as Duo said his goodbyes, neither boy, the one awake or the one feigning sleep, would know.
Heero let out another breath slowly, unwilling to pull into the seductive realms of Dream, where would no longer have control, where he would see Duo lingering in the folds and crevices of his mind. And it hurt, not in the sense of a broken bone, something which was, yes, painful but could be set right again, and the pain would fade into a wisp of a memory. This pain was deep, lingering and made his teeth stand on end, as every nerve wept, as his hands reached, reached for something that was just out of his grasp, something which could not be grapsed or claimed, something... which he could not even envision or voice. But something which was represented by his lover, by his love.
And Heero slept, his dreams haunted, or blessed, with dreams of his love. In this dream, here was the future he wished for, that perhaps Duo had wished for. A future together, buildling something together, an 'ever after'.