In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. If we denote by SPACE, the set of all problems that can be solved by Turing machines using O space for some function t of the input size n we can define PSPACE formally as P S P A C E = ⋃ k ∈ N S P A C E. PSPACE is a strict superset of the set of context-sensitive languages, it turns out. Because of Savitch's theorem, NPSPACE is equivalent to PSPACE because a deterministic Turing machine can simulate a non-deterministic Turing machine without needing much more space; the complements of all problems in PSPACE are in PSPACE, meaning that co-PSPACE = PSPACE. The following relations are known between PSPACE and the complexity classes NL, P, NP, PH, EXPTIME and EXPSPACE: N L ⊆ P ⊆ N P ⊆ P H ⊆ P S P A C E P S P A C E ⊆ E X P T I M E ⊆ E X P S P A C E N L ⊊ P S P A C E ⊊ E X P S P A C E P ⊊ E X P T I M E It is known that in the first and second line, at least one of the set containments must be strict, but it is not known which.

It is suspected that all are strict. The containments in the third line are both known to be strict; the first follows from direct diagonalization and the fact that PSPACE = NPSPACE via Savitch's theorem. The second follows from the space hierarchy theorem; the hardest problems in PSPACE are the PSPACE-Complete problems. See PSPACE-Complete for examples of problems that are suspected to be in PSPACE but not in NP; the class PSPACE is closed under operations union and Kleene star. An alternative characterization of PSPACE is the set of problems decidable by an alternating Turing machine in polynomial time, sometimes called APTIME or just AP. A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in second-order logic with the addition of a transitive closure operator. A full transitive closure is not needed, it is the addition of this operator that distinguishes PSPACE from PH. A major result of complexity theory is that PSPACE can be characterized as all the languages recognizable by a particular interactive proof system, the one defining the class IP.

In this system, there is an all-powerful prover trying to convince a randomized polynomial-time verifier that a string is in the language. It should be able to convince the verifier with high probability if the string is in the language, but should not be able to convince it except with low probability if the string is not in the language. PSPACE can be characterized as the quantum complexity class QIP. PSPACE is equal to PCTC, problems solvable by classical computers using closed timelike curves, as well as to BQPCTC, problems solvable by quantum computers using closed timelike curves. A language B is PSPACE-complete if it is in PSPACE and it is PSPACE-hard, which means for all A ∈ PSPACE, A ≤ p B, where A ≤ p B means that there is a polynomial-time many-one reduction from A to B. PSPACE-complete problems are of great importance to studying PSPACE problems because they represent the most difficult problems in PSPACE. Finding a simple solution to a PSPACE-complete problem would mean we have a simple solution to all other problems in PSPACE because all PSPACE problems could be reduced to a PSPACE-complete problem.

An example of a PSPACE-complete problem is the quantified Boolean formula problem. Arora, Sanjeev.

While earlier Jewish immigrants from Germany tended to be politically conservative, the wave of Eastern European Jews, starting in the early 1880s, were more liberal or left-wing, became the political majority. Many of the latter came to America with experience in the socialist and communist movements as well as the Labor Bund, emanating from Eastern Europe. Many Jews rose to leadership positions in the early 20th century American labor movement, helped to found unions that played a major role in left-wing politics and, after 1936, in Democratic Party politics. For most of the 20th century since 1936, the vast majority of Jews in the United States have been aligned with the Democratic Party. Towards the end of the 20th century, at the beginning of the 21st century, Republicans have launched initiatives to persuade American Jews to support Republican policies. Over the past century, Jews in Europe and the Americas have traditionally tended towards the political left, played key roles in the birth of the labor movement as well as socialism.

While Diaspora Jews have been represented in the conservative side of the political spectrum politically conservative Jews have tended to support pluralism more than many other elements of the political right. The divide between right and left correlates to the various religious movements among American Jews; the more conservative movements in American Judaism tend to be politically conservative, while the majority movements tend to be more politically liberal or left-leaning. There are a number of Jewish secular organizations at the local and international levels; these organizations play an important part in the Jewish community. Most of the largest groups, such as Hadassah and the United Jewish Communities, have an elected leadership. No one secular group represents the entire Jewish community, there is significant internal debate among Jews about the stances these organizations take on affairs dealing with the Jewish community as a whole, such as anti-Semitism and Israeli policies. In the United States and Canada today, the secular United Jewish Communities known as the United Jewish Appeal, represents over 150 Jewish Federations and 400 independent communities across North America.

Every major American city has its local "Jewish Federation", many have sophisticated community centers and provide services health care-related. They raise record sums of money for philanthropic and humanitarian causes in North America and Israel. Other organizations such as the Anti-Defamation League, American Jewish Congress, American Jewish Committee, American Israel Public Affairs Committee, Zionist Organization of America, Americans for a safe Israel, B'nai B'rith, Agudath Israel represent different segments of the American Jewish community on a variety of issues. With the influx of Jews from Central and Eastern Europe many members of the Jewish community were attracted to labor and socialist movements and numerous Jewish newspapers such as Forwerts and Morgen Freiheit had a socialist or communist orientation. Left wing organizations such as the Arbeter Ring and the Jewish People's Fraternal Order played an important part in Jewish community life until World War II. Liberal Jewish Americans were not just involved in nearly every important social movement, but in the forefront of promoting such issues as workers rights, civil rights, woman's rights, gay rights, freedom of religion, freedom from religion, peace movements, various other progressive causes.

Although American Jews leaned Republican in the second half of the 19th century, the majority has voted Democratic or leftist since at least 1916, when they voted 55% for Woodrow Wilson. In 1940 and 1944, 90% of American Jews voted for Franklin D. Roosevelt, 75% voted for Harry S. Truman in 1948, despite both party platforms supporting the creation of a Jewish state in the latter two elections. During the 1952 and 1956 elections, they voted 60% or more for Adlai Stevenson, while General Eisenhower garnered 40% for his reelection. In 1960, 83% voted for Democrat John F. Kennedy, the first Catholic, versus Richard Nixon, in 1964, 90% of American Jews voted for Lyndon Johnson. Hubert Humphrey garnered 81% of the Jewish vote in the 1968 elections, in his losing bid for president against Richard Nixon. During the Nixon re-election campaign of 1972, Jewish voters were apprehensive about George McGovern, only favored the Democrat by 65%, while Nixon more than doubled Jewish support for Republicans to 35%.

In the election of 1976, Jewish voters supported Democrat Jimmy Carter by 71% over incumbent president Gerald Ford's 27%, but during the Carter re-election campaign of 1980, Jewish voters abandoned the Democrat, with only 45% support, while Republican winner, Ronald Reagan, garnered 39%, 14% went to independent John Anderson. During the Reagan re-election campaign of 1984, the Republican retained 31% of the Jewish vote, while 67% voted for Democrat Walter Mondale; the 1988 election saw Jewish voters favor Democrat Michael Dukakis by 64%, while George Bush Sr. polled a respectable 35%, but during his re-election in 1992, Jewish support dropped to just 11%, with 80%, voting for Bill Clinton and 9% going to independent Ross Perot. Clinton's re-election campaign in 1996 maintained high Jewish support at 78%, with 16% supporting Robert Dole and 3% for Perot; the elections of 2000 and 2004 saw continued Jewish support for Democrats Al

Craig Electronics is an American brand that specializes in consumer electronics sold at pharmacies, big-box stores and through online retailers. Craig Electronics was founded in 1963 as a manufacturer of car stereos; the company had many celebrity sponsorships for these stereo systems, which included The Beach Boys, Ray Charles, Ringo Starr, Emerson, Lake & Palmer. In 2001, the company was re-established as a manufacturer of low-cost consumer electronics. Craig has a brand licensing deal to sell certain consumer goods under the Magnavox brand. In 2013, Craig Electronics, along with Curtis International and ViewSonic were sued for patent infringement by MPEG LA. Official website