query: where did the phrase mind your p 's and q 's come from

pos: Mind your Ps and Qs According to Michael Quinion, "[i]nvestigations by the Oxford English Dictionary in 2007 when revising the entry turned up early examples of the use of Ps and Qs to mean learning the alphabet. The first is in a poem by Charles Churchill, published in 1763: “On all occasions next the chair / He stands for service of the Mayor, / And to instruct him how to use / His As and Bs, and Ps and Qs.” The conclusion must be that this is the true origin." When pupils were taught to write lowercase alphabet, the positioning of the vertical line before

neg: Mind your Ps and Qs or after the circle represented different letters: d & b, p & q. Pupils also had to mind the placement of the letters when writing the alphabet eg. p comes before q. As noted by W. D. Henkle in "Educational Notes and Queries " in 1876, in this case the proper spelling of the phrase should be "note your "p"s and "q"s", because the distinction of majuscule "P" and "Q" does not pose a problem. Nevertheless, a number of alternative explanations have been considered as more or less plausible. Another explanation suggests that "Ps and Qs" is short for "pleases"

neg: Mind your Ps and Qs Mind your Ps and Qs Mind your Ps and Qs is an English language expression meaning "mind your manners", "mind your language", "be on your best behaviour" or similar. Attempts at explaining the origin of the phrase go back to the mid-19th century. One explanation favoured in a letter to the editors of "Notes and Queries" dated 1851, as well as by the Oxford English Dictionary upon their revision of the relevant entry in 2007, is literal interpretation of the saying, concerning the distinction of the lowercase letters "p" and "q" in the context of the school-room or the printing-office.

neg: Transition system state automata in several ways: Transition systems can be represented as directed graphs. Formally, a transition system is a pair ("S", →) where "S" is a set of states and → is a set of state transitions (i.e., a subset of "S" × "S"). A transition from state "p" to state "q", i.e. ("p", "q") ∈ →, is written as "p" → "q". A labelled transition system is a tuple ("S", Λ, →) where "S" is a set of states, Λ is a set of labels and → is a set of labelled transitions (i.e., a subset of "S" ×

neg: Mind your Ps and Qs quality" used in 17th-century English: "The Oxford English Dictionary has a citation from Rowlands’ "Knave of Harts" of 1612: “Bring in a quart of Maligo, right true: And looke, you Rogue, that it be Pee and Kew.” Nobody is really sure what either P or Q stood for. To say they’re the initials of “Prime Quality” seems to be folk etymology, because surely that would make “PQ” rather than “P and Q”. Mind your Ps and Qs Mind your Ps and Qs is an English language expression meaning "mind your manners", "mind your language", "be on your best behaviour" or

neg: Fundamental theorem of arithmetic must be at least two primes in each factorization of "s": If any "p" = "q" then, by cancellation, "s"/"p" = "s"/"q" would be another positive integer, different from s, which is greater than 1 and also has two distinct factorizations. But "s"/"p" is smaller than "s", meaning "s" would not actually be the smallest such integer. Therefore every "p" must be distinct from every "q". Without loss of generality, take "p" < "q" (if this is not already the case, switch the "p" and "q" designations.) Consider and note that 1 < "q" ≤ "t" < "s". Therefore "t"

neg: Stable roommates problem (where reduced list means a list in the stable table): (i) "p" is first on "q"s reduced list if and only if "q" is last on "p"s (ii) "p" is not on "q"s reduced list if and only if "q" is not on "p"s if and only if "q" prefers the last person on their list to "p"; or "p", the last person on their list to "q" (iii) no reduced list is empty Stable tables have several important properties, which are used to justify the remainder of the procedure: 1. Any stable table must be a subtable of the

neg: Simulation preorder and "q" are said to be "similar", written p ≤≥ q, if "p" simulates "q" and "q" simulates "p". Similarity is an equivalence relation, but it is coarser than bisimilarity. When comparing two different transition systems (S', Λ', →') and (S", Λ", →"), the basic notions of simulation and similarity can be used by forming the disjoint composition of the two machines, (S, Λ, →) with S = S' ∐ S", Λ = Λ' ∪ Λ" and → = →' ∪ →", where ∐ is the disjoint union operator between sets. Simulation preorder In theoretical computer science a simulation preorder

neg: Ratio "p", "q" and "r" are in proportion then "p":"r" is the "duplicate ratio" of "p":"q" and if "p", "q", "r" and "s" are in proportion then "p":"s" is the "triplicate ratio" of "p":"q". If "p", "q" and "r" are in proportion then "q" is called a "mean proportional" to (or the geometric mean of) "p" and "r". Similarly, if "p", "q", "r" and "s" are in proportion then "q" and "r" are called two mean proportionals to "p" and "s". In general, a comparison of the quantities of a two-entity ratio can be expressed as a fraction derived from the

neg: Converse (logic) Converse (logic) In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. For the implication "P" → "Q", the converse is "Q" → "P". For the categorical proposition "All S are P", the converse is "All P are S". In neither case does the converse necessarily follow from the original statement. Let "S" be a statement of the form "P implies Q" ("P" → "Q"). Then the converse of "S" is the statement "Q implies P" ("Q" → "P"). In general, the verity of "S" says nothing about the verity of its

neg: Padovan sequence two complex conjugate roots "q" and "r". Given these three roots, the Padovan sequence can be expressed by a formula involving "p", "q" and "r": where "a", "b" and "c" are constants. Since the magnitudes of the complex roots "q" and "r" are both less than 1 (and hence "p" is a Pisot–Vijayaraghavan number), the powers of these roots approach 0 for large "n", and formula_23 tends to zero. For all formula_24, P(n) is the integer closest to formula_25, where "s" = "p"/"a" = 1.0453567932525329623... is the only real root of "s" − 2"s" + 23"s" − 23 = 0.

neg: Durand–Kerner method coefficients. Let the (complex) numbers "P,Q,R,S" be the roots of this polynomial ƒ. Then for all "x". One can isolate the value "P" from this equation, So if used as a fixed point iteration it is strongly stable in that every initial point "x" ≠ "Q,R,S" delivers after one iteration the root "P=x". Furthermore, if one replaces the zeros "Q", "R" and "S" by approximations "q" ≈ "Q", "r" ≈ "R", "s" ≈ "S", such that "q,r,s" are not equal to "P", then "P" is still a fixed point of the perturbed fixed point iteration since Note that the denominator

neg: P-variation known as the "Young integral" because it comes from . The value of this definite integral is bounded by the Young-Loève estimate as follows where "C" is a constant which only depends on "p" and "q" and ξ is any number between "a" and "b". If "f" and "g" are continuous, the indefinite integral formula_17 is a continuous function with finite "q"-variation: If "a" ≤ "s" ≤ "t" ≤ "b" then formula_18, its "q"-variation on ["s","t"], is bounded by formula_19 where "C" is a constant which only depends on "p" and "q". A function from ℝ to "e" × "d"

neg: Golomb coding to say when "q" ends and "r" begins ; both the qcode and rcode are self-delimited). Given an alphabet of two symbols, or a set of two events, "P" and "Q", with probabilities "p" and (1 − "p") respectively, where "p" ≥ 1/2, Golomb coding can be used to encode runs of zero or more "P"<nowiki>'</nowiki>s separated by single "Q"<nowiki>'</nowiki>s. In this application, the best setting of the parameter "M" is the nearest integer to formula_33. When "p" = 1/2, "M" = 1, and the Golomb code corresponds to unary ("n" ≥ 0 "P"'s followed by a "Q" is encoded

neg: Simulation preorder R ⊆ S × S) such that for every pair of elements (p,q) ∈ R, for all α ∈ Λ, and for all p' ∈ S, implies that there is a q' ∈ S such that and (p',q') ∈ R. Equivalently, in terms of relational composition: Given two states p and q in S, q "simulates" p, written p ≤ q if there is a simulation R such that (p, q) ∈ R. The relation ≤ is a preorder, and is usually called the "simulation preorder". It is the largest simulation relation over a given transition system. Two states "p"

neg: Necessity and sufficiency for "N", and from the second that "S" is a necessary condition for "N". This is expressed as ""S" is necessary and sufficient for "N" ", ""S" if and only if "N" ", or formula_4. The assertion that "Q" is necessary for "P" is colloquially equivalent to ""P" cannot be true unless "Q" is true" or "if Q is false, then P is false". By contraposition, this is the same thing as "whenever "P" is true, so is "Q"". The logical relation between "P" and "Q" is expressed as "if "P", then "Q"" and denoted ""P" ⇒ "Q"" ("P" implies

neg: Gosper's algorithm Gosper's algorithm In mathematics, Gosper's algorithm is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose we have "a"(1) + ... + "a"("n") = "S"("n") − "S"(0), where "S"("n") is a hypergeometric term (i.e., "S"("n" + 1)/"S"("n") is a rational function of "n"); then necessarily "a"("n") is itself a hypergeometric term, and given the formula for "a"("n") Gosper's algorithm finds that for "S("n"). Step 1: Find a polynomial "p" such that, writing "b"("n") = "a"("n")/"p"("n"), the ratio "b"("n")/"b"("n" − 1) has the form "q"("n")/"r"("n") where "q" and "r" are polynomials and no "q"("n")

neg: Idiom dictionary Idiom dictionary An idiom dictionary is a dictionary or phrase book that lists and explains idioms – distinctive words or phrases having a figurative meaning that goes beyond the original semantics of the word(s). For example, the phrase "keep your breath to cool your porridge" is more likely to be a rebuke to mind your own business than literal advice at breakfast. An idiom is a phrase whose meaning could not be readily deduced from the meaning of its individual words. The word comes from the Greek "ἰδίωμα" (idioma) – the distinctive style of a particular person. The traditional example

neg: Gomory–Hu tree "s"-"t" cut in "G"<nowiki>'</nowiki> is also a minimum "s"-"t" cut in "G" for any "s", "t" ∈ "X". To show that for all ("P", "Q") ∈ "E", "w"("P","Q") = λ for some "p" ∈ "P", "q" ∈ "Q" throughout the algorithm, one makes use of the following Lemma, The Lemma can be used again repeatedly to show that the output "T" satisfies the properties of a Gomory–Hu Tree. The following is a simulation of the Gomory–Hu's algorithm, where Gusfield's algorithm can be used to find a Gomory–Hu tree without any vertex contraction in the same running time-complexity, which simplifies the

neg: Complete metric space space Q of "p"-adic numbers is complete for any prime number "p". This space completes Q with the "p"-adic metric in the same way that R completes Q with the usual metric. If "S" is an arbitrary set, then the set "S" of all sequences in "S" becomes a complete metric space if we define the distance between the sequences ("x") and ("y") to be , where "N" is the smallest index for which "x" is distinct from "y", or 0 if there is no such index. This space is homeomorphic to the product of a countable number of copies

neg: Divisor function "n" excluding "n" itself. This function is the one used to recognize perfect numbers which are the "n" for which "s"("n") = "n". If "s"("n") > "n" then "n" is an abundant number and if "s"("n") < "n" then "n" is a deficient number. If n is a power of 2, for example, formula_26, then formula_27 and "s(n) = n - 1", which makes "n" almost-perfect. As an example, for two distinct primes "p" and "q" with "p < q", let Then and where formula_33 is Euler's totient function. Then, the roots of: allow us to express "p" and "q"

neg: Verifiable secret sharing "p", along with a generator "g" of "G", is chosen publicly as a system parameter. The group "G" must be chosen such that computing discrete logarithms is hard in this group. (Typically, one takes a subgroup of (Z), where "q" is a prime such that "p" divides "q"-1.) The dealer then computes (and keeps secret) a random polynomial "P" of degree "t" with coefficients in Z, such that "P"(0)="s", where "s" is the secret. Each of the "n" share holders will receive a value "P"(1), ..., "P"("n") modulo "p". Any "t"+1 share holders can recover the secret "s" by using

neg: Probabilistic metric space Probabilistic metric space A probabilistic metric space is a generalization of metric spaces where the distance has no longer values in non-negative real numbers, but in distribution functions. Let "D+" be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from formula_1 into [0, 1] such that max(F) = 1). The ordered pair (S,F) is said to be a probabilistic metric space if S is a nonempty set and F: S×S → D+ (F(p, q) is denoted by F for every (p, q) ∈ S × S) satisfies the

neg: Ratio notation it says that given quantities "p", "q", "r" and "s", then "p":"q">"r":"s" if there are positive integers "m" and "n" so that "np">"mq" and "nr"≤"ms". As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms "p", "q" and "r" to be in proportion when "p":"q"::"q":"r". This is extended to 4 terms "p", "q", "r" and "s" as "p":"q"::"q":"r"::"r":"s", and so on. Sequences that have the property that the ratios of consecutive terms are equal are called geometric progressions. Definitions 9 and 10 apply this, saying that if

neg: Feit–Thompson theorem of the proof now focuses on these two types of maximal subgroup "S" and "T" and the relation between them. More character-theoretic arguments show that they cannot be of types IV or V. The two subgroups have a precise structure: the subgroup "S" is of order "p"×"q"×("p"–1)/("p"–1) and consists of all automorphisms of the underlying set of the finite field of order "p" of the form "x"→"ax"+"b" where "a" has norm 1 and "σ" is an automorphism of the finite field, where "p" and "q" are distinct primes. The maximal subgroup "T" has a similar structure with "p" and "q"

neg: Minakshisundaram–Pleijel zeta function Neumann boundary conditions. More generally one can define for "P" and "Q" on the manifold, where the "f" are normalized eigenfunctions. This can be analytically continued to a meromorphic function of "s" for all complex "s", and is holomorphic for "P"≠"Q". The only possible poles are simple poles at the points "s" = "N"/2, "N"/2−1, "N"/2−2..., 1/2,−1/2, −3/2... for "N" odd, and at the points "s" = "N"/2, "N"/2−1, "N"/2−2, ...,2, 1 for "N" even. If "N" is odd then "Z"("P","P","s") vanishes at "s" = 0, −1, −2... If "N" is even its values can be explicitly by Wiener-Ikehara theorem

neg: Epistemic closure theory of knowledge, in which the x was said to know P if x's belief in P tracked the truth of P through the relevant modal scenarios. A subject may not actually believe q, for example, regardless of whether he or she is justified or warranted. Thus, one might instead say that knowledge is closed under "known" deduction: if, while knowing p, S believes q because S knows that p entails q, then S knows q. An even stronger formulation would be as such: If, while knowing various propositions, S believes p because S knows that these propositions entail p,

neg: Kripke structure (model checking) sequence of sets of the atomic propositions "w"="L"(s),"L"(s),"L"(s)..., which is an ω-word over alphabet 2. With this definition, a Kripke structure (say, having only one initial state "i∈I") may be identified with a Moore machine with a singleton input alphabet, and with the output function being its labeling function. Let the set of atomic propositions "AP" = {p,q}. p and q can model arbitrary boolean properties of the system that the Kripke structure is modelling. The figure at right illustrates a Kripke structure "M" = ("S", "I", "R", "L"), where "M" may produce a path ρ = s,s,s,s,s,s,s... and "w"

neg: Oligopoly firm 1's demand function is "P" = ("M" − "Q") − "Q" where "Q" is the quantity produced by the other firm and Q is the amount produced by firm 1, and M=60 is the market. Assume that marginal cost is C=12. Firm 1 wants to know its maximizing quantity and price. Firm 1 begins the process by following the profit maximization rule of equating marginal revenue to marginal costs. Firm 1's total revenue function is "R" = "Q" "P" = "Q"("M" − "Q" − "Q") = "MQ" − "Q" "Q" − "Q". The marginal revenue function is formula_1. Equation

neg: Exact completion a reflexivity morphism "r" from "X" to "X" such that "x""r" = "x""r" = 1; a symmetry morphism "s" from "X" to itself such that "x""s" = "x" and "x""s" = "x"; and a transitivity morphism "t" from "X" × "X" to "X" such that "x""t" = "x""p" and "x""t" = "x""q", where "p" and "q" are the two projections of the aforementioned pullback. A morphism from ("X", "X", "x", "x") to ("Y", "Y", "y", "y") in "C" is given by an equivalence class of morphisms "f" from "X" to "Y" such that there exists a morphism "f" from "X"

neg: Supermatrix a fixed superalgebra (assumed to be unital and associative). Often one requires "R" be supercommutative as well (for essentially the same reasons as in the ungraded case). Let "p", "q", "r", and "s" be nonnegative integers. A supermatrix of dimension ("r"|"s")×("p"|"q") is a matrix with entries in "R" that is partitioned into a 2×2 block structure with "r"+"s" total rows and "p"+"q" total columns (so that the submatrix "X" has dimensions "r"×"p" and "X" has dimensions "s"×"q"). An ordinary (ungraded) matrix can be thought of as a supermatrix for which "q" and "s" are both zero. A "square" supermatrix is

neg: Supermatrix linear map can be written as a ("r"|"s")×("p"|"q") supermatrix relative to the chosen bases. The components of the associated supermatrix are determined by the formula The block decomposition of a supermatrix "T" corresponds to the decomposition of "M" and "N" into even and odd submodules: Many operations on ordinary matrices can be generalized to supermatrices, although the generalizations are not always obvious or straightforward. The supertranspose of a supermatrix is the Z-graded analog of the transpose. Let be a homogeneous ("r"|"s")×("p"|"q") supermatrix. The supertranspose of "X" is the ("p"|"q")×("r"|"s") supermatrix where "A" denotes the ordinary transpose of "A". This can

neg: P. D. Q. Bach "Einstein on the Beach", provides an example. The underlying music is J.S. Bach's first prelude from "The Well-Tempered Clavier", but at double the normal speed, with each phrase repeated interminably in a minimalist manner that parodies Glass's. On top of this mind-numbing structure is added everything from jazz phrases to snoring to heavily harmonized versions of "Three Blind Mice" to the chanting of a meaningless phrase ("Koy Hotsy-Totsy," alluding to the art film "Koyaanisqatsi" for which Glass wrote the score). Through all these mutilations, the piece never deviates from Bach's original harmonic structure. The humor in P. D. Q. Bach

neg: Pattern language (formal languages) distinct variables (each of which may occur several times) is denoted by "P", the set of all patterns at all by "P". A substitution is a mapping "f": "P" → "P" such that If "p" = "f"("q") for some patterns "p", "q" ∈ "P" and some substitution "f", then "p" is said to be less general than "q", written "p"≤"q"; in that case, necessarily |"p"| ≥ |"q"| holds. For a pattern "p", its language is defined as the set of all less general patterns that are built from constants only, formally: "L"("p") = { "s" ∈ Σ : "s" ≤

neg: MISSPSTAR Katamomoi". P-Star released two versions of herself singing in English and Japanese. She appeared in the Off-Broadway production of "Take the Train to Maine", 2005's short film "Sita" and as Jessica Ruiz in the 2009 version of "The Electric Company". Diaz has modeled clothing for Royal Addiction, Baby Phat, J. Lo, and Roca Wear. In 2008 P-Star released her exercise DVD "P-Star Workout DVD: Exercise Your Body and Mind, Vol. 1" to help fight childhood obesity. In 2017 P-Star released her come back single "Don't Sleep On Me" through SoundCloud. Diaz released the single "Broke" under the new stage name

neg: Limit-preserving function (order theory) The categorical concept of limit-preserving and limit-reflecting functors is in complete harmony with order theory, since orders can be considered as small categories defined as poset categories with defined additional structure. Consider two partially ordered sets "P" and "Q", and a function "f" from "P" to "Q". Furthermore, let "S" be a subset of "P" that has a least upper bound "s". Then "f" preserves the supremum of "S" if the set "f"("S") = {"f"("x") | "x" in "S"} has a least upper bound in "Q" which is equal to "f"("s"), i.e. Note that this definition consists of two requirements:

neg: Surgery theory that the space S × S can be understood either as the boundary of D × S or as the boundary of S × D. In symbols, ∂(S × D) = S × S = ∂(D × S), where D is the "q"-dimensional disk, i.e., the set of "q"-dimensional points that are at distance one-or-less from a given fixed point (the center of the disk); for example, then, D is (equivalent, or homeomorphic to), the unit interval, while D is a circle together with the points in its interior. Now, given a manifold "M" of dimension "n" = "p" +

neg: Epistemic closure or deny the conclusion, and how to do each. Ernest Sosa says that there are three possibilities in responding to the skeptic: In the seminal 1963 paper, “Is Justified True Belief Knowledge?”, Edmund Gettier gave an assumption (later called the “principle of deducibility for justification” by Irving Thalberg, Jr.) that would serve as a basis for the rest of his piece: “for any proposition P, if S is justified in believing P and P entails Q, and S deduces Q from P and accepts Q as a result of this deduction, then S is justified in believing Q.” This was

neg: Don't Come Home a Drinkin' (With Lovin' on Your Mind) on Your Mind)" was the only single released from the album and it peaked at No. 1 on the US "Billboard" Hot Country Singles chart, becoming Lynn's first No. 1 single. Recording for the album took place at Bradley's Barn in Mount Juliet, TN, over four sessions, beginning on July 7, 1966. Three more sessions would follow on October 3, October 5, and November 17, 1966. "Saint to a Sinner" was recorded during the November 15, 1966 session for 1966's "I Like 'Em Country", at Columbia Recording Studio, Nashville, TN. Don't Come Home a Drinkin' (With Lovin' on Your Mind)

neg: Stable roommates problem i = 0, ..., k-1 where the indices are taken modulo k. It follows that in any stable table with a reduced list containing at least two individuals, such a rotation always exists. To find it, start at such a "p" containing at least two individuals in their reduced list, and define recursively "q" to be the second on "p"'s list and "p" to be the last on "q"'s list, until this sequence repeats some "p", at which point a rotation is found: it is the sequence of pairs starting at the first occurrence of ("p", "q") and ending at

neg: Weyl–Brauer matrices 2"k" is even, let for "i" = 1,2...,"k" (where the "P" or "Q" is considered to occupy the "i"-th position). The operation formula_5 is the tensor product of matrices. It is no longer important to distinguish between the "P"s and "Q"s, so we shall simply refer to them all with the symbol "P", and regard the index on "P" as ranging from "i" = 1 to "i" = 2"k". For instance, the following properties hold: Thus the algebra generated by the "P" is the Clifford algebra of euclidean "n"-space. Let "A" denote the algebra generated by these matrices. By counting

neg: Fundamental theorem of arithmetic But then "n" = "ab" = "p""p"..."p""q""q"..."q" is a product of primes. Assume that "s" > 1 is the product of prime numbers in two different ways: We must show "m" = "n" and that the "q" are a rearrangement of the "p". As "p" divides "s", Euclid's lemma implies that "p" divides one of the "q"; relabeling the "q" if necessary, say that "p" divides "q". But "q" is prime, so its only divisors are itself and 1. Therefore, "p" = "q", so that Reasoning the same way, "p" must equal one of the remaining "q". Relabeling again if

neg: Difference in differences by the dotted line "Q". (Notice that the slope from "P" to "Q" is the same as the slope from "S" to "S".) The treatment effect is the difference between the observed outcome and the "normal" outcome (the difference between P and Q). Consider the model where formula_2 is the dependent variable for individual formula_3 and formula_4, formula_5 is the group to which formula_3 belongs, and formula_7 is short-hand for the dummy variable equal to 1 when the event described in formula_8 is true, and 0 otherwise. In the plot of time versus formula_9 by group, formula_10 is the vertical

neg: Udi language Ć ć, D d, E e, Ĕ ĕ, F f, G g, Ğ ğ, H h, I ı, İ i, Ĭ ĭ, J j, Ĵ ĵ, K k, K' k', L l, M m, N n, O o, Ö ö, Ŏ ŏ, P p, P' p', Q q, Q' q', R r, S s, Ś ś, S' s', Ŝ ŝ, Ş ş, T t, T' t', U u, Ü ü, Ŭ ŭ, V v, X x, Y y, Z z, Ź ź. In 2013 in Russia, an Udi primer, "Nanay muz" (), was published with a Cyrillic-based alphabet, a modified

neg: Quartic reciprocity They all assume that "p" = "a" + "b" and "q" = "c" + "d" are primes where "b" and "d" are even, and that formula_16 Gosset's version is Letting "i" ≡ −1 (mod "p") and "j" ≡ −1 (mod "q"), Frölich's law is Burde stated his in the form: Note that Let "p" ≡ "q" ≡ 1 (mod 4) be primes and assume formula_21. Then "e" = "p f" + "q g" has non-trivial integer solutions, and Let "p" ≡ "q" ≡ 1 (mod 4) be primes and assume "p" = "r" + "q s". Then Let "p" =

neg: Fundamental theorem of arithmetic necessary, say "p" = "q". Then This can be done for each of the "m" "p"'s, showing that "m" ≤ "n" and every "p" is a "q". Applying the same argument with the formula_7's and formula_8's reversed shows "n" ≤ "m" (hence "m" = "n") and every "q" is a "p". The fundamental theorem of arithmetic can also be proved without using Euclid's lemma, as follows: Assume that "s" > 1 is the smallest positive integer which is the product of prime numbers in two different ways. If "s" were prime then it would factor uniquely as itself, so there

neg: Polynomial greatest common divisor coefficient of degree "i" of "S"("P", "Q"). They have the property that the GCD of "P" and "Q" has a degree "d" if and only if In this case, "S"("P" ,"Q") is a GCD of "P" and "Q" and Every coefficient of the subresultant polynomials is defined as the determinant of a submatrix of the Sylvester matrix of "P" and "Q". This implies that subresultants "specialize" well. More precisely, subresultants are defined for polynomials over any commutative ring "R", and have the following property. Let "φ" be a ring homomorphism of "R" into another commutative ring "S". It extends to

neg: Pattern language (formal languages) index grammar can be constructed from any pattern. Given a sample set "S" of strings, a pattern "p" is called descriptive of "S" if "S" ⊆ "L"("p"), but not "S" ⊆ "L"("q") ⊂ "L"("p") for any other pattern "q". Given any sample set "S", a descriptive pattern for "S" can be computed by Based on this algorithm, the class of pattern languages can be identified in the limit from positive examples. Pattern language (formal languages) In theoretical computer science, a pattern language is a formal language that can be defined as the set of all particular instances of a string

neg: Kripke structure (model checking) = {p,q},{q},{p,q},{q},{p},{p},{p}... is the execution word over the path ρ. "M" can produce execution words belonging to the language ({p,q}{q})*({p})∪({p,q}{q}). Although this terminology is widespread in the model checking community, some textbooks on model checking do not define "Kripke structure" in this extended way (or at all in fact), but simply use the concept of a (labelled) transition system, which additionally has a set "Act" of actions, and the transition relation is defined as a subset of "S" × "Act" × "S", which they additionally extend to include a set of atomic propositions and a labeling function for the states

neg: Pagodane of connecting eight atoms of this cage in pairs by four alkane chains. The general member is denoted ["m"."n"."p"."q"]pagodane where "m", "n", "p" and "q" are the number of carbons of those four chains. The general formula is then where "s"= "m"+"n"+"p"+"q". In particular, the basic compound has those carbons connected by four methylene bridges ("m"="n"="p"="q"=1), and its name within that family is therefore [1.1.1.1]pagodane. The compound was first synthesized by Horst Prinzbach and his associates in 1987, by a 14-step sequence starting from isodrin. In the process they also synthesized [2.2.1.1]pagodane and several derivatives. Prinzbach remarked that "the obvious

neg: Payment ["X→P→CB→R→Y"]. In practice, not every payment agent might have a direct nostro with the central bank, thus rather weird payment constellations can exist, which are hard to describe. If we define (p,q) where "p" is number of parties on the sender side and "q" on the receiver side, we can classify the directness of payments. We denote with "S" the senders and with "R" the receivers in a payment process, such that "R=S=CB" is the central bank of the currency which has 'no' distance ("n=m=0") to the payment process, "m,n=1": the "direct" nostro agents of the sender / receiver: S

neg: Polynomial greatest common divisor polynomials "P", "Q" is a polynomial function of the coefficients of "P" and "Q" which has the value zero if and only if the GCD of "P" and "Q" is not constant. The subresultants theory is a generalization of this property that allows characterizing generically the GCD of two polynomials, and the resultant is the 0-th subresultant polynomial. The "i"-th "subresultant polynomial" "S"("P" ,"Q") of two polynomials "P" and "Q" is a polynomial of degree at most "i" whose coefficients are polynomial functions of the coefficients of "P" and "Q", and the "i"-th "principal subresultant coefficient" "s"("P" ,"Q") is the

neg: Typesetting stick, a lower case 'q' looks like a 'd', a lower case 'b' looks like a 'p', a lower case 'p' looks like a 'b' and a lower case 'd' looks like a 'q'. This is reputed to be the origin of the expression "mind your p's and q's". It might just as easily have been "mind your b's and d's". The diagram at right illustrates a cast metal sort: a face, b body or shank, c point size, 1 shoulder, 2 nick, 3 groove, 4 foot. Wooden printing sorts were in use for centuries in combination with metal type.

neg: Polytope compound the compound of 10 tetrahedra. Each of the tetrahedral compounds is self-dual, and the compound of 5 cubes is dual to the compound of 5 octahedra. Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols. The material inside the square brackets, ["d"{"p","q"}], denotes the components of the compound: "d" separate {"p","q"}'s. The material "before" the square brackets denotes the vertex arrangement of the compound: "c"{"m","n"}["d"{"p","q"}] is a compound of "d" {"p","q"}'s sharing the vertices of an {"m","n"} counted "c" times. The material "after" the square brackets denotes the facet arrangement of the compound: ["d"{"p","q"}]"e"{"s","t"} is

neg: Morphism of schemes "S"-morphism from "p":"X" →"S" to "q":"Y" →"S" is a morphism ƒ:"X" →"Y" of schemes such that "p" = "q" ∘ ƒ. Given an "S"-scheme formula_14, viewing "S" as an "S"-scheme over itself via the identity map, an "S"-morphism formula_15 is called a "S"-section or just a section. All the "S"-schemes form a category: an object in the category is an "S"-scheme and a morphism in the category an "S"-morphism. (Succinctly, this category is the slice category of the category of schemes with the base object "S".) Let formula_16 be a ring homomorphism and let formula_17 be the induced map. Let

neg: Durand–Kerner method is still different from zero. This fixed point iteration is a contraction mapping for "x" around "P". The clue to the method now is to combine the fixed point iteration for "P" with similar iterations for "Q,R,S" into a simultaneous iteration for all roots. Initialize "p, q, r, s": There is nothing special about choosing 0.4 + 0.9 i except that it is neither a real number nor a root of unity. Make the substitutions for "n" = 1,2,3,··· Re-iterate until the numbers "p, q, r, s" stop essentially changing in relative to the desired precision. Then they have the

neg: Completeness (knowledge bases) inconsistent: In the case where KB := { P(a), Q(a), Q(b) }, KB ⊭ P(b) and KB ⊭ ¬P(b), so, with the closed world assumption, KB' = { P(a), ¬P(b), Q(a), Q(b) }, where KB' ⊨ ¬P(b). In data management, completeness is metaknowledge that can be asserted for parts of the KB via completeness assertions. As example, a knowledge base may contain complete information for predicates R and S, while nothing is asserted for predicate T. Then consider the following queries: For Query 1, the knowledge base would return a complete answer, as only predicates that are themselves complete are

neg: Jordan normal form to this basis, is in Jordan normal form. Next consider the subspace Ker("A" − λ I). If the desired result follows immediately from the rank–nullity theorem. This would be the case, for example, if "A" was Hermitian. Otherwise, if let the dimension of "Q" be "s" ≤ "r". Each vector in "Q" is an eigenvector of "A' " corresponding to eigenvalue "λ". So the Jordan form of "A' " must contain "s" Jordan chains corresponding to "s" linearly independent eigenvectors. So the basis {"p", ..., "p"} must contain "s" vectors, say {"p", ..., "p"}, that are lead vectors in these

neg: Projective geometry (2*) every two planes intersect in a unique line and a dual version of (3*) to the effect: if the intersection of plane P and Q is coplanar with the intersection of plane R and S, then so are the respective intersections of planes P and R, Q and S (assuming planes P and S are distinct from Q and R). In practice, the principle of duality allows us to set up a "dual correspondence" between two geometric constructions. The most famous of these is the polarity or reciprocity of two figures in a conic curve (in 2 dimensions) or

neg: Gas-s-s-s ahead, write something, just keep the title."" "Gas-s-s-s" was an idea of Corman's, about a world where everybody over thirty had died. Corman later said "my first thought was to do a science fiction film with allegorical overtones." Armitage remembers the concept just being "a sentence, and that's what we went with ... He let you make it your own, and I did." Corman said that although "there was some good work in" Armitage's first draft, "the points I was trying to make in the script either did not come through or came through too obviously different parts, and it

neg: Todi (raga) d N S' or S r g m P, m d N S' Avarohana S' N d P m g r S or S' N d P m d m g r g r S Komal Dha and Komal Ga. Re, ga and dha are intoned slightly low, and tivra ma is very sharp. The distinctive phrase is r/g-\r\S, where r may be subtly oscillated. Pa is omitted in ascent, but present and often sustained. Kaufmann mentions that some musicians would call Todi with Pa Miyan Ki Todi, but others would see no difference between Todi and Miyan Ki Todi.

neg: Feit–Thompson theorem reversed.The subgroups "S" and "T" are closely linked. Taking "p">"q", one can show that the cyclic subgroup of "S" of order ("p"–1)/("p"–1) is conjugate to a subgroup of the cyclic subgroup of "T" of order ("q"–1)/("q"–1). (In particular, the first number divides the second, so if the Feit–Thompson conjecture is true, it would assert that this cannot happen, and this could be used to finish the proof at this point. The conjecture is still unproven, however.) The conclusion from applying character theory to the group "G" is that "G" has the following structure: there are primes "p">"q" such that ("p"–1)/("p"–1)

neg: Beltrami equation your "u" (real) axis coincides with the original "x" axis, you can swap "x" with "y", apply the following, and then swap "u" with "v".) Gauss lets "q"("t") be some complex-valued function of a real variable "t" that satisfies the following ordinary differential equation: where "E", "F", and "G" are here evaluated at "x" = "t" and "y" = "q"("t"). If we specify the value of "q"("s") for some start value "s", this differential equation determines the values of "q"("t") for "t" either less than or greater than "s". Gauss then defines his isothermal coordinate system "h" by setting "h"("x",

neg: Polynomial greatest common divisor require the GCD to be monic (i.e. it has 1 as coefficient of the highest degree), but in more general cases there is no general convention. Therefore, equalities like "d" = gcd("p", "q") or gcd("p", "q") = gcd("r", "s") are usual abuses of notation which should be read ""d" is a GCD of "p" and "q"" and ""p", "q" has the same set of GCD as "r", "s"". In particular, gcd("p", "q") = 1 means that the invertible constants are the only common divisors, and thus that "p" and "q" are coprime. There are several ways to find the greatest

neg: Hermitian symmetric space subgroup "P" can be described explicitly in these classical cases. For AIII in SL("p"+"q",C). "P"("p","q") is the stabilizer of a subspace of dimension "p" in C. The other groups arise as fixed points of involutions. Let "J" be the "n" × "n" matrix with 1's on the antidiagonal and 0's elsewhere and set Then Sp("n",C) is the fixed point subgroup of the involution θ("g") = "A" ("g") "A" of SL(2"n",C). SO("n",C) can be realised as the fixed points of ψ("g") = "B" ("g") "B" in SL("n",C) where "B" = "J". These involutions leave invariant "P"("n","n") in the cases DIII and

neg: Propositional formula rule that allows the assignment of the formula to a variable. In general there is no stipulation (either axiomatic or truth-table systems of objects and relations) that forbids this from happening. The simplest case occurs when an OR formula becomes one its own inputs e.g. p = q. Begin with (p ∨ s) = q, then let p = q. Observe that q's "definition" depends on itself "q" as well as on "s" and the OR connective; this definition of q is thus impredicative. Either of two conditions can result: oscillation or memory. It helps to think of the formula

neg: Spin representation "n" = 6): The complex spin representations of so("n",C) yield real representations "S" of so("p","q") by restricting the action to the real subalgebras. However, there are additional "reality" structures that are invariant under the action of the real Lie algebras. These come in three types. The type of structure invariant under so("p","q") depends only on the signature "p" − "q" modulo 8, and is given by the following table. Here R, C and H denote real, hermitian and quaternionic structures respectively, and R + R and H + H indicate that the half-spin representations both admit real or quaternionic structures

neg: Lindsay, Bobcaygeon and Pontypool Railway time the rush to service the Haliburton and Hastings meant the area was well served by other railways. The southern section of the LB&P between Lindsay and the O&Q was made redundant with the completion of the Georgian Bay and Seaboard Railway (GB&S) in 1912. The GB&S had considered using the LB&P line out of Lindsay when their original plans to connect to Peterborough were scuttled. In keeping with the original goal of providing a shortcut to the east, backtracking along the LB&P did not make sense, and a new connection to the O&Q was made near Bethany, originally known

neg: Polytope compound a compound of "d" {"p","q"}'s sharing the faces of {"s","t"} counted "e" times. These may be combined: thus "c"{"m","n"}["d"{"p","q"}]"e"{"s","t"} is a compound of "d" {"p","q"}'s sharing the vertices of {"m","n"} counted "c" times "and" the faces of {"s","t"} counted "e" times. This notation can be generalised to compounds in any number of dimensions. A dual compound is composed of a polyhedron and its dual, arranged reciprocally about a common intersphere or midsphere, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five such compounds of the regular polyhedra. The core is the

neg: Conway triangle notation Some useful trigonometric conversions: Some useful formulas: Some examples using Conway triangle notation: Let "D" be the distance between two points P and Q whose trilinear coordinates are "p" : "p" : "p" and "q" : "q" : "q". Let "K" = "ap" + "bp" + "cp" and let "K" = "aq" + "bq" + "cq". Then "D" is given by the formula: Using this formula it is possible to determine OH, the distance between the circumcenter and the orthocenter as follows: For the circumcenter "p" = "aS" and for the orthocenter "q" = "S""S"/"a" Hence: This gives: Conway triangle

neg: Supermatrix ("r"|"s")×("p"|"q") can then be thought of as a right "R"-linear map where the action of "T" on "R" is just supermatrix multiplication (this action is not generally left "R"-linear which is why we think of "R" as a "right" supermodule). Let "M" be free right "R"-supermodule of rank "p"|"q" and let "N" be a free right "R"-supermodule of rank "r"|"s". Let ("e") be a free basis for "M" and let ("f") be a free basis for "N". Such a choice of bases is equivalent to a choice of isomorphisms from "M" to "R" and from "N" to "R". Any (ungraded)

neg: Saint Benedict Medal be my overlord!"). The large C S P B stand for "Crux Sancti Patris Benedicti" ("The Cross of [our] Holy Father Benedict"). Surrounding the back of the medal are the letters V R S N S M V - S M Q L I V B, in reference to "Vade retro satana": "Vade retro Satana! Nunquam suade mihi vana! Sunt mala quae libas. Ipse venena bibas!" ("Begone Satan! Never tempt me with your vanities! What you offer me is evil. Drink the poison yourself!") and finally, located at the top is the word PAX which means "peace". Lay Anglicans, Catholics,

neg: Hoare logic and "y" refer to the same variable (aliasing), although it is a proper instance of the assignment axiom scheme (with both {"P"} and {"P"[2/"x"]} being {"y"=3}). Hoare's rule of composition applies to sequentially executed programs "S" and "T", where "S" executes prior to "T" and is written "S;T" ("Q" is called the "midcondition"): For example, consider the following two instances of the assignment axiom: and By the sequencing rule, one concludes: Another example is shown in the right box. The conditional rule states that a postcondition "Q" common to then and else part is also a postcondition of the whole

neg: Stable roommates problem from "x"s list, so that "q" is first in "p"s list; and "p", last in "q"s, since "q" and any "x" cannot be partners in any stable matching. The resulting reduced set of preference lists together is called the Phase 1 table. In this table, if any reduced list is empty, then there is no stable matching. Otherwise, the Phase 1 table is a "stable table". A stable table, by definition, is the set of preference lists from the original table after members have been removed from one or more of the lists, and the following three conditions are satisfied

neg: Qualitative variation between the two sites. The Canberra distance is a weighted version of the "L" metric. It was introduced by introduced in 1966 and refined in 1967 by G. N. Lance and W. T. Williams. It is used to define a distance between two vectors – here two sites with "K" categories within each site. The Canberra distance "d" between vectors p and q in an "K"-dimensional real vector space is where "p" and "q" are the values of the "i" category of the two vectors. This is used to measure similarities between communities. where "s" and "s" are the number

neg: Hasse–Weil zeta function "E" then takes the form Here, ζ("s") is the usual Riemann zeta function and "L"("s", "E") is called the "L"-function of "E"/Q, which takes the form where, for a given prime "p", where, in the case of good reduction "a" is "p" + 1 − (number of points of "E" mod "p"), and in the case of multiplicative reduction "a" is ±1 depending on whether "E" has split or non-split multiplicative reduction at "p". The Hasse–Weil conjecture states that the Hasse–Weil zeta function should extend to a meromorphic function for all complex "s", and should satisfy a functional equation similar

neg: Young's lattice is a partially ordered set "Y" formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers diagrams). The traditional application of Young's lattice is to the description of the irreducible representations of symmetric groups "S" for all "n", together with their branching properties, in characteristic zero. The equivalence classes of irreducible representations may be parametrized by partitions or Young diagrams, the restriction from "S" to "S" is multiplicity-free, and the representation of "S" with partition "p" is contained in the representation of "S" with partition "q" if and only if "q" covers "p" in Young's lattice.

neg: Tachyphylaxis stimulus causing the activation, p may be variable (non-constant). More complete is an open system, namely, in its simplest form, formula_2 where R stands for the "rate of production" of A, p(S) is the "activation" rate coefficient explicitly expressing its dependence on the stimulus intensity S and q represents the rate coefficient of "removal" from the state B. In this elementally open system the steady state of B always equal to R/q. The above scheme is only the necessary condition for the rate sensitivity phenomenon, and other pathways of deactivation of B may be considered, with the subsequent return to

neg: Elliptic curve "P" and "Q"; i.e., Since "K" is a field, "s" is well-defined. Then we can define "R" = ("x", "y") = −("P" + "Q") by If "x" = "x", then there are two options: if "y" = −"y" (third and fourth panes below), including the case where "y" = "y" = 0 (fourth pane), then the sum is defined as 0; thus, the inverse of each point on the curve is found by reflecting it across the "x"-axis. If "y" = "y" ≠ 0, then "Q" = "P" and "R" = ("x", "y") = −("P" + "P") = −2"P" =

neg: Propositional formula back into "p". Given that the formula is first evaluated (initialized) with p=0 & q=0, it will "flip" once when "set" by s=1. Thereafter, output "q" will sustain "q" in the "flipped" condition (state q=1). This behavior, now time-dependent, is shown by the state diagram to the right of the once-flip. The next simplest case is the "set-reset" flip-flop shown below the once-flip. Given that r=0 & s=0 and q=0 at the outset, it is "set" (s=1) in a manner similar to the once-flip. It however has a provision to "reset" q=0 when "r"=1. And additional complication occurs if both

neg: Arch of Septimius Severus · DOMI · FORISQVE · S · P · Q · R Imp(eratori) Caes(ari) Lucio Septimio M(arci) fil(io) Severo Pio Pertinaci Aug(usto) patri patriae Parthico Arabico et Parthico Adiabenico pontific(i) maximo tribunic(ia) potest(ate) XI imp(eratori) XI, co(n)s(uli) III proco(n)s(uli) et imp(eratori) Caes(ari) M(arco) Aurelio L(ucii) fil(io) Antonino Aug(usto) Pio Felici tribunic(ia) potest(ate) VI co(n)s(uli) proco(n)s(uli) (p(atri) p(atriae) optimis fortissimisque principibus) ob rem publicam restitutam imperiumque populi Romani propagatum insignibus virtutibus eorum domi forisque S(enatus) P(opulus) Q(ue) R(omanus). In English: "To the emperor Caesar Lucius Septimius Severus Pius Pertinax Augustus Parthicus Arabicus Parthicus Adiabenicus, son of Marcus, father of his country,

neg: Structural holes relation between two individuals if there is no indirect connection between them through mutual contacts. Burt introduced the measure of network’s redundancy. He aims to estimate to what extent contact "j" is redundant with other contacts of node "i". Redundancy is understood as an investment of time and energy in a relationship with another node "q", with whom node "j" is strongly connected. formula_1 Where "p" is proportion of "i"’s energy invested in relationship with "q", And "m" is calculated as "j"’s interaction with "q" divided by "j"’s strongest relationship with anyone. The redundancy in network is calculated by summing

neg: Simply connected space any loop in "X" defined by "f" : S → "X" can be contracted to a point: there exists a continuous map "F" : D → "X" such that "F" restricted to S is "f". Here, S and D denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: "X" is simply connected if and only if it is path-connected, and whenever "p" : [0,1] → "X" and "q" : [0,1] → "X" are two paths (i.e.: continuous maps) with the same start and endpoint ("p"(0) = "q"(0) and "p"(1) = "q"(1)),

neg: Möbius configuration of the symmetric group formula_4. One obtains a permutation from the four points of one tetrahedron S = ABCD to itself as follows: each point P of S is on a plane containing three points of the second tetrahedron T. This leaves the other point of T, which is on three points of a plane of S, leaving another point Q of S, and so the permutation maps P → Q. The five conjugacy classes have representatives e, (12)(34), (12), (123), (1234) and, of these, the Möbius configuration corresponds to the conjugacy class e. It could be denoted Ke. It

neg: Fundamental theorem of arithmetic "p" from appearing. Thus ("q" - "p") is not 1, but is positive, so it factors into primes: ("q" - "p") = ("r" ... "r"). This yields a prime factorization of which we know is unique. Now, "p" appears in the prime factorization of "t", and it is not equal to any "q", so it must be one of the "r"'s. That means "p" is a factor of ("q" - "p"), so there exists a positive integer "k" such that "p""k" = ("q" - "p"), and therefore But that means "q" has a proper factorization, so it is not a

neg: Stumpage value method where the value of the stock is obtained by multiplying the current volume of standing timber by its stumpage price, assuming the rate of discount is equal to the natural growth rate. "S" = "A" * "p" * "Q" Value of standing timber = "A"rea * "p"rice per m³ * "Q"uantity of timber. Consumption value method. A variant of the stumpage value method. Different prices are used for different ages or diameter classes. The consumption value method measures the value of the stock as if it were all cut now. "S" = ∑ "At" * "pt" * "Qt"

neg: Quantitative genetics large (s → ∞ ), then p → p and q → q. It might be queried whether panmixia would effectively re-appear under these circumstances. However, the sampling of allele frequencies has "still occurred", with the result that σ ≠ 0. In fact, as s → ∞, the formula_31, which is the "variance" of the "whole binomial distribution". Furthermore, the "Wahlund equations" show that the progeny-bulk "homozygote" frequencies can be obtained as the sums of their respective average values (p or q) "plus" σ. Likewise, the bulk "heterozygote" frequency is (2 p q) "minus" twice the σ. The variance arising

neg: Parshall flume to 0.7, it is a submerged flow. Q = Q – Q Q = CH From Table 1: Throat Width = 36 in = 3 ft, C = 12, and n = 1.57. Q = 12(1.5 ft)1.57 = 22.68 ft3/s Q = M (0.000132 Ha2.123 e9.284 S) Where S = H/H = 1.4 ft/1.5 ft = 0.93 From Table 3, M = 2.4 for a flume size of 3 ft Q = 2.4(0.000132(1.5 ft)2.123e9.284(0.93)) = 4.21 ft3/s Q = 22.68 ft3/s – 4.21 ft3/s = 18.5 ft3/s A wide variety of materials are used to make Parshall flumes, including: Smaller

neg: Transposition (logic) cannot be inferred by conversion from the original 'A' type proposition "All S is P". All that can be inferred is the type "A" proposition "All non-P is non-S" (Note that (P → Q) and (~Q → ~P) are both 'A' type propositions). Grammatically, one cannot infer "all mortals are men" from "All men are mortal". An 'A' type proposition can only be immediately inferred by conversion when both the subject and predicate are distributed, as in the inference "All bachelors are unmarried men" from "All unmarried men are bachelors". In traditional logic the reasoning process of transposition as a

neg: Pivot language then to the target language; and (3) "synthesis", which builds a corpus of its own for system training. The triangulation method (also called "phrase table multiplication") calculates the probability of both translation correspondences and lexical weight in s-p and p-t, to try to induce a new s-t phrase table. The transfer method (also called "sentence translation strategy") simply carries a straightforward translation of s into p and then another translation of p into t without using probabilistic tests (as in triangulation). The synthetic method uses an existing corpus of s and tries to build an own synthetic corpus out of

neg: Four-bar linkage the condition is satisfied if "S" + "L" ≤ "P" + "Q", where "S" is the shortest link, "L" is the longest, and "P" and "Q" are the other links. The movement of a quadrilateral linkage can be classified into eight cases based on the dimensions of its four links. Let a, b, g and h denote the lengths of the input crank, the output crank, the ground link and floating link, respectively. Then, we can construct the three terms: The movement of a quadrilateral linkage can be classified into eight types based on the positive and negative values for

neg: Divergence (statistics) "D*" is defined as Many properties of divergences can be derived if we restrict "S" to be a statistical manifold, meaning that it can be parametrized with a finite-dimensional coordinate system "θ", so that for a distribution we can write . For a pair of points with coordinates "θ" and "θ", denote the partial derivatives of "D"("p" || "q") as Now we restrict these functions to a diagonal , and denote By definition, the function "D"("p" || "q") is minimized at , and therefore where matrix "g" is positive semi-definite and defines a unique Riemannian metric on the manifold "S".

neg: Ratio compare a ratio with a rational number. Specifically, given two quantities, "p" and "q", and a rational number "m"/"n" we can say that the ratio of "p" to "q" is less than, equal to, or greater than "m"/"n" when "np" is less than, equal to, or greater than "mq" respectively. Euclid's definition of equality can be stated as that two ratios are equal when they behave identically with respect to being less than, equal to, or greater than any rational number. In modern notation this says that given quantities "p", "q", "r" and "s", then "p":"q"::"r":"s" if for any positive

neg: Constructive dilemma Constructive dilemma Constructive dilemma is a valid rule of inference of propositional logic. It is the inference that, if "P" implies "Q" and "R" implies "S" and either "P" or "R" is true, then "Q or S" has to be true. In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too. "Constructive dilemma" is the disjunctive version of modus ponens, whereas, destructive dilemma is the disjunctive version of "modus tollens". The rule can be stated: where the rule is that whenever instances of "formula_2", "formula_3",

neg: Omnitruncated polyhedron mixed Wythoff symbols "p q (r s) |", and bow-tie shaped vertex figures, 2p.2q.-2q.-2p. They are not true omnitruncated polyhedra: the true omnitruncates "p q r |" or "p q s |" have coinciding 2"r"-gonal faces that must be removed to form a proper polyhedron. All these polyhedra are one-sided, i.e. non-orientable. The "p q r |" degenerate Wythoff symbols are listed first, followed by the actual mixed Wythoff symbols. Omnitruncations are also called cantitruncations or truncated rectifications (tr), and Conway's bevel (b) operator. When applied to nonregular polyhedra, new polyhedra can be generated, for example these 2-uniform polyhedra: Omnitruncated

neg: Destructive dilemma Destructive dilemma Destructive dilemma is the name of a valid rule of inference of propositional logic. It is the inference that, if "P" implies "Q" and "R" implies "S" and either "Q" is false or "S" is false, then either "P" or "R" must be false. In sum, if two conditionals are true, but one of their consequents is false, then one of their antecedents has to be false. "Destructive dilemma" is the disjunctive version of "modus tollens". The disjunctive version of "modus ponens" is the constructive dilemma. The rule can be stated: where the rule is that wherever instances

neg: Bhairavi (Carnatic) In descent as well as in the phrase P D N D P the D1 of the parent scale is used. Only R2 and M1 can be prolonged without gamakam. In the ascent G2 is held in conjunction with R2 with short or wide range of oscillation-the later as S G R G M. The gamakam on G ends in R. In descent the phrase M P G R is also used and G2 slides from M1 and oscillates. The phrase S G R G in the ascent and the ending of gamakam on G at R help to avoid

neg: Binomial distribution "r"/("r" + "s") where "p" is the probability of success and "r" and "s" are positive integers. Blaise Pascal had earlier considered the case where "p" = 1/2. Binomial distribution </math> In probability theory and statistics, the binomial distribution with parameters "n" and "p" is the discrete probability distribution of the number of successes in a sequence of "n" independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability "p") or failure/no/false/zero (with probability "q" = 1 − "p"). A single success/failure experiment is

neg: Polynomial long division then polynomial long division can be used to factor "P"("x") into the form where "Q"("x") is a polynomial of degree "n" − 1. "Q"("x") is simply the quotient obtained from the division process; since "r" is known to be a root of "P"("x"), it is known that the remainder must be zero. Likewise, if more than one root is known, a linear factor in one of them ("r") can be divided out to obtain "Q"("x"), and then a linear term in another root, "s", can be divided out of "Q"("x"), etc. Alternatively, they can all be divided out at once:

neg: Smith–Minkowski–Siegel mass formula all the "p"-masses have their standard value, then the total mass is the standard mass where The values of the Riemann zeta function for an even integers "s" are given in terms of Bernoulli numbers by So the mass of "ƒ" is given as a finite product of rational numbers as If the form "f" has a p-adic Jordan decomposition where "q" runs through powers of "p" and "f" has determinant prime to "p" and dimension "n"("q"), then the "p"-mass is given by Here "n"(II) is the sum of the dimensions of all Jordan components of type 2 and "p"

neg: Twisted: The Game Show Wormington is a used car salesman. He is seen in the game with his dog, Otis. His catch phrase is, "Come on down to Wormington's, where service is a slogan!" Madame Elaine is a late-night fortune teller. Her catch phrase is, "Call me at 1-900-Elaine! I can read your mind...and you should be ashamed!" Johnny Pow is a popular guy in the form of Fonzie. His catch phrase is, "Keep cool, daddy o!" Humble Howard is a televangelist. His catch phrase is, "Eternal security in the kingdom of Humble Howard. Operators are standing by." Major Steel is a military-esque fitness