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Talk:Bedlam

Moved
I moved the article from Bedlam to Bethlem Royal Hospital and put a redirect to it at Bedlam

as per the quote from http://en2.wikipedia.org/wiki/Wikipedia:Redirect "In accordance with wikipedia:naming conventions (precision) it's best to have an article at a well-defined, unambiguous term, with redirects from looser colloquial terms, rather than vice versa."

Steeev 08:00am 20 Nov 2003 UTC

 * Agreed that the article on "Bethlem Royal Hospital" should be at "Bethlem Royal Hospital".

 * "Bedlam" redirect turned into disambiguation page for the numerous other meanings of "Bedlam".

 * —Lowellian (reply) 05:31, 1 January 2010 (UTC)

 * Hi Lowellian, I don't agree with this move. "Bedlam" is known as the word for the old hospital, and/or for the chaos associated with it. It should be redirected to the hospital, or ideally a separate article should be written for the history of the hospital, but it shouldn't be the title of the disambig page. SlimVirgin  TALK  contribs 03:47, 3 January 2010 (UTC)
 * Agree with Slim. The hospital is the original and primary use.--Cúchullain t/ c 01:08, 4 January 2010 (UTC)
 * I'm going to restore the original redirect. SlimVirgin  TALK  contribs 22:36, 11 January 2010 (UTC)

Page moves
Keith, if you want pages to be moved, please be explicit, because this is already a mess. If you look at the history of Bedlam, it has been moved around enough already. Simple redirects seem to make most sense now. SlimVirgin TALK  contribs 23:41, 11 January 2010 (UTC)

 * I reverted the move as it was done by a cut and paste method which destroys the attribution. I would have expected any move back to this title would have been done by a move. Keith D (talk) 01:32, 12 January 2010 (UTC)

 * If you look at the history of Bedlam, it has been a redirect, an article, and a disambiguation page, so I would say the history's already messed up. I think it's best to stick to a simple redirect. I can't think where it would be moved to now that would make the history make sense. SlimVirgin  TALK  contribs 01:35, 12 January 2010 (UTC)

 * The page history was not a mess until your cut-and-paste move, which is against Wikipedia policy. Even if you had consensus that "Bedlam" should be a redirect rather than a disambiguation page (and you did not, as both User:Steev and I have expressed opposition), you should have moved this disambiguation page ("Bedlam") over to the disambiguation position ("Bedlam (disambiguation)") to preserve the page history and then created a redirect at "Bedlam", instead of doing a cut-and-paste move that separated the page history and as User:Keith D stated, destroyed the attribution. I've reverted the cut-and-paste move. —Lowellian (reply) 18:13, 14 May 2010 (UTC)

Move discussion in progress
There is a move discussion in progress on Talk:Bethlem Royal Hospital which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 17:00, 13 November 2018 (UTC)

title: Talk:Bedlam


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User talk:Ockrn

Hello Ockrn, welcome to Wikipedia. Here are some useful links in case you haven't already found them;

 * How to edit a page
 * How to write a great article
 * Naming conventions
 * Manual of Style

If you have any questions, see the help pages, add a question to the village pump or ask me on my talk page. I hope you enjoy editing here and being a Wikipedian!

Angela 07:54, 20 Nov 2003 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Ockrn


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Broker-dealer

In financial services, a broker-dealer is a natural person, company or other organization that engages in the business of trading securities for its own account or on behalf of its customers. Broker-dealers are at the heart of the securities and derivatives trading process.

Although many broker-dealers are "independent" firms solely involved in broker-dealer services, many others are business units or subsidiaries of commercial banks, investment banks or investment companies.

When executing trade orders on behalf of a customer, the institution is said to be acting as a broker. When executing trades for its own account, the institution is said to be acting as a dealer. Securities bought from clients or other firms in the capacity of dealer may be sold to clients or other firms acting again in the capacity of dealer, or they may become a part of the firm's holdings.

In addition to execution of securities transactions, broker-dealers are also the main sellers and distributors of mutual fund shares.

Main points of activity

 * Professional participant in securities market who carries out dealer activity shall be called dealer.
 * Announcing the price, the dealer is committed to announce other essential conditions of the buy-sell contract of securities: minimum and maximum number of securities subject to purchase and/or sale, as well as the term of announced price's validity.

Functions

 * All the functions of stockbroker including financial consulting
 * Organization and support of turnover (liquidity), or market-making (price announcing, duty of sell and buy of security at announced price, announcing of min and max number of securities that can be bought/sold at announced price, implementing time periods when announced prices are available)
 * Dealers are large financial institutions that sell securities to end users and then hedge their risk by partaking in the interdealer market. Interdealers facilitate price discovery and execution between dealers.

United States
In the United States, broker-dealers are regulated under the Securities Exchange Act of 1934 by the Securities and Exchange Commission (SEC), a unit of the U.S. government. All brokers and dealers that are registered with the SEC (pursuant to ), with a number of exceptions, are required to be members of the Securities Investor Protection Corporation (SIPC) (pursuant to ) and are subject to its regulations. Some regulatory authority is further delegated to the Financial Industry Regulatory Authority (FINRA), a self-regulatory organization. Many states also regulate broker-dealers under separate state securities laws (called "blue sky laws").

The 1934 Act defines "broker" as "any person engaged in the business of effecting transactions in securities for the account of others", and defines "dealer" as "any person engaged in the business of buying and selling securities for his own account, through a broker or otherwise". Under either definition, the person must be performing these functions as a business; if conducting similar transactions on a private basis, they are considered a trader and subject to different requirements. When acting on behalf of customers, broker-dealers have a duty to obtain "best execution" of transactions, which generally means achieving the best economic price under the circumstances.

On April 28, 2004, the SEC voted unanimously to change the net capital rule which applies to broker-dealers, thus allowing those with "tentative net capital" of more than $5 billion to increase their leverage ratios. The rule change remains in effect, though subject to modifications.

Although broker-dealers often provide investment advice to their clients, in many situations they are exempt from registration under the U.S. Investment Advisers Act of 1940, so long as (i) the investment advice is "solely incidental" to brokerage activities; and (ii) the broker-dealer receives no "special compensation" for providing the investment advice. Both elements of this exemption must be met to rely on it.

Many broker-dealers also serve primarily as distributors for mutual fund shares. These broker-dealers may be compensated in numerous ways and, like all broker-dealers, are subject to compliance with requirements of the Securities and Exchange Commission and one or more self-regulatory organizations, such as the Financial Industry Regulatory Authority (FINRA). The forms of compensation may be sales loads from investors, or Rule 12b-1 fees or servicing fees paid by the mutual funds. There are several online portals that offer broker dealer assistance and search capabilities.

United Kingdom
UK securities law uses the term intermediary to refer to businesses involved in the purchase and sale of securities for the account of others.

The Financial Conduct Authority authorises and regulates companies engaging in such activity as "regulated activities" under the Financial Services and Markets Act 2000.

Japan
The common Japanese term for a broker-dealer is "securities company" (証券会社). Securities companies are regulated by the Financial Services Agency under the Financial Instruments and Exchange Law. The "big five" are Nomura Securities, Daiwa Securities, SMBC Nikko Securities, Mizuho Securities, and Mitsubishi UFJ Securities. Most major commercial banks in Japan also maintain broker-dealer subsidiaries, as do many foreign commercial banks and investment banks.

Securities companies must be organized as kabushiki kaisha with a statutory auditor or auditing committee, and must maintain minimum shareholder equity of ¥50 million.

Largest dealer banks

 * Bank of America Merrill Lynch
 * Barclays
 * BNP Paribas
 * Citigroup
 * Credit Suisse
 * Deutsche Bank
 * Goldman Sachs
 * HSBC
 * JPMorgan Chase
 * Morgan Stanley
 * Nomura
 * Commerzbank AG
 * Societe Generale
 * The Royal Bank of Scotland
 * UBS
 * Wells Fargo
 * Bank of New York Mellon

title: Broker-dealer


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Brahmarshi

In Hinduism, a Brahmarshi (Sanskrit ', a tatpurusha compound of ' and ) is a member of the highest class of Rishis ("seers" or "sages"). A Brahmarshi is a sage who has attained enlightenment (Kaivalya or Moksha) and became a Jivanmukta by completely understanding the meaning of Brahman and has attained the highest divine knowledge, infinite knowledge (omniscience) and self knowledge called Brahmajnana. When a Brahmarshi dies he attains Paramukti and frees himself from Samsara, the cycle of birth and death.

Order
The superlative title of Brahmarshi is not attested in the Vedas themselves and first appears in the Sanskrit epics.

According to this classification, a Brahmarshi is the ultimate expert of religion and spiritual knowledge known as 'Brahmajnana'. Below him are the Maharishis (Great Rishis).

The Saptarishis created out of Brahma's thoughts are perfect brahmarshis. They are often cited to be at par with the Devas in power and piety in the Puranas.

Bhrigu, Angiras, Atri, Vishwamitra, Kashyapa, Vasishta, and Shandilya are the seven brahmarshis. But there is another list of Saptarishi also who are also Gotra-pravartakas, i.e.,founders of Brahamanical clans, and this second list appeared somewhat later, but belongs to ancient period.

All the hymns of third mandala of the Rig Veda, including the Gayatri mantra, are ascribed to Vishwamitra, who is mentioned as the son of Gaadhi. Vishwamitra is also referred to as Kaushika, because he attained Brahmajnana on the banks of the river Koshi.

Parasurama and Jamadagni have also been credited the title of Brahmarishi by Bhishma as in Mahabharata.

The Period of the Manusmriti
Brahmarshi-desha, 'the country of the holy sages,' includes the territories of the Kurus, Matsyas, Panchalas and Surasenas (i.e. the eastern half of the State of Patiala and of the Delhi division of the Punjab, the Alwar State and adjacent territory in Rajputana, the region which lies between the Ganges and the Jumna, and the Mathura District in the United Provinces).

title: Brahmarshi


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Polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers.

Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings.

A closely related notion is that of the ring of polynomial functions on a vector space, and, more generally, ring of regular functions on an algebraic variety.

Definition (univariate case)
Let $K$ be a field or (more generally) a commutative ring.

The polynomial ring in $X$ over $K$, which is denoted $K[X]$, can be defined in several equivalent ways. One of them is to define $K[X]$ as the set of expressions, called polynomials in $X$, of the form
 * $$p = p_0 + p_1 X + p_2 X^2 + \cdots + p_{m - 1} X^{m - 1} + p_m X^m,$$

where $p_{0}, p_{1}, …, p_{m}$, the coefficients of $p$, are elements of $K$, $pm ≠ 0$ if $m > 0$, and $X, X, …,$ are symbols, which are considered as "powers" of $X$, and follow the usual rules of exponentiation: $X = 1$, $X = X$, and $$ X^k\, X^l = X^{k+l}$$ for any nonnegative integers $k$ and $l$. The symbol $X$ is called an indeterminate or variable. (The term of "variable" comes from the terminology of polynomial functions. However, here, $X$ has no value (other than itself), and cannot vary, being a constant in the polynomial ring.)

Two polynomials are equal when the corresponding coefficients of each $X$ are equal.

One can think of the ring $K[X]$ as arising from $K$ by adding one new element $X$ that is external to $K$, commutes with all elements of $K$, and has no other specific properties. This can be used for an equivalent definition of polynomial rings.

The polynomial ring in $X$ over $K$ is equipped with an addition, a multiplication and a scalar multiplication that make it a commutative algebra. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if
 * $$p = p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m,$$

and
 * $$q = q_0 + q_1 X + q_2 X^2 + \cdots + q_n X^n,$$

then
 * $$p + q = r_0 + r_1 X + r_2 X^2 + \cdots + r_k X^k,$$

and
 * $$pq = s_0 + s_1 X + s_2 X^2 + \cdots + s_l X^l,$$

where $k = max(m, n), l = m + n$,
 * $$r_i = p_i + q_i$$

and
 * $$s_i = p_0 q_i + p_1 q_{i-1} + \cdots + p_i q_0.$$

In these formulas, the polynomials $p$ and $q$ are extended by adding "dummy terms" with zero coefficients, so that all $p_{i}$ and $q_{i}$ that appear in the formulas are defined. Specifically, if $m < n$, then $p_{i} = 0$ for $m < i ≤ n$.

The scalar multiplication is the special case of the multiplication where $p = p_{0}$ is reduced to its constant term (the term that is independent of $X$); that is
 * $$p_0\left(q_0 + q_1 X + \dots + q_n X^n\right) = p_0 q_0 + \left(p_0 q_1\right)X + \cdots + \left(p_0 q_n\right)X^n$$

It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over $K$. Therefore, polynomial rings are also called polynomial algebras.

Another equivalent definition is often preferred, although less intuitive, because it is easier to make it completely rigorous, which consists in defining a polynomial as an infinite sequence $(p_{0}, p_{1}, p_{2}, …)$ of elements of $K$, having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some $m$ so that pn = 0 for $n > m$. In this case, $p0$ and $X$ are considered as alternate notations for the sequences $(p0, 0, 0, …)$ and $(0, 1, 0, 0, …)$, respectively. A straightforward use of the operation rules shows that the expression
 * $$p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m$$

is then an alternate notation for the sequence

Terminology
Let
 * $$p = p_0 + p_1 X + p_2 X^2 + \cdots + p_{m - 1} X^{m - 1} + p_m X^m,$$

be a nonzero polynomial with $$p_m\ne 0$$

The constant term of $(p_{0}, p_{1}, p_{2}, …, p_{m}, 0, 0, …)$ is $$p_0.$$ It is zero in the case of the zero polynomial.

The degree of $p$, written $p$ is $$m,$$ the largest $deg(p)$ such that the coefficient of $k$ is not zero.

The leading coefficient of $Xk$ is $$p_m.$$

In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined, defined to be $p$, or defined to be a $−1$.

A constant polynomial is either the zero polynomial, or a polynomial of degree zero.

A nonzero polynomial is monic if its leading coefficient is $$1.$$

Given two polynomials $p$ and $q$, if the degree of the zero polynomial is defined to be $$-\infty,$$ one has
 * $$\deg(p+q) \le \max (\deg(p), \deg (q)),$$

and, over a field, or more generally an integral domain,
 * $$\deg(pq) = \deg(p) + \deg(q).$$

It follows immediately that, if $−∞$ is an integral domain, then so is $K$.

It follows also that, if $K[X]$ is an integral domain, a polynomial is a unit (that is, it has a multiplicative inverse) if and only if it is constant and is a unit in $K$.

Two polynomials are associated if either one is the product of the other by a unit.

Over a field, every nonzero polynomial is associated to a unique monic polynomial.

Given two polynomials, $p$ and $q$, one says that $p$ divides $q$, $p$ is a divisor of $q$, or $q$ is a multiple of $p$, if there is a polynomial $r$ such that $K$.

A polynomial is irreducible if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree.

Polynomial evaluation
Let $K$ be a field or, more generally, a commutative ring, and $R$ a ring containing $K$. For any polynomial $P$ in $q = pr$ and any element $a$ in $R$, the substitution of $X$ with $a$ in $P$ defines an element of $K[X]$, which is denoted $R$. This element is obtained by carrying on in $R$ after the substitution the operations indicated by the expression of the polynomial. This computation is called the evaluation of $P(a)$ at $P$. For example, if we have
 * $$P = X^2 - 1,$$

we have
 * $$\begin{align}

P(3) &= 3^2-1 = 8, \\ P(X^2+1) &= \left(X^2 + 1\right)^2 - 1 = X^4 + 2X^2 \end{align}$$

(in the first example $a$, and in the second one $R = K$). Substituting $R = K[X]$ for itself results in
 * $$P = P(X),$$

explaining why the sentences "Let $P$ be a polynomial" and "Let $X$ be a polynomial" are equivalent.

The polynomial function defined by a polynomial $P$ is the function from $K$ into $K$ that is defined by $$x\mapsto P(x).$$ If $K$ is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if $K$ is a field with $q$ elements, then the polynomials $P(X)$ and $0$ both define the zero function.

For every $X^{q} − X$ in $a$, the evaluation at $a$, that is, the map $$P \mapsto P(a)$$ defines an algebra homomorphism from $R$ to $K[X]$, which is the unique homomorphism from $R$ to $K[X]$ that fixes $K$, and maps $X$ to $a$. In other words, $R$ has the following universal property:
 * For every ring $R$ containing $K$, and every element $a$ of $R$, there is a unique algebra homomorphism from $K[X]$ to $R$ that fixes $K$, and maps $X$ to $a$.

As for all universal properties, this defines the pair $K[X]$ up to a unique isomorphism, and can therefore be taken as a definition of $(K[X], X)$.

The image of the map $$P \mapsto P(a)$$, that is, the subset of $R$ obtained by substituting $a$ for $X$ in elements of $K[X]$, is denoted $K[X]$. For example, $$\Z[\sqrt{2}]=\{P(\sqrt{2})\mid P(X)\in\Z[X]\}$$, and the simplification rules for the powers of a square root imply $$\Z[\sqrt{2}]= \{a+b\sqrt 2 \mid a\in \Z, b\in \Z\}.$$

Univariate polynomials over a field
If $K$ is a field, the polynomial ring $K[a]$ has many properties that are similar to those of the ring of integers $$\Z.$$ Most of these similarities result from the similarity between the long division of integers and the long division of polynomials.

Most of the properties of $K[X]$ that are listed in this section do not remain true if $K$ is not a field, or if one considers polynomials in several indeterminates.

Like for integers, the Euclidean division of polynomials has a property of uniqueness. That is, given two polynomials $a$ and $K[X]$ in $b ≠ 0$, there is a unique pair $K[X]$ of polynomials such that $(q, r)$, and either $a = bq + r$ or $r = 0$. This makes $deg(r) < deg(b)$ a Euclidean domain. However, most other Euclidean domains (except integers) do not have any property of uniqueness for the division nor an easy algorithm (such as long division) for computing the Euclidean division.

The Euclidean division is the basis of the Euclidean algorithm for polynomials that computes a polynomial greatest common divisor of two polynomials. Here, "greatest" means "having a maximal degree" or, equivalently, being maximal for the preorder defined by the degree. Given a greatest common divisor of two polynomials, the other greatest common divisors are obtained by multiplication by a nonzero constant (that is, all greatest common divisors of $a$ and $b$ are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to $1$).

The extended Euclidean algorithm allows computing (and proving) Bézout's identity. In the case of $K[X]$, it may be stated as follows. Given two polynomials $p$ and $q$ of respective degrees $m$ and $n$, if their monic greatest common divisor $g$ has the degree $d$, then there is a unique pair $K[X]$ of polynomials such that
 * $$ap + bq = g,$$

and
 * $$\deg (a) \le n-d, \quad \deg(b) < m-d.$$

(For making this true in the limiting case where $(a, b)$ or $m = d$, one has to define as negative the degree of the zero polynomial. Moreover, the equality $$\deg (a)= n-d$$ can occur only if $p$ and $n = d$ are associated.) The uniqueness property is rather specific to $q$. In the case of the integers the same property is true, if degrees are replaced by absolute values, but, for having uniqueness, one must require $K[X]$.

Euclid's lemma applies to $a > 0$. That is, if $a$ divides $bc$, and is coprime with $b$, then $a$ divides $c$. Here, coprime means that the monic greatest common divisor is $1$. Proof: By hypothesis and Bézout's identity, there are $e$, $p$, and $q$ such that $K[X]$ and $ae = bc$. So $$c=c(ap+bq)=cap+aeq=a(cp+eq).$$

The unique factorization property results from Euclid's lemma. In the case of integers, this is the fundamental theorem of arithmetic. In the case of $1 = ap + bq$, it may be stated as: every non-constant polynomial can be expressed in a unique way as the product of a constant, and one or several irreducible monic polynomials; this decomposition is unique up to the order of the factors. In other terms $K[X]$ is a unique factorization domain. If $K$ is the field of complex numbers, the fundamental theorem of algebra asserts that a univariate polynomial is irreducible if and only if its degree is one. In this case the unique factorization property can be restated as: every non-constant univariate polynomial over the complex numbers can be expressed in a unique way as the product of a constant, and one or several polynomials of the form $K[X]$; this decomposition is unique up to the order of the factors. For each factor, $r$ is a root of the polynomial, and the number of occurrences of a factor is the multiplicity of the corresponding root.

Derivation
The (formal) derivative of the polynomial
 * $$a_0+a_1X+a_2X^2+\cdots+a_nX^n$$

is the polynomial
 * $$a_1+2a_2X+\cdots+na_nX^{n-1}.$$

In the case of polynomials with real or complex coefficients, this is the standard derivative. The above formula defines the derivative of a polynomial even if the coefficients belong to a ring on which no notion of limit is defined. The derivative makes the polynomial ring a differential algebra.

The existence of the derivative is one of the main properties of a polynomial ring that is not shared with integers, and makes some computations easier on a polynomial ring than on integers.

Factorization
Except for factorization, all previous properties of $X − r$ are effective, since their proofs, as sketched above, are associated with algorithms for testing the property and computing the polynomials whose existence are asserted. Moreover these algorithms are efficient, as their computational complexity is a quadratic function of the input size.

The situation is completely different for factorization: the proof of the unique factorization does not give any hint for a method for factorizing. Already for the integers, there is no known algorithm running on a classical computer for factorizing them in polynomial time. This is the basis of the RSA cryptosystem, widely used for secure Internet communications.

In the case of $K[X]$, the factors, and the methods for computing them, depend strongly on $K$. Over the complex numbers, the irreducible factors (those that cannot be factorized further) are all of degree one, while, over the real numbers, there are irreducible polynomials of degree 2, and, over the rational numbers, there are irreducible polynomials of any degree. For example, the polynomial $$X^4-2$$ is irreducible over the rational numbers, is factored as $$(X - \sqrt[4]2)(X+\sqrt[4]2)(X^2+\sqrt 2)$$ over the real numbers and, and as $$(X-\sqrt[4]2)(X+\sqrt[4]2)(X-i\sqrt[4]2)(X+i\sqrt[4]2)$$ over the complex numbers.

The existence of a factorization algorithm depends also on the ground field. In the case of the real or complex numbers, Abel–Ruffini theorem shows that the roots of some polynomials, and thus the irreducible factors, cannot be computed exactly. Therefore, a factorization algorithm can compute only approximations of the factors. Various algorithms have been designed for computing such approximations, see Root finding of polynomials.

There is an example of a field $K[X]$ such that there exist exact algorithms for the arithmetic operations of $K$, but there cannot exist any algorithm for deciding whether a polynomial of the form $$X^p - a$$ is irreducible or is a product of polynomials of lower degree.

On the other hand, over the rational numbers and over finite fields, the situation is better than for integer factorization, as there are factorization algorithms that have a polynomial complexity. They are implemented in most general purpose computer algebra systems.

Minimal polynomial
If $K$ is an element of an associative $K$-algebra $θ$, the polynomial evaluation at $L$ is the unique algebra homomorphism $θ$ from $φ$ into $K[X]$ that maps $L$ to $X$ and does not affect the elements of $θ$ itself (it is the identity map on $K$). It consists of substituting $K$ with $X$ in every polynomial. That is,

\varphi\left(a_m X^m + a_{m - 1} X^{m - 1} + \cdots + a_1 X + a_0\right) = a_m \theta^m + a_{m - 1} \theta^{m - 1} + \cdots + a_1 \theta + a_0. $$

The image of this evaluation homomorphism is the subalgebra generated by $θ$, which is necessarily commutative. If $θ$ is injective, the subalgebra generated by $θ$ is isomorphic to $φ$. In this case, this subalgebra is often denoted by $K[X]$. The notation ambiguity is generally harmless, because of the isomorphism.

If the evaluation homomorphism is not injective, this means that its kernel is a nonzero ideal, consisting of all polynomials that become zero when $X$ is substituted with $θ$. This ideal consists of all multiples of some monic polynomial, that is called the minimal polynomial of $θ$. The term minimal is motivated by the fact that its degree is minimal among the degrees of the elements of the ideal.

There are two main cases where minimal polynomials are considered.

In field theory and number theory, an element $θ$ of an extension field $L$ of $K$ is algebraic over $K$ if it is a root of some polynomial with coefficients in $K$. The minimal polynomial over $K$ of $θ$ is thus the monic polynomial of minimal degree that has $θ$ as a root. Because $L$ is a field, this minimal polynomial is necessarily irreducible over $K$. For example, the minimal polynomial (over the reals as well as over the rationals) of the complex number $i$ is $$X^2 + 1$$. The cyclotomic polynomials are the minimal polynomials of the roots of unity.

In linear algebra, the $K[θ]$ square matrices over $K$ form an associative $K$-algebra of finite dimension (as a vector space). Therefore the evaluation homomorphism cannot be injective, and every matrix has a minimal polynomial (not necessarily irreducible). By Cayley–Hamilton theorem, the evaluation homomorphism maps to zero the characteristic polynomial of a matrix. It follows that the minimal polynomial divides the characteristic polynomial, and therefore that the degree of the minimal polynomial is at most $n$.

Quotient ring
In the case of $n×n$, the quotient ring by an ideal can be built, as in the general case, as a set of equivalence classes. However, as each equivalence class contains exactly one polynomial of minimal degree, another construction is often more convenient.

Given a polynomial $p$ of degree $d$, the quotient ring of $K[X]$ by the ideal generated by $p$ can be identified with the vector space of the polynomials of degrees less than $d$, with the "multiplication modulo $p$" as a multiplication, the multiplication modulo $p$ consisting of the remainder under the division by $p$ of the   (usual) product of polynomials. This quotient ring is variously denoted as $$K[X]/pK[X],$$ $$K[X]/\langle p \rangle,$$ $$K[X]/(p),$$ or simply $$K[X]/p.$$

The ring $$K[X]/(p)$$ is a field if and only if $p$ is an irreducible polynomial. In fact, if $p$ is irreducible, every nonzero polynomial $q$ of lower degree is coprime with $p$, and Bézout's identity allows computing $r$ and $s$ such that $K[X]$; so, $r$ is the multiplicative inverse of $q$ modulo $p$. Conversely, if $p$ is reducible, then there exist polynomials $a, b$ of degrees lower than $sp + qr = 1$ such that $deg(p)$ ; so $a, b$ are nonzero zero divisors modulo $p$, and cannot be invertible.

For example, the standard definition of the field of the complex numbers can be summarized by saying that it is the quotient ring
 * $$\mathbb C =\mathbb R[X]/(X^2+1),$$

and that the image of $X$ in $$\mathbb C$$ is denoted by $i$. In fact, by the above description, this quotient consists of all polynomials of degree one in $i$, which have the form $ab = p$, with $a$ and $b$ in $$\mathbb R.$$ The remainder of the Euclidean division that is needed for multiplying two elements of the quotient ring is obtained by replacing $a + bi$ by $i2$ in their product as polynomials (this is exactly the usual definition of the product of complex numbers).

Let $−1$ be an algebraic element in a $K$-algebra $A$. By algebraic, one means that $θ$ has a minimal polynomial $p$. The first ring isomorphism theorem asserts that the substitution homomorphism induces an isomorphism of $$K[X]/(p)$$ onto the image $θ$ of the substitution homomorphism. In particular, if $A$ is a simple extension of $K$ generated by $K[θ]$, this allows identifying $A$ and $$K[X]/(p).$$ This identification is widely used in algebraic number theory.

Modules
The structure theorem for finitely generated modules over a principal ideal domain applies to K[X], when K is a field. This means that every finitely generated module over K[X] may be decomposed into a direct sum of a free module and finitely many modules of the form $$K[X]/\left\langle P^k \right\rangle$$, where P is an irreducible polynomial over K and k a positive integer.

Definition (multivariate case)
Given $n$ symbols $$X_1, \dots, X_n,$$ called indeterminates, a monomial (also called power product)
 * $$X_1^{\alpha_1}\cdots X_n^{\alpha_n}$$

is a formal product of these indeterminates, possibly raised to a nonnegative power. As usual, exponents equal to one and factors with a zero exponent can be omitted. In particular, $$X_1^0\cdots X_n^0 =1.$$

The tuple of exponents $θ$ is called the multidegree or exponent vector of the monomial. For a less cumbersome notation, the abbreviation
 * $$X^\alpha=X_1^{\alpha_1}\cdots X_n^{\alpha_n}$$

is often used. The degree of a monomial $α = (α_{1}, …, α_{n})$, frequently denoted $X^{α}$ or $deg α$, is the sum of its exponents:
 * $$ \deg \alpha = \sum_{i=1}^n \alpha_i. $$

A polynomial in these indeterminates, with coefficients in a field $K$, or more generally a ring, is a finite linear combination of monomials
 * $$ p = \sum_\alpha p_\alpha X^\alpha$$

with coefficients in $K$. The degree of a nonzero polynomial is the maximum of the degrees of its monomials with nonzero coefficients.

The set of polynomials in $$X_1, \dots, X_n,$$ denoted $$K[X_1,\dots, X_n],$$ is thus a vector space (or a free module, if $K$ is a ring) that has the monomials as a basis.

$$K[X_1,\dots, X_n]$$ is naturally equipped (see below) with a multiplication that makes a ring, and an associative algebra over $K$, called the polynomial ring in $n$ indeterminates over $K$ (the definite article the reflects that it is uniquely defined up to the name and the order of the indeterminates. If the ring $K$ is commutative, $$K[X_1,\dots, X_n]$$ is also a commutative ring.

Operations in $|α|$
Addition and scalar multiplication of polynomials are those of a vector space or free module equipped by a specific basis (here the basis of the monomials). Explicitly, let $$p=\sum_{\alpha\in I}p_\alpha X^\alpha,\quad q=\sum_{\beta\in J}q_\beta X^\beta,$$ where $I$ and $J$ are finite sets of exponent vectors.

The scalar multiplication of $p$ and a scalar $$c\in K$$ is
 * $$cp = \sum_{\alpha\in I}cp_\alpha X^\alpha.$$

The addition of $p$ and $q$ is
 * $$p+q = \sum_{\alpha\in I\cup J}(p_\alpha+q_\alpha) X^\alpha,$$

where $$p_\alpha=0$$ if $$\alpha \not\in I,$$ and $$q_\beta=0$$ if $$\beta \not\in J.$$ Moreover, if one has $$p_\alpha+q_\alpha=0$$ for some $$\alpha \in I \cap J,$$ the corresponding zero term is removed from the result.

The multiplication is
 * $$pq = \sum_{\gamma\in I+J}\left(\sum_{\alpha, \beta\mid \alpha+\beta=\gamma} p_\alpha q_\beta\right) X^\gamma,$$

where $$I+J$$ is the set of the sums of one exponent vector in $I$ and one other in $J$ (usual sum of vectors). In particular, the product of two monomials is a monomial whose exponent vector is the sum of the exponent vectors of the factors.

The verification of the axioms of an associative algebra is straightforward.

Polynomial expression
A polynomial expression is an expression built with scalars (elements of $K$), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers.

As all these operations are defined in $$K[X_1,\dots, X_n]$$ a polynomial expression represents a polynomial, that is an element of $$K[X_1,\dots, X_n].$$ The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called the canonical form, normal form, or expanded form of the polynomial. Given a polynomial expression, one can compute the expanded form of the represented polynomial by expanding with the distributive law all the products that have a sum among their factors, and then using commutativity (except for the product of two scalars), and associativity for transforming the terms of the resulting sum into products of a scalar and a monomial; then one gets the canonical form by regrouping the like terms.

The distinction between a polynomial expression and the polynomial that it represents is relatively recent, and mainly motivated by the rise of computer algebra, where, for example, the test whether two polynomial expressions represent the same polynomial may be a nontrivial computation.

Categorical characterization
If $K$ is a commutative ring, the polynomial ring $K[X1, ..., Xn]$ has the following universal property: for every commutative $K$-algebra $A$, and every $n$-tuple $K[X_{1}, …, X_{n}]$ of elements of $A$, there is a unique algebra homomorphism from $(x_{1}, …, x_{n})$ to $A$ that maps each $$X_i$$ to the corresponding $$x_i.$$ This homomorphism is the evaluation homomorphism that consists in substituting $$X_i$$ with $$x_i$$ in every polynomial.

As it is the case for every universal property, this characterizes the pair $$(K[X_1, \dots, X_n], (X_1, \dots, X_n))$$ up to a unique isomorphism.

This may also be interpreted in terms of adjoint functors. More precisely, let $K[X_{1}, …, X_{n}]$ and $SET$ be respectively the categories of sets and commutative $K$-algebras (here, and in the following, the morphisms are trivially defined). There is a forgetful functor $$\mathrm F: \mathrm{ALG}\to \mathrm{SET}$$ that maps algebras to their underlying sets. On the other hand, the map $$X\mapsto K[X]$$ defines a functor $$\mathrm{POL}: \mathrm{SET}\to \mathrm{ALG}$$ in the other direction. (If $X$ is infinite, $ALG$ is the set of all polynomials in a finite number of elements of $X$.)

The universal property of the polynomial ring means that $K[X]$ and $F$ are adjoint functors. That is, there is a bijection
 * $$\operatorname{Hom}_{\mathrm {SET}}(X,\operatorname{F}(A))\cong \operatorname{Hom}_{\mathrm {ALG}}(K[X], A). $$

This may be expressed also by saying that polynomial rings are free commutative algebras, since they are free objects in the category of commutative algebras. Similarly, a polynomial ring with integer coefficients is the free commutative ring over its set of variables, since commutative rings and commutative algebras over the integers are the same thing.

Univariate over a ring vs. multivariate
A polynomial in $$K[X_1, \ldots, X_n]$$ can be considered as a univariate polynomial in the indeterminate $$X_n$$ over the ring $$K[X_1, \ldots, X_{n-1}],$$ by regrouping the terms that contain the same power of $$X_n,$$ that is, by using the identity
 * $$\sum_{(\alpha_1, \ldots, \alpha_n)\in I} c_{\alpha_1, \ldots, \alpha_n} X_1^{\alpha_1} \cdots X_n^{\alpha_n}=\sum_i\left(\sum_{(\alpha_1, \ldots, \alpha_{n-1})\mid (\alpha_1, \ldots, \alpha_{n-1}, i)\in I} c_{\alpha_1, \ldots, \alpha_{n-1}} X_1^{\alpha_1} \cdots X_{n-1}^{\alpha_{n-1}}\right)X_n^i,$$

which results from the distributivity and associativity of ring operations.

This means that one has an algebra isomorphism
 * $$K[X_1, \ldots, X_n]\cong (K[X_1, \ldots, X_{n-1}])[X_n]$$

that maps each indeterminate to itself. (This isomorphism is often written as an equality, which is justified by the fact that polynomial rings are defined up to a unique isomorphism.)

In other words, a multivariate polynomial ring can be considered as a univariate polynomial over a smaller polynomial ring. This is commonly used for proving properties of multivariate polynomial rings, by induction on the number of indeterminates.

The main such properties are listed below.

Properties that pass from $POL$ to $R$
In this section, $R$ is a commutative ring, $K$ is a field, $X$ denotes a single indeterminate, and, as usual, $$\mathbb Z$$ is the ring of integers. Here is the list of the main ring properties that remain true when passing from $R$ to $R[X]$.

 * If $R$ is an integral domain then the same holds for $R[X]$ (since the leading coefficient of a product of polynomials is, if not zero, the product of the leading coefficients of the factors).
 * In particular, $$K[X_1,\ldots,X_n]$$ and $$\mathbb Z[X_1,\ldots,X_n]$$ are integral domains.
 * If $R$ is a unique factorization domain then the same holds for $R[X]$. This results from Gauss's lemma and the unique factorization property of $$L[X],$$ where $L$ is the field of fractions of $R$.
 * In particular, $$K[X_1,\ldots,X_n]$$ and $$\mathbb Z[X_1,\ldots,X_n]$$ are unique factorization domains.
 * If $R$ is a Noetherian ring, then the same holds for $R[X]$.
 * In particular, $$K[X_1,\ldots,X_n]$$ and $$\mathbb Z[X_1,\ldots,X_n]$$ are Noetherian rings; this is Hilbert's basis theorem.
 * If $R$ is a Noetherian ring, then $$\dim R[X] = 1+\dim R,$$ where "$$\dim$$" denotes the Krull dimension.
 * In particular, $$\dim K[X_1,\ldots,X_n] = n$$ and $$\dim \mathbb Z[X_1,\ldots,X_n] = n+1.$$
 * If $R$ is a regular ring, then the same holds for $R[X]$; in this case, one has $$\operatorname{gl}\, \dim R[X]= \dim R[X]= 1 + \operatorname{gl}\, \dim R=1+\dim R,$$ where "$$\operatorname{gl}\, \dim$$" denotes the global dimension.
 * In particular, $$K[X_1,\ldots,X_n]$$ and $$\mathbb Z[X_1,\ldots,X_n]$$ are regular rings, $$\operatorname{gl}\, \dim \mathbb Z[X_1,\ldots,X_n] = n+1,$$ and $$\operatorname{gl}\, \dim K[X_1,\ldots,X_n] = n.$$ The latter equality is Hilbert's syzygy theorem.

Several indeterminates over a field
Polynomial rings in several variables over a field are fundamental in invariant theory and algebraic geometry. Some of their properties, such as those described above can be reduced to the case of a single indeterminate, but this is not always the case. In particular, because of the geometric applications, many interesting properties must be invariant under affine or projective transformations of the indeterminates. This often implies that one cannot select one of the indeterminates for a recurrence on the indeterminates.

Bézout's theorem, Hilbert's Nullstellensatz and Jacobian conjecture are among the most famous properties that are specific to multivariate polynomials over a field.

Hilbert's Nullstellensatz
The Nullstellensatz (German for "zero-locus theorem") is a theorem, first proved by David Hilbert, which extends to the multivariate case some aspects of the fundamental theorem of algebra. It is foundational for algebraic geometry, as establishing a strong link between the algebraic properties of $$K[X_1, \ldots, X_n]$$ and the geometric properties of algebraic varieties, that are (roughly speaking) set of points defined by implicit polynomial equations.

The Nullstellensatz, has three main versions, each being a corollary of any other. Two of these versions are given below. For the third version, the reader is referred to the main article on the Nullstellensatz.

The first version generalizes the fact that a nonzero univariate polynomial has a complex zero if and only if it is not a constant. The statement is: a set of polynomials $S$ in $$K[X_1, \ldots, X_n]$$ has a common zero in an algebraically closed field containing $K$, if and only if $R[X]$ does not belong to the ideal generated by $S$, that is, if $1$ is not a linear combination of elements of $S$ with polynomial coefficients.

The second version generalizes the fact that the irreducible univariate polynomials over the complex numbers are associate to a polynomial of the form $$X-\alpha.$$ The statement is: ''If $K$ is algebraically closed, then the maximal ideals of $$K[X_1, \ldots, X_n]$$ have the form $$\langle X_1 - \alpha_1, \ldots, X_n - \alpha_n \rangle.$$

Bézout's theorem
Bézout's theorem may be viewed as a multivariate generalization of the version of the fundamental theorem of algebra that asserts that a univariate polynomial of degree $n$ has $n$ complex roots, if they are counted with their multiplicities.

In the case of bivariate polynomials, it states that two polynomials of degrees $d$ and $e$ in two variables, which have no common factors of positive degree, have exactly $de$ common zeros in an algebraically closed field containing the coefficients, if the zeros are counted with their multiplicity and include the zeros at infinity.

For stating the general case, and not considering "zero at infinity" as special zeros, it is convenient to work with homogeneous polynomials, and consider zeros in a projective space. In this context, a projective zero of a homogeneous polynomial $$P(X_0, \ldots, X_n)$$ is, up to a scaling, a $1$-tuple $$(x_0, \ldots, x_n)$$ of elements of $K$ that is different from $(n + 1)$, and such that $$P(x_0, \ldots, x_n) = 0 $$. Here, "up to a scaling" means that $$(x_0, \ldots, x_n)$$ and $$(\lambda x_0, \ldots, \lambda x_n)$$ are considered as the same zero for any nonzero $$\lambda\in K.$$ In other words, a zero is a set of homogeneous coordinates of a point in a projective space of dimension $n$.

Then, Bézout's theorem states: Given $n$ homogeneous polynomials of degrees $$d_1, \ldots, d_n$$ in $(0, …, 0)$ indeterminates, which have only a finite number of common projective zeros in an algebraically closed extension of $K$, the sum of the multiplicities of these zeros is the product $$d_1 \cdots d_n.$$

Generalizations
Polynomial rings can be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, noncommutative polynomial rings, skew polynomial rings, and polynomial rigs.

Infinitely many variables
One slight generalization of polynomial rings is to allow for infinitely many indeterminates. Each monomial still involves only a finite number of indeterminates (so that its degree remains finite), and each polynomial is a still a (finite) linear combination of monomials. Thus, any individual polynomial involves only finitely many indeterminates, and any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates. This generalization has the same property of usual polynomial rings, of being the free commutative algebra, the only difference is that it is a free object over an infinite set.

One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree. This ring is larger than the usual polynomial ring, as it includes infinite sums of variables. However, it is smaller than the ring of power series in infinitely many variables. Such a ring is used for constructing the ring of symmetric functions over an infinite set.

Generalized exponents
A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents: X ⋅ X = X. A set for which addition makes sense (is closed and associative) is called a monoid. The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a + b, then cn = an + bn for every n in N. The multiplication is defined as the Cauchy product, so that if c = a ⋅ b, then for each n in N, cn is the sum of all aibj where i, j range over all pairs of elements of N which sum to n.

When N is commutative, it is convenient to denote the function a in R[N] as the formal sum:
 * $$\sum_{n \in N} a_n X^n$$

and then the formulas for addition and multiplication are the familiar:
 * $$\left(\sum_{n \in N} a_n X^n\right) + \left(\sum_{n \in N} b_n X^n\right) = \sum_{n \in N} \left(a_n + b_n\right)X^n$$

and
 * $$\left(\sum_{n \in N} a_n X^n\right) \cdot \left(\sum_{n \in N} b_n X^n\right) = \sum_{n \in N} \left( \sum_{i+j=n} a_i b_j\right)X^n$$

where the latter sum is taken over all i, j in N that sum to n.

Some authors such as go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials in several variables simply take N to be the direct product of several copies of the monoid of non-negative integers.

Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negative rational numbers,. See also Puiseux series.

Power series
Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can also be seen as the ring completion of the polynomial ring with respect to the ideal generated by $x$.

Noncommutative polynomial rings
For polynomial rings of more than one variable, the products X⋅Y and Y⋅X are simply defined to be equal. A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained. Formally, the polynomial ring in n noncommuting variables with coefficients in the ring R is the monoid ring R[N], where the monoid N is the free monoid on n letters, also known as the set of all strings over an alphabet of n symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other.

Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.

Differential and skew-polynomial rings
Other generalizations of polynomials are differential and skew-polynomial rings.

A differential polynomial ring is a ring of differential operators formed from a ring R and a derivation δ of R into R. This derivation operates on R, and will be denoted X, when viewed as an operator. The elements of R also operate on R by multiplication. The composition of operators is denoted as the usual multiplication. It follows that the relation δ(ab) = aδ(b) + δ(a)b may be rewritten as
 * $$X\cdot a = a\cdot X +\delta(a).$$

This relation may be extended to define a skew multiplication between two polynomials in X with coefficients in R, which make them a noncommutative ring.

The standard example, called a Weyl algebra, takes R to be a (usual) polynomial ring k[Y ], and δ to be the standard polynomial derivative $$\tfrac{\partial}{\partial Y}$$. Taking a = Y in the above relation, one gets the canonical commutation relation, X⋅Y − Y⋅X = 1. Extending this relation by associativity and distributivity allows explicitly constructing the Weyl algebra. .

The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X⋅r = f(r)⋅X to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism F from the monoid N of the positive integers into the endomorphism ring of R, the formula X n⋅r = F(n)(r)⋅X n allows constructing a skew-polynomial ring. Skew polynomial rings are closely related to crossed product algebras.

Polynomial rigs
The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a field or a ring to the requirement that R only be a semifield or rig; the resulting polynomial structure/extension R[X] is a polynomial rig. For example, the set of all multivariate polynomials with natural number coefficients is a polynomial rig.

title: Polynomial ring


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Alcohol and Tobacco Tax and Trade Bureau

The Alcohol and Tobacco Tax and Trade Bureau, statutorily named the Tax and Trade Bureau and frequently shortened to TTB, is a bureau of the United States Department of the Treasury, which regulates and collects taxes on trade and imports of alcohol, tobacco, and firearms within the United States.

Overview
TTB was created on January 24, 2003, when the Homeland Security Act of 2002 split the Bureau of Alcohol, Tobacco and Firearms (ATF) into two new organizations with separate functions. Specifically, the Act transferred ATF and its law enforcement functions from the Department of the Treasury to the Department of Justice. ATF's other functions, dealing with tax collection and regulation of legitimate trade, remained within the Treasury Department and became part of the new TTB.

Organization
TTB's Field Operations are organized into five divisions:
 * 1) National Revenue Center: reconciles returns, reports, and claims; screens applications and promptly issues permits; and provides expert technical assistance for industry, the public and government agencies to ensure fair and proper revenue collection and public safety.
 * 2) Risk Management: develops, implements, and maintains monitoring programs for collecting the revenue due the Federal Government and protecting the public, and ensures resources are effectively used.
 * 3) Tax Audit: verifies the proper payment of alcohol, tobacco, firearms and ammunition excise taxes and ensures compliance with laws and regulations by taxpayers in a manner that protects the revenue, protects the consumer, and promotes voluntary compliance.
 * 4) Trade Investigations: comprises investigators who ensure industry compliance with the laws and regulations administered by the TTB.
 * 5) Tobacco Enforcement Division: protects the revenue and promotes voluntary compliance by monitoring the domestic tobacco trade, ensuring only qualified applicants enter the tobacco trade, ensuring compliance with the tax laws relating to tobacco, and facilitating TTB's enforcement functions in cases of non-compliance.

The Advertising, Labeling, and Formulation Division (ALFD) implements and enforces a broad range of statutory and compliance provisions of the Internal Revenue Code and the Federal Alcohol Administration Act. This act requires importers and bottlers of beverage alcohol to obtain certificates of label approval or certificates of exemption from label approval (COLAs) for most alcohol beverages prior to their introduction into interstate commerce. ALFD acts on these COLAs to ensure that products are labeled in accordance with federal laws and regulations. ALFD also examines formulas for wine and distilled spirits, statements of process, and pre-import applications filed by importers and proprietors of domestic distilleries, wineries, and breweries for proper tax classification and to ensure that the products are manufactured in accordance with federal laws and regulations.


text:
Sannyasa

Sannyasa (Sanskrit: संन्यास; IAST: ), sometimes spelled Sanyasa or Sanyasi (for the person), is life of renunciation and the fourth stage within the Hindu system of four life stages known as Ashramas, with the first three being Brahmacharya (on the path of Brahma), Grihastha (householder) and Vanaprastha (forest dweller, retired). Sannyasa is traditionally conceptualized for men or women in late years of their life, but young brahmacharis have had the choice to skip the householder and retirement stages, renounce worldly and materialistic pursuits and dedicate their lives to spiritual pursuits.

Sannyasa, a form of asceticism marked by renunciation of material desires and prejudices, is represented by a state of disinterest in and detachment from material life, whose purpose is spending one's life in peaceful, spiritual pursuits. An individual in Sanyasa is known as a sannyasi (male) or sannyasini (female) in Hinduism. Sannyasa shares similarities with the Sadhu and Sadhvi traditions of Jain monasticism, and the sannyasi and sannyasini share similarity with the bhikkhus and bhikkhunis of Buddhism.

Sannyasa has historically been a stage of renunciation, ahimsa (non-violence), a peaceful and simple life and spiritual pursuit in Indian traditions. However, this has not always been the case. After the invasions and establishment of Muslim rule in India, from the 12th century through the British Raj, parts of the Shaiva (Gossain) and Vaishnava (Bairagi) ascetics metamorphosed into a military order, where they developed martial arts, created military strategies, and engaged in guerrilla warfare. These warrior sanyasi (ascetics) played an important role in helping European colonial powers establish themselves in the Indian subcontinent.

Etymology and synonyms
 in Sanskrit nyasa means purification, sannyasa means "Purification of Everything". It is a composite word of copulative a which means "together, all", ni- which means "down" and ' from the root ', meaning "to throw" or "to put". A literal translation of Sannyāsa is thus "to put down everything, all of it". Sannyasa is sometimes spelled as Sanyasa.

The term Saṃnyasa makes appearance in the Samhitas, Aranyakas and Brahmanas, the earliest layers of Vedic literature (2nd millennium BCE), but it is rare. It is not found in ancient Buddhist or Jaina vocabularies, and only appears in Hindu texts of the 1st millennium BCE, in the context of those who have given up ritual activity and taken up non-ritualistic spiritual pursuits discussed in the Upanishads. The term Sannyasa evolves into a rite of renunciation in ancient Sutra texts, and thereafter became a recognized, well discussed stage of life (Ashrama) by about the 3rd and 4th century CE.

Sanyasis are also known as Bhiksu, Pravrajita/Pravrajitā, Yati, Sramana and Parivrajaka in Hindu texts.

History
Jamison and Witzel state early Vedic texts make no mention of Sannyasa, or Ashrama system, unlike the concepts of Brahmacharin and Grihastha which they do mention. Instead, Rig Veda uses the term Antigriha (अन्तिगृह) in hymn 10.95.4, as still a part of the extended family, where older people lived in ancient India, with an outwardly role. It is in later Vedic era and over time, that Sannyasa and other new concepts emerged, while older ideas evolved and expanded. A three-stage Ashrama concept, along with Vanaprastha, emerged about or after 7th Century BC, when sages such as Yājñavalkya left their homes and roamed around as spiritual recluses and pursued their Pravrajika (wanderer) lifestyle. The explicit use of the four-stage Ashrama concept appeared a few centuries later.

However, early Vedic literature from 2nd millennium BC mentions Muni (मुनि, monks, mendicants, holy men), with characteristics that mirror those found in later Sannyasins and Sannyasinis. For example, the Rig Veda, in Book 10 Chapter 136, mentions Munis as those with Kesin (केशिन्, long haired) and Mala clothes (मल, soil-colored, yellow, orange, saffron), engaged in the affairs of Mananat (mind, meditation). The Rigveda, however, refers to these people as Muni and Vati (वति, monks who beg).

"केश्यग्निं केशी विषं केशी बिभर्ति रोदसी । केशी विश्वं स्वर्दृशे केशीदं ज्योतिरुच्यते ॥१॥

मुनयो वातरशनाः पिशङ्गा वसते मला । वातस्यानु ध्राजिं यन्ति यद्देवासो अविक्षत ॥२॥

He with the long loose locks (of hair) supports Agni, and moisture, heaven, and earth; He is all sky to look upon: he with long hair is called this light. The Munis, girdled with the wind, wear garments of soil hue; They, following the wind's swift course, go where the Gods have gone before."

These Munis, their lifestyle and spiritual pursuit, likely influenced the Sannyasa concept, as well as the ideas behind the ancient concept of Brahmacharya (bachelor student). One class of Munis were associated with Rudra. Another were Vratyas.

Lifestyle and goals
Hinduism has no formal demands nor requirements on the lifestyle or spiritual discipline, method or deity a Sanyasin or Sanyasini must pursue – it is left to the choice and preferences of the individual. This freedom has led to diversity and significant differences in the lifestyle and goals of those who adopt Sannyasa. There are, however, some common themes. A person in Sannyasa lives a simple life, typically detached, itinerant, drifting from place to place, with no material possessions or emotional attachments. They may have a walking stick, a book, a container or vessel for food and drink, often wearing yellow, saffron, orange, ochre or soil colored clothes. They may have long hair and appear disheveled, and are usually vegetarians. Some minor Upanishads as well as monastic orders consider women, children, students, fallen men (those with a criminal record) and others as not qualified to become Sannyasa; while other texts place no restrictions. The dress, the equipage and lifestyle varies between groups. For example, Sannyasa Upanishad in verses 2.23 to 2.29, identifies six lifestyles for six types of renunciates. One of them is described as living with the following possessions,

"Pot, drinking cup and flask – the three supports, a pair of shoes,

a patched robe giving protection – in heat and cold, a loin cloth,

bathing drawers and straining cloth, triple staff and coverlet."

Those who enter Sannyasa may choose whether they join a group (similar to Christian mendicant orders). Some are anchorites, homeless mendicants preferring solitude and seclusion in remote parts, without affiliation. Others are cenobites, living and traveling with kindred fellow-Sannyasi in the pursuit of their spiritual journey, sometimes in Ashramas or Matha/Sangha (a Hermitage, the practice of seclusion known generally as monasticism).

Most Hindu ascetics adopt celibacy when they begin Sannyasa. However, there are exceptions, such as the Saiva Tantra school of asceticism where ritual sex is considered part of liberation process. Sex is viewed by them as a transcendence from a personal, intimate act to something impersonal and ascetic.

The goal
The goal of the Hindu Sannyasin is moksha (liberation). The idea of what that means varies from tradition to tradition.

"Who am I, and in what really do I consist? What is this cage of suffering?"

For the Bhakti (devotion) traditions, liberation consists of being an eternal servant to the Divine and release from Saṃsāra (rebirth in future life); for Yoga traditions, liberation is the experience of the highest Samādhi (deep awareness in this life); and for the Advaita tradition, liberation is jivanmukti – the awareness of the Supreme Reality (Brahman) and Self-realization in this life. Sannyasa is a means and an end in itself. It is a means to decreasing and then ultimately ending all ties of any kind. It is a means to the soul and meaning, but not ego nor personalities. Sannyasa does not abandon the society, it abandons the ritual mores of the social world and one's attachment to all its other manifestations. The end is a liberated, content, free and blissful existence.

The behaviors and characteristics
The behavioral state of a person in Sannyasa is described by many ancient and medieval era Indian texts. Bhagavad Gita discusses it in many verses, for example:

"ज्ञेयः स नित्यसंन्यासी यो न द्वेष्टि न काङ् क्षति । निर्द्वन्द्वो हि महाबाहो सुखं बन्धात्प्रमुच्यते ॥५-३॥

He is known as a permanent Sannyasin who does not hate, does not desire, is without dualities (opposites). Truly, Mahabaho (Arjuna), he is liberated from bondage."

Other behavioral characteristics, in addition to renunciation, during Sannyasa include: ahimsa (non-violence), akrodha (not become angry even if you are abused by others), disarmament (no weapons), chastity, bachelorhood (no marriage), avyati (non-desirous), amati (poverty), self-restraint, truthfulness, sarvabhutahita (kindness to all creatures), asteya (non-stealing), aparigraha (non-acceptance of gifts, non-possessiveness) and shaucha (purity of body, speech and mind). Some Hindu monastic orders require the above behavior in form of a vow, before a renunciate can enter the order. Tiwari notes that these virtues are not unique to Sannyasa, and other than renunciation, all of these virtues are revered in ancient texts for all four Ashramas (stages) of human life.

Baudhayana Dharmasūtra, completed by about 7th century BC, states the following behavioral vows for a person in Sannyasa

"These are the vows a Sannyasi must keep –

Abstention from injuring living beings, truthfulness, abstention from appropriating the property of others, abstention from sex, liberality (kindness, gentleness) are the major vows. There are five minor vows: abstention from anger, obedience towards the guru, avoidance of rashness, cleanliness, and purity in eating. He should beg (for food) without annoying others, any food he gets he must compassionately share a portion with other living beings, sprinkling the remainder with water he should eat it as if it were a medicine."

Types
Ashrama Upanishad identified various types of Sannyasi renouncers based on their different goals: Kutichaka – seeking atmospheric world; Bahudaka – seeking heavenly world; Hamsa – seeking penance world; Paramahamsa – seeking truth world; and Turiyatitas and Avadhutas seeking liberation in this life.

In some texts, such as Sannyasa Upanishad, these were classified by the symbolic items the Sannyasins carried and their lifestyle. For example, Kutichaka sannyasis carried triple staffs, Hamsa sannyasis carried single staffs, while Paramahamsas went without them. This method of classification based on emblematic items became controversial, as anti-thematic to the idea of renunciation. Later texts, such as Naradaparivrajaka Upanishad stated that all renunciation is one, but people enter the state of Sannyasa for different reasons – for detachment and getting away from their routine meaningless world, to seek knowledge and meaning in life, to honor rites of Sannyasa they have undertaken, and because he already has liberating knowledge.

There were many groups of Hindu, Jain and Buddhist Sannyasis co-existing in pre-Maurya Empire era, each classified by their attributes, such as: Achelakas (without clothes), Ajivika, Aviruddhaka, Devadhammika, Eka-satakas, Gotamaka, Jatilaka, Magandika, Mundasavaka, Nigrantha (Jains), Paribbajaka, Tedandikas, Titthiya and others.
 * Other classifications

Literature
The Dharmasūtras and Dharmaśāstras, composed about mid 1st millennium BC and later, place increasing emphasis on all four stages of Ashrama system including Sannyasa. The Baudhayana Dharmasūtra, in verses 2.11.9 to 2.11.12, describes the four Ashramas as "a fourfold division of Dharma". The newer Dharmaśāstra vary widely in their discussion of Ashrama system.

The Dharmasūtras and Dharmaśāstras give a number of detailed but widely divergent guidelines on renunciation. In all cases, Sannyasa was never mandatory and was one of the choices before an individual. Only a small percentage chose this path. Olivelle posits that the older Dharmasūtras present the Ashramas including Sannyasa as four alternative ways of life and options available, but not as sequential stage that any individual must follow. Olivelle also states that Sannyasa along with the Ashrama system gained mainstream scholarly acceptance about 2nd century BC.

Ancient and medieval era texts of Hinduism consider Grihastha (householder) stage as the most important of all stages in sociological context, as human beings in this stage not only pursue a virtuous life, they produce food and wealth that sustains people in other stages of life, as well as the offspring that continues mankind. However, an individual had the choice to renounce any time he or she wanted, including straight after student life.

When can a person renounce?
Baudhayana Dharmasūtra, in verse II.10.17.2 states that anyone who has finished Brahmacharya (student) life stage may become ascetic immediately, in II.10.17.3 that any childless couple may enter Sannyasa anytime they wish, while verse II.10.17.4 states that a widower may choose Sannyasa if desired, but in general, states verse II.10.17.5, Sannyasa is suited after the completion of age 70 and after one's children have been firmly settled. Other texts suggest the age of 75.

The ' and Āpastamba Dharmasūtras, and the later ' describe the āśramas as sequential stages which would allow one to pass from Vedic studentship to householder to forest-dwelling hermit to renouncer. However, these texts differ with each other. Yājñavalkya Smṛti, for example, differs from Manusmṛti and states in verse 3.56 that one may skip Vanaprastha (forest dwelling, retired) stage and go straight from the Grihastha (householder) stage to Sannyasa.

Who may renounce?
The Jabala Upanishad mentions one who gets vairagya of any class or gender can renounce or take sanyasa. Nevertheless, Dharmaśāstra texts document people of all castes as well as women, entered Sannyasa in practice.

What happened to renouncers' property and human rights?
After renouncing the world, the ascetic's financial obligations and property were adjudicated by the state, in the manner of a decedent's estate. Viṣṇu Smriti in verse 6.27, for example, states that if a debtor takes Sannyasa, his sons or grandsons should settle his debts. As to the little property a Sannyasin may collect or possess after renunciation, Book III Chapter XVI of Kautiliya's Arthashastra states that the property of hermits (vánaprastha), ascetics (yati, sannyasa), and student bachelors (Brahmachári) shall on their death be taken by their guru, disciples, their dharmabhratri (brother in the monastic order), or classmates in succession.

Although a renouncer's practitioner's obligations and property rights were reassigned, he or she continued to enjoy basic human rights such as the protection from injury by others and the freedom to travel. Likewise, someone practicing Sannyasa was subject to the same laws as common citizens; stealing, harming, or killing a human being by a Sannyasi were all serious crimes in Kautiliya's Arthashastra.

Renunciation in daily life
Later Indian literature debates whether the benefit of renunciation can be achieved (moksha, or liberation) without asceticism in the earlier stages of one's life. For example, Bhagavad Gita, Vidyaranya's Jivanmukti Viveka, and others believed that various alternate forms of yoga and the importance of yogic discipline could serve as paths to spirituality, and ultimately moksha. Over time, four paths to liberating spirituality have emerged in Hinduism: Jñāna yoga, Bhakti yoga, Karma yoga and Rāja yoga. Acting without greed or craving for results, in Karma yoga for example, is considered a form of detachment in daily life similar to Sannyasa. Sharma states that, "the basic principle of Karma yoga is that it is not what one does, but how one does it that counts and if one has the know-how in this sense, one can become liberated by doing whatever it is one does", and "(one must do) whatever one does without attachment to the results, with efficiency and to the best of one's ability".

Warrior ascetics
Ascetic life was historically a life of renunciation, non-violence and spiritual pursuit. However, in India, this has not always been the case. For example, after the Mongol and Persian Islamic invasions in the 12th century, and the establishment of Delhi Sultanate, the ensuing Hindu-Muslim conflicts provoked the creation of a military order of Hindu ascetics in India. These warrior ascetics formed paramilitary groups called ‘‘Akharas'’ and they invented a range of martial arts.

Nath Siddhas of the 12th century AD, may have been the earliest Hindu monks to resort to a military response after the Muslim conquest. Ascetics, by tradition, led a nomadic and unattached lifestyle. As these ascetics dedicated themselves to rebellion, their groups sought stallions, developed techniques for spying and targeting, and they adopted strategies of war against Muslim nobles and the Sultanate state. Many of these groups were devotees of Hindu deity Mahadeva, and were called Mahants. Other popular names for them was Sannyasis, Yogis, Nagas (followers of Shiva), Bairagis (followers of Vishnu) and Gosains from the 16th to the 19th centuries; in some cases, these Hindu monks cooperated with Muslim fakirs who were Sufi and also persecuted.

Warrior monks continued their rebellion through the Mughal Empire, and became a political force during the early years of British Raj. In some cases, these regiments of soldier monks shifted from guerrilla campaigns to war alliances, and these Hindu warrior monks played a key role in helping British establish themselves in India. The significance of warrior ascetics rapidly declined with the consolidation of British Raj in late 19th century, and with the rise in non-violence movement by Mahatma Gandhi.

Novetzke states that some of these Hindu warrior ascetics were treated as folk heroes, aided by villagers and townspeople, because they targeted figures of political and economic power in a discriminatory state, and some of these warriors paralleled Robin Hood's lifestyle.

Upanishads
Sannyasa, or the renunciant way of life, is discussed in various Upanishads.

Major Upanishads
Among the thirteen major or Principal Upanishads, all from the ancient era, many include sections related to Sannyasa. The Mundaka Upanishad discusses the path of Sannyasa as a means to attain spiritual knowledge and liberation. It emphasizes the renunciant's simple and austere lifestyle in pursuit of wisdom. The motivations and state of a Sannyasi are mentioned in Maitrāyaṇi Upanishad, a classical major Upanishad that Robert Hume included among his list of "Thirteen Principal Upanishads" of Hinduism. Maitrāyaṇi starts with the question, "given the nature of life, how is joy possible?" and "how can one achieve moksha (liberation)?"; in later sections it offers a debate on possible answers and its views on Sannyasa. "In this body infected with passions, anger, greed, delusion, fright, despondency, grudge, separation from what is dear and desirable, attachment to what is not desirable, hunger, thirst, old age, death, illness, sorrow and the rest - how can one experience only joy? – Hymn I.3

The drying up of great oceans, the crumbling down of the mountains, the instability of the pole-star, the tearing of the wind-chords, the sinking down, the submergence of the earth, the tumbling down of the gods from their place - in a world in which such things occur, how can one experience only joy ?! – Hymn I.4"

"Dragged away and polluted by the river of the Gunas (personality), one becomes rootless, tottering, broken down, greedy, uncomposed and falling in the delusion of I-consciousness, he imagines: 'I am this, this is mine' and binds himself, like a bird in the net. – Hymn VI.30

Just as the fire without fuel comes to rest in its place,

so also the passive mind comes to rest in its source;

When it (mind) is infatuated by the objects of sense, he falls away from truth and acts;

Mind alone is the Samsara, one should purify it with diligence;

You are what your mind is, a mystery, a perpetual one;

The mind which is serene, cancels all actions good and bad;

He, who, himself, serene, remains steadfast in himself - he attains imperishable happiness. – Hymn VI.34"

Sannyasa Upanishads
Of the 108 Upanishads of the Muktika, the largest corpus is dedicated to Sannyasa and to Yoga, or about 20 each, with some overlap. The renunciation-related texts are called the Sannyasa Upanishads. These are as follows:

Six of the Sannyasa Upanishads – Aruni, Kundika, Kathashruti, Paramahamsa, Jabala and Brahma – were composed before the 3rd-century CE, likely in the centuries before or after the start of the common era, states Sprockhoff; the Asrama Upanishad is dated to the 3rd-century, the Naradaparivrajaka and Satyayaniya Upanishads to around the 12th-century, and about ten of the remaining Sannyasa Upanishads are dated to have been composed in the 14th- to 15th-century CE well after the start of Islamic Sultanates period of South Asia in late 12th-century.

The oldest Sannyasa Upanishads have a strong Advaita Vedanta outlook, and these pre-date Adi Shankara. Most of the Sannyasa Upanishads present a Yoga and nondualism (Advaita) Vedanta philosophy. This may be, states Patrick Olivelle, because major Hindu monasteries of early medieval period (1st millennium CE) belonged to the Advaita Vedanta tradition. The 12th-century Shatyayaniya Upanishad is a significant exception, which presents qualified dualistic and Vaishnavism (Vishishtadvaita Vedanta) philosophy.

title: Sannyasa


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Rajarshi

Rajarshi (राजर्षि) is a title in Hinduism and Hindu mythology, referring to a sage who hails from a royal background.

Description
A rajarshi may be described to be a king (raja) who adopted a path of devotion, thereby becoming a royal sage (rishi). A rajarshi does not have to leave the kingship to become rishi, as in the example of Vishvamitra (who later becomes a Brahmarishi), but could attain the status of a sage through self-realisation during his reign. A rajarshi still performs the duties of their kshatriya class, and remain similar to most rishis, maharishis, and brahmarishis descendants in their level of spiritual knowledge. Another example of Rajarshi is King Janaka, who is said to have attained self-Knowledge from the ascetic sage Astavakra.

They belong to the four types of rishis mentioned in Hinduism and Vedas.
 * 1) Rajarshi
 * 2) Maharishi
 * 3) Brahmarshi
 * 4) Devarishi

Literature
In the Ramayana, Brahma proclaims Vishvamitra to be a rajarshi in response to the sage performing austerities for a thousand years.

title: Rajarshi


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List of Swedish clergy and theologians

This is a list of Swedish clergy and theologians.

Bishops

 * Mikael Agricola, bishop and reformer
 * Hans Brask, bishop
 * Bishop Henry, bishop
 * Lars Levi Laestadius, founder of the Laestadian movement
 * Christina Odenberg, first woman to become a bishop in the Church of Sweden
 * Laurentius Petri, archbishop and reformer
 * Olaus Petri, reformer
 * Nathan Söderblom, archbishop, Nobel Peace Prize laureate

Other clergy

 * Israel Acrelius (1714–1800), pastor in Rialla, Wilmington, Delaware, and Fellingsboro
 * Johan Campanius (1601–1683), clergyman assigned to New Sweden, translated the Lutheran Catechism into the Lenape language
 * Elisabeth Djurle, one of the first three women to be ordained as priests of the Church of Sweden
 * Israel Israelsson Näslund III (1796–1858), pastor at Torsåker
 * Ingrid Persson (1912–2000), one of the first three women to be ordained as priests of the Church of Sweden
 * Lewi Pethrus (1884–1974), founder of the Swedish Pentecostal movement
 * Margit Sahlin (1914–2003), one of the first three women to be ordained as priests of the Church of Sweden
 * Reorus Torkillus (1608–1643), first Lutheran clergyman assigned to New Sweden

title: List of Swedish clergy and theologians


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Carl Hellmuth Hertz

Carl Hellmuth Hertz (also written Carl Helmut Hertz, 15 October 1920 – 29 April 1990) was a German physicist known primarily for being involved in the development of inkjet technology and ultrasound technology. He was the son of Gustav Ludwig Hertz and great nephew of Heinrich Hertz.

Biography
Hellmuth Hertz was born on 15 October 1920 in Berlin, Germany. His father was Gustav Hertz who, along with James Franck, was awarded the Nobel Prize in Physics 1925 for their experiments on inelastic electron collisions in gases. Gustav Hertz's uncle was in turn Heinrich Hertz, who first conclusively proved the existence of the electromagnetic waves.

Hellmuth graduated from the elite Schule Schloss Salem boarding school in 1939 at the age of 19 years with the highest grade in mathematics and physics. The same year, he was conscripted into the German Army (Wehrmacht) and served as a soldier for Nazi Germany in World War II. In 1943 he was captured in the North African theatre by US troops and brought to America, where he was placed in a prisoner-of-war camp until 1946. Because his father did some research in the Soviet Union at that time, he could not get a job in the US. Instead, he got a job at the Department of Physics in Lund University in Sweden with the assistance of James Franck and the Nobel laureate Niels Bohr, who were both friends of Gustav Hertz.

From 1961 he was a teacher at Lund University, and from 1963 he was Professor of Electrical Measurement Technology in Lund. He was involved in the development of both the inkjet and the ultrasound technology. He produced the first echocardiographs together with the Swedish physician Inge Edler. He was married to Birgit Nordbring and was the father of Thomas and Hans Hertz, and he died on 29 April 1990.

title: Carl Hellmuth Hertz


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List of crossings of the River Severn

This is a list of crossings of the River Severn in Great Britain (including bridges, tunnels, ferries and fords), in order from source to mouth.

The Severn has historically been a very important and busy river, and has been bridged throughout history. The bridges that stand today are often of great historical and/or engineering interest. For example, the world's first iron bridge, The Iron Bridge, built from cast iron, crosses the River Severn at Ironbridge Gorge. The Iron Bridge is one of three bridges on the River Severn that are listed as grade I structures, including Bewdley Bridge and the Severn Bridge, which was opened in 1966. In total, 31 bridges that cross the River Severn are listed, either grade I, II* or II. Four bridges are scheduled monuments, including The Iron Bridge, which are nationally important archaeological bridges.

Many reaches of the Severn are prone to severe flooding, prompting the design of some unique bridges to cope with this.

There were historically also ferry crossings on the tidal river at:
 * Minsterworth
 * Framilode
 * Purton, Lydney
 * Newnham on Severn

Crossings
In order, moving downstream:

title: List of crossings of the River Severn