query: do n 't let the pigeon drive the bus author

pos: Don't Let the Pigeon Drive the Bus! Don't Let the Pigeon Drive the Bus! Don't Let the Pigeon Drive the Bus! is a children's picture book by Mo Willems. Released by Disney-Hyperion in 2003, it was Willems' first book for children, and received the Caldecott Honor. The plot is about a bus driver who has to leave so he asks the reader to not allow the pigeon to drive the bus. The pigeon tries many excuses and tries to finagle readers into letting it drive the bus. An animated adaptation of the book won the 2010 Carnegie Medal for Excellence in Children's Video. Willems has created further

pos: Don't Let the Pigeon Drive the Bus! salesperson problem by taking the shortest route to visit multiple feeders in a laboratory. The authors found that pigeons "appeared to plan ahead multiple steps," which provided "clear and strong evidence that animals other than primates are capable of planning sophisticated travel routes." A parody, "Don't Let The Republican Drive The Bus" (not by Willems) replaces the pigeon with a vulture and advocates Democratic ideals. Don't Let the Pigeon Drive the Bus! Don't Let the Pigeon Drive the Bus! is a children's picture book by Mo Willems. Released by Disney-Hyperion in 2003, it was Willems' first book for children, and

pos: Mo Willems have twice been awarded the Carnegie Medal ("Knuffle Bunny", 2007, and "Don't Let the Pigeon Drive the Bus", 2010), and been developed into theatrical musical productions. His illustrations, wire sculpture, and carved ceramics have been exhibited in galleries and museums across the nation. Mo Willems currently resides in Massachusetts with his wife, Cher, and his daughter, Trixie. In 2003, Willems left his career in television and became a stay-at-home father for his daughter. In an interview with Sherry Joe Crosby of the "Los Angeles Daily News", Willems shared that there were two reasons for making this decision. Willems explained: In

pos: Don't Let the Pigeon Stay Up Late! Don't Let the Pigeon Stay Up Late! Don't Let the Pigeon Stay Up Late! is a bestselling children's picture book by Mo Willems. Released by Disney-Hyperion in 2006, it is part of Willems's "Pigeon" series. It won a National Parenting Publications Award in 2006 and was named one of the top three books for kindergarteners and first graders in a 2006 poll by Scholastic Books. Reviews have been positive. In the "New York Times Book Review", Bruce Handy called the title pigeon "one of this decade's contributions to the pantheon of great picture book characters". He added, "The Pigeon is

pos: Don't Let the Pigeon Stay Up Late! enhancing the growing sense of drowsiness. At last, the hyperactive Pigeon succumbs to slumber, but sleep-resistant and savvy readers will likely plead to read this again." Don't Let the Pigeon Stay Up Late! Don't Let the Pigeon Stay Up Late! is a bestselling children's picture book by Mo Willems. Released by Disney-Hyperion in 2006, it is part of Willems's "Pigeon" series. It won a National Parenting Publications Award in 2006 and was named one of the top three books for kindergarteners and first graders in a 2006 poll by Scholastic Books. Reviews have been positive. In the "New York Times

neg: Don't Let the Pigeon Drive the Bus! the "Top 100 Picture Books" of all time in a 2012 poll by "School Library Journal". "The Pigeon Finds a Hot Dog!" is a "Publishers Weekly" Best Book of the Year. "Don't Let the Pigeon Stay Up Late!" is a "Parenting Magazine" Best Book. "The Pigeon Wants a Puppy!" is the 2009 Children’s Choice Kindergarten to Second Grade Book of the Year. Each of the four of the standard-format Pigeon books has been on the New York Times best-seller list. A 2011 scientific study entitled "Let the Pigeon Drive the Bus" examined the ability of pigeons to solve the traveling

neg: Don't Let the Pigeon Drive the Bus! books about the Pigeon's adventures: Cameo appearances by the Pigeon can also be found in Willems's "Knuffle Bunny", "Elephant and Piggie", and "Cat the Cat" book series. In addition to the Caldecott Honor, "Don't Let the Pigeon Drive the Bus!" was an American Library Association Notable Book, a National Council of Teachers of English Notable Book, a "Bulletin of the Center for Children's Books" Blue Ribbon Book, and a South Carolina Picture Book Award winnerBased on a 2007 online poll, the National Education Association named the book one of its "Teachers' Top 100 Books for Children." It was one of

neg: Jordan–Chevalley decomposition In general the terms in the Iwasawa decomposition "g" = "kan" do not commute. If the ground field is not perfect, then a Jordan–Chevalley decomposition may not exist. Example: Let "p" be a prime number, let "k" be imperfect of characteristic "p", and choose "a" in "k" that is not a "p"th power. Let "V = k[x]/(x-a)", and let "T" be the "k"-linear operator given by multiplication by "x" on "V". This has as its stable "k"-linear subspaces precisely the ideals of "V" viewed as a ring. Suppose "T=S+N" for commuting "k"-linear operators "S" and "N" that are respectively semisimple

neg: O'Nan–Scott theorem of "M" is primitive and if we take "α" = 1 we have "M" = {("t","t")|"t" ∈ "T"}, which includes Inn("T") on Ω. In fact any automorphism of "T" will act on Ω. A primitive group of type HS is then any group "G" such that "M" ≅ "T".Inn("T") ≤ "G" ≤ "T".Aut("T"). All such groups have "N" and "N" as minimal normal subgroups. HC (holomorph of a compound group): Let "T" be a nonabelian simple group and let "N" ≅ "N" ≅ "T" for some integer "k" ≥ 2. Let Ω = "T". Then "M" = "N" × "N"

neg: Convergent matrix T is semi-convergent. We call an "n" × "n" matrix T a convergent matrix if for each "i" = 1, 2, ..., "n" and "j" = 1, 2, ..., "n". Let Computing successive powers of T, we obtain and, in general, Since and T is a convergent matrix. Note that "ρ"(T) = , where "ρ"(T) represents the spectral radius of T, since is the only eigenvalue of T. Let T be an "n" × "n" matrix. The following properties are equivalent to T being a convergent matrix: A general iterative method involves a process that converts the system of linear

neg: Edwina, the Dinosaur Who Didn't Know She Was Extinct Eagle" declared, "Willems has written a unique story every bit as good as the classic "Danny and the Dinosaur". His illustrations are playful with bright colors and bold expressions." Lana Berkowitz of the "Houston Chronicle" called it "a winner", while "The Courier-Journal"<nowiki>'</nowiki>s Jill Johnson Keeney said it was a "worthy successor" to Willems' earlier books, such as "Don't Let the Pigeon Drive the Bus!" and "". Explaining Reginald's last name to an interviewer, Willems said, "'Hoobie-Doobie' is my generic fill-in for when I can't think of a word (as in 'Hand me the Hoobie-Doobie, please.'). Because I consistently blank on

neg: Kannika Kannika Kannika is a 1947 Tamil-language film starring T. E. Varadan, M. S. Sarojini, D. Balasubramaniam, M. R. Santhanalakshmi, N. S. Krishnan, T. A. Mathuram, Kali N. Rathnam, Harini, Hemamalini and Lalitha-Padmini. It was directed and produced by South Indian movie pioneer S. M. Sriramulu Naidu. The film is about a demon whose life is imprisoned inside a pigeon. A young man destroys the demon inside the pigeon and marries the demon's daughter. Adapted from The Hindu article. Dance (Choreographer: Vazhuvoor B. Ramaiyah Pillai) Music was composed by Papanasam Sivan who also penned the lyrics. Singers are S. S. Mani

neg: Geometric separator subset ("inside "R"" and "outside "R""), with a relatively small "loss" (the squares intersected by R are considered "lost" because they do not belong to any of the two subsets). Define a "2-fat rectangle" as an axis-parallel rectangle with an aspect ratio of at most 2. Let "R" be a minimal-area 2-fat rectangle that contains the centers of at least "n"/3 squares. Thus every 2-fat rectangle smaller than "R" contains fewer than "n"/3 squares. For every "t" in [0,1), let "R" be a 2-fat rectangle with the same center as "R", inflated by 1 + "t". Now it remains to

neg: Suburbia I've Given You All and Now I'm Nothing for "Don't Let Me Cave In". The track was also released as a single. In May, the group supported Parkway Drive on their headlining tour of Australia, dubbed The Mix N Mash Tour. On June 1, "Coffee Eyes" was made available for streaming on the "Alternative Press" website. "Suburbia I've Given You All and Now I'm Nothing" was made available for streaming on June 10, before being released on June 14 through Hopeless Records. To celebrate its release, the group performed a record release show in Philadelphia, Pennsylvania. The artwork features the band's mascot, Hank the Pigeon. Later in the

neg: Rigid body dynamics known as the "principle of virtual work." This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is "Q"=0. Let a mechanical system be constructed from n rigid bodies, B, i=1...,n, and let the resultant of the applied forces on each body be the force-torque pairs, F and T, i=1...,n. Notice that these applied forces do not include the reaction forces where the bodies are connected. Finally, assume that the velocity V and angular velocities ω, i=,1...,n, for each rigid body, are defined by a single generalized coordinate q. Such a system of

neg: Coupon collector's problem grows as formula_1. For example, when "n" = 50 it takes about 225 trials on average to collect all 50 coupons. Let "T" be the time to collect all "n" coupons, and let "t" be the time to collect the "i"-th coupon after "i" − 1 coupons have been collected. Think of "T" and "t" as random variables. Observe that the probability of collecting a coupon is "p" = ("n" − ("i" − 1))/"n". Therefore, "t" has geometric distribution with expectation 1/"p". By the linearity of expectations we have: Here "H" is the "n"-th harmonic number. Using the asymptotics of

neg: O'Nan–Scott theorem is, "G" is an almost simple group and so the name. We are not told anything about what the action is, other than that it is primitive. Analysis of this type requires knowing about the possible primitive actions of almost simple groups, which is equivalent to knowing the maximal subgroups of almost simple groups. SD (simple diagonal): Let "N" = "T" for some nonabelian simple group "T" and integer "k" ≥ 2 and let "H" = {("t...,t")| "t" ∈ "T"} ≤ "N". Then "N" acts on the set Ω of right cosets of "H" in "N" by right multiplication. We

neg: Banach fixed-point theorem easily follows that the fixed point is the limit of any sequence of iterations of "T". "Remark 3." When using the theorem in practice, the most difficult part is typically to define "X" properly so that "T"("X") ⊆ "X". Let "x" ∈ "X" be arbitrary and define a sequence {"x"} by setting "x" = "T"("x"). We first note that for all "n" ∈ N, we have the inequality This follows by induction on "n", using the fact that "T" is a contraction mapping. Then we can show that {"x"} is a Cauchy sequence. In particular, let "m", "n" ∈ N

neg: Shared register disjoint sets of size (n-f) among the n processes. So 2*(n-f)≤ n leading to n≤2f, which contradicts the fact that n>2f. So the "READ" must read at least one value written by that "WRITE". A SWMR register can be written by only one process but can be read by multiple processes. Let n be the number of processes which can read the SWMR register. Let R, 0i when 0j. The implementations of "Read" and "Write" are shown below. The t-value of an operation is the value of t it writes and the operations are linearized by t-values. If "Write" and

neg: Reflexive space vertex, the two children are "t"-separated in the given space norm: Theorem. The Banach space "X" is super-reflexive if and only if for every , there is a number "n"("t") such that every "t"-separated tree contained in the unit ball of "X" has height less than "n"("t"). Uniformly convex spaces are super-reflexive. Let "X" be uniformly convex, with modulus of convexity δ and let "t" be a real number in . By the properties of the modulus of convexity, a "t"-separated tree of height "n", contained in the unit ball, must have all points of level contained in the ball

neg: Isomorphism theorems for vector spaces (modules over a field) and abelian groups (modules over formula_102) are special cases of these. For finite-dimensional vector spaces, all of these theorems follow from the rank–nullity theorem. In the following, “module” will mean “"R"-module” for some fixed ring "R". Let "M" and "N" be modules, and let "φ": "M" → "N" be a module homomorphism. Then: In particular, if "φ" is surjective then "N" is isomorphic to "M" / ker("φ"). Let "M" be a module, and let "S" and "T" be submodules of "M". Then: Let "M" be a module, "T" a submodule of "M". To

neg: Town Bus Town Bus Town Bus is a 1955 Indian Tamil language drama film directed by K. Somu and produced by M. A. Venu under his banner M. A. V. Pictures. The film was written by A. P. Nagarajan, and the soundtrack was composed by K. V. Mahadevan. The film stars N. N. Kannappa and Anjali Devi, A. Karunanidhi, T. K. Ramachandran in the lead roles with V. K. Ramasamy M. N. Rajam T. P. Muthulakshmi portraying supporting roles. The plot of the film was built around a bus transport company based in Coimbatore. The film performed well at the box office.

neg: Finite topological space topologies on a set with "n" elements. It also lists the number of inequivalent (i.e. nonhomeomorphic) topologies. Let "T"("n") denote the number of distinct topologies on a set with "n" points. There is no known simple formula to compute "T"("n") for arbitrary "n". The Online Encyclopedia of Integer Sequences presently lists "T"("n") for "n" ≤ 18. The number of distinct T topologies on a set with "n" points, denoted "T"("n"), is related to "T"("n") by the formula where "S"("n","k") denotes the Stirling number of the second kind. Finite topological space In mathematics, a finite topological space is a topological space

neg: Proofs of Fermat's little theorem ≥ 2. Lemma 2. For any positive integers "n" and "m", and any 0 ≤ x ≤ 1, In other words, "T"("x") is the composition of "T"("x") and "T"("x"). Proof. The proof of this lemma is not difficult, but we need to be slightly careful with the endpoint "x" = 1. For this point the lemma is clearly true, since So let us assume that 0 ≤ "x" < 1. In this case, so "T"("T"("x")) is given by Therefore, what we really need to show is that To do this we observe that {"nx"} = "nx" − "k", where "k"

neg: Karhunen–Loève theorem bases formula_147 are generated by kernel formula_148, i.e., solution to Do the expansion: where formula_151, then under H and formula_153 under K. Let formula_154, we have Hence, the log-LR is given by and the optimum detector is Define then formula_164 Since k(t) is the solution to If "N"("t")is wide-sense stationary, which is known as the Wiener–Hopf equation. The equation can be solved by taking fourier transform, but not practically realizable since infinite spectrum needs spatial factorization. A special case which is easy to calculate "k"("t") is white Gaussian noise. The corresponding impulse response is "h"("t") = "k"("T" − "t") =

neg: Light's associativity test For each "t" in "T", let "L"("t") be the "m" × "n" matrix of elements of "X" whose "i" - th row is and let "R"("t") be the "m" × "n" matrix of elements of "X", the elements of whose "j" - th column are According to the generalised test (due to Bednarek), that the property to be verified holds if and only if "L"("t") = "R"("t") for all "t" in "T". When "X" = "T", Bednarek's test reduces to Light's test. There is a randomized algorithm by Rajagopalan and Schulman to test associativity in time proportional to the input

neg: Identical particles have the same physical properties, the "n"'s run over the same range of values. Let "ε"("n") denote the energy of a particle in state "n". As the particles do not interact, the total energy of the system is the sum of the single-particle energies. The partition function of the system is where "k" is Boltzmann's constant and "T" is the temperature. This expression can be factored to obtain where If the particles are identical, this equation is incorrect. Consider a state of the system, described by the single particle states ["n", ..., "n"]. In the equation for "Z", every possible

neg: Sparse distributed memory it received sufficient excitatory input (beyond the threshold) and not too much of inhibitory input within a certain period. The formal model of neuron makes further simplifying assumptions. An "n"-input neuron is modeled by a "linear threshold function" formula_13 as follows : For formula_14 where n is the number of inputs, let formula_15 be the output at time "t": formula_16, and let formula_17 be the "i"-th input at time "t": formula_18. Let formula_19 be the weight of the "i"-th input and let formula_20 be the threshold. The "weighted sum" of the inputs at time "t" is defined by formula_21 The

neg: Level ancestor problem with a preprocessing algorithm of complexity O("n" log "n"). The preprocessing time is improved to O("n") by an application of the Method of Four Russians. Level ancestor problem In graph theory and theoretical computer science, the level ancestor problem is the problem of preprocessing a given rooted tree "T" into a data structure that can determine the ancestor of a given node at a given distance from the root of the tree. More precisely, let "T" be a rooted tree with "n" nodes, and let "v" be an arbitrary node of "T". The level ancestor query LA("v","d") requests the ancestor

neg: Velocity filter Consider an array of "N" sensors that receive one desired and "M" undesired broadband interferences. Let the measurement of "n" sensor be modeled by the expression: where n=1,2...,N; m=0,1...,M; s(t) are signals travelling across the array, and ŋ(t) represents zero-mean white random noise at the "n" sensor, uncorellated from sensor to sensor. The parameters a and T are amplitude gain and time delays of the signal s(t) when received at "n" sensor. Without loss of generality, we shall assume that s(t) is the desired signal and s(t),s(t)...,s(t) are the undesired interferences. Additionally we shall assume that T=0, and a =1.

neg: Benoor AKG library Kundadukkam. "benoor" Transportation Nearest Railway station is Kalanad [do have stop only for passenger trains] The major railway station is Kasaragod. Nearest airport is at Mangalore known asBajpe/ Managalore(IXE) international Airport. Kerala state road transport corporation ( K S R T C ) bus is the only bus transportation covering benoor which have stops at kappana[750m] and kundadukkam[200m].Private bus service is not available Benoor Benoor /b ae n u: r/ is a small village located at Kasaragod district of Kerala state India. Popularly known as Benoor Perumbala. Perumbala is the nearby village and benoor is a part of

neg:
Cyclic number the primes (for "b" in ). Cyclic numbers can be constructed by the following procedure: Let "b" be the number base (10 for decimal)
 Let "p" be a prime that does not divide "b".
 Let "t" = 0.
 Let "r" = 1.
 Let "n" = 0.
 loop: if "t" = "p" − 1 then "n" is a cyclic number. This procedure works by computing the digits of 1 /"p" in base "b", by long division. "r" is the remainder at each step, and "d" is the digit produced. The step serves simply to collect the digits. For computers not capable

neg: Proof of O(log*n) time complexity of union–find contain vertices with ranks from interval [R, 2 − 1]. We can make two observations about the buckets. Let "F" represent the list of "find" operations performed, and let Then the total cost of "m" finds is "T" = "T" + "T" + "T" Since each find operation makes exactly one traversal that leads to a root, we have "T" = "O"("m"). Also, from the bound above on the number of buckets, we have "T" = "O"("m"log"n"). For T, suppose we are traversing from "u" to "v", where "u" and "v" have rank in the bucket ["B", 2 − 1].

neg: Inner model Inner model In set theory, a branch of mathematical logic, an inner model for a theory "T" is a substructure of a model "M" of a set theory that is both a model for "T" and contains all the ordinals of "M". Let formula_1 be the language of set theory. Let "S" be a particular set theory, for example the ZFC axioms and let "T" (possibly the same as "S") also be a theory in formula_2. If "M" is a model for "S", and "N" is a substructure of "M" such that then we say that "N" is an inner

neg: Mad Pigeon beaten women - The day Santa died - They shall not pass - Blunder bus - Photo wedding - Boabab terminus - Madam! Madam? - Water of the Fish Men - The really fast food show - Clean fight - Super Spotless and Grimy Girl - The impossible fritters Mad Pigeon Mad Pigeon is an international series of short films created and directed by Julien Vergne. These brief (usually 2 min, but up to 6 min) and humorous films do not use any specific language: every actor must talk what he thinks to be Danish (no real Danish-speaking actor has

neg: Fuglede's theorem such that "xz = zy". Then it follows that "x* z = z y*". Fuglede's theorem In mathematics, Fuglede's theorem is a result in operator theory, named after Bent Fuglede. Theorem (Fuglede) Let "T" and "N" be bounded operators on a complex Hilbert space with "N" being normal. If "TN = NT", then "TN*" = "N*T", where "N*" denotes the adjoint of "N". Normality of "N" is necessary, as is seen by taking "T"="N". When "T" is self-adjoint, the claim is trivial regardless of whether "N" is normal: Tentative Proof: If the underlying Hilbert space is finite-dimensional, the spectral theorem

neg: Fuglede's theorem Fuglede's theorem In mathematics, Fuglede's theorem is a result in operator theory, named after Bent Fuglede. Theorem (Fuglede) Let "T" and "N" be bounded operators on a complex Hilbert space with "N" being normal. If "TN = NT", then "TN*" = "N*T", where "N*" denotes the adjoint of "N". Normality of "N" is necessary, as is seen by taking "T"="N". When "T" is self-adjoint, the claim is trivial regardless of whether "N" is normal: Tentative Proof: If the underlying Hilbert space is finite-dimensional, the spectral theorem says that "N" is of the form where "P" are pairwise orthogonal projections. One

neg: Poincaré duality a triangulated manifold, there is a corresponding dual polyhedral decomposition. The dual polyhedral decomposition is a cell decomposition of the manifold such that the "k"-cells of the dual polyhedral decomposition are in bijective correspondence with the ("n" − "k")-cells of the triangulation, generalising the notion of dual polyhedra. Precisely, let "T" be a triangulation of an "n"-manifold "M". Let "S" be a simplex of "T". We denote the dual cell (to be defined precisely) corresponding to "S" by "DS". Let formula_13 be a top-dimensional simplex of "T" containing "S". So we can think of "S" as a subset of the

neg: S-algol constant, this is a cfloat (constant float) ! if and case can have values and be used in expressions let no.of.lives := if animal = "cat" then 9 else 1 ! Sieve of Eratosthenes write "Find primes up to n = ?" let n = readi ! constant values can be set during the program run let p = vector 2::n of true ! vector of bool with bounds 2 to n for i = 2 to truncate(sqrt(n)) do ! for indexes are constants so they use = rather than := for i = 2 to n do ! structure

neg: Mo Willems on the first four seasons of "", created by one of his colleagues from "Sheep", Tom Warburton. He left the show to pursue his writing career. Since 2003, Willems has authored numerous books for young children, many of which have garnered significant critical acclaim. "The New York Times Book Review" referred to Willems as "the biggest new talent to emerge thus far in the 00's" — and to his pigeon character as "one of this decade's contributions to the pantheon of great picture book characters." Three of Willems' books have been awarded a Caldecott Honor: "Don't Let the Pigeon Drive

neg: Quantum t-design Using the fact that for any N there exists a set of formula_18 functions {0...,N-1} formula_19 {0...,N-1} such that for any distinct formula_20 {0...,N-1} the image under f, where f is chosen at random from S, is exactly the uniform distribution over tuples of d elements of {0...,N-1}. Let formula_21 be drawn from the Haar measure. Let formula_22 be the probability distribution of formula_23 and let formula_24. Finally let formula_25 be drawn from P. If we define formula_26 with probability formula_27 and formula_28 with probability formula_1 then: formula_30 for odd j and formula_31 for even j. Using this and Gaussian

neg: Classifying space for U(n) are in one-to-one correspondence with the homotopy classes of maps from "M" to CP. One also has the relation that that is, BU(1) is the infinite-dimensional projective unitary group. See that article for additional discussion and properties. For a torus "T", which is abstractly isomorphic to U(1) × ... × U(1), but need not have a chosen identification, one writes B"T". The topological K-theory "K"(B"T") is given by numerical polynomials; more details below. Let "F"(C) be the space of orthonormal families of "n" vectors in C and let "G"(C) be the Grassmannian of "n"-dimensional subvector spaces of C. The total

neg: Foliation is compact, nonempty, and a union of flow lines. If formula_84 there is a value "t"* ∈ R such that Φ("z") ∈ "N" and it follows that Since "N" is compact and formula_50 is transverse to "N", it follows that the set {"t" > 0 | Φ("y") ∈ N} is a monotonically increasing sequence formula_87 that diverges to infinity. As "y" ∈ "N" varies, let τ("y") = τ("y"), defining in this way a positive function τ ∈ "C"("N") (the first return time) such that, for arbitrary "y" ∈ "N", Φ("y") ∉ "N", 0 < "t" < τ("y"), and Φ("y") ∈

neg: Middleton bus station Middleton bus station Middleton bus station is a bus station located in the town of Middleton in Greater Manchester. The bus station is found next to the Middleton Arndale shopping centre on the site of the old bus station. The new bus station was opened in 2005 and is run by TfGM. There are thirteen stands in the bus station, A to N, with some of the stands operating a drive-in/reverse out system with the others operating a drive-through system. There are two entrances into the bus station, one from Park Road/Manchester New Road junction, and the other from Middleton

neg: Banach fixed-point theorem such that "m" > "n": Let ε > 0 be arbitrary, since "q" ∈ [0, 1), we can find a large "N" ∈ N so that Therefore, by choosing "m" and "n" greater than "N" we may write: This proves that the sequence {"x"} is Cauchy. By completeness of ("X","d"), the sequence has a limit "x*" ∈ "X". Furthermore, "x*" must be a fixed point of "T": As a contraction mapping, "T" is continuous, so bringing the limit inside "T" was justified. Lastly, "T" cannot have more than one fixed point in ("X","d"), since any pair of distinct fixed points

neg: Interval edge coloring t, w(G) ≤ t ≤ W(G). (1) w ("K") = m + n − gcd(m, n), (2) W ("K") = m + n − 1, (3) if w ("K") ≤ t ≤ W ("K"), then "K" has an interval t-coloring. w (G["K"]) ≤ (w(G) + 1)(m + n) − 1 and W (G["K"]) ≥ (W(G) + 1)(m + n) − 1. Interval edge-colorings of outerplanar graphs were investigated by Giaro and Kubale and proved all the outer planar bipartite graphs are interval colorable. Proof: Let "c" be an interval coloring of 'G' such that "e=xy" gets the smallest label among

neg: Tarski's undefinability theorem "N" be the standard structure for "L". Thus ("L", "N") is the "interpreted first-order language of arithmetic." Each sentence "x" in "L" has a Gödel number "g"("x"). Let "T" denote the set of "L"-sentences true in "N", and "T"* the set of Gödel numbers of the sentences in "T". The following theorem answers the question: Can "T"* be defined by a formula of first-order arithmetic? "Tarski's undefinability theorem": There is no "L"-formula "True"("n") that defines "T"*. That is, there is no "L"-formula "True"("n") such that for every "L"-formula "A", "True"("g"("A")) ↔ "A" holds. Informally, the theorem says that given some

neg: Z-channel (information theory) Define the sphere formula_16 of radius "t" around a word formula_17 of length "n" as the set of all the words at distance "t" or less from formula_18, in other words, A code formula_20 of length "n" is said to be "t"-asymmetric-error-correcting if for any two codewords formula_21, one has formula_22. Denote by formula_23 the maximum number of codewords in a "t"-asymmetric-error-correcting code of length "n". The Varshamov bound. For "n"≥1 and "t"≥1, The constant-weight code bound. For "n > 2t ≥ 2", let the sequence "B, B, ..., B" be defined as Then formula_27 Z-channel (information theory) A Z-channel

neg: Pregroup grammar the core principles behind pregroups and their use in linguistic domains. Let "L" = {John, Mary, the, dog, cat, met, barked, at}, let "T" = {"N, S, N"}, and let the following typing relation holds: A sentence "S" that has type "T" is said to be grammatical if formula_19. We can prove this by use of a chain of formula_11. For example, we can prove that formula_21 is grammatical by proving that formula_22: by first using contraction on formula_24 and then again on formula_25. A more convenient notation exists, however, that indicates contractions by connecting them with a drawn link

neg: Poisson wavelet a family of complex wavelets indexed by positive integers which are connected with the derivatives of the Poisson integral kernel. For each positive integer "n" the Poisson wavelet formula_1 is defined by To see the relation between the Poisson wavelet and the Poisson distribution let "X" be a discrete random variable having the Poisson distribution with parameter (mean) "t" and, for each non-negative integer "n", let Prob("X" = "n") = "p"("t"). Then we have The Poisson wavelet formula_1 is now given by The Poisson wavelet family can be used to construct the family of Poisson wavelet transforms of functions defined

neg: New Zealand pigeon are three subspecies of New Zealand pigeon; of these, only two survive: "H. n. novaseelandiae" of mainland New Zealand and "H. n. chathamensis" of the Chatham Islands. The other subspecies, Norfolk pigeon ("H. n. spadicea") of Norfolk Island, is now extinct. The subspecies differ in their plumage colour and physical morphology. In 2001, it was proposed that "H. n. chathamensis", the Chatham pigeon, was distinct enough to be raised to full species status, "H. chathamensis", instead of the traditional subspecies status, "H. n. chathamensis". Few authorities outside New Zealand have followed this, with most still considering it a subspecies. The

neg: Reductive group reductive if and only if "G" is of multiplicative type and "G"/"G" has order prime to "p". The classification of reductive algebraic groups is in terms of the associated root system, as in the theories of complex semisimple Lie algebras or compact Lie groups. Here is the way roots appear for reductive groups. Let "G" be a split reductive group over a field "k", and let "T" be a split maximal torus in "G"; so "T" is isomorphic to ("G") for some "n", with "n" called the rank of "G". Every representation of "T" (as an algebraic group) is a

neg: Schur functor of "G" for any Schur functor formula_2. Schur functors are indexed by partitions and are described as follows. Let "R" be a commutative ring, "E" an "R"-module and λ a partition of a positive integer "n". Let "T" be a Young tableau of shape λ, thus indexing the factors of the "n"-fold direct product, "E" × "E" × ... × "E", with the boxes of "T". Consider those maps of "R"-modules formula_3 satisfying the following conditions (1) formula_4 is multilinear, (2) formula_4 is alternating in the entries indexed by each column of "T", (3) formula_4 satisfies an exchange condition stating

neg: Perspectivity in a plane determined by a point of that plane not on either line has higher-dimensional analogues which will also be called perspectivities. Let "S" and "T" be two distinct "m"-dimensional projective spaces contained in an "n"-dimensional projective space "R". Let "P" be an ("n" − "m" − 1)-dimensional subspace of "R" with no points in common with either "S" or "T". For each point "X" of "S", the space "L" spanned by "X" and "P" meets "T" in a point . This correspondence "f" is also called a perspectivity. The central perspectivity described above is the case with and

neg: Loop-erased random walk the loop-erased random walk until time "T" is LE("R"([1,"T"])). In other words, take "R" from its beginning until "T" — that's a (random) path — erase all the loops in chronological order as above — you get a random simple path. The stopping time "T" may be fixed, i.e. one may perform "n" steps and then loop-erase. However, it is usually more natural to take "T" to be the hitting time in some set. For example, let "G" be the graph Z and let "R" be a random walk starting from the point (0,0). Let "T" be the time when

neg: Characterizations of the exponential function the other direction, note that by the above expression of "t", if 2 ≤ "m" ≤ "n", we have Fix "m", and let "n" approach infinity. We get (again, we must use liminf's because we don't yet know that "t" converges). Now, take the above inequality, let "m" approach infinity, and put it together with the other inequality. This becomes so that We can then extend this equivalence to the negative real numbers by noting formula_43 and taking the limit as n goes to infinity. The error term of this limit-expression is described by where the polynomial's degree (in "x")

neg: Rotational diffusion (or Einstein–Smoluchowski relation): where formula_9 is the Boltzmann constant and formula_10 is the absolute temperature. These relationships are in complete analogy to translational diffusion. The rotational frictional drag coefficient for a sphere of radius formula_11 is where formula_13 is the dynamic (or shear) viscosity. A rotational version of Fick's law of diffusion can be defined. Let each rotating molecule be associated with a vector n of unit length n·n=1; for example, n might represent the orientation of an electric or magnetic dipole moment. Let "f(θ, φ, t)" represent the probability density distribution for the orientation of n at time "t".

neg: Positive-definite function on a group minimality condition holds: where formula_4 denotes the closure of the linear span. Identify "H" as elements (possibly equivalence classes) in "V", whose support consists of the identity element "e" ∈ "G", and let "P" be the projection onto this subspace. Then we have "PUP" = "F"("a") for all "a" ∈ "G". Let "G" be the additive group of integers Z. The kernel "K"("n", "m") = "F"("m" − "n") is called a kernel of "Toeplitz" type, by analogy with Toeplitz matrices. If "F" is of the form "F"("n") = "T" where "T" is a bounded operator acting on some Hilbert space.

neg: Cut-insertion theorem Cut-insertion theorem The Cut-insertion theorem, also known as Pellegrini's theorem, is a linear network theorem that allows transformation of a generic network N into another network N' that makes analysis simpler and for which the main properties are more apparent. Let "e", "h", "u", "w", "q=q"', and " t=t' " be six arbitrary nodes of the network N and "formula_1" be an independent voltage or current source connected between "e" and "h", while "formula_2" is the output quantity, either a voltage or current, relative to the branch with immittance formula_3, connected between "u" and "w". Let us now cut the

neg: Japaridze's polymodal logic And the rules of inference are: Consider a “sufficiently strong” first-order theory "T" such as Peano Arithmetic PA. Define the series "T","T","T"... of theories as follows: For each "n", let Pr("x") be a natural arithmetization of the predicate “"x" is the Gödel number of a sentence provable in "T". A "realization" is a function which sends each nonlogical atom "a" of the language of GLP to a sentence a of the language of "T". It extends to all formulas of the language of GLP by stipulating that commutes with the Boolean connectives, and that (["n"]"F") is Pr_"n"(‘"F"’), where ‘"F"’ stands

neg: Be Thou My Vision n-aingel. Rop tú cech maithius dom churp, dom anmain; rop tú mo flaithius i n-nim 's i talmain. Rop tussu t' áenur sainserc mo chride; ní rop nech aile acht Airdrí nime. Co talla forum, ré n-dul it láma, mo chuit, mo chotlud, ar méit do gráda. Rop tussu t' áenur m' urrann úais amra: ní chuinngim daíne ná maíne marba. Rop amlaid dínsiur cech sel, cech sáegul, mar marb oc brénad, ar t' fégad t' áenur. Do serc im anmain, do grád im chride, tabair dam amlaid, a Rí secht nime. Tabair dam amlaid, a Rí secht nime, do

neg: Polynomial greatest common divisor "X". Similarly, the "i"-subresultant polynomial is defined in term of determinants of submatrices of the matrix of formula_42 Let us describe these matrices more precisely; Let "p" = 0 for "i" < 0 or "i" > "m", and "q" = 0 for "i" < 0 or "i" > "n". The Sylvester matrix is the ("m" + "n") × ("m" + "n")-matrix such that the coefficient of the "i"-th row and the "j"-th column is "p" for "j" ≤ "n" and "q" for "j" > "n": The matrix "T" of formula_44 is the ("m" + "n" − "i") × ("m" +

neg:
Ratio distribution as formula_144 with formula_145

The distribution of Y is limited here to the interval [0,1]. It can be generalized by scaling such that if formula_146 then
where formula_148

 Though not ratio distributions of two variables, the following identities are useful: This result was first derived by Katz et al in 1978. Let p and p be the probabilities of success in the binomial distributions B(X,n) and B(Y,m) respectively. Let T = (X/n)/(Y/m). Then log(T) is approximately normally distributed with mean log(p/p) and variance (1/x) - (1/n) + (1/y) - (1/m). Ratio distributions also appear in multivariate analysis. If

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: Amiga custom chips Amiga custom chips In addition to the Amiga chipsets, various specially designed chips have been used in Commodore Amiga computers that do not belong to the 'Amiga chipset' in a tight sense. "Gary", short for "Gate Array", has been used in the Amiga 500, 2000(B) and CDTV. Gary provides glue logic for bus control and houses supporting functions for the floppy disk drive. It integrates many functions built discretely in the earlier Amiga 1000 in order to reduce costs. "Fat Gary" was Gary's upgrade for the 32-bit A3000/T and A4000/T. "Gayle" replaced Gary in the A600 and A1200. It also

neg: Second moment method neighbors and every other vertex has three neighbors. The second moment method can be used to show that at every parameter "p" ∈ (1/2, 1] with positive probability the connected component of the root in the percolation subgraph of "T" is infinite. Let "K" be the percolation component of the root, and let "T" be the set of vertices of "T" that are at distance "n" from the root. Let "X" be the number of vertices in "T" ∩ "K". To prove that "K" is infinite with positive probability, it is enough to show that formula_7 with positive probability. By

neg: Battle of Middle Boggy Depot Army of the Frontier Fort Gibson, C. N., January 30, 1864 Soldiers! I take you with me to clean out the Indian Nation south of the river and drive away and destroy the rebels. Let me say a few words to you that you are not to forget. Do not begin firing in battle until you are ordered. When you fire, aim low, about the knee, or at the lower part of a man's body, if on horseback. Never fire in the air. Fire slowly and never until you see something to shout at that you may hit. Do not

neg: Time-slot interchange correctly connected. The switching data entered into the ICs is double-buffered. That is, a new switch set-up is shifted-in, and then a single pulse applies the new configuration instantly to all the connected ICs. In a time-slot interchange (TSI) switch, two memory accesses are required for each connection (one to read and one to store). Let T be the time to access the memory. Therefore, for a connection, 2T time will be taken to access the memory. If there are n connections and t is the operation time for n lines, then t=2nT which givesn=t/2T t and n normally come

neg: Diagonal lemma in the proof is not literally the same as F(°#(ψ)), but is provably equivalent to it in the theory "T". Let "f": N→N be the function defined by: for each "T"-formula θ in one free variable, and "f"("n") = 0 otherwise. The function "f" is computable, so there is a formula Γ"" representing "f" in "T". Thus for each formula θ, "T" proves which is to say Now define the formula β("z") as: Then "T" proves which is to say Now take θ = β and ψ = β(°#(β)), and the previous sentence rewrites to ψ ↔ F(°#(ψ)), which is

neg: Ambient construction scaling action. Then the following properties are immediate: Let "N" = "N" × (-1,1), with the natural inclusion "i" : "N" → "N". The dilations δ extend naturally to "N", and hence so does the generator "X" of dilation. An ambient metric on "N" is a Lorentzian metric "h" such that Suppose that a fixed representative of the conformal metric "g" and a local coordinate system "x" = ("x") are chosen on "M". These induce coordinates on "N" by identifying a point in the fibre of "N" with ("x","t""g"("x")) where "t" > 0 is the fibre coordinate. (In these coordinates,

neg: Distance oracle Distance oracle In computing, a distance oracle (DO) is a data structure for calculating distances between vertices in a graph. Let "G"("V","E") be an undirected, weighted graph, with "n" = |"V"| nodes and "m" = |"E"| edges. We would like to answer queries of the form "what is the distance between the nodes "s" and "t"?". One way to do this is just run the Dijkstra algorithm. This takes time formula_1, and requires no extra space (besides the graph itself). In order to answer many queries more efficiently, we can spend some time in pre-processing the graph and creating an

neg: Kostant's convexity theorem the coordinates of Λ. Kostant used this to generalize the Golden–Thompson inequality to all compact groups. Let "K" be a connected compact Lie group with maximal torus "T" and Weyl group "W" = "N"("T")/"T". Let their Lie algebras be formula_1 and formula_2. Let "P" be the orthogonal projection of formula_1 onto formula_2 for some Ad-invariant inner product on formula_1. Then for "X" in formula_2, "P"(Ad("K")⋅"X") is the convex polytope with vertices "w"("X") where "w" runs over the Weyl group. Let "G" be a compact Lie group and σ an involution with "K" a compact subgroup fixed by σ and containing

neg: Lambda calculus as "T"("x","N") removes all occurrences of "x" from "N", while still allowing argument values to be substituted into the positions where "N" contains an "x". The conversion function "T" can be defined by: In either case, a term of the form "T"("x","N") "P" can reduce by having the initial combinator I, K, or S grab the argument "P", just like β-reduction of (λ"x"."N") "P" would do. I returns that argument. K throws the argument away, just like (λ"x"."N") would do if "x" has no free occurrence in "N". S passes the argument on to both subterms of the application, and

neg: International Drive located in the area. Wet n' Wild was a water park opened on March 13, 1977, formerly located on International Drive, considered to be America's first water park until it closed on December 31, 2016 to make way for a new water park built by Universal called Volcano Bay that opened up on May 25, 2017. The I-Ride Trolley system operated by the International Drive Master Transit & Improvement District provides bus transportation along the I-Drive Resort Area primarily for tourists, convention attendees, and locals alike. Several Lynx public bus routes also serve the I-Drive area with routes to Orlando

neg: Compact operator on Hilbert space strong operator topology and "T" is compact, then "ST" converges to "ST" in norm. For example, consider the Hilbert space "l"(N), with standard basis {"e"}. Let "P" be the orthogonal projection on the linear span of {"e" ... "e"}. The sequence {"P"} converges to the identity operator "I" strongly but not uniformly. Define "T" by "Te" = (1/"n") · "e". "T" is compact, and, as claimed above, "PT" → "I T" = "T" in the uniform operator topology: for all "x", Notice each "P" is a finite-rank operator. Similar reasoning shows that if "T" is compact, then "T" is the

neg: Ω-consistent theory is not needed). However, PA + ¬Con(PA) proves that, for "some" natural number "n", "n" "is" the Gödel number of such a proof (this is just a direct restatement of the claim ¬Con(PA) ). In this example, the axiom ¬Con(PA) is Σ, hence the system PA + ¬Con(PA) is in fact Σ-unsound, not just ω-inconsistent. Let "T" be PA together with the axioms "c" ≠ "n" for each natural number "n", where "c" is a new constant added to the language. Then "T" is arithmetically sound (as any nonstandard model of PA can be expanded to a model of "T"),

neg: Reductive group "T" is exactly the Lie algebra formula_11 of "T". Therefore, the Lie algebra of "G" decomposes into formula_11 together with 1-dimensional subspaces indexed by the set Φ of roots: For example, when "G" is the group "GL"("n"), its Lie algebra formula_14 is the vector space of all "n" × "n" matrices over "k". Let "T" be the subgroup of diagonal matrices in "G". Then the root-space decomposition expresses formula_14 as the direct sum of the diagonal matrices and the 1-dimensional subspaces indexed by the off-diagonal positions ("i", "j"). Writing "L"...,"L" for the standard basis for the weight lattice "X"("T") ≅

neg: Direct multiple shooting method "t"] by introducing additional grid points The method starts by guessing somehow the values of "y" at all grid points "t" with 0 ≤ "k" ≤ "N" − 1. Denote these guesses by "y". Let "y"("t"; "t", "y") denote the solution emanating from the "k"th grid point, that is, the solution of the initial value problem All these solutions can be pieced together to form a continuous trajectory if the values "y" match at the grid points. Thus, solutions of the boundary value problem correspond to solutions of the following system of "N" equations: The central "N"−2 equations are the

neg: Bunce–Deddens algebra the general result. It is also known that, in general, crossed products arising from minimal homeomorphism on the Cantor set are simple AT-algebras of real rank zero. Let "C"( T ) denote continuous functions on the circle and "M"("C"(T)) be the C*-algebra of "r" × "r" matrices with entries in "C"(T). For a supernatural number {"n"}, the corresponding Bunce–Deddens algebra "B"({"n"}) is the direct limit: One needs to define the embeddings These imbedding maps arise from the natural embeddings between C*-algebras generated by shifts with periodic weights. For integers "n" and "m", we define an embedding "β" : "M"("C"(T)) →

neg: T-structure if formula_191 is also left (resp. right) exact, then formula_192. If formula_174 is left (resp. right) "t"-exact, and if formula_109 is in formula_7 (resp. formula_8), then there is a natural isomorphism formula_197 (resp. formula_198). If formula_199 are exact functors with formula_200 left adjoint to formula_201, then formula_200 is right "t"-exact if and only if formula_201 is left "t"-exact, and in this case, formula_204 are a pair of adjoint functors formula_205. Let formula_2 be a "t"-structure on formula_1. If "n" is an integer, then the translation by "n" "t"-structure is formula_208. The dual "t"-structure is the "t"-structure on the opposite category

neg: OKMO or . A slightly more practical "land battleship" was the T-35 heavy tank, designed by N. Tseits's team and inspired by the British Vickers Independent. In 1934, OKMO designed the T-43-2, a design for an amphibious tank with convertible drive—being able to run on tracks or wheels—as a possible replacement for the T-37 amphibious scout tank. It, and Moscow Factory no. 37's competing T-43-1 were both rejected in favour of continuing T-37 production. Also in 1934, N. Barykov and N. Tseits worked with graduate students from the Leningrad Technical Institute to modernize the multi-turreted T-28 medium tank by adding a

neg: Distribution (number theory) to "D". Let "F" be the "D"-module of all functions on rational numbers with denominator coprime to "N". For any prime "l" not dividing "N" we define the "Hecke operator" "T" by Let "f" be an eigenfunction for "T" with eigenvalue λ in "D". The quadratic equation "X" − λ"X" + "p" = 0 has roots π, π with π a unit and π divisible by "p". Define a sequence "a" = 2, "a" = π+π = "λ" and so that Distribution (number theory) In algebra and number theory, a distribution is a function on a system of finite sets

neg: Atheist Bus Campaign "serious or widespread offence". By the end of the year, the ASA had received 392 complaints about the adverts. The campaign has also received criticism from leading clergy including George Carey, the former Archbishop of Canterbury, and Peter Price, the current Bishop of Bath and Wells, who said "the campaign lacked both judgement and a sense of reality." In Southampton, a bus driver refused to drive a bus displaying the advert. His employers, First Bus, undertook to find him another bus to drive. Canadian author, Margaret Atwood, has said of the campaign: "That's religion! Once you're paying money to put

neg: Compact operator on Hilbert space under "T". Let "S" denote the restriction of "T" to "F". If the process was stopped after finitely many steps, with a last vector "e", then "F"= "H" and "S" = "T" = 0 by construction. In the infinite case, compactness of "T" and the weak-convergence of "e" to 0 imply that , therefore . Since "F" is contained in "H" for every "n", it follows that "m"("S") ≤ "m"({"T"}) = |λ| for every "n", hence "m"("S") = 0. This implies again that . The fact that "S" = 0 means that "F" is contained in the kernel of "T".

neg: De Bruijn–Erdős theorem (incidence geometry) a projective plane, not all on a line. Let "t" be the number of lines determined by "P". Then, The theorem is clearly true for three non-collinear points. We proceed by induction. Assume "n" > 3 and the theorem is true for "n" − 1. Let "P" be a set of "n" points not all collinear. The Sylvester–Gallai theorem states that there is a line containing exactly two points of "P". Such two point lines are called "ordinary lines". Let "a" and "b" be the two points of "P" on an ordinary line. If the removal of point "a" produces

neg: Two-body problem be solved exactly, the corresponding two-body problem can also be solved. By contrast, the three-body problem (and, more generally, the "n"-body problem for "n" ≥ 3) cannot be solved in terms of first integrals, except in special cases. Let x and x be the vector positions of the two bodies, and "m" and "m" be their masses. The goal is to determine the trajectories x("t") and x("t") for all times "t", given the initial positions x("t" = 0) and x("t" = 0) and the initial velocities v("t" = 0) and v("t" = 0). When applied to the two masses, Newton's

neg: O'Nan–Scott theorem the subgroup of all translations, and the stabilizer G in "G" of the zero vector must be an irreducible subgroup of GL("d,p"). Primitive groups of type HA are characterized by having a unique minimal normal subgroup which is elementary abelian and acts regularly. HS (holomorph of a simple group): Let "T" be a finite nonabelian simple group. Then "M" = "T"×"T" acts on Ω = "T" by "t" = "t""tt". Now "M" has two minimal normal subgroups "N", "N", each isomorphic to "T" and each acts regularly on Ω, one by right multiplication and one by left multiplication. The action

neg: Envelope (mathematics) by arc length. Consider the one-parameter family of normal lines to γ("I"). A line is normal to γ at γ("t") if it passes through γ("t") and is perpendicular to the tangent vector to γ at γ("t"). Let T denote the unit tangent vector to γ and let N denote the unit normal vector. Using a dot to denote the dot product, the generating family for the one-parameter family of normal lines is given by where Clearly (x − γ)·T = 0 if and only if x − γ is perpendicular to T, or equivalently, if and only if x −

neg: Casual Sex? Stacy finds the fortitude within herself to take the drive back to LA to tell Nick that their relationship will not work and that he must leave her apartment and her life. Leaving the resort, she comes across Vinny waiting near the exit with his luggage in tow. He flags her down and begs her to let him in the car so she can drop him off at the nearest bus station. He wants to leave the health resort as well and, without prompting, Vinny promises not to come on to her or do anything else of the sort. Reluctantly,

neg: Mission Bay, San Francisco Old Navy brand of The Gap clothing retailer. It is the location of a new research campus of the University of California, San Francisco, UCSF Mission Bay Mission Bay is served by the N Judah and T Third Street lines of San Francisco's Muni Metro. The N Judah links the neighborhood to Downtown, BART, Hayes Valley and the Sunset District, and the T Third Street links to downtown, BART, and the Bayview and Visitacion Valley neighborhoods. Several other Muni bus and trolley bus lines link the area to neighborhoods to the north, west and south. The Caltrain commuter rail system

neg: Ω-consistent theory but ω-inconsistent (as it proves formula_1, and "c" ≠ "n" for every number "n"). Σ-sound ω-inconsistent theories using only the language of arithmetic can be constructed as follows. Let "I"Σ be the subtheory of PA with the induction schema restricted to Σ-formulas, for any "n" > 0. The theory "I"Σ is finitely axiomatizable, let thus "A" be its single axiom, and consider the theory "T" = "I"Σ + ¬"A". We can assume that "A" is an instance of the induction schema, which has the form If we denote the formula by "P"("n"), then for every natural number "n", the theory

neg: Binomial distribution in the binomial distributions B(X,n) and B(Y,m) respectively. Let T = (X/n)/(Y/m). Then log(T) is approximately normally distributed with mean log(p/p) and variance ((1/p) - 1)/n + ((1/p) - 1)/m. If "X" ~ B("n", "p") and, conditional on "X", "Y" ~ B("X", "q"), then "Y" is a simple binomial variable with distribution "Y" ~ B("n", "pq"). For example, imagine throwing "n" balls to a basket "U" and taking the balls that hit and throwing them to another basket "U". If "p" is the probability to hit "U" then "X" ~ B("n", "p") is the number of balls that hit "U".

neg: C*-algebra Hilbert spaces {"H"} such that where the (C*-)direct sum consists of elements ("T") of the Cartesian product Π "K"("H") with ||"T"|| → 0. Though "K"("H") does not have an identity element, a sequential approximate identity for "K"("H") can be developed. To be specific, "H" is isomorphic to the space of square summable sequences "l"; we may assume that "H" = "l". For each natural number "n" let "H" be the subspace of sequences of "l" which vanish for indices "k" ≤ "n" and let "e" be the orthogonal projection onto "H". The sequence {"e"} is an approximate identity for "K"("H").

neg: Bézier curve 1/3 and "t" = 2/3, the control points for the original Bézier curve can be recovered. The derivative of the cubic Bézier curve with respect to "t" is The second derivative of the Bézier curve with respect to "t" is Bézier curves can be defined for any degree "n." A recursive definition for the Bézier curve of degree "n" expresses it as a point-to-point linear combination (linear interpolation) of a pair of corresponding points in two Bézier curves of degree "n" − 1. Let formula_10 denote the Bézier curve determined by any selection of points P, P, ..., P. Then

neg: Anthropic principle already provide a framework for maximizing our confidence in any theory, given a limited sequence of physical observations, and some prior distribution on the set of possible explanations of the universe. There are two kinds of dimensions, spatial (bidirectional) and temporal (unidirectional). Let the number of spatial dimensions be "N" and the number of temporal dimensions be "T". That "N" = 3 and "T" = 1, setting aside the compactified dimensions invoked by string theory and undetectable to date, can be explained by appealing to the physical consequences of letting "N" differ from 3 and "T" differ from 1. The

neg: Fuglede's theorem result as a special case. The proof by Rosenblum pictured below is just that presented by Fuglede for his theorem when assuming "N=M". Theorem (Calvin Richard Putnam) Let "T", "M", "N" be linear operators on a complex Hilbert space, and suppose that "M" and "N" are normal, M is bounded and "MT" = "TN". Then "M"*"T" = "TN"*. First proof (Marvin Rosenblum): By induction, the hypothesis implies that "M""T" = "TN" for all k. Thus for any λ in formula_7, Consider the function This is equal to where formula_11 and formula_12. However we have so U is unitary, and hence

neg: Analytically unramified ring "R" and the field of fractions "K" of "R", the integral closure of "S" in "K" is a finitely generated module over "S". The second follows from the first. Let "K" be a perfect field of characteristic 2, such as F. Let "K" be "K"({"u", "v" : "n" ≥ 0}), where the "u" and "v" are indeterminates. Let "T" be the subring of the formal power series ring "K" generated by "K" and "K" and the element ∑("u""x"+ "v""y"). Nagata proves that "T" is a normal local noetherian domain whose completion has nonzero nilpotent elements, so "T" is analytically ramified.

neg: Moschovakis coding lemma the s - m - n theorem, let s : ( ω ω ) 2 → ω ω be continuous such that for all ,x,t,w, U ( s (,x ) ,t,w ) ↔ ∃ y ∃ z [ y ≺ x ∧ U (,y,z ) ∧ U ( z,t,w )]. By the recursion theorem, let 0 be such that U (0 ,x,z ) ↔ z = τ ( s (0 ,x )). A straightforward induction on | x | ≺ for x ∈ dom( ≺ ) shows that ∀ x ∈ dom( ≺ ) ∃ ! z U ( 0

neg: Dowling geometry "E". In his second paper (1973b) Dowling gave an intrinsic definition of the rank-"n" Dowling lattice of any finite group "G". Let "S" be the set {1...,"n"}. A "G"-labelled set ("T", α) is a set "T" together with a function α: "T" → "G". Two "G"-labelled sets, ("T", α) and ("T", β), are equivalent if there is a group element, "g", such that β = "g"α. An equivalence class is denoted ["T", α]. A partial "G"-partition of "S" is a set γ = {["B",α], ..., ["B",α]} of equivalence classes of "G"-labelled sets such that "B", ..., "B" are nonempty subsets